the dixmier unitarisability problem and ℓ -betti numbers · the second chapter deals with the...

The Dixmier Unitarisability Problem andℓ2-Betti Numbers

Henrik Densing Petersen [email protected]

ResumeDette speciale i matematik omhandler hovedsageligt spørgsmalet om hvorvidt

enhver unitariserbar gruppe er amenabel. Efter en gennemgang af relevant teorifor ℓ2-Betti tal for grupper diskuteres nylige partielle løsninger, specielt arbe-jde af Monod-Epstein der viser at residualt endelige unitariserbare grupper harforsvindende første ℓ2-Betti tal.

Endelig gives ogsa et par lette kommentarer vedrørende en mulig udvidelseaf dette resultat til højere ℓ2-Betti tal.

• Speciale for cand.scient graden i matematik. Institut for matematiske fag,Københavns Universitet.

• Thesis for the Master degree in Mathematics. Department of Mathemat-ical Sciences, University of Copenhagen.

• Supervisor (Vejleder): Ryszard Nest

• Date (Afleveringsdato): 22/07 - 2009

Abstract

This master’s thesis in mathematics deals mainly with the problem of whetherevery countable, unitarisable group is amenable. After an exposition of somebackground theory on ℓ2-invariants, some recent partial solutions, in particularwork of Monod-pstein showing that residually finite unitarisable groups havevanishing first ℓ2-Betti number, are discussed.

Finally, there are some lightweight musings on how to extend this result tocover also higher ℓ2-betti numbers

Introduction

A group G is unitarisable if all of its uniformly bounded (with respect to theoperator norm) representations as invertible operators on a Hilbert space, aresimilar to unitary representations. B. Sz.-Nagy showed in 1947 that Z is unitaris-able, and in 1950 Dixmier pointed out that this readily extended to all amenablegroups. In this thesis, ’group’ means countable group unless explicitly stated,and of course we always equip such a group with the discrete topology.

The Dixmier problem asks whether the converse is true.In fact, Dixmier showed that amenability implies unitarisability for more

general, locally compact groups, but the consensus among experts [Pis05] seemsto be that if a counterexample to the converse exists, one should look for itamong countable groups.

Recently the problem has received some interest, and several authors haveproved partial results. In 2005, [Pis05], Pisier states that he believes it to havea negative solution in general, in particular he believes that unitarisability doesnot behave well under direct limits. Whether this is still the case given recentprogress, in particular [MO09], I would not presume to know.

Statement of the main results. The main interest of this thesis is thefollowing result, by Monod-Epstein.

5.2 Theorem. Any countable residually finite group G with first ℓ2-Betti num-

ber β(2)1 (G) > 0 is non-unitarisable.

This is proved in [ME08], which was the original starting point of this thesis.There is a similar theorem, 5.3, dealing with the cost instead of ℓ2-Betti numbers,but even though it is a priori a stronger result, it has not really led me anywhereand so I consider the above more relevant for what will happen during the restof this thesis.

Theorem 5.2 should be viewed in perspective to the neo-classical fact thatfor amenable groups, all the ℓ2-Betti numbers vanish. The converse, however, isnot true. In fact, fairly recently, in the very influential paper [Gab02], Gaboriauhas shown that this also requires a treeable action on a standard probabilityspace - and finding out, even in specific examples, whether such an action existsis not a straightforward problem.

The hope was that one could generalize Monod-Epstein’s proof of 5.2 tohigher dimensions without too much drama. It basically consists of two parts -

1

2

one uniformly bounding the first ℓ2-Betti number of the actions the subgroups ofa unitarisable group on simplicial complexes, in terms of numbers of orbits. Thesecond part is the construction of the Cayley graph, utilizing that a subgroupof a finitely generated group is finitely generated.

Originally I was fairly positive concerning the generalization of the first part,but having failed it horribly now I am not so positive. In chapter 7 there are afew questions dealing with some of the issues I encountered.

In chapter 6 the following generalization of the first half of the Monod-Epstein proof is proved:

6.10 Theorem. Suppose that G is unitarisable (and countable). Then thereis a constant, M , such that for any subgroup H of G and any H-invariantsimplicial complex K ⊆ E0H with the action of H on K (free,) cocompact wehave

1√♯F (1)

· infL:K⊆L⊆E0H

dimL(H) Im(H

(2)

1 (K,H) → H(2)

1 (L,H))≤ ♯F (0) ·M,

where F (i) is a(ny) fundamental domain for the action of Si+1×H on K(i) andthe infimum is over H-invariant subcomplexes with (free,) cocompact action ofH, as usual.

Initially the idea was that this should have a strong generalization to higherdimensions and that this would give a guide to what to do with the Cayleygraph part in higher dimensions. This idea has not succeeded and appears tobe at the very least, troublesome in terms of calculations.

Description of contents. An annoying fact is that the only comprehen-sive reference on ℓ2-Betti numbers is [Luc02] which is written in ’high style’, andso the first chapter deals mainly with establishing the basic theory and mak-ing sure were are all speaking the same language. In particular, I have given amore indepth construction of a classifying complex than “take an infinite Milnorjoin”.

In general, I have tried to keep things at a fairly un-sophisticated level, fo-cusing on combinatorial and geometrically flavoured ideas, rather than powerfulhigh-style algebraic tools. This is not because I somehow believe the former tobe a universally better approach, but mainly because the restrictions of time ona master’s thesis does not match very well with the latter - especially given mylow level of previous exposure to many of these things (geometric and measur-able group theory is not a big deal at Copenhagen). I have also strived to focuson giving just enough that we can use the theory and not get sidetracked toomuch, but one perhaps unfortunate side effect of this is that there are not somany examples.

The second chapter deals with the measurable version of ℓ2-Betti numbersand also briefly introduces the cost and its (known) relation to ℓ2-Betti numbers.When I was learning this stuff I was expecting it to be an important part of mylater work, especially because of the important paper [GL07] and statements to

3

that effect in [ME08]. One reason that I have not really pursued this, I guess,is that just as I was starting to think about what to do with my time, [MO09]appeared on the arXiv, and I decided I could probably not push this approachfurther than what is done there.

Thus, this chapter is not really too relevant in terms of what is actually donehere, but the things in it still is in the greater picture so I have left it in.

Chapter 3 contains a quick introduction to random forests. There is a pos-sibly new, though to what degree I do not know, derivation of the probabilityof the free uniform spanning forest containing some given edge. The result hasbeen known since 1847. The proof I give is somehow based on “rearrangements”,and though none of the involved constructions are new, for instance Lyons con-siders matroids with a certain transfer property in his work [Lyo03, Lyo08] ondeterminantal measures which reduces in the case of graphs exactly to the com-binatorial observation I use, the specific way of doing things may not have beennoticed before.∗

Chapter 4 gives a short introduction to the Dixmier problem, proving anextension of Dixmier’s result and discussing the relation of the Dixmier problemto the von Neumann problem.

Chapter 5 gives an exposition on Monod-Epsteins work in [ME08].Chapter 6 describes what turned out to be a failed attempt to do a proof

of 5.2 without using the Cayley graph. Originally it was a roaring success, butthen I found a mistake. Maybe I should have labelled it a failure and left it out,but on the other hand it is a slight generalization of the central idea of the proofof 5.2 and some of the ideas of the proof are, perhaps, not entirely useless.

Finally, chapter 7 is dedicated to listing and briefly commenting on someopen problems.

Reading Guide. The knowledgeable reader can safely skip the first two,maybe even three, chapters and refer to them only if needed. The rest shouldbe read in the order in which it appears.

In any case do not forget, under any circumstances, to check out the stand-ing assumptions below.

A word on language. Since it is often misinterpreted I feel obliged toexplain the meanings of a few common words. First of all, the difference between“I” and “we” is that when I say “I blahblah” I do not necessarily expect thereader’s understanding or support, whereas use of “we” means that the readeris supposed to agree or something like that.

Also, I may sometimes use phrases like “it is easily seen that”. These phrasesare there not to mock the reader or state my own prowess, but as a service. If

∗I did the proof mainly because I did not really understand (nor did I try very hard to) theones I could find, using Wilson’s algorithm, and I only later found out about determinantalmeasures. In the case of determinantal measures, however, an extended version of the resultis taken as the definition, and using formulas from linear algebra one can then show that onegets indeed a probability measure.

4

the reader does not see it easily, then either he or she did not understand thepreceding material properly, or I did not write it properly.

Finally let me mention that there is a difference between “trivial”, “obvi-ous”, and “easy”: The first means that things are just a matter of plugging indefinitions; The second that any monkey can immediately see exactly what todo besides plugging in the definitions; The third means that, whereas thingsmight not be obvious, if one just writes them down and follow the smell one willget there with little trouble.

Acknowledgements. I would like to thank Ryszard Nest for many usefuldiscussions. Getting things on a whiteboard helps!

I would also like to thank the 4th floor coffee machine for all the coffee, andwhatever genius decided that the afforementioned coffee machine should be freeto use.

Contents

Statement of the main results . . . . . . . . . . . . . . . . . . 1Description of contents . . . . . . . . . . . . . . . . . . . . . . . 2Reading Guide . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3A word on language . . . . . . . . . . . . . . . . . . . . . . . . . 3Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . 4

1 A prologue in three parts 8Basic definitions concerning graphs . . . . . . . . . . . . . . . . . . . . 8Cycles and their (co)boundaries . . . . . . . . . . . . . . . . . . . . . . 9Measure preserving group actions . . . . . . . . . . . . . . . . . . . . . 11The reduced ℓ2-homology of group actions on simplicial complexes . . 14The chain complex of a simplicial complex . . . . . . . . . . . . . . . . 15Actions by groups and the ℓ2-chain complex . . . . . . . . . . . . . . . 16The L(G)-dimension of an L(G)-module . . . . . . . . . . . . . . . . . 17Reduced ℓ2-homology of group actions . . . . . . . . . . . . . . . . . . 18INTERMEZZO: The classifying simplicial complex of a group . . . . . 21END INTERMEZZO . . . . . . . . . . . . . . . . . . . . . . . . . . 23

2 A tutorial on invariants of (SP1) equivalence relations 25A generalized definition of simplicial complex . . . . . . . . . . . . . . 25Representations and chain spaces . . . . . . . . . . . . . . . . . . . . . 28Basic definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30Basic theorems and connections with the classical theory . . . . . . . . 31Graphings and the cost . . . . . . . . . . . . . . . . . . . . . . . . . . 32

3 Random graphs and their associated equivalence relations 36Random graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36The free uniform spanning forest . . . . . . . . . . . . . . . . . . . . . 37INTERMEZZO: On the number of spanning trees . . . . . . . . . . . 38END INTERMEZZO . . . . . . . . . . . . . . . . . . . . . . . . . . 42

The free minimal spanning forest . . . . . . . . . . . . . . . . . . . . . 43Properties of the FUSF and the FMSF . . . . . . . . . . . . . . . . . . 44The full and cluster equivalence relations . . . . . . . . . . . . . . . . 45

5

CONTENTS 6

4 Amenable and unitarisable groups 46Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46The von Neumann problem . . . . . . . . . . . . . . . . . . . . . . . . 49

5 The cost of unitarisability 50Statement of the results . . . . . . . . . . . . . . . . . . . . . . . . . . 50INTERMEZZO: Some combinatorial group theory . . . . . . . . . . . 50END INTERMEZZO . . . . . . . . . . . . . . . . . . . . . . . . . . 51

An application of the FUSF . . . . . . . . . . . . . . . . . . . . . . . . 51The proof of theorem 5.3 . . . . . . . . . . . . . . . . . . . . . . . . . 53

6 Classifying complexes and a different proof of theorem 5.2 56Induced complexes, subgroups, and EG . . . . . . . . . . . . . . . . . 56Special automorphisms of E0G: The cyclic shift of order n . . . . . . . 58A baby-baby Hodge-de Rahm theorem and the FUSF . . . . . . . . . 59

7 An annotated list of problems 63Behavior under exact sequences . . . . . . . . . . . . . . . . . . . . . . 63The Dixmier problem . . . . . . . . . . . . . . . . . . . . . . . . . . . 63Transitive unimodular graphs . . . . . . . . . . . . . . . . . . . . . . . 64The fixed price question . . . . . . . . . . . . . . . . . . . . . . . . . . 64cost vs. ℓ2-Betti numbers . . . . . . . . . . . . . . . . . . . . . . . . . 64Treeability vs. anti-treeabiility . . . . . . . . . . . . . . . . . . . . . . 65Determinantal measures and a bound in dimension n . . . . . . . . . . 65Determinantal measures and uniformity . . . . . . . . . . . . . . . . . 66A hands on proof of theorem 7.8 . . . . . . . . . . . . . . . . . . . . . 66

Bibliography 68

STANDING

ASSUMPTIOS

I will list here some assumptions and conventions that will be in effect through-out this thesis except where explicitly stated.

• Except where explicitly stated, all groups are countable, equiped with thediscrete topology.

• All hilbert spaces of the form ℓ2(A) are real. In particular, ℓ2(G) for anygroup G is for us a real Hilbert space, and the group von Neumann algebraL(G) is therefore also real.

• the term “standard Borel space” here really means a standard Borel spacewithout atoms.

• A cycle in a graph will always mean a simple cycle, whereas a cycle inboundary map context is just a finitely supported element in the kernel.

• If S is a generating set for the group G, we always take it to not containthe identity, and also there are no repetition of elements. (In other words,the Cayley graph has no loops or multiple unoriented edges.)

• “free action” in the context of actions on simplicial complexes alwaysmeans free on unordered simplices, not just on ordered ones.

• The symbol ’π’ is used throughout the text as a projection onto some-thing, typically specified in some sense by a subscript. Whenever it isrelatively clear from context, or maybe doesn’t matter all that much, Ihave not bothered with specifying domains and codomains, chiefly since Ifelt this would slow down the text even more. This is especially prevalentin chapter 3, and is maybe also relevant to note when I deal with fibredspaces.

7

Chapter 1

A prologue in three parts

The first rule about fight club is you do not talk about fight club.key words:

graph theory basics - groups acting on measure spaces - ℓ2-homology of groupactions.

Below I give some basic definitions and results of graph theory. Most of thiswill be either known or trivial to the reader, but to my knowledge there is nogood, concise, comprehensive reference giving what we need and only what weneed, and so I shall do it myself.

1.1 Basic definitions concerning graphs. We shall, as the rest of thepack, follow Serre’s conventions for graphs, meaning that a (oriented) graph (onV ) is a pair G = (V,E) of sets, where V is the set of vertices of G and E ∈ 2V ×V

is the set of edges. We usually suppress V if it is understood, allowing us toregard a set of graphs on V as a subset of 2V ×V .

We will often disregard the orientation of the edges and work with unorientedgraphs, meaning that E will be symmetric - that is, invariant under the canonicalinvolution on 2V ×V given by∗

(v, v′) = e 7→ e = (v′, v).

In this case the involution restricts to a bijection on E, where it acts by “re-versing arrows.”

A geometric (or unoriented) edge (associated to e) is a pair of opposed edgese, e, and e, e are representatives of the geometric edge.

We also have vertex maps ± : E → V given by

(v, v′) = e 7→ e+ = v′

(v, v′) = e 7→ e− = v,

∗Here and everywhere else, we identify 2V ×V with the set of subsets of V ×V in the usualmanner, and we use notations interchangably.

8

CHAPTER 1. A PROLOGUE IN THREE PARTS 9

and we say that the geometric edge associated to e is incident on a vertex u ∈ Vif u ∈ e+, e−. note that the choice of representative of the geometric edge isirrelevant here.

A finite graph is one with finite set of vertices, and all other graphs weencounter will have countable set of vertices. Further, unless explicitly stated,our graphs shall be simple meaning they have no loops - i.e. that E is disjointfrom the diagonal (and that there are no multiple geometric edges between anypair of vertices, but this is already included in our definition above).

Definition. • A path p (of length l ≥ 1) in G is a sequence p = (e1, e2, . . . , el)of edges in E such that

e+i = e−i+1, i = 1, 2, . . . , l− 1.

We say that p is a path from e−1 to e+l .

By convention there is also a path of length 0 from each vertex to itself,called the empty path.

• We define a relation ∼ on V by v ∼ u if and only if there is a path from v tou. If G is unoriented, this is clearly an equivalence relation, and we shallcall the equivalence class containing v the component of v. The clusterg(v) of v is the subgraph with vertices the component of C = C(v) of vand edges E ∩2C×C, i.e. the (maximal) subgraph spanning the componentof v.

• A graph is connected if it has exactly one component.

1.2 Cycles and their (co)boundaries. A very powerful way to studygraphs is by studying associated sets of spanning trees/forests. For infinitegraphs we shall later also consider “random” versions of these objects, usuallyobtained by passing to limits of random forests on finite subgraphs. Translatingbasic graph-theoretic notions to properties of certain Hilbert spaces, we will beable to write explicitly the projection onto d(C0

(2)(G)) ⊆ C1(2)(G) when G is finite,

in terms of its spanning trees. Passing to limits, this will give a link betweenℓ2-(co)homology and random spanning forests of the Cayley graph of a group.

Definition. • A cycle c (of length l) in G is a sequence c = (e1, e2, . . . , el),modulo rotation, of edges in E such that

ei 6= ei+1, e+i = e−i+1, i = 1, 2, . . . , l,

where el+1 := e1, and such that the e+i are pairwise distinct.

If G is unoriented, a geometric cycle of length l is again a sequence of lgeometric edges such that there is a choice of representatives yielding acycle as defined above.

• A forest on V is a graph without cycles. It is a tree if it is connected, anda spanning tree if it is maximal among all trees on V .

If G is a connected graph on V , a spanning tree of G is a subgraph of Gwhich is a spanning tree on V .


Remark. • Often cycles are allowed to intersect themselves, and what wecall a cycle would then be a simple cycle. A cycle is thus just a path,without “backtracking” and proper self-intersection, from a vertex to itself.

• If G is not connected, clearly it has no spanning trees. On the other hand,if it is connected a standard maximality argument shows that it does in-deed have a spanning tree: Since an increasing union of trees is again atree, Zorn’s lemma gives a maximal subtree T of G. If this is not a span-ning tree, there is some vertex, u, not connected to any vertex in V (T ).However, choosing any v ∈ V (T ) we get by connectedness of G a pathp = (e1, e2, . . . , el) from u to v with ei ∈ E(G) for all i. Then, choosing i0minimal such that e+i0 ∈ V (T ) we may add this edge to T without creatingany cycles, contradicting maximality.

Definition. Let G is an unoriented graph. For v ∈ V (G) we define the degreeof v as

deg v =∑

e∈E(G):e+=v

1.

Suppose now that G is unoriented and consider the vector spaces F(V )respectively Falt(E) of (real-valued) functions on V respectively alternatingfunctions on E i.e. functions f satisfying f(e) = −f(e) for all edges e. Wedefine the coboundary map d : F(V ) → Falt(E) by

(df)(e) = f(e+) − f(e−).

Proposition. If G is an unoriented graph with bounded degree, then the re-striction of d to ℓ2(V ) is linear, has image in ℓ2alt(E), and is bounded with respectto the standard Hilbert space norms on these spaces.

When G is the Cayley graph of a finitely generated group, or more generallythe 1-skeleton of a simplicial complex with a cocompact aaction of G, this isjust a special case of proposition 1.16.

Proof. Indeed, suppose that each vertex in V has degree at most M , and con-sider an f =

∑ni=1 aiδvi

. Then, using |a− b|2 ≤ 2(|a|2 + |b|2),

‖df‖22 ≤ 4

n∑

i=1

∑

e:e+=vi

|ai|2 ≤ 4M‖f‖2,

proving the claim. ////

Note that the converse is obviously true.

1.3 Definition. If G is an unoriented graph with bounded degree, we definethe star- respectively cycle-spaces of ℓ2alt(E) by

ℓ2⋆(E) = d(ℓ2(V )),


respectively

ℓ2(E) = spanχe1 + χe2 + · · · + χel| (e1, e2, . . . , el) is a cycle,

where χe := δe − δe.

The star space is named such since the image dδv is a star-shaped coloringof the graph.

1.4 Theorem. If G is an unoriented graph with bounded degree, then

(i) ℓ2⋆(E) is orthogonal to ℓ2(E).

(ii) If g1, g2, . . . are the clusters of G then ℓ2♠(E) =⊕

i ℓ2♠(E(gi)) where ♠ is

either of ⋆, , alt.

(iii) If further G is finite, actually ℓ2alt(E) = ℓ2⋆(E) ⊕ ℓ2(E).

Proof. It is clear that for any v ∈ V we have dδv ∈ ℓ2(E)⊥. Indeed, if(e1, e2, . . . , el) is a cycle and, say e+k = e−k+1 = v, then

〈dδv, χek+ χek+1

〉 = 〈χek− χek+1

, χek+ χek+1

〉 = 0

and (i) follows.(ii) is completely straightforward.For (iii), note that we need only prove it for connected G by (ii). In this case,

choose a spanning tree T for G and some base vertex ρ, and let f ∈ ℓ2(E)⊥.Then, putting f(ρ) = 0 and successively working our way out to the neighboursof ρ, then their neighbours etc., it is easy to see that we can construct f ∈ ℓ2(V )such that

(df)(e) = f(e), e ∈ E(T ).

Note that we do not need finiteness to construct f , but to ensure that it isin ℓ2(V ).

Now simply note, that if e1 is not in E(T ), then both f and df sum to zeroalong the fundamental cycle CT (e1) = (e1, e2, . . . , el) (see def. 3.4) since theyare both orthogonal to ℓ2(E). Thus, recalling that ei ∈ E(T ) for i = 2, 3, . . . , lwe get

(df)(e1) = −l∑

i=2

(df)(ei) = −l∑

i=2

f(ei) = f(e1).

////

1.5 Measure preserving group actions. The subject of measurablegroup theory traces its beginnings to two papers of H.A. Dye [Dye59,Dye63].In these he studies ergodic actions of Z by measure preserving automorphismson a standard Borel probability space. The most famous result is that any twosuch actions of Z are orbit equivalent.


The theory reached a climax in the late 70’s and early 80’s, where it de-veloped a complete understanding of the measurable aspect of the theory ofamenable groups:†

1.6 (Connes-Feldman-Weiss ’80) Theorem. Let G1, G2 be amenable groups.Then any two ergodic measure preserving actions of G1 respectively G2 on stan-dard Borel probability spaces are orbit equivalent.

Let us take this opportunity to actually explain some terminology.

Definition. Let (X,B, µ) be a standard borel space. We shall usually assumethat µ is a probability measure, or at least finite. Let also G be a countablegroup, acting by Borel auutomorphisms on X.

• We say that the action is measure preserving, or that G preserves themeasure, if µ is G-invariant: ∀g ∈ G∀B ∈ B : µ(g.B) = µ(B).

• The action of G is (essentially) free if the set of points fixed by g ∈ G hasmeasure zero for every non-identity g.

• The action of G induces an equivalence relation RG on X, defined byx ∼RG y ⇔ ∃g ∈ G : y = g.x.

A guiding principle underlying much of the work described in chapter 2, andmany of its applications, is that one should not study group actions directly,but instead study a sufficiently specific class of equivalence relations and theneither apply this to group theory afterwards, or apply it directly to a problemby translating it into equivalence relation terms. In fact, this is probably moregeneral or at least more practical, since it is apparently unknown whether thestronger version of theorem 1.7 providing a free action of G holds.

Definition. Let again (X,B, µ) be a standard Borel space, and R an equiva-lence relation on X.

• We say that R is measurable if it is measurable as a subset of X×X. It isstandard if it is measurable and each equivalence class (also called orbits,a priori since we want to think of R as induced by a group action, and afortiori because it usually is - see theorem 1.7) is countable.

• R preserves the measure if every partial Borel automorphism with graphcontained in R (again as a subset of X ×X) preserves µ.

R quasi-preserves the measure if every partial Borel automorphism withgraph contained in R sends nullsets to nullsets.

If R is standard and preserves the measure, we say that it is an (SP )relation. It is (SP1) if µ is a probability measure, and these are therelations we usually deal with.

†See chapter 4 for the definition of amenability. The point here is that Z is the standardexample of an amenable group.


• If B is a (Borel) set, we denote by R[B] the R-saturation of B - theunion of all orbits containing points in B. We say that B is R-invariantif R[B] = B, i.e. B is a union of orbits, i.e. if x ∼ y and x ∈ B theny ∈ B.

If every R-invariant Borel set is a either nullset or a co-nullset, then wesay that R is ergodic.

• If R,S are (SP) equivalence relations on (X,µ) resp. (Y, ν), we say thatthey are orbit equivalent if there is a Borel isomorphism f : X → Y suchthat x1 ∼R x2 ⇔ f(x1) ∼S f(x2), and f∗µ is equivalent to (i.e. has thesame nullsets as) ν.

They are isometrically orbit equivalent, or isomorphic, if actually f∗µ = ν.

Two specific papers worth mentioning at this point, illustrating the com-ments above, are the ’77 papers of Feldman and Moore [FM77a,FM77b]. Theprincipal result in in these papers is that, extending the group measure spaceconstruction, every standard equivalence relation quasi-preserving the (prob-ability) measure gives rise to an inclusion of a Cartan subalgebra in a vonNeumann algebra with a faithful normal state, and vice versa.‡ The measure ispreserved if and only if the state is a trace. This is what allows Popa to calculatefundamental groups of certain II1 factors (namely those with a unique Cartansubalgebra, up to conjugacy), using invariants of (SP1) equivallence relations.

Another result from the first of these papers is worth mentioning, justifyinguse of the term orbit for the equivalence classes of a standard equivalnce relation:

1.7 ( [FM77a]) Theorem. Let R be a standard equivalence relation on (X,B, µ).Then there is a countable group G of Borel automorphisms of X, preserving µ,such that R = RG.

Remark. If (X,B, µ) is a standard Borel probability space with an action ofthe countable group G, then RG is an (SP1) equivalence relation on X.

Indeed, this result tells us that it is not unreasonable to expect invariants andconstructions for groups to carry over to (SP1) equivalence relations. Of course,different groups may give rise to the same equivalence relation, so this requiresa bit of “rigidity,” in some sense, of the invariant/construction. In chapters2 and 5 we will do this for ℓ2-homology, and in particular -Betti numbers, to(SP1) equivalence relations, and apply a related invariant - the cost - to group-theoretical questions.

Finally we list the following lemma, which although we shall have no occasionto use it, is of fundamental technical importance both for theoretic results andfor specific calculations (for example of the cost.)

1.8 (Levitt ’95) Lemma. If R is an (SP1) equivalence relation with infiniteorbits a.e. on the standard Borel probability space (X,µ), then there is a se-quence (Bn) of Borel sets satisfying

‡For this to be a reversible construction, one need also keep track of a so called 2-cocycleof the equivalence relation.


• ∀x ∈ X : Bn ∩R[x] 6= ∅.

• µ(Bn) →n 0.

1.9 The reduced ℓ2-homology of group actions on simplicial complexes.In the following paragraphs we shall give a fast account of the “classical”

theory of the ℓ2-homology of group actions on topological spaces. Actually,We couldn’t care less about general topological spaces and shall restrict ourattention to actions on simplicial complexes, since the ℓ2-homology theory ofequivalence relations we develop later is the analogue of this. We also note thatthis is equivalent to considering cellular complexes (see [Luc98]), but again be-cause we shall only work with the analogues of simplicial complexes later, wewill not pursue this further.

The hallmark paper in this area is the 1986 paper [CG86] of J. Cheegerand M. Gromov, in which they define singular ℓ2-cohomology and prove that itvanishes in all dimensions for amenable groups. (The original notion of L2-Bettinumbers comes from geometry and is due to Atiyah, published in a 1976 paper.)

The standard modern day tome of reference is the book [Luc02], which isreally big and scary. The point of Luck’s work on L2-invariants is to put thetheory in a completely algebraic framework. In particular he shows that onecan in fact calculate ℓ2-Betti numbers using the homology group Hn(G,L(G))which offers some advantages over using reduced homology groups as we shallbelow.§

To develop and use ℓ2-cohomology one needs a certain technical requirementin order to get anywhere. This is not required for ℓ2-homology, so we shall stickto this theory. In any case, when cohomology does work, the ℓ2-Betti numbersof each of the two theories coincide.

Definition. • A simplicial complex K is the disjoint union of a sequenceof sets K(0),K(1), . . . , such that

– For every n ≥ 1, K(n) ⊆∏ni=0K

(0).

– Each K(n) is invariant under the canonical action of Sn+1 permutingthe coordinates.

– The coordinates of each n-simplex s ∈ K(n) are pairwise distinct.

– For every n-simplex (v0, v1, . . . vn) ∈ K(n) and every 0 ≤ i ≤ n, wehave (v0, . . . , vi, . . . vn) ∈ K(n−1), this notation signifying the removalof the i’th coordinate so to speak.¶

All simplicial complexes will be countable, so that in particular the topologyon them is just the discrete topology.

The corresponding unordered simplicial complex is the disjoint union ofthe sets Sn+1 \K(n).

§One disadvantage, though, is that vanishing of ℓ2-Betti numbers does not imply vanish-ing of these homology groups, but this can in fact be corrected for by considering differentcoefficients. But I digress.

¶That is, (v0, . . . , bvi, . . . vn) = (v0, v1, . . . vi−1, vi+1, . . . vn).


• A simplicial map f : K → L is a map between simplicial complexes map-ping simplices to simplices, but not necessarily of the same dimension.If f is of the form‖ f : (v0, v1, . . . , vn) 7→ (f(v0), f(v1), . . . , f(vn)) thenit is a simplicial homomorphism, and if it is also invertible (the inverseautomatically has the same form) then it is a simplicial isomorphism.

• For a simplicial complex K and for each n ≥ 0, the n’th chain groupCn(K) is defined as the free abelian group on the n-simplicies in K, i.e.on K(n), modulo the relations

(v0, v1, . . . vn) = sign(σ)(vσ(0) , vσ(1), . . . , vσ(n)), σ ∈ Sn+1. (1.1)

1.10 The chain complex of a simplicial complex. Let K be asimplicial complex and consider for n ≥ 0 the Z-linear maps∗∗

∂n : Z(K(n)) → Z

(K(n−1))

(induced by) : (v0, v1, . . . , vn) 7→n∑

i=0

(−1)i(v0, . . . , vi, . . . , vn)

Observation. The maps ∂n preserve the relations defining the chain groups,and thus induce homomorphisms ∂n : Cn(K) → Cn−1(K).

To see this, note that to permute

(vσ(0), . . . , vσ(i)︸︷︷︸i’th spot

, . . . , vσ(n))

back into(v0, v1, . . . , vσ(i)︸︷︷︸

σ(i)’th spot

, vσ(i)+1, . . . , vn)

we neeed to apply a permutation in Sn which is either the restriction of σ−1, inthe case where σ(i) = i, or the restriction of (σ(i) − 1 σ(i)) · · · (i+ 1 i+ 2)(i+1 i)σ−1 in the case where σ(i) > i, and similarly for σ(i) < i.

This gives a difference in sign if and only if one of i, σ(i) is odd, the othereven, which is exactly what we want.

Furthermore, if v = vk is a vertex in the simplex s = (v0, v1, . . . , vn), thendenoting by ∐v the operation

∐v(v0, . . . , v︸︷︷︸k’th spot

, . . . , vn) 7→ (−1)k(v0, . . . , v, . . . , vn)

we see that∐vj

∐vis = −∐vi

∐vjs,

‖This is equivalent to f mapping simplices to simplices of the same dimension, commutingwith boundary maps and the action of the Sn+1 in every dimension.

∗∗It is implicit in definition that in particular ∂0 = 0 into C−1(K) = 0.


owing simply to the fact that (if, say i > j) on the right-hand side, removingthe j’th vertex shifts the i’th one spot down (similarly for the left-hend side ifi < j). Since then

∂n−1∂ns =∑

0≤i6=j≤n

∐vi∐vj

s = 0

we have made the following

Observation. For every n ≥ 1, ∂n∂n−1 = 0, whence the chain groups of Kconstitute a chain complex of abelian groups

· · · ∂3 // C2(K)∂2 // C1(K)

∂1 // C0(K)∂0 // 0

1.11 Actions by groups and the ℓ2-chain complex. Let G be a count-

able group acting on K by simplicial automorphisms. Then ZG acts on Z(K(n))

by shifting simplicies, i.e.

((n∑

i=1

aigi

).f

)(s) =

n∑

i=1

aif(g−1i .s), f ∈ Z

(K(n)), s ∈ K(n).

This clearly preserves the relations (1.1) and so induces an action of ZG onCn(K). Suppose that G acts freely on Sn+1 \ K(n) for all n, and considerthe tensor product ℓ2(G)⊗ZGCn(K) where ZG acts on ℓ2(G) from the right by(f.g)(h) = f(hg−1). Choose a fundamental domain for the action ofG×Sn+1 onK(n) and a labelling sjj∈I (note that I is countable, so we may, for notationalconvenience, take it to be a hereditary subset of N)

Observation. By freeness of the action, each element s in the abelian groupℓ2(G) ⊗ZG Cn(K) can be written uniquely as a sum s =

∑nk=1 fk ⊗ sjk

.

Now, ℓ2(G) ⊗ZG Cn(K) is given a vector space structure by letting R actby multiplication on ℓ2(G), this obviously commuting with the action of ZG.Further, the observation above allows us to define an inner product on ℓ2(G)⊗ZG

Cn(K) by

〈m∑

k=1

fk ⊗ sjk,

n∑

l=1

gl ⊗ sjl〉 =

∑

k,l

〈fk, gl〉δsjk,sjl

where the δ is Kronecker δ.

Definition. The n’th ℓ2 chain space C(2)n (K;G), or just C

(2)n (K) when G is

understood, is the completion of ℓ2(G) ⊗ZG Cn(K) in the norm induced by theinner product defined above.

Thus it is a real Hilbert space. Furthermore, since the action of L(G) onℓ2(G) commutes with the right action of ZG, it induces an action on ℓ2(G)⊗ZG

Cn(K) which extends by continuity to an action on C(2)n (G). Further, if we


denote by (⊕j∈I)ℓ2(G) the space of finite sums fj1 ⊕fj2 ⊕· · ·⊕fjl

∈ ⊕j∈Iℓ2(G),

then the map

in : (⊕

j∈I)ℓ2(G) → C(2)

n (K)

fj1 ⊕ fj2 ⊕ · · · ⊕ fjl7→ fj1 ⊗ sj1 + fj2 ⊗ sj2 + · · · + fjl

⊗ sjl

is (linear,) isometric and L(G)-equivariant, and thus extends to an L(G)-equivariant

isometric isomorphism in : ⊕j∈Iℓ2(G) → C

(2)n (K) of real Hilbert spaces.

In particular, the specific choice of fundamental domain sjj∈I above isirrelevant, since the ℓ2 chain spaces produced by two different choices are L(G)-equivariantly isometrically isomorphic.

If τ is a faithful normal tracial state on L(G), then by uniqueness of thestandard form

⊕

j∈N

ℓ2(L(G), τ)L(G)-equivariant

≃⊕

j∈N

ℓ2(G)

and so we have shown the following

1.12 Theorem. For every n ≥ 0, C(2)n (K) is a L(G)-module, i.e. it is a real

Hilbert space and embeds isometrically and L(G)-equivariantly in ⊕j∈Nℓ2(L(G), τ)

for any faithful normal trace τ .

1.13 The L(G)-dimension of an L(G)-module. Suppose that H is anL(G)-module and consider two embeddings

ik : H →⊕

j∈N

ℓ2(L(G), τk), k = 1, 2.

In each case, ikH is a closed L(G)-invariant subspace of ⊕j∈Nℓ2(L(G), τk). Thus

πjPikHπj is in L(G)′ ∩B(ℓ2(L(G), τk)), where πj is projection on the j’th coor-dinate. We claim that

Tr1(Pi1H) :=

∞∑

j=1

τ ′1(πjPi1Hπj) =

∞∑

j=1

τ ′2(πjPi2Hπj) =: Tr2(Pi2H),

where τ ′k is the canonical trace on L(G)′ ∩ ℓ2(L(G), τk) induced by the modularconjugation. Indeed, this is nothing but

τ ′k(A) = 〈A1L(G),1L(G)〉ℓ2(LR(G),τk),

so if U : ℓ2(L(G), τ1) → ℓ2(L(G), τ2) is the L(G)-equivariant isometrically iso-morphic equivalence of the standard form uniqueness theorem and U0 = ⊕j∈NU ,then for all j we have U−1πjPi2HπjU ∈ L(G)′ ∩ B(ℓ2(L(G), τ1)) and

τ ′1(U−1πjPi2HπjU) = τ ′2(πjPi2Hπj).

Further, U−10 Pi2HU0 is the orthogonal projection onto H ′ := U−1

0 i2H and bythe composition i1i

−12 U0 this is L(G)-equivariantly isometrically isomoprhic to


i1H , whence the two projections are Murray-von Neumann equivalent, and theclaim follows since now

Tr1(Pi1H) = Tr1(PH′ ) = Tr1(U−10 Pi2HU0) =

∑

j∈N

τ ′1(πjU−10 Pi2HU0πj)

and using that U0 is diagonal, this is

=∑

j∈N

τ ′1(U−1πjPi2HπjU) =

∑

j∈N

τ ′2(πjPi2Hπj) = Tr2(Pi2H).

Thus we have shown

1.14 Theorem. If H is a L(G)-module, there is a well defined L(G)-dimensionof H, given by

dimL(G)H =∑

j∈N

τ ′(πjPiHπj)

=∑

j∈N

〈πjPiH |j’th summand1L(G),1L(G)〉ℓ2(L(G),τ),

where i is any embedding i : H → ⊕j∈Nℓ2(L(G), τ) for a faithful normal tracial

state τ on L(G), πj is projection onto the j’th coordinate, and PiH the orthogonalprojection onto iH.

1.15 Reduced ℓ2-homology of group actions. Suppose that the actionof G on K is cocompact, i.e. that for every n ≥ 0 the orbit space G \K(n) isfinite.

Note first that the boundary maps ∂n : Cn(K) → Cn−1(K) are ZG-equivariant,and hence we may extend these to linear maps on ℓ2(G) ⊗ZG Cn(K) in the ob-vious manner:

∂(2)n : ℓ2(G) ⊗ZG Cn(K) → ℓ2(G) ⊗ZG Cn−1(K)

f ⊗ s 7→ f ⊗ (∂ns).

It is clear that the ∂(2)n are also L(G)-equivariant.

1.16 Proposition. Keeping in mind that the action is cocompact, the maps

∂(2)n are bounded.

Proof. By cocompactness and the freeness assumption, the number of sum-mands of ∂nsj in any given G×Sn-orbit in K(n−1) is bounded, and this in turngives a bound on the norm, using the geometric-algebraic inequality. (See alsothe proof of proposition 1.2) ////

Thus the ∂(2)n ’s extend by continuity to L(G)-equivariant, bounded linear

maps ∂(2)n : C

(2)n (K) → C

(2)n−1(K).

Again, ∂(2)n−1∂

(2)n = 0, so that we have a chain complex of L(G)-modules.

· · · ∂(2)3 // C

(2)2 (K)

∂(2)2 // C

(2)1 (K)

∂(2)1 // C

(2)0 (K)

∂(2)0 // 0


1.17 Definition. When the action of G is cocompact (still free as above), wedefine for each n ≥ 0 the n’th reduced ℓ2-homology of the pair (K,G) to be theHilbert space

H(2)

n (K,G) := ker ∂(2)n

/Im ∂

(2)n+1 .

Via. the canonical embedding in ker ∂(2)n and the L(G)-equivariance of ∂n+1,

this is an L(G)-module. We define the n’th ℓ2-Betti number

β(2)n (K,G) := dimL(G)H

(2)

n (K,G).

If the action is not cocompact, the boundary maps will not all be boundedand we run into trouble. To remedy this, we shall in this case define the ℓ2-homology as a direct limit of homologies of subcomplexes on which the actionis cocompact. This construction (rather its analogue for cohomology) is thecentral technical point of the infamous paper [CG86].

First we note that if we have an inclusion K0 ⊆ K of simplicial complexeswith a (cocompact, free) action of G, with K0 invariant, we get an inclusion ofchain groups j : Cn(K0, G) → Cn(K,G) for all n. Since j is G-equivariant this

gives an inclusion j : C(2)n (K0, G) → C

(2)n (K,G), and since j commutes with the

boundary maps, a map (no longer necessarily an inclusion) j : H(2)

n (K0, G) →H

(2)

n (K,G). This map is continuous, but not necessarily closed.Suppose that the action of G on K is free, fix n and denote by C(K)

the (small) category with objects H(2)

n (Ka, G) for G-invariant subcomplexesKa ≤ K on which the restricted action is cocompact, and morphisms the maps

ja,b : H(2)

n (Ka, G) → H(2)

n (Kb, G) for Ka ⊆ Kb, induced by this very inclusion.The direct limit is then a vector space, and one can equip this with the final

topology, i.e. the strongest topology making all the maps ja : H(2)

n (Ka, G) →lim→H

(2)

n (Kb, G) continuous. This is not in general an L(G)-module, exactlybecause you cannot induce an inclusion in ⊕i∈Nℓ

2(G) by lack of closedness ofthe images of the ja,b. In fact, though it has a topology, it need not even be atopological vector space.

By equivariance of the ja,b one can, however, equip the direct limit with anaction of G, however we shall not really use this for anything.

1.18 Definition. For an action (still free in the sense above) of G on K, then’th reduced ℓ2-homology is the space

H(2)

n (K,G) := lim→H

(2)

n (Ka, G).

Further, considering†† Imja,bH(2)

n (Ka, G) as an L(G)-submodule of H(2)

n (Kb, G),we denote its L(G)-dimension by ∇n(Ka,Kb). Then the n’th ℓ2-Betti numberis defined by

β(2)n (K,G) := supinf∇n(Ka,Kb) | Ka ⊆ KbKa.

††Here and in the sequel, Im f means the closure of the image of f .


Remark. • Note that ∇n(Ka,Kb) is decreasing in b for fixed a, and in-creasing in a for fixed b (whence also infb ∇n(Ka,Kb) is increasing in a.)

• In case the action of G on K is cocompact, K is a final object of C(K)and so the two definitions of homology coincide. So do the two definitionsof Betti-numbers, by the remark above.

• One reason we will be more interrested in the ℓ2-Betti numbers than in theactual homology groups, is that they are easier to compute. In fact, onehas the following striking

1.19 (essentially [CG86]) Theorem. If (Kk)k∈N is an exhausting se-quence of G-invariant subcomplexes of K with the restricted action cocom-pact, then

β(2)n (K,G) = lim

k∈N

liml≥k

∇n(Ka,Kb).

It is, furthermore, clear that such a sequence always exists.

A priori, H(2)

n (K,G) is just a vector space with a topology and an action ofG. However, it could happen that it is in fact an L(G)-module.

1.20 Lemma. In case H(2)

n (K,G) is an L(G)-module, the n’th ℓ2-Betti number

β(2)n (K,G) coincides with the L(G)-dimension dimL(G)H

(2)

n (K,G).

Proof. Denote by ja the map ja : H(2)

n (Ka, G) → H(2)

n (K,G). Then, since thelatter is an L(G)-module, it makes sense to talk about the L(G)-dimension of

the image Im ja. In fact, considering the commutative diagram below, withKa ⊆ Kb,

H(2)

n (Ka, G)ja,b //

ja ''OOOOOOOOOOOH

(2)

n (Kb, G)

jbwwooooooooooo

H(2)

n (K,G)

we have

dimL(G) ker ja,b + dimL(G) Im ja,b = dimL(G)H(2)

n (Ka, G)

= dimL(G) ker ja + dimL(G) Im ja.

Hence, what we need to show is that ker ja = ∪b≥a ker ja,b, but this is true bythe very construction of the direct limit, even without taking the closure. ////

Remark. The proof of theorem 1.19 is actually somewhat similar, with someextra technical details needed.

The ℓ2-Betti numbers of G are defined in terms of a special simplicial com-plex (actually class of complexes), namely the classifying complex of G. This isbasically the usual construction from topology, adapted slightly to the case at


hand (see the remark at the end of the intermezzo below.) This is recalled inthe following

1.21 INTERMEZZO: The classifying simplicial complex of a group.

To develop the ℓ2-theory of group actions, it will be useful to have some kindof space to classify all (simplicial) actions of G.

Definition. Let G be a countable group.

• Let K,L be simplicial complexes, equiped with actions of G, and considerthe corresponding chain complexes

K : · · · ∂n+1// Cn(K)∂n // · · · ∂2 // C1(K)

∂1 // C0(K)∂0 // 0

L : · · · ∂n+1 // Cn(L)∂n // · · · ∂2 // C1(L)

∂1 // C0(L)∂0 // 0

Then we say that the G-equivariant f, g : K → L chain morphisms areG-(chain-)homotopic if there is a sequence of G-equivariant maps (i.e.morphisms)

Dn : Cn(K) → Cn+1(L), n ≥ 0

(we take D−1 = 0 from 0 into C0(L)) such that

∂n+1Dn +Dn−1∂n = gn − fn, n ≥ 0.

Observe that this defines an equivalence relation on the set of G-equivariantmorphisms K → L .

• If EG is a simplicial complex with a free action of G, we say that EGis a model of the classifying (space) simplicial complex of G, if for everysimplicial complex K with a free action of G there is a unique, up toG-homotopy, injective simplicial homomorphism f : K → EG.

Remark. Note that, in the definition of EG, the uniqueness of f up to ho-motopy really concerns uniqueness up to chain homotopy, of the induced chainmap.

With respect to terminology, we will use the term “classifying complex,” orwe may actually abuse this and say something like “let K be a model for EG,”using EG here not with reference to a specific model but to the class of modelsof the classifying complex.

A priori it is not clear that a model for EG always exists. To construct it,we need the notion of taking the Milnor join of simplicial complexes.

Definition. Let K,L be simplicial complexes. The join, K∗L, is the simplicialcomplex defined by (K ∗ L)(0) = K(0)∪L(0) and for −1 ≤ m,n, m + n ≥ 0 we


let for v0, . . . , vm ∈ K, u0, . . . , un ∈ L

(v0, . . . , vm, u0, . . . , un) ∈ (K ∗ L)(m+n+1) ⇔

v0, . . . , vm ∈ K(m)uo

u0, . . . , un ∈ L(n)uo

(this does not exclude the possibility that m = −1 (n = −1) so that the set ofvi’s (ui’s) is empty.)

This is clearly a simplicial complex, and if the countable group G acts onK,L, then it also acts on K ∗L, the action being free if and only if both actionson K and L are free.

1.22 Theorem. For every countable group G, a model for EG exists.

Proof. Consider the simplicial complex E0 with (E0)(k)uo the set of k + 1-point

subsets of G such that no subset (i.e. no subset of the k+1-point subset) is fixed

by the left action of any non-identity element of G. (In particular, E(0)0 = G.)

Now let E0G be the infinite join of copies of E0, either defined by the obviousgeneralization of the definition above, or as the inductive limit of the system offinite joins. Alternatively, we can actually write this down explicitly: Namelywe have E0G

(0) = G × Z, and the higher dimensional simplices consist of setsof points such that the projection on each coordinate is not invariant under leftmultiplication by any non-identity element of G.

That any simplicial complexK on whichG acts freely embeds in the requiredmanner is easy, requiring simply a choice of fundamental domain for the actionof G on K(0). Note though, that this is of course not canonical.

Now let f, g be two such embeddings (i.e. injective simplicial homomor-

phisms), and let s(0)i be a fundamental domain for the action on K(0), and

let for each i, D0s(0)i be a path in (EG)(1) from f(s

(0)i ) to g(s

(0)i ) (such a path

clearly always exists.)We extend D0 to a map D0 : C0(K) → C1(E0G) by ZG-linearity. Then by

construction, for any v ∈ K(0)

(∂1D0)(v) = g0(v) − f0(v).

Suppose that we have defined D0, . . . , Dn−1 such that for k = 0, . . . , n− 1

∂k+1Dk +Dk−1∂k = gk − fk.

Choose a fundamental domain s(n)i for the action of G×Sn+1 on K(n) and fix

i for convenience. Write Dn−1(∐js(n)i ) = t

(j)1 + · · ·+ t

(j)lj

for j = 0, . . . , n (recall

that ∐j(v0, . . . , vm) = (−1)j(v0, . . . , vj , . . . , vn).)Then

∂n(n∑

j=0

t(j)1 + · · · + t

(j)lj

) = (∂nDn−1∂n)(s(n)i )

= gn−1(∂ns(n)i ) − fn−1(∂ns

(n)i ) − (Dn−2∂n−1∂n)(s

(n)i )

= ∂n

(gn(s

(n)i ) − fn(s

(n)i )).


Now the simplices in this equation contain only finitely many vertices inE0G

(0), and so for M sufficiently large (i.e. large enough that none of thesevertices have second coordinate M) and putting ρ = (1,M) ∈ E0G

(0), wedefine (here, ρ ∨ (v0, . . . , vm) denotes the simplex (ρ, v0, . . . , vm))

Dn(s(n)i ) = ρ ∨

gn(s

(n)i ) − fn(s

(n)i ) −

n∑

j=0

t(j)1 + · · · + t

(j)lj

.

This is clearly well-defined and satisfies the required equation at s(n)i , using the

identity

∂

(ρ ∨

∑

ν

rν

)=∑

ν

rν , when∑

ν

∂rν = 0. (1.2)

Finally, proceeding analogously for the other s(n)k (i was arbitrary) and extend-

ing by ZG-linearity, we get a Dn satisfying the required equation, and applyinginduction we are done. ////

Remark. • The uniqueness part only uses that E0G is contractible.

• We will refer to the specific model for EG constructed in the proof as thestandard model for EG, and will still denote it E0G.

• The construction of E0G may deviate from the reader’s expectations ofthe usual construction of the classifying space‡‡ in that E0 is not simplythe zero-dimensional complex G. This is due to the fact that, say, linesbetween vertices in the same orbit can appear, and with our definition ofsimplicial homomorphisms not easily gotten rid of. One can, however,change K so that this does not happen, without changing the homology, sothat this is more of a technical glitch than an actual difference.

END INTERMEZZO.

1.23 Definition. For a discrete group G, the n’th ℓ2-Betti number is definedas

β(2)n (G) := β(2)

n (K,EG)

where EG is any model for the classifying complex for G (note that this isactually well-defined, though we leave out the proof.)

We state without proof the following baby version of the ℓ2-Hogde-de Rahmtheorem, showing how to calculate the first ℓ2-Betti number of finitely generatedgroups. The proof is not particularly hard but it requires machinery able tohandle cellular complexes with non-free actions - machinery that we don’t have.The theorem makes an appearance in my exposition of the result of Monod-Epstein in chapter 5.

‡‡Given how “usual” this is, it is amazingly hard to actually track it down in the litterature.


1.24 Theorem. Suppose that G is a (countable) finitely generated group, S afinite symmetric generating set and G = (V,E) the Cayley graph of G wrt. S.Then

β(2)1 (G) = dimL(G)

(ℓ2⋆(E) ⊕ ℓ2(E)

)⊥.

Remark. Note that in the case where G is finite, this just reduces to thestatement that the first ℓ2-homology is trivial (see theorem 1.4.)

Chapter 2

A tutorial on invariants of

(SP1) equivalence relations

The second rule about fight club is you do not talk about fightclub.

key words:ℓ2-Betti numbers - cost.

In this chapter I give a tutorial on the invariants of (SP1) equivalence rela-tions, the cost respectively the ℓ2-Betti numbers, exposed by Gaboriau in hispapers [Gab00] respectively [Gab02].

Note that it is a tutorial, not a full-blown exposition. This is due mainlyto the fact that the two Gaboriau papers are so well written and self-containedthat, without anything significant to add, this is neither the time nor the placeto go through it all in detail. Instead I will simply define the concepts in questionand rehash some (not all) of the main properties and results. Defining the costis no big deal, but defining the ℓ2-Betti numbers takes some work.

I shall give a presentation focusing on the definition of ℓ2-Betti numbers,and define the cost in this high-brow context. This probably discards some ofthe “flavour” of the cost, but we will just have to live with that.

Naturally I had to leave out a lot of interesting stuff, in particular calcu-lations and constructions from the cost paper [Gab00], and from the ℓ2 pa-per [Gab02] especially the omission of the ME (measure equivalence) stuff isa shame, as ME seems to be a fairly hot topic these days. The prime crimethough is probably leaving out the theorems concerning induced representa-tions [Gab00]II.6 [Gab02]5.5.

2.1 A generalized definition of simplicial complex. In the follow-ing, we fix some (SP1) equivalence relation R on a standard Borel probabilityspace (X,B, µ). We denote by πl respectively πr the projections R → X ontothe first coordinate respectively the second.

25

CHAPTER 2. INVARIANTS OF (SP1) EQUIVALENCE RELATIONS 26

Definition. • A standard Borel space V is called a fibred space over X ifit is equiped with a Borel map π : V → X such that the fiber π−1(x) overeach point x ∈ X is countable. Actually, what is called a fibred space isthe pair (V, π) but we may leave out π if it is understood.

In this case, there is a canonical measure ν on V on V defined as countingmeasure along each fiber:

ν(C) =

∫

X

♯(π−1(x) ∩C)dµ.

• The fibred space V is called a (left) R-space if, for every (x, y) ∈ R thereis a (Borel) map (x, y).· : π−1(y) → π−1(x) such that

∀x ∈ X∀v ∈ π−1(x) : (x, x).v = v,

∀x ∼ y ∼ z ∈ X∀v ∈ π−1(z) : (x, y). ((y, z).v) = (x, z).v.

We emphasize that π is not required to be surjective.

• The R-space V is called discrete∗ if it has a Borel fundamental domain.

Example. R itself, with π = πl, is a discrete R-space, the action being theobvious one: (x, y).(y, z) = (x, z). The fundamental domain is nothing but thediagonal.

Definition. Let (V, π1) and (U, π2) be fibred spaces over X. The fibred productV ∗ U of V and U is the set

V ∗ U = (v, u) ∈ V × U | π1(v) = π2(u).

It is itself a fibred space over X with map π(v, u) = π1(v) = π2(u).

If both V and U in the definition above are R-spaces, then so is V ∗U withthe diagonal action. Further, if V is discrete with fundamental domain F , thenV ∗U is discrete with fundamental domain F ∗U , and similarly if U is discrete.

The following example explains the choice of the terminology “discrete” andsays just about everything there is to say about finite (SP1) equivalence rela-tions.

2.2 Example. Notice that X is itself an R-space with map π the identity. Itis discrete if and only if (a.e.) the orbits of R are finite.

The point here is that X splits into a disjoint union X = (∪n∈NXn) ∪Xℵ0

of sets such that the orbit of each x ∈ Xn has cardinality n.To get this splitting, consider the fibred product Rn = R ∗ R ∗ · · · ∗ R with

n ∈ N factors. Then the map π : Rn → X is countable-to-one, and Borel,whence it maps Borel sets to Borel sets (Rn splits into a disjoint union of Borelsets such that π restricted to each is injective whence maps Borel sets to Borelsets.)

∗these days apparently also smooth.


Now put Xn = π(Rn \ ∆n) where ∆n is the diagonal set

∆n =

((x, x1 = x), (x, x2), . . . , (x, xn)) ∈ R ∗ R ∗ · · · ∗ R︸︷︷︸

n factors

| ∃i 6= 1 : xi = x

.

Then Xn is the Borel set of points x ∈ X such that the orbit of x has cardinalityat least n, and taking set differences gives the splitting above.

Thus, if the orbits are finite, we may assume they all have the same length,say n > 1. Let ϕ : X → [0, 1] be a Borel isomorphism. Then the map

ϕn : Rn → [0, 1]n

((x, x1), (x, x2), . . . , (x, xn)) 7→ (ϕ(x1), ϕ(x2), . . . , ϕ(xn))

is Borel, and so the inverse image of the Borel set of points in [0, 1]n with firstcoordinate strictly larger than the rest is Borel. Let U be the intersection of thisset with the Borel subset of points in Rn satisfying x = x1. Then U is Borelwhence so is π(U), and this is clearly a fundamental domain.

On the other hand, suppose that we have some Borel fundamental domainF . Let S(x) =

∑n∈N

χXn(x), x ∈ X. Then S is measurable, and

∫

F

Sdµ =∑

n∈N

µ(Xn) ≤ µ(X) = 1,

so that S is finite a.e.This actually uses that nµ(Xn ∩ F ) = µ(Xn). This identity is shown by

noting that Xn ∩F is a fundamental domain of the relation induced on Xn andthat the induced relation on Xn \ F has a fundamental domain D by the above,and finally that there’s a partial Borel isomorphism with domain F and rangeD. In notation above, this would correspond, for instance, to the flip in Rn

interchanging the first 2 coordinates etc.Hence R has only finite orbits.

2.3 Definition. • A simplicial R-complex Σ is a disjoint union of R-spaces Σ(0),Σ(1), . . . such that

– Σ(0) is discrete and Σ(n) ⊆ Σ(0) ∗ Σ(0) ∗ · · · ∗ Σ(0)︸︷︷︸

n+1 factors

for all n ≥ 0, and

is furthermore invariant under the diagonal R-action.

– Each Σ(n) is invariant under the canonical action of Sn+1 permutingthe coordinates.

– The coordinates of each n-simplex s ∈ Σ(n) are pairwise distinct.

– If (x0, x1, . . . , xn) ∈ Σ(n), then (x0, . . . , xi, . . . , xn) ∈ Σ(n−1).

• Σ is called uniformly locally bounded, abreviated ULB, if the fundamentaldomain of Σ(0) has finite measure, and if there is some N ∈ N such thatfor each point v ∈ Σ(0) there are at most N simplicies is Σ containing v.


• For every x ∈ Σ, the disjoint union of the sets π−1n (x) ∩ Σ(n) (where π0

is the map in the definition of Σ(0) as a fibred space and the πn are thecorresponding maps on the fibred products) is a simplicial complex, denotedΣx.

We say that Σ is n-connected, contractible etc. if the property in questionholds for Σx a.e.

2.4 Representations and chain spaces. † We shall now define the chainspaces of a ULB simplicial R-complex and describe how to attach a dimensionto a subspace of one of these.

First of all, if (U, π) is a discrete R-space with fundamental domain F , thenwe have the following “representation” of R:

For every Borel automorphism α of X , with graph contained in R, we geta unitary operator Lα on L2(U, ν) (recall that ν is the canonical measure on Udefined above), defined by (Lαη)(u) = η((α−1(π(u)), π(u)).u). We show thatit is (real) unitary by calculating the adjoint: (the same calculation also showsthat Lα is well-defined and bounded)

〈Lαξ, η〉 =

∫

X

∑

u∈π−1(x)

ξ((α−1(x), x).u)η(u)dµ(x)

=

∫

X

∑

u∈π−1(α−1(x))

ξ(u)η((x, α−1(x)).u)dµ(x)

=

∫

X

∑

u∈π−1(x)

ξ(u)η((α(x), x).u)dµ(x)

= 〈ξ, Lα−1η〉,

so that L∗α = Lα−1 from which the claim follows. In the calculation we used first

that (y, x).· is a bijection, the inverse being (x, y).·, and then that α preservesµ.

Further, for φ ∈ L∞(X,µ) we define a bounded operator Lφ on L2(U, ν) by(Lφη)(u) = φ(π(u))η(u).

In particular, the von Neumann algebra L(R) associated to R by Feldman-Moore is the one generated by this representation of R on L2(R, ν) and thetrace is just τ(L) = 〈Lχ∆, χ∆〉, where χ∆ is the characteristic function of thediagonal in X ×X .

The construction above then gives a representation of L(R) on L2(U, ν). Fur-ther, this actually makes L2(U, ν) an L(R)-module. Indeed, since π is countable-to-one, we can write F as a (countable) disjoint union F = ∪iFi such that π isinjective on each Fi. With ∆i = (π(f), π(f)) | f ∈ Fi ⊆ R, we have

L2(U, ν) = L2(∪iR.Fi, ν) ≃⊕

i

L2(R.Fi, ν|R.Fi).

†Below, we abuse notation a bit and write ν for several different measures. They are allthe measures induced on the relevant fibred spaces by µ, so this should not create too muchconfusion.


As is seen from the definitions, each summand L2(R.Fi, ν|R.Fi) is L(R) invari-

ant, and equivariantly, isometrically isomorphic to L2(R.∆i, ν|R.∆i), itself an

invariant subspace of L2(R, ν). This shows the claim.Now, suppose that Σ is a simplicial R-complex and fix some n. Choose a

fundamental domain D for the action of R× Sn+1 on Σ(n). One way to do thisis to note that we have a fundamental domain for the action of R by definition,and then use a Borel isomorphism ϕ : X → [0, 1] to order the points in eachfiber π−1(x) in Σ(0).‡

Then we can retain only the simplices with, say, strictly increasing points toget D. Now, write as usual D as a disjoint union D = ∪iDi with πn injectiveon each Di. Then put for every i and x ∈ X

si(x) =

πn|−1

Di(x) , x ∈ πn(Di)

0 , otherwise∈ C(2)

n (Σx).

Finally, choose some countable group G of automorphisms of X such that R =RG.

Then, if we define (Lαsi)(x) = si(α−1(x)), we have that for each x ∈ X , the

rational linear combinations of the (Lαsi)(x)α∈G,all i constitute a dense set

in C(2)n (Σx). This allows us to define the n’th ℓ2-chain space of Σ as the direct

integral

C(2)n (Σ) :=

∫ ⊕

X

C(2)n (Σx,1)dµ(x),

using the family Lαsi as the fundamental family.Now, there is an isometric isomorphism L2(Σ(n)/Sn+1, ν) = L2(R[D], ν) ≃

C(2)n (Σ), almost by definition. Indeed, suppose that s ∈ πn|−1

R[D](x). Then if

η ∈ L2(R[D], ν) we put[(Uη)(x)] (s) = η(s).

This uniquely determines an element Uη in C(2)n (Σ), and it is easy to see that

U is a (real) unitary.

Thus we can import an L(R)-module structure to C(2)n (Σ). Further,

Sn+1[D] =⋃

i

⋃

σ∈Sn+1

σ.Di

is a fundamental domain for the action of R on Σ(n) and since the σ.Di have thesame measure for different σ, the following lemma follows directly by applyingsuccessive embeddings.

2.5 Lemma. If E is an L(R)-invariant subspace of C(2)n (Σ), then

dimL(R)E =∑

i

〈Psi, si〉C(2)n (Σ)

,

where P is the orthogonal projection onto E.

‡If π(u) = π(v) = x and (y, x).u, (z, x).v ∈ fundamental domain for Σ(0) we order u, v

according to the order of y, z, as prescribed by ϕ.


Note that since the L(R)-dimension is independent of the embedding cho-sen, the formula above is independent of the choice of the si, in particular onthe choice of fundamental domain.

2.6 Basic definitions. We work still with the same objects as above - wehave some (SP1) equivalence relation R on a standard Borel probability spaceX , a simplicial R-complex Σ, and some fixed but arbitrary n ≥ 0.

Now, for each x ∈ X , we have a boundary map ∂(2)n,x : C

(2)n (Σx,1) →

C(2)n−1(Σx,1). If Σ is ULB, then these are uniformly bounded (indeed, this

is the whole point of the definition of being ULB) and so we can integrate§ to

get a boundary map ∂(2)n : C

(2)n (Σ) → C

(2)n (Σ), and it is not hard to see that

this is L(R)-equivariant.

Definition. • The n’th reduced ℓ2-homology of the pair (Σ,R), where Ris an (SP1) equivalence relation on (X,µ) and Σ a ULB simplicial R-complex, is the Hilbert space

H(2)

n (Σ,R, µ) = ker ∂(2)n /Im∂

(2)n+1.

By the usual embedding of this as ker ∂(2)n ⊖ Im ∂

(2)n+1 it is an L(R)-module,

and the n’th ℓ2-Betti number of the pair (Σ,R) is exactly its dimension:

β(2)n (Σ,R, µ) = dimL(R)H

(2)

n (Σ,R, µ).

• Further, this is extended to the case where Σ is not ULB by taking directlimits and dimensions of embeddings exactly as in the classical case. Weshall not even bother writing it down!

The homology and Betti numbers actually do depend on µ, which is notunique in its equivalence class in general, and so it is slightly abusive to say“...of the pair...” but the particular µ is usually either understood or irrelevant(as in statements of general theorems, for instance), so we shall stick to thisterminology.

We remark that, at first glance the class of ULB simplicial R-complexes mayseem smaller somehow than the class of simplicial complexes with cocompactG-action, but this is an illusion, as we shall see next.

Indeed, let K be a simplicial complex and G a countable group acting freelyon it (that is, it acts freely onK(n)/Sn+1) by simplicial automorphisms. Supposealso that G acts freely, preserving the measure on the standard Borel probabilityspace (X,µ).

Then we can define a simplicial RG-complex ΣK as follows: We let (ΣK)(0) =X ×K(0) with map π : X ×K(0) → X the projection on the first coordinate,and with action by RG given by (g.x, x).(x, v) = (g.x, g.v). Further, (ΣK)(n)

§One needs of course to first note that this will be well defined, i.e. that it sends measurablefields to measurable fields, at least densely, but this is clear. In fact, the secret here is to useonly fundamental domains of the form F ∗ Σ(0) ∗ · · · ∗ Σ(0) ∩ Σ(k).


is the obvious embedding of X × K(n) in the fibred product - i.e. we embed(x, (v0, v1, . . . , vn)) 7→ ((x, v0), (x, v1), . . . , (x, vn)).

It is clear that this is a simplicial RG-complex, the fundamental domain of(ΣK)(0) being X × F with F a fundamental domain for the action of G on K.

( [Gab02]) Lemma. Suppose that K is a simplicial complex with a free actionof G, and that (X,µ) is a standard Borel probability space, also with a freemeasure-preserving action of G. Then the action of G on K is cocompact if andonly if ΣK is ULB.

Proof. Suppose first that the action is cocompact. Then in particular the fun-damental domain F for the action of G on K(0) is finite, so that the measure ofX × F is finite (indeed it is exactly ♯F .)

Also, if v ∈ K(0), then v is a vertex of only finitely many simplices. For ifnot, this implies that there are infinitely many lines (v, un), n ∈ N containingit, and no three of these can be in the same orbit, this requiring an elementof G leaving v fixed, contradicting the freeness assumption. The same is thentrue of (x, v) for any x, and it is an easy observation that this argument appliesuniformly so that ΣK is ULB.

On the other hand, if ΣK is ULB, let F be a fundamental domain of (ΣK)(0)

with finite measure. Suppose that G \ K(0) is infinite and let (vi)i∈N be afundamental domain. Then E = ∪i∈NX × vi is a fundamental domain for(ΣK)(0). Now put

Ei,j = (x, vi) ∈ X × vi | (gj .x, gj .vi) ∈ F,where G = gjj∈N. Then

X × N =⋃

i,j∈N

Ei,j , and F =⋃

i,j∈N

gj.Ei,j ,

contradicting finiteness of the measure, since the action of G on (ΣK)(0) pre-serves the measure.

Thus, there is a fundamental domain H = (ui)i∈I for the action of G on K(0)

such that I is a finite set. Then H ×K(0) × · · · ×K(0)

︸︷︷︸n factors

∩K(n) is a fundamental

domain for the action of G on K(n), and it is finite by the ULB assumption,whence the action is cocompact. ////

2.7 Basic theorems and connections with the classical theory.The first results we recount tell us that everything is as it should be in

that we really do extend the ℓ2-Betti numbers from a group setting to an (SP1)setting. Firstly, the ℓ2-Betti numbers are again computable by countable means:

( [Gab02]) Theorem. Suppose that R is an (SP1) equivalence relation on(X,µ) and Σ a simplicial R-complex. If (Σn)n∈N is an exhausting sequence ofULB, invariant subcomplexes, then

β(2)n (Σ,R, µ) = lim

alim

b∇n(Σa,Σb).


Remark. We remark that again such a sequence always exists. This is seenby applying the countable-to-one splitting of fundamental domains of the Σ(n).

2.8 ( [Gab02]) Theorem. If K is a simplicial complex, G acting freely on itby automorphisms, and (X,µ) is a standard Borel probability space on which Gacts (freely) by measure-preserving automorphisms. Then for all n ≥ 0

β(2)n (ΣK,RG, µ) = β(2)

n (K,G). (2.1)

By the above, all one has to do is take an exhausting sequence (Kn) ofinvariant subcomplexes of K with the restricted action of G cocompact, andthen show that ∇n(Ka,Kb) = ∇n(ΣKa,ΣKb), and this is just chalk.

2.9 ( [Gab02]) Theorem. If Σ,Ξ are n- respectively (n− 1)-connected simpli-cial R-complexes, R an (SP1) equivalence relation on (X,µ), then

β(2)n (Σ,R, µ) ≤ β(2)

n (Ξ,R, µ).

In particular, if they are both n-connected we have equality, and we define this

to be the n’th ℓ2-Betti number β(2)n (R, µ) of R (again depends on µ.)

Corollary. By the previous two theorems, if G is a countable group actingfreely, preserving the measure, on the standard Borel probability space (X,µ)then

β(2)1 (G) = β

(2)1 (RG, µ).

Actually, the original motivation for introducing ℓ2-Betti numbers in thecontext of equivalence relations seems to have been the following

( [Gab02]) Theorem. Suppose that G,H are countable groups acting by mea-sure preserving Borel automorphisms on a standard Borel probability space (X,µ).

If RG = RH then β(2)n (G) = β

(2)n (H) for all n ≥ 0.

2.10 Graphings and the cost. We shall now see another way in whichone can construct special simplicial R-complexes.

Definition. Let R be an (SP ) equivalence relation on a standard Borel (usu-ally probability) space.

• An unoriented graphing of R is a Borel subset Ψ ⊆ R ⊆ X × X suchthat Ψ is symmetric, i.e. invariant under the flip θ : (x, y) 7→ (y, x), andgenerates R (i.e. R is the smallest equivalence relation containing Ψ.)

• An oriented graphing of R is a countable set Φ = (φi)i of partial Borelautomorphisms φi : Ai → Bi such that the relation defined by x ∼ y ⇔∃i : y = φi(x) generates R.

Usually, what kind of graphing we are dealing with is clear from the contextand we suppress the (un)oriented part. Further, it is clear how to get an unori-ented graphing from an oriented one. The following example shows how to go


the other way. An important feature is that while the transition from orientedto unoriented is canonical, the transition in the other direction, of course isnot. In certain cases, this makes it more desirable to work with the unorientedconcept, for instance if one can induce this canonically somehow, but cannotinduce an oriented graphing canonically (see example 2.2 in [Gab05].)

Example. Suppose that Ψ is an unoriented graphing of the (SP1) equivalencerelation R on (X,µ). Let ϕ : X → [0, 1] be a Borel isomorphism, and putΨro = (x, y) ∈ Ψ | ϕ(x) < ϕ(y).

Then πl|Ψrois countable-to-one, so we get a decomposition Ψro = ∪iΨi such

that Ψi is Borel and πl|Ψiis injective for all i. Similarly, we get for each i a

decomposition Ψi = ∪jΨi,j such that Ψi,j is Borel and πr |Ψi,jis injective for all

j.Put now Ai,j = πl(Ψi,j), Bi,j = πr(Ψi,j) for all i, j. These are Borel since

πl|Ψi,jis injective and similarly the πr|Ψi,j

. For all i, j we now define maps ψi,j :

Ai,j → Bi,j by ψi,j = πr πl|−1Ψi,j

. These are clearly partial Borel automorphisms

(the inverse of ψi,j being πl πr|−1Ψi,j

and both obviously Borel.)

Finally, it is clear that the oriented graphing (ψi,j)i,j generates R since thesymmetric closure of its associated relation is Ψ.

In the above example, we note also that

µ(Ai,j) = ν(Ψi,j) = µ(Bi,j).

2.11 Definition. Let R be an (SP1) equivalence relation on the standardBorel probability space (X,µ). If Ψ is

• An unoriented graphing of R we define its cost as costµ(Ψ) = 12ν(Ψ).

• An oriented graphing of R we define its cost as costµ(Ψ) =∑

i µ(Ai).

The example above shows that the transition from an unoriented graphing to thecorresponding oriented one preserves the cost, and its trivial that the transitionin the other direction does so as well. We can then define the cost of R as theinfimum over either type of graphing, or just the infimum over all graphings,unoriented or not since we are lazy:

costµ(R) = infΨ:graphing of R

costµ(Ψ).

Finally, for a countable group G, we define its cost as the infimum of thecost of its free actions by measure-preserving automorphisms on a standard Borelprobability space:

cost(G) = infR(SP1):R=RG

costµ(R).

We emphasize that the actions considered in the infimum are free.


Now, if Ψ is an unoriented graphing of R we can construct a one-dimensionalsimplicial R-complex ΣΨ as follows: We let (ΣΨ)(0) = (R, πl) with the usualaction of R, and

(ΣΨ)(1) = ((x, y), (x, z)) ∈ R ∗ R | (y, z) ∈ Ψ.

A fundamenntal domain is ∆ ∗ (ΣΨ)(0) ∩ (ΣΨ)(1) where ∆ is the diagonal in R(i.e. a fundamental domain for R). But this is just (Borel isomorphic to, via. ameasure preversing isomorphism) Ψ.

Then, in lemma 2.5 the Di (used in the definition of the si appearing in thelemma) are exactly, by the very example above, (by the isomorphism above)the Ψi,j. Then, by the lemma

dimL(R) C(2)1 (ΣΨ) =

∑

i,j

〈si,j , si,j〉C(2)1 (ΣΨ)

=∑

i,j

µ(πl(Ψi,j))

=∑

i,j

µ(Ai,j) =∑

i,j

ν(Ψi,j) = costµ(Ψ).

Now let (Σk)k∈N be an exhausting sequence of R-invariant ULB subcom-plexes of ΣΨ such that (Σk)(0) = (ΣΨ)(0) for all k (it is clear that such asequence always exists, simply adding one orbit of the Ψi,j at a time, say.)Then we have, for all k ≤ l a commutative diagram

C(2)1 (ΣΨ)

C(2)1 (Σl)

∂(2)1,b //

?

jl

OO

C(2)0 (ΣΨ)

C(2)1 (Σk)

∂(2)1,k //

?

jk,l

OO

C(2)0 (ΣΨ)

Since everything is one-dimensional we have

H(2)

1 (Σl,R, µ) = ker∂(2)1,l H

(2)

0 (Σl,R, µ)

H(2)

1 (Σk,R, µ) = ker ∂(2)1,k

?

jk,l|ker ∂

(2)1,k

OO

H(2)

0 (Σk,R, µ)

OOOO

It follows that

β(2)1 (ΣΨ,R, µ) = lim

aր dimL(R) ker ∂

(2)1,a,

and also that

β(2)0 (ΣΨ,R, µ) = 1 − lim

aց dimL(R) Im ∂

(2)1,a.


Now, for all a, we have dimL(R) C(2)1 (Σa) = dimL(R) ker∂

(2)1,a + dimL(R) Im ∂

(2)1,a

and since the left-hand side is ≤ dimL(R) C(2)1 (ΣΨ) it follows by the above

calculation that

costµ(Ψ) ≥ β(2)1 (ΣΨ,R, µ) + 1 − β

(2)0 (ΣΨ,R, µ).

≥ β(2)1 (R, µ) + 1 − β

(2)0 (R, µ),

where the second inequality is due to ΣΨ being connected. Taking the infimumover all graphings gives then

costµ(R) − 1 ≥ β(2)1 (R, µ) − β

(2)0 (R, µ).

Further, if R is produced by a free measure preserving action of a countably

infinite group G on (X,µ), then the right-hand side is just β(2)1 (G)−0,¶ so that

taking the infimum over all such actions leaves us with

2.12 ( [Gab02]) Theorem. If G is a countably infinite group then

cost(G) − 1 ≥ β(2)1 (G).

Remark. One can show that actually, if G has a free measure-preservingaction which is treeable (i.e. can be generated by a graphing Ψ such that (almost)every fiber of ΣΨ is a tree), then we have equality above. A major open questionis whether we always have equality (see chapter 7.)

Finally, it would hardly be seemly to claim giving a tutorial on cost and notmention the following

2.13 ( [Gab00]) Theorem. If Ψ is a graphing of the (SP1) equivalence relationR (on (X,µ) standard Borel probability space) such that (ΣΨ)x is a tree foralmost every x, then

costµ(R) = costµ(Ψ),

and the converse is also true.Also, if G is a countable group acting freely on (X,µ), preserving the mea-

sure, such that R = RG then actually

cost(G) = costµ(Ψ) (2.2)

¶In general, β(2)0 (R, µ) = 0 if the equivalence classes of R are infinite a.e. - see [Gab02] for

this.

Chapter 3

Random graphs and their

associated equivalence

relations

The third rule about fight club is only two men per fight.key words:

random graphs - the uniform spanning tree on finite graphs - free minimal anduniform spanning forests - full and cluster equivalence relations.

3.1 Random graphs. We denote by G+V respectively GV the set of simple

graphs, respectively the set of simple unoriented graphs on V. Note that thereis a canonical quotient map from the former onto the latter.

As in the prologue then, these are just subsets of 2V ×V . This is a very nicecompact, separable, metrizable topological space (in particular, it is a Polishspace), and it is not hard to see that GV ,G+

V are in fact closed subsets. Forinstance, the latter is just the set subsets not intersecting the diagonal, so forany subset A not in G+

V , some π(v,v)(A) is non-zero, giving a basis neighbourhood

containing A and not intersecting G+V .

Also, we denote by FV ,F+V the set of unoriented respectively oriented forests

on V , and by TV , T +V the set of unoriented respectively oriented spanning trees.∗

We have then the following diagram of (continuous) inclusion and quotients:

T +V

//

F+V

//

G+V

TV

// FV // GV

∗For an oriented graph, we can consider connectivity of its corresponding unoriented graph,so that this does make sense. However, note that one may also consider connectivity of orientedgraphs as they are, and this is probably standard. Thus, what we are really considering hereare sections of unoriented forests respectively trees.

36

CHAPTER 3. RANDOM GRAPHS 37

Finally we note that FV and F+V are also closed. Indeed, if G is some

graph with a cycle, say (e1, e2, . . . , el) in it, then π−1e1

(1)∩ π−1e2

(1)∩ · · · ∩ π−1el

(1)is a neighbourhood of G in which every graph contains a cycle, whence notintersecting F+

V . The unoriented case follows readily from this.We can relativize all this with respect to some fixed graph G0 on V . Indeed,

the set of subgraphs, denoted G+(G0), is simply

G+(G0) =⋂

e∈2V ×V

G′ ∈ G+V | πe(G′) ≤ πe(G0.

Similarly for F+ and T +, and the corresponding unoriented cases.

Definition. A random spanning forest (respectively tree) on an unorientedgraph G is a probability Borel measure supported on F(G) (respectively T (G)).

We shall primarily be interested in two random spanning forests, arising asweak limits of “free” random spanning trees on finite subgraphs of G - this iswhy we call them random spanning forests instead of just random forests. Thesewill be the uniform and minimal spanning trees.

There are also so called “wired” cases. For the free/wired uniform span-ninng forests see the survey [BLPS01], and for the minimal ones see [LPS06].

3.2 The free uniform spanning forest. Suppose G is some (countablyinfinite) connected, unoriented graph. Label its vertices v1, v2, . . . in such a waythat the subgraph Gn spanned by v1, v2, . . . , vn is connected for all n.

Consider for each n the uniform spanning tree measure un on Gn.† For edgese1, e2, . . . , el in G, define

u(π−1

e1(ǫ1) ∩ π−1

e2(ǫ2) ∩ · · · ∩ π−1

el(ǫl))

= limn

un

(πe1 |−1

Gn(ǫ1) ∩ πe2 |−1

Gn(ǫ2) ∩ · · · ∩ πel

|−1Gn

(ǫl)),

(3.1)where the ǫi ∈ 0, 1. (Also, here and throughtout, the πe are restrictions tothe set of unoriented subgraphs of G.)

We will show that this makes sense, i.e. that the limit actually exists, andthat it extends to a Borel probability measure supported on F(G). By proposi-tion 3.8 below it is also independent of the particular exhaustion (Gn).

Definition. The measure u is called the free uniform spanning forest, abre-viated FUSF, on G.

In fact, we shall deal with the extension right away since this is fairly stan-dard.

Let V1, V2, . . . , Vn be basis neighbourhoods (i.e. of the form π−1e1

(ǫ1) ∩π−1

e2(ǫ2) ∩ · · · ∩ π−1

el(ǫl)) and note that the intersection of any number of them

is again a basis neighbourhood. Thus we may extend u to finite unions by

†That is, un is the uniformly distributed probability measure on the finite set T (Gn).


inclusion-exclusion:

u

n⋃

j=1

Vj

= u(V1) + u(V2) − u(V2 ∩ V1) +

+u(V3) − u(V3 ∩ V1) − u(V3 ∩ V2) + u(V3 ∩ V1 ∩ V2) + · · ·

=∑

∅6=I⊆1,2,...,n

(−1)♯I−1u

⋂

j∈I

Vj

.

It is clear that this is always non-negative. Then we can define for compactK

u(K) = infK⊆∪jVj

u

n⋃

j=1

Vj

.

This is easily seen to be finitely additive on compact sets, since we can separatethem by open sets. Now for any open set V , define u(V ) as the supremum ofu(K) for K ⊆ V , and this is countably additive by an ε/2k type argument.Finally, define for every general Borel set A, u(A) = infA⊆V u(V ) the infimumbeing taken over open V . It is straightforward to check that this is indeed ameasure.

Further, it is completely straightforward to see that u is supported on F(G).Indeed, if C = (e1, e2, . . . , el) is a cycle, then the set of unoriented subgraphs ofG containing C is just π−1

e1(1)∩π−1

e2(1)∩· · ·∩π−1

el(1) and this clearly has measure

zero.Finally, it is clear that u is a probability measure. Indeed, fix any edge e in

G. Then F(G) is (contained in) the disjoint union π−1e (1)∪π−1

e (0), and so

u(F(G)) = u(π−1e (1)) + u(π−1

e (0)) = limn

un(πe|−1Gn

(1)) + limn

un(πe|−1Gn

(0))

= limn

un(T (G)) = 1.

Thus we need only justify equation (3.1). This is quite heavy, requiring usto actually determine the probability un(π−1

e (1)). Intuitively though, one needonly notice that adding edges to a graph can never increase the probability thate is in the uniform spanning tree. For us, this will come out quite naturallyfrom the ordering of projections on a Hilbert space

3.3 INTERMEZZO: On the number of spanning trees. We shallnow essentially calculate the expected degree of the free uniform spanning treeon a finite connected graph. In the literature ( [ME08], [BLPS01]), the proofstypically use or refer to techniques developed by Kirchoff in 1847. We shallinstead give a more intuitive and direct proof,‡ writing down explicitly the

‡I imagine this is known to the powers that be, but I have no reference (there is a proofin [BLPS01] but it uses something called Wilson’s algorithm. There is also the approach ofdeterminantal measures [Lyo03,Lyo08] but this approach is sort of backwards.)


projection of ℓ2alt(E) onto ℓ2⋆(E) in terms of the set T (G) of unoriented spanningtrees of G.

Suppose G is a finite, connected, unoriented graph and T ∈ T (G) is a span-ning tree for G. If e0 ∈ E(G) \ E(T ) is an edge og G not in the tree, there isa uniquely determined cycle C = (e0, e1, . . . , el) in G such that ei ∈ E(T ) forall 0 < i ≤ l. To see this, note that by maximality there is at least one suchcycle. However, if there were two (that is, at least two) cycles in T ∪ e0, sayC = (e0, e1, . . . , el) and D = (e0, f1, f2, . . . , fk), then

(er, e2, . . . , es, ft, ft−1, . . . , fr),

where r is the smallest index i such that e+i 6= f+i and s, t are the smallest

indices i, j > r such that e+i = f+j , is a cycle in T .

3.4 Definition. The uniquely determined cycle above is denoted CT (e0) andwe call it the fundamental cycle of e0 with respect to T .

Now, for each spanning tree T ∈ T (G) we consider the cycle- respectivelystar-maps P,T , P⋆,T : ℓ2alt(E) → ℓ2alt(E) defined by§

P,T f =∑

e∈E(G)\E(T )

∑

e′∈CT (e)

f(e′)

δe,

P⋆,T f = (1− P,T )f =∑

e∈E(T )

f(e)δe −∑

e∈E(G)\E(T )

∑

e′∈CT (e)\e

f(e′)

δe.

What happens here is that P⋆,T takes the restriction of f to the spanning tree Tand then pertubes it in the obvious manner so that the result lies in ℓ2⋆(E), andP,T keeps track of these pertubations. Taking averages og this operation overall (unoriented) spanning trees actually gives the desired projection. It is quitestraightforward to see that this will give an idempotent, but self-adjointness is alittle more tricky. Our strategy will be to show that the corresponding averageof the P,T ’s is also an idempotent, and deduce self-adjointness from this.

3.5 Lemma. Suppose that G is a finite, connected, unoriented graph. Then

(i) dim ℓ2⋆(E) = ♯V − 1.

(ii) For any spanning tree T ∈ T (G) there are, up to involution, exactly n =♯E − ♯V + 1 fundamental cycles CT (e1), CT (e2), . . . , CT (en) wrt. T , andthe associated functions

cei=

∑

e∈CT (ei)

χe, i = 1, 2, . . . , n

constitute a basis of ℓ2(E).

§We abuse notation a little, and write for a cycle C = (e0, e1, . . . , el), C \ e0 for the path(e1, e2 . . . , el).


The point of this lemma, is that each function in the star-space is givenuniquely by its values on any spanning tree (we remark that it is not hard tosee that a subgraph is a spanning tree if and only if it has exactly ♯V −1 edges),and by (ii) that each funtion on any spanning tree actually extends uniquelyto a function in the starspace (and this extension is exactly the value of thestar-map).

Proof. Noting that ker d = R · 1 gives (i).For (ii), it is then enough to show linear independence. But this is just

noting that, trivially, the edge ei is not in CT (ej) for any j 6= i, so that forα1, α2, . . . , αn ∈ R we have

n∑

j=1

αjcej

(ei) = αi.

////

Corollary. If G is a finite, connected, unoriented graph, then for each T ∈T (G) we have

(i) P⋆,T |ℓ2⋆(E) = 1.

(ii) P⋆,T (ℓ2alt(E)) = ℓ2⋆(E).

Proof. (i) is seen directly from the definition of P⋆,T , and (ii) then follows fromthe lemma. ////

On the other hand, P,T does not in general (that is, when G is not itself atree) have image contained in ℓ2(E), equivalently P⋆,T is not self-adjoint. Thefollowing lemma corrects this deficiency, as stated above, by taking averagesover all spanning trees. For the statement, we set

P =1

♯T (G)

∑

T ∈T (G)

P,T ,

P⋆ =1

♯T (G)

∑

T ∈T (G)

P⋆,T .

3.6 Lemma. If G is a finite, connected, unoriented graph, we have

(i) P(ℓ2alt) = ℓ2(E).

(ii) P|ℓ2(E) = 1.

The key observation of the proof is that the set of spanning trees havingsome cycle C as fundamental cycle is evenly distributed with respect to whichedge of C is left out, and thus averaging over these the alternating functioncorresponding to C is left invariant.

We can now give the


Proof of lemma 3.6. Indeed, let C be a cycle with corresponding geometric cycleC. Let e be an edge in C. We want to show first that

(PχC)(e) = 1, where χC =∑

f∈C

χf .

To do this we (sort of) count the total number of spanning trees of G. The setof spanning trees T not containing e are partitioned according to CT (e). Foreach such T write i(T ) respectively o(T ) for the set of edges in (CT (e)\e)∩Crespectively CT (e) ∩ C. Then

(P,T χC)(e) = 1 + ♯i(T ) − ♯o(T ).

For every such T we count T itself once with positive coefficient, and we alsocount the spanning tree obtained by removing any geometric edge correspond-ing to an edge in i(T ) and adding e, e, and similarly except with negativecoefficient for edges in o(T ).

Now, suppose S is a spanning tree containing e and write χC as a linearcombination of the analogous functions corresponding to fundamental cycles ofS. Let these fundamental cycles be CS(fj), j = 1, 2, . . . , n. We may suppose byreversing directions that all the fundamental cycles intersecting e, e actuallycontain e. Now, S can be constructed from several T above. Indeed, for everyj, either fj or fj is in C by linear independence of the fundamental cycles, andwe may remove e, e and add fj, fj for any j such that e ∈ CS(fj), to get aspanning tree not containing e. Then the reverse construction gives S startingwith a T .

Let us write J = j | e ∈ CS(fj). By the above, we have counted S exactly♯J times with any coefficient.

The point is then that fj ∈ i(T ) for the T above if and only if fj ∈ C, forj ∈ J .

Further,∑

j∈J χC(fj) = χC(e) = 1, so that in total S is counted once morewith positive sign than with negative sign. Thus we end up counting eachspanning tree exactly once, so that (PχC)(e) = 1 as desired.

It is not hard to see that if e, e ∩ C = ∅ then the value at e is zero. Incase no fundamental cycle of e with respect to a spanning tree intersects C orits reverse there is no problem. If there is such a fundamental cycle, this allowsus to construct two cycles, C2, C3 such that

χC =

3∑

k=2

χCk.

and such that, say, C2 contains e and C3 contains e. ////

Personally I like the proof above because it illustrates very well a sort ofrearrangement property of the set of spanning trees on a finite graph. For thereader’s convenience, here is a much faster


Proof of lemma 3.6. Consider edges e, e′. Then we have

♯T (G) · 〈P⋆χe, χe′〉 = ♯T | e ∈ T , e′ = e or e′ /∈ T , e ∈ CT (e′)−♯T | e ∈ T , e′ = e or e′ /∈ T , e ∈ CT (e′),

and also

♯T (G) · 〈χe, P⋆χe′〉 = ♯T | e′ ∈ T , e = e′ or e /∈ T , e′ ∈ CT (e)−♯T | e′ ∈ T , e = e′ or e /∈ T , e′ ∈ CT (e),

and these are equal by the observations before the statement of the lemma.////

Now, by the corollary of lemma 3.5 we have P⋆(ℓ2alt(E)) = ℓ2⋆(E) and

P⋆|ℓ2⋆(E) = 1, so that kerP = ℓ2⋆(E) from which it now follows that P isthe orthogonal projection onto ℓ2(E).

Furthermore, noting that for T ∈ T (G)

(P⋆,T χe)(e) =

1 if e ∈ T0 if not

we have shown

3.7 Theorem. If G is a finite, connected, unoriented graph, then P⋆ is the or-thogonal projection onto ℓ2⋆(E), and P is the orthogonal projection onto ℓ2(E).Further, for any e ∈ E(G) we have u(π−1

e (1)) = (P⋆χe)(e) where u is the freeuniform spanning tree on G.

For the definition of the free uniform spanning tree, see 3.1. In more downto earth terms, this means that

(P⋆χe)(e) =♯T ∈ T (G) | e ∈ T

♯T (G).

Remark. We note that in Hilbert space terms, the result above can be writtenas

u(π−1e (1)) =

1

2〈P⋆χe, χe〉

and this is the formulation we shall use henceforth.

END INTERMEZZO.Now, by theorem 3.7 it is a simple matter to deduce that un(πe|−1

Gn(ǫ))

converges. Indeed, suppose ǫ = 1 and consider the natural “projection” P :ℓ2alt(E(Gn+1)) → ℓ2alt(E(Gn)). Its adjoint is the corresponding embedding, andit is clear that PP⋆,n+1P

∗ ≤ P⋆,n, (if something sums to zero along every cy-cle in Gn+1 it particularly sums to zero along every cycle in Gn) whence fore ∈ E(Gn)

1

2〈P⋆,n+1χe, χe〉 =

1

2〈PP⋆,n+1P

∗χe, χe〉 ≤1

2〈P⋆,nχe, χe〉.


Thus, the sequence in question is decreasing, giving convergence. It followsthen that when ǫ = 0, we have convergence also, either directly, or noticing thatsimilarly the sequence is in this case increasing.

The general case really follows from this: for a graph G′ and a set S of edgeswe denote by ∐SG′ the graph obtained by removing S (but leaving all verticesin place, so that in particular a connected graph may become disconnected.)

Then if e1, e2, . . . , ek, f1, f2, . . . , fl are edges in G we have

un

(πe1 |−1

Gn(1) ∩ πe2 |−1

Gn(1) ∩ · · · ∩ πek

|−1Gn

(1) ∩ πf1 |−1Gn

(0) ∩ πf2 |−1Gn

(0) ∩ · · · ∩ πfl|−1Gn

(0))

=

= un

(πf1 |−1

Gn(0) ∩ πf2 |−1

Gn(0) ∩ · · · ∩ πfl

|−1Gn

(0))×

×u∐f1,f2,...,flGn

(πe1 |−1

∐f1,f2,...,flGn

(1) ∩ πe2 |−1∐f1,f2,...,fl

Gn(1) ∩ · · · ∩ πek

|−1∐f1,f2,...,fl

Gn(1)),

whre n is large enough that ∐f1,f2,...,flGn is connected (if there is no such n,the probability is clearly 0.)

Then (∐f1,f2,...,flGn) is an exhaustion of ∐f1,f2,...,flG and we need only tohandle the two factors separately for arbitrary graphs. In fact, taking comple-ments in the second factor and using inclusion-exclusion, we need only handlethe first. For this, we simply have

un

(πf1 |−1

Gn(0) ∩ πf2 |−1

Gn(0) ∩ · · · ∩ πfl

|−1Gn

(0))

=

= un

(πf1 |−1

Gn(0))u∐f1Gn

(πf2 |−1

∐f1Gn(0))· · · u∐f1,f2,...,fl−1Gn

(πfl

|−1∐f1,f2,...,fl−1Gn

(0)).

Thus we have shown that any exhaustion of G by connected subgraphs will giverise to a unifrom spanning forest measure. However, we still need to justifycalling u the free uniform spanning forest:

3.8 Proposition. The free uniform spanning measure on a graph G is in-dependent of the particular exhaustion of G by finite subgraphs used to obtainit.

Proof. This is just the fact that un

(πe|−1

Gn(1))

is decreasing, giving uss a “com-mon refinement” type argument. Thus, if (G′

n) is some other exhaustion byconnected subgraphs, then for any n we may choose n′ such that Gn ⊆ G′

n′

whenceuGn

(πe|−1

Gn(1))≥ uG′

n′

(πe|−1

G′n′

(1))≥ u′

(π−1

e (1)),

where u′ is the free uniform spanning forest obtained using the (G′n). Since this

is true for all n we get inequality in one direction. The other is of course entirelysimillar. ////

3.9 The free minimal spanning forest. Here we define another randomforest on infinite graphs, namely the free minimal spanning forest, abbreviatedFMSF. Unlike its uniform counterpart, it can be defined directly on infinitegraphs.

Given an infinite (connected) graph G = (V,E) consider the product Lebesguemeasure on X =

∏e[0, 1], the direct product over the set of geometric edges.


The measure assigned to a set F of forests on G is then simply the productmeasure on X of the set of points x such that, retaining exactly the geometricedges e such that for any path in G connecting the endpoints, there is an edgee′ such that x(e′) ≥ x(e), one gets a graph in F .

Definition. The random spanning forest obtained above is denoted m and iscalled the free minimal spanning forest, abreviated FMSF, on G.

We will not get too involved with the FMSF as it is not heavily used in thefollowing, but only makes a crucial appearance. We recall the basic propertiesof the FUSF and FMSF that we need below.

3.10 Properties of the FUSF and the FMSF. In terms of later appli-cations, the basic property shared by the free uniform/minimal spanning forestsis invariance under automorphisms of the graph.

Proposition. The FUSF as well as the FMSF on G are both invariant underany automorphism of G.

Proof. Suppose h is an automorphism of G and that (Gn) is an exhaustion of Gby connected subgraphs. Then so is (h(Gn)) and the claim for the FUSF followssince it is independent of the particular exhaustion used.

Similarly, one needs simply note that the product Lebesgue measure is pre-served under the permutation of edges defined by h. ////

This allows us to consider the two random forests as G-invariant randomforests on the Cayley graph of a finitely generated group G wrt. any finitesymmetric generating set S. The degree of a G-invariant random forest µ onsuch a Cayley graph is the expected degree of any vertex (they all have the sameexpected degree, by the G-invariance):

deg(µ) =

∫

µ

deg1(F)dµ(F) =

∑

s∈S

∫

µ

π(1,s)(F)dµ(F).

In fact, we will later also consider the FUSF on the 1-skeleton of a simplicialcomplex, on which G acts freely.

For later reference if not for independent interest, we list the following com-parison, a special case of corollary [LPS06]3.24, in the case of Cayley graphs:

3.11 Proposition. If G is the Cayley graph of a finitely generated group Gwrt. some finite, symmetric generating set S, then

deg(uG) ≤ deg(mG).

Finally, we shall need the so called standard coupling m(ε) of the FMSF andthe Bernoulli bond percolation with parameter 1 ≥ ε ≥ 0. This is the randomgraph of G obtained as the FMSF, except now m retains edges that do not havemaximal weight in any cycle or have weight ≤ ε. The following theorem showsthat the FMSF is “thick” in G, a property apparently not shared by the FUSF.


3.12 (see [LPS06]3.22) Theorem. If G is the Cayley graph wrt. a finite,symmetric generating set S of the finitely generated group G, then for any ε > 0the standard coupling m(ε) is connected a.e.

3.13 The full and cluster equivalence relations. Suppose that(X,B, µ) is a standard Borel probability space and that the finitely generatedgroup G acts on it freely and preserving the measure. Let S be a finite, sym-metric generating set of G and G the Cayley graph of G wrt. S. Suppose wehave a G-equivariant Borel map g : X → G(G).¶

We define the full equivalence relation RG on X to be simply the orbit equiv-alence relation given by the action of G. The reason we give it a special name isthat it is actually defined more generally for graphs with some unimodular auto-morphism group acting transitively on them, in which case the full equivalencerelation is different, due to the possibility of non-trivial isotropy groups.

Secondly, we define the cluster equivalence relation RclG on X by letting

x ∼RclG y if and only if there is some g ∈ G such that y = g.x and 1 and g are

connected in g(y). By the G-equivariance this is an equivalence relation, andsince g is Borel it is an (SP1) equivalence relation. We denote by Ug the setof points x ∈ X for which g(x) has a unique infinite cluster, and this clustercontains the identity.

The following simple, yet very useful, result is due to Gaboriau. In [Gab05]this is the basic technical result needed to pass between knowledge of a graphand its infinite clusters under Bernoulli percolation.

3.14 ( [Gab05]) Theorem. The induced relations RG|Ug and RclG|Ug are equal.

We may as well include the short proof.

Proof. The inclusion RclG|Ug ⊆ RG|Ug is trivial. For the other one, suppose

x ∼RG y with both points in Ug. Then there is some g ∈ G with y = g.x, andapplying g to this, we see that g maps the infinite cluster containing the identityof g(x) to the one of g(y), by uniqueness. This shows that 1 and g are indeedconnected in g(y). ////

¶yeah yeah bad notation...

Chapter 4

Amenable and unitarisable

groups

The fourth rule about fight club is one fight at a time.key words:

unitarisability - amenability - history of the Dixmier problem - the von Neu-mann problem - examples (Fn, B(p, q), F ).

In this chapter I shall, at long last, state the name of the game. I will de-fine unitarisability and amenability for groups, and discuss some basics and alittle history. I will also give a few examples, and finally I will shortly present arecent result of Gaboriau-Lyons that describes a positive solution to a weakenedversion of the von Neumann problem.

4.1 Definition. Let G be a countable group.

• We say that G is amenable if it admits an invariant mean, i.e. if there isa left-invariant positive linear functional on ℓ∞(G).

• We say that G is unitarisable if whenever π : G → B(H) is a represen-tation of G such that supg∈G‖π(g)‖ < ∞, there is an invertible operatorT ∈ B(H) suuch that T−1π(g)T is unitary for all g ∈ G.

A representation satisfying the condition supg∈G‖π(g)‖ <∞ as above is said tobe uniformly bounded.

In fact, the above definition extends word for word to the context of locallycompact groups, with the addition that we only consider representations that arecontinuous wrt. the strong-operator topology. Dixmier and Day independentlyin 1950 proved the following

Theorem. If G is a locally compact amenable group, then G is unitarisable.

46

CHAPTER 4. AMENABLE AND UNITARISABLE GROUPS 47

Their work was based on a 1947 proof due to Sz.-Nagy in the case whereG = Z. The question we are interrested in now is the converse, asked by Dixmierin 1950 - i.e. whether every unitarisable locally compact group is amenable.

We will need a slightly stronger result, namely the following (stated here forcountable groups for convenience)

4.2 Theorem. ∗ Let G,A be countable groups with G unitarisable and Aamenable. Then G×A is unitarisable.

Proof. Suppose that ϕ : G×A→ B(H) is a uniformly bounded representation.In particular then, ϕ|G×1 is a uniformly bounded representation of G and sothere is some invertible S ∈ B(H) such that S−1ϕ(g,1)S is unitary for all g ∈ G.Furhter, by uniform boundedness, for all x, y ∈ H the function f(x,y) : A → C

given byf(x,y) : a 7→ 〈S−1ϕ(1, a)Sx, S−1ϕ(1, a)Sy〉

is bounded, i.e. f(x,y) ∈ ℓ∞(A). Let ρ be a right-invariant state on ℓ∞(A) anddefine a functional b on H×H by

b(x, y) = ρ(f(x,y)).

Clearly this is conjugate-bilinear and bounded (by ‖b‖ ≤ (supa‖ϕ(1, a)‖)2 ·‖S‖2·‖S−1‖2), so there is a T ∈ B(H) such that for all x, y we have b(x, y) = 〈Tx, y〉.Now, T is invertible (since there is a bounded R such that 〈RT ·, ·〉 = 〈·, ·〉), andsince ρ is positive so is T . Then for (g, a0) ∈ G×A we have

b(S−1ϕ(g, a0)Sx, S−1ϕ(g, a0)Sy) = ρ(a 7→ 〈S−1ϕ(g, aa0)Sx, S

−1ϕ(g, aa0)Sy〉)= ρ(a 7→ 〈S−1ϕ(1, aa0)Sx, S

−1ϕ(1, aa0)Sy〉)= b(x, y),

where the second equality is the fact that S−1ϕ(g,1)S is unitary and the third

equality is the right-invariance of ρ. It follows now that T12S−1ϕ(g, a0)ST

−12 is

unitary and since g, a0 were arbitrary we are done. ////

In [Pis05] it is proved that if G is a discrete (possibly uncountable) group,then it is unitarisable if and only if all its countable subgroups are. Further,one has

Theorem. If G is a locally compact group, Γ is a normal subgroup which isamenable and such that the quotient G/Γ is amenable, then G is amenable.

Since any connected locally compact group is unitarisable ( [Pis05]), if onewere able to prove the analogous theorem for unitarisability, then taking thequotient G/G1 where G1 is the connected component containing the identity,the problem would be completely reduced to the countable case.

∗As far as I can determine, this result first appears in [NW99] but I imagine it was knownbefore that. Also, since I don’t read japanese I have not been able to acertain that it actuallydoes appear.


The requested theorem is indeed stated in [Pis05], but Pisier here writes thathe was not able to actually prove it. “Porting” simple properties of amenablegroups to unitarisable groups seems to be a problem in general.

From this point on, all groups will once more be countable.

4.3 Example. • Any abelian group is amenable. A directed union of amenablegroups is amenable. Any subgroup or quotient of an amenable group isamenable.

( [CG86]) Theorem. If G is a countably infinite amenable group, then

β(2)n (G) = 0 for all n.

See also the very neat proof in [Lyo08].

( [Gab00]) Theorem. If G is a countably infinite amenable group, thencost(G) = 1 and G is treeable.

• F2 is not amenable. Neither is B(p, q) for p ≥ 2, q ≥ 665.

• We have β(2)1 (Fn) = n for all n, including n = ∞. Similarly the cost.

• It is not known whether Thompson’s group F is amenable.†

• It is, however, known that cost(F ) = 1.

( [Pis05]) Theorem. Unitarisability passes to subgroups and quotients.

The proof for quotients is trivial. The proof for subgroups uses that theinduced representation of a uniformly bounded representation is again uniformlybounded.

Remark. It is currently not known whether a directed union of unitarisablegroups is unitarisable.

In fact, Dixmier also asked whether there exists any group at all which is notunitarisable. A counter example was provided in 1955. We shall give a simpleexample, showing that F2 is not unitarisable. For this, let f be a complex-valuedfunction on G and consider the set F of pairs (f+, f−) of functions on G suchthat f(g−1h) = f+(g, h) + f−(g, h) for all g, h ∈ G. Then we define

‖f‖T1 = inf(f+,f−)∈F

sup

g∈G

∑

h∈G

|f+(g, h)| + suph∈G

∑

g∈G

|f−(g, h)|

.

Then T1(G) is the space of functions f such that ‖f‖T1 <∞. This is a Banachspace, and in fact we have the following

†There is at the time of writing two papers [Akh09,Sha09] on the arXiv - the first claiminga proof that it is not and the second claiming a proof that it is. Word on the street is thatthe second paper is being taken very seriously, while the first is somewhat abscure and is nottaken seriously at all.


4.4 ( [Pis01,ME08]) Theorem. Let G be a countable group. If G is unitaris-able, there is a constant K such that, for any subgroup H of G

‖f‖ℓ2(H) ≤ K‖f‖T1(H), f ∈ T1(H). (4.1)

In particular T1(G) ⊆ ℓ2(G) and this inclusion is continuous.

Remark. In fact, the inclusion T1(G) ⊆ ℓ1(G) holds if and only if G isamenable. See [Pis01].

It follows from the theorem that F∞ is not unitarisable. Indeed if we letf be the characteristic function of the set of words of length 1 in F∞, thenf ∈ T1(F∞) (see [Pis01]) and clearly f /∈ ℓ2(F∞).

Hence, since one can embed F∞ in F2 (map the n’th generator of F∞ toanbn where a, b are the generatoors of F2) it follows that any group containinga subgroup isomorphic to F2 is non-unitarisable.

4.5 The von Neumann problem.Actually, until the 1980s it was an open problem whether any non-amenable

group must contain a subgroup isomorphic to F2. This became known as thevon Neumann problem. It was finally resolved in the negative by Ol’shanskii andAdian, showing that the Burnside groups B(p, q), if infinite, are non-amenable(this was Adian. Ol’shanskii gave examples of non-amenable torsion groups,but with elements of arbitrarily large order.)

Indeed, they are known to be for p ≥ 2, q ≥ 665 (not everything is knownabout these though. For instance, it is apparently not known whether B(2, 5)is infinite or not.) Very recently, Monod-Ozawa [MO09] has shown that theB(p, q) also are not unitarisable.

However, there is a recent paper of Gaboriau-Lyons wherein a weaker versionof the von Neumann problem is shown to hold in the positive:

4.6 ( [GL07]) Theorem. Suppose that G is a countable non-amenable group.Then there is an ergodic, essentially free action of F2 by measure-preservingautomorphisms on the Bernoulli shift ([0, 1]G, µ) such that RF2 ⊆ RG (a.e.).

The idea of the proof is to consider the action of G on 2E(G) × [0, 1]G whereG is a Cayley graph of G (we gloss over the fact that G may not be finitelygenerated) and a random graphing obtained by a certain map f : 2E(G) ×[0, 1]G → 2E(G), yielding a cluster equivalence relation which is ergodic and has∞ > cost > 1. By some set-theoretic mumbo-jumbo this gives the result.

We shall later use this construction with f the projection on the first coor-dinate to give a proof a main theorem of this thesis.

Chapter 5

The cost of unitarisability

The fifth rule about fight club is no shoes or shirts.key words:

Expose of Monod-Epstein paper - Another look at Monod-Epstein paper.

5.1 Statement of the results. In [ME08] the authors prove the fol-lowing

5.2 Theorem. Any countable residually finite group G with first ℓ2-Betti num-

ber β(2)1 (G) > 0 is non-unitarisable.

As well as the similar

5.3 Theorem. Any (countable) finitely generated residually finite group G withcost(G) > 1 is non-unitarisable.

Remark. By work of D. Osin, this has led to examples of non-unitarisablegroups not containing F2.

Note that by theorem 2.12 the second theorem is stronger than the firstwhenever G is finitely generated. Actually, it is always stronger:

Observation. Theorem 5.3 implies theorem 5.2.

Proof. Suppose that G is a countable residually finite group with β(2)1 (G) >

0. Numbering the elements of G, we write it as the increasing union G =

∪∞n=1〈g1, . . . , gn〉 and it is not hard to see that β

(2)1 (G) ≤ lim infn β

(2)1 (〈g1, . . . , gn〉)

so that we get a finitely generated subgroup G0 with β(2)1 (G0) > 0.

Then by theorem 2.12 we have cost(G0) ≥ β(2)1 (G0) + 1 > 1 and induction

of representations (see chapter 4) proves the claim. ////

In the following intermezzo we cover a bit of classical combinatorial grouptheory.5.4 INTERMEZZO: Some combinatorial group theory.

50

CHAPTER 5. THE COST OF UNITARISABILITY 51

Definition. A (countable) group G is residually finite if for every non-identityg ∈ G there is a homomorphism ϕ : G → F into a finite group F such thatϕ(g) 6= 1.

5.5 Proposition. If G is a residually finite countable group.

• Then there are subgroups of G with arbitrarily large (finite) index.

• Every subgroup of G is residually finite.

Proof. Indeed, suppose that H is a subgroup of index n ≥ 1 and choose a g ∈ Hnot the identity. There is a homomorphism ϕ : G → F where ϕ(g) 6= 1 and Fis finite, and putting H ′ = H ∩kerϕ, H ′ has index ∞ > [G : H ′] = [G : H ] · [H :kerϕ|H ] > n.

This proves the first claim, and the second is trivial. ////

5.6 Theorem. Suppose that H is a subgroup of finite index in the finitelygenerated countable group G0. Then

rkH ≤ [G0 : H ]rkG0.

We leave out the proof.

Remark. In the statement above, “rkG” denotes the rank of the group G,which is the minimal cardinality of a generating set.

END INTERMEZZO.5.7 An application of the FUSF. We start by giving an account of themain focus of the paper [ME08], namely the proof of theorem 5.2 by use of theFUSF on Cayley graphs of finitely generated groups. This choice seems perhapsa bit peculiar compared to using the FMSF which is conjectured to induce thecluster equivalence relation with cost in (1,∞), of the full orbit equivalencerelation for the Bernoulli shift action of any (finitely generated) non-amenablegroup.

Later we give a proof of theorem 5.3 which applies the FMSF. Our proofwill deviate slightly from the proof in the paper, but on the technical level theyare basically the same proof.

5.8 Theorem. For any (countable) finitely generated group H with (finite,symmetric) generating set S, the free uniform spanning forest u on the Cayley

graph G of the pair H = 〈S〉 has degree deg(u) ≥ 2β(2)1 (H).

Proof. Choose a connected exhaustion (gn) of G, and let un be the FUSF on


gn. Denoting by es the edge (1, s), s ∈ S, we get using theorem 3.7

deg(u) =

∫degT (1)du(T )

=

∫ ∑

s∈S

χT ′∈FH |es∈T ′(T )du(T )

=∑

s∈S

u(T ∈ FH | es ∈ T )

=∑

s∈S

limn

un(T ∈ T (gn) | es ∈ T )

=∑

s∈S

limn

1

2〈P⋆,E(gn)χes

, χes〉.

We now make the observation that∗

limn〈P,E(gn)χes

, χes〉 = 〈Pχes

, χes〉,

owing simply to the fact that every cycle has finite length and so is eventuallyin gn, whence span ∪n ℓ

2(E(gn)) is dense in ℓ2(E).

Thus, continuing the above calculation, we get

deg(u) =∑

s∈S

limn

1

2〈(1− P,E(gn))χes

, χes〉

=∑

s∈S

1

2〈(1− P)χes

, χes〉

≥∑

s∈S

1

2〈Pχes

, χes〉,

where P is the orthogonal projection onto(ℓ2⋆(E) ⊕ ℓ2(E)

)⊥. Since the χes

constitute a fundamental domain for the action of H on ℓ2alt(E), the above issimply ∑

s∈S

1

2〈Pχes

, χes〉 = 2 dimL(H) P (ℓ2alt(E)) = 2β

(2)1 (H),

where the final equality is theorem 1.24.† ////

Now one uses this property to violate a bound on the degree, given in thefollowing result.

5.9 Lemma. Suppose that H is a finitely generated countable group. Then ifµ is a random forest on G, the Cayley graph of H wrt. a finite symmetric

∗For infinite graphs we extend the notation from the finite case and write P for theorthogonal projection of ℓ2

alt(E) onto ℓ2(E) and similarly for the star space.

†We glossed over the fact that there might be a generator of order two. This goes awaywith a definition covering actions with compact stabilizers.


generating set S as above, and we denote fµ(g) = µT ∈ T (G) | (1, g) ∈ T , wehave

deg(µ)√♯S

≤ ‖fµ‖2, ‖fµ‖T1(H) ≤ 2.

Proof. The first inequality is a direct application of Schwarz’ inequality. Thesecond relies, basically, upon choosing a nice Borel section O : FG → FG

+

such that at each vertex, there is at most one outgoing arrow. This can beaccomplished, for instance, by choosing one vertex in each tree of the forest andthen directing every edge along the unique path from one of its endpoints tothe chosen vertex in its tree. Then with f+(g, g′) = µT | (g, g′) ∈ O(T ) andf−(g, g′) = f+(g′, g) one gets that the sum

f+(g, g′) + f−(g, g′) = µT | (g, g′) ∈ T .

Then by the H-invariance of µ one gets that this is actually equal to fµ(g−1g′),and since there is only one outgoing arrow at each vertex, the sets in the defi-nition of f+ are disjoint for different g′ giving the bound.

We refer to the paper ( [ME08]) for the specific choice of section. ////

Combining this and theorem 5.8 with theorem 4.4 we get the

5.9.1 Corollary. If G is a (countable) unitarisable group. Then there is apositive M such that for any finitely generated subgroup H of G,

β(2)1 (H)2

rkH≤M.

Now we can give Monod-Epstein’s proof of theorem 5.2:

Proof of theorem 5.2. As above we get a finitely generated subgroup G0 of G

with β(2)1 (G0) > 0. By proposition 5.5 this has a subgroup H of index

[G0 : H ] >rkG0

β(2)1 (G0)2

M

where M is as in the corollary. Then, applying theorems 5.6 and 6.4 we get

β(2)1 (H)2

rkH≥ [G0 : H ]

β(2)1 (G0)

2

rkG0> M,

a contradiction. ////

5.10 The proof of theorem 5.3. The proof of theorem 5.3 runs alongessentially the same lines as the proof of 5.2 did. The difference lies, as is thennatural, in the the following analogue of theorem 5.8.

5.11 Theorem. Let G be the Cayley graph of some finitely generated group Gwrt. a finite, symmetric generating set S, and let m be the free minimal spanningforest on G. Then deg m ≥ 2cost(G).


Remark. The attentive reader will compare theorems 5.8 and 5.11 and wonderhow this corresponds to the relation between the FUSF (see proposition 3.11) and

the relation between cost and β(2)1 (see theorem 2.12). In fact, the statement of

5.8 is not optimal. It is known that it can be sharpened to an equality deg u =

2(β(2)1 (G)+1), so that everything is indeed in order. For this, see the forthcoming

book [LP09].

Our proof will differ from the one of Monod-Epstein in its starting point,namely that we are, as promised, inspired by [GL07] and will not use [KM04].A casual glance though, tells us that the proof of the theorem in [KM04] usedby Monod-Epstein rests upon the same ideas that Gaboriau’s techniques do.

The real difference here is then philosophical rather than technical: Neverread anything written by a set theorist if you can avoid it, and never ever admitto use anything done by a set theorist if you can get away with it.‡

Proof. Consider the Bernoulli action of G on X =∏

g∈G[0, 1] equipped withthe product Lebesgue measure m, as well as its canonical action on Ω = G(G)equiped with the standard coupling measure m(ε) where ε > 0 is fixed butarbitrary.§ Then Ω×X is a standard Borel probability space, with free diagonalaction of G, inducing an (SP1) equivalence relation on it.

Further, we consider the projection g : Ω ×X → Ω. This is G-equivariantand gives a cluster equivalence relation on Ω×X . Now the crucial observation isthat Rcl

G is graphed by the graphing Φcl = (ϕcls )s∈S , where ϕcl

s is the restrictionof the action of s−1 to the set As of points z ∈ Ω×X for which the edge (1, s)is in g(z). Then

costm(ε)×m(Φcl) =∑

s∈S

(m(ε) ×m)(As)

=∑

s∈S

m(ε)(ω ∈ Ω | (1, s) ∈ E(ω))

= deg m(ε).

However, our generating set S is symmetric and this is about twice as symmetricas we need.¶ Thus

2costm(ε)×m(RclG) ≤ deg m(ε).

Further, combining theorems 3.12 and 3.14 we get

2cost(G) ≤ 2costm(ε)×m(RG) = 2costm(ε)×m(RclG) ≤ deg m(ε).

Finally, the right-hand side of this is equal to deg m + ε · ♯S and since ε wasarbitrary we are done. ////

‡This provocation, if taken to heart is somewhat hypochritical (is that a word?) - Indeed,the main theorem of [GL07], though not used here, relies on [KM04].

§yeah yeah bad notation again...¶In fact, there is a small annoyance here in that S could contain an element s′ of order

two. However, in that case we can cut As′ in half and still have a graphing, so there is noactual problem here.


From hereon in the proof is then entirely similar to that of theorem 5.2: Theabove gives a bound M such that

cost(H)2

rkH≤M, H ≤ G finitely generated subgroup.

Then by the analogue of theorem 6.4 for cost (see [Gab00]) and using residualfiniteness we once again violate this for a contradiction.

Chapter 6

Classifying complexes and a

different proof of theorem

5.2

The sixth rule about fight club is fights go on as long as theyhave to.

key words:Classifying complex - Cayley graph - actions arbitrary complexes

In this chapter I will give a proof of theorem 5.2 that was originally supposedto leave the niceties of the Cayley graph behind. As we shall see, this wasnot quite possible, but the result does generalize 5.2 slightly. The originalmotivation was to do completely without the Cayley graph so that the proofwould generalize to higher dimensions, but as also indicated by the results ofthe next chapter, the message turns out to be that one should perhaps insteadlook for higher dimensional analogues of the Cayley graph, or something thatdoes a similar job.

We start out by proving some general theorems for simplicial complexes withfree action(s) of group(s). It turns out that one computational nicety of usingthe Cayley graph to prove the theorem is that there is exactly one orbit of points,and that this actually determines explicitly a fundamental domain for the actionon the 1-skeleton. Thus the central idea∗ of this chapter is the construction in6.5 of a simplicial complex with just one orbit of points, from one with morethan one orbit. Coupled with theorem 4.2 the rest, actually, is just chalk. Forgood measure we mention that one can take a more direct approach without thisbusiness of trying to reduce the number of orbits, but we will not give this proof.

6.1 Induced complexes, subgroups, and EG.

∗As mentioned, not a particularly successful one.

56

CHAPTER 6. CLASSIFYING COMPLEXES 57

Definition. Suppose the K,L are simplicial complexes, and H is a groupacting on K from the right and on L from the left. Define the simplicial productof K and L over H, deonted K∆HL by letting (K∆HL)(0) = K × L/ ∼ where∼ is the equivalence relation generated by (v, u) ∼ (v.h, h.u). Further, we let([(v0, u0)], . . . , [(vn, un)]) ∈ (K∆HL)(n) if and only if the equivalence classes arepairwise distinct, and that for the set (vi, ui) of representatives, the sets virespectively ui are unordered simplices in some K(n′) resp. some L(n′′).

When H is trivial we write simply K∆L.

Note that we do not exclude the possibility that vi = vj (ui = uj) for somei 6= j as long as the equivalence classes are distinct.

It is clear that the simplicial product actually is a simplicial complex.

Observation. If K,L are equiped with actions of G, then G acts on K∆L bythe diagonal action, and this is free if and only if at least one of the actions onK,L is free.

This allows us to calculate an ℓ2-Betti number for K with a non-free actionby considering K∆EG for some model of the classifying complex of G. Onefeature of Luck’s approach to ℓ2 invariants is that this awkwardness is avoided.

Now, let H be a subgroup of G and let K be a simplicial complex withan action of H . We want to define an induced complex IndG

HK containing Kand such that G acts on IndG

HK, and of course in such a manner that whenrestricting the action to H on K ⊆ IndG

HK we recover the original action.To do this, consider G as a zero-dimensional complex with the left-action

by multiplication of itself and the right-action by multiplication of H , and thenconsider that simplicial product G∆HK. Since the left action of G on itselfcommutes with the right action of H , this has a well-defined action of G, byleft-multiplication on the first coordinate.

Definition. For H a subgroup og G and K a simplicial complex with an actionof H, the induced complex is the complex IndG

HK := G∆HK as constructedabove.

Note the obvious, but important, fact that if the action of H on K is free,then so is the action of G on IndG

HK.The induced complex has exactly the main property that we wanted be-

sides an action of G, namely that we can H-equivariantly identify K with thesubcomplex consisting of simplices with vertices in [(1, u)] | u ∈ K(0).

As an application we get

Theorem. Let H be a subgroup of the countable group G. Then any modelEG of the classifying complex for G, when equiped with the obvious action ofH, is also a model of the classifying complex for H.

Proof. Indeed, if K is a simplicial complex with a free action of H , then theaction of G on IndG

HK is free, and so there is a G-equivariant embeddingIndG

HK → EG. By the H-equivariant identification K ⊆ IndGHK this restricts

to an H-equivariant embedding of K.


for the uniqueness, simply run through the proof of theorem 1.22 again to getuniqueness for EG = E0G and the general cases follows since any two modelsof the classifying complex are (chain-)homotopic. ////

6.2 Lemma. Suppose that H is a subgroup of G with finite index, and that Kis a simplicial complex with a free cocompact action of G.

Then the action of H on K is (free,) cocompact, and for all n ≥ 0 we have

an L(H)-equivariant isometric isomorphism fn : C(2)n (K,G) → C

(2)n (K,H) such

that for every closed, G-invariant subspace S of C(2)n ,

dimL(H) fn(S) = [G : H ] dimL(G) S.

Further, the fn can be taken to commute with boundary maps and inclusion ofG-invariant complexes.

Proof. Fix n and let s1, . . . , sm be a fundamental domain for the action ofSn+1 × G on K(n). Choose representatives g1, . . . , g[G:H] for the cosets H \ G.Then put

fn

(m∑

k=1

ak ⊗ sk

)=

m∑

k=1

[G:H]∑

l=1

(ak.gl)|H ⊗ gl.sk

,

noting that gi.sj is a fundamental domain for the action of Sn+1×H on K(n),and where ak.gl denotes the usual right action of gl on the function ak ∈ ℓ2(G).

The claims are now easily checked. ////

6.3 Lemma. Let E0G be the standard model of the classifying complex of G andH a finite index subgroup of G. Suppose that K ⊆ E0G is a simplicial complexwith a free, cocompact action of G. Then for any H-equivariant embedding Kof HK in E0H we have

[G : H ] · infK⊆L⊆E0G

dimL(G) Im(H

(2)

n (K,G) → H(2)

n (L,G))

=

= infK⊆L⊆E0H

dimL(H) Im(H

(2)

n (K,H) → H(2)

n (L,H)).

We leave out the relatively straightforward proof.Without too much work (basically the argument we didn’t give, only applied

to the K, K instead of L, L) one gets the following

6.4 Theorem. For H a finite index subgroup of the countable group G we havefor all n ≥ 0

β(2)n (H) = [G : H ]β(2)

n (G).

6.5 Special automorphisms of E0G: The cyclic shift of order n.

Consider the standard model E0G of EG, and recall that (E0G)(0) = G × Z.We define for n ≥ 1 a map cn : (E0G)(0) → (E0G)(0) by

cn(g, i) = cn(g, q(i) · n+ r(i)) = (g, q(i) · n+ r(i+ 1))


where r(i) is the remainder of i wrt. division by n, i.e. i = q(i) · n + r(i), 0 ≤r(i) < n for all i ∈ Z.

Obviously cn is a simplicial automorphism og E0G, and cmn acts freely on(E0G)k

uo for k < n whenever 0 < m < n, and cnn = 1.

Observation. It follows straight from the definitions that cn is G-equivariant.Thus we get an action of Cn ×G on E0G, where Cn is the cyclic group of ordern.

Definition. Suppose that K ⊆ E0G is a G-invariant subcomplex. Then itsn’th suspension in dimension zero is the smallest Cn ×G-invariant subcomplex

S(0)n K ⊆ E0G containing K.

6.6 Lemma. If the action of G on the invariant subcomplex K ⊆ E0G is

cocompact, then so is the action of Cn ×G on S(0)n K, whence so is the action

of G on S(0)n K.

Proof. Indeed, let s ∈ (S(0)n K)(m) for some m. If Cn ×G.s does not intersect

K(m) then we may remove this entire orbit from S(0)n K and have a strictly

smaller Cn×G-invariant subcomplex, still containing K, a contradiction. (Note

that we also need to remove every simplex in (S(0)n K)(m

′),m′ > m, containinga simplex in Cn × G.s, but it is clear that we can actually do this since if oneof these were in K then so would be all its faces.)

It follows that if siki=1 is a fundamental domain for the action of G on

K(m) then it actually intersects every Cn ×G-orbit on (S(0)n K)(m). ////

6.7 Observation. Note that the freeness of the action of Cn ×G is certainlynot to be taken for granted. However, when the action of G on K is cocompactwe can make it free, basically by choosing n big enough. Indeed, since the actionis cocompact, there is some m such the the second coordinates of K(0) (recallthat K ⊆ E0G) are in −m,−m+ 1, . . . ,m. In fact, we may as well assumethat they are in 0, . . . ,m.

Indeed, let p be the smallest (or just any) prime, p > m. To get the freeness,we have to show that for any set S of ≤ m numbers in 0, . . . , p− 1 we haveS 6≡ S + j (mod p) where 0 < j < p. (The j is the action of cjp.)

But this is just a simple exercise in algebra. Indeed, if we do have equality,then letting S = aik

i=1 where 1 ≤ k ≤ m we have, summing over i

k∑

i=1

ai ≡k∑

i=1

(ai + j) (mod p).

This implies that kj ≡ 0 (mod p), which is impossible.

6.8 A baby-baby Hodge-de Rahm theorem and the FUSF.

6.9 (E0G-Baby-baby ℓ2-Hodge-de Rahm) Lemma. Suppose that K is a G-

invariant subcomplex of E0G with (free,) cocompact action of G. Let Hn be the


closed subspace of C(2)n (K) generated by elements in the kernel of ∂

(2)n with finite

support and integer coefficients.Then (H

n is G-invariant and)

dimL(G)

(ker∂(2)

n ⊖Hn

)≥ inf

L:K⊆L⊆E0GdimL(G) Im

(H

(2)

n (K,G) → H(2)

n (L,G)).

Compare this with theorem 1.24. Also note that Hn contains the closure of

the image of ∂(2)n+1.

Proof. The idea is to apply the same trick as in the construction of E0G.Namely, let c = s1 + · · · sk be an element in H

n. We may assume that thevertices in K(0) have second coordinates in −m. . . 0, and putting ρ1 = (1, 1)we can consider the simplicial complex L1 = K ∪ ρ1 ∨ K where ρ1 ∨ K is tobe understood as (ρ1 ∨ K)(j) = Sj+1.ρ1 ∨ s | s ∈ K(j−1) for j ≥ 1 and(ρ1 ∨K)(0) = ρ1.

Then clearly L1 is a simplicial complex with cocompact action of G, and by

the identity eq. (1.2) we have that c = ∂(2)n+1(ρ1 ∨ c) in C

(2)n (L1).

Since there are only countably many possibilities for c, it is clear how toproceed to get an increasing sequence (Lm)m of simplicial complexes with co-compact action of G such that

ker(H

n → H(2)

n (Lm, G))րm H

n,

from which the claim follows. ////

The point in the Baby-baby Hodge-de Rahm lemma above is of course thatH is exactly the cycle space of ℓ2alt(K

(1)), so that we can now use the FUSF toprove the following analogue of theorem 4.4.

6.10 Theorem. Suppose that G is unitarisable (and countable). Then thereis a constant, M , such that for any subgroup H of G and any H-invariantsimplicial complex K ⊆ E0H with the action of H on K (free,) cocompact wehave

1√♯F (1)

· infL:K⊆L⊆E0H

dimL(H) Im(H

(2)

1 (K,H) → H(2)

1 (L,H))≤ ♯F (0) ·M,

where F (i) is a(ny) fundamental domain for the action of Si+1×H on K(i) andthe infimum is over H-invariant subcomplexes with (free,) cocompact action ofH, as usual.

Proof. Since G is unitarisable, so is the group

G :=

∏

j∈N

Cj

×G,


whence by theorem 4.4 there is an M such that for any subgroup H of G

∀f ∈ T1(H) : ‖f‖ℓ2(H) ≤M‖f‖T1(H).

Now let K be given as in the statement of the theorem, and as usual we mayassume that the second coordinates of the vertices in K(0) are in 0, . . . ,m.Choosing a prime p > m the suspension S

(0)p K has free, cocompact action of

Cp ×H . Since K is contained in this, we get

1√♯F (1)

· infL:K⊆L⊆E0H

dimL(H) Im(H

(2)

1 (K,H) → H(2)

1 (L,H))≤

≤ 1√♯F (1)

· infL: H(S

(0)p K)⊆L⊆E0H

dimL(H) Im(H

(2)

1 ( H(S (0)p K), H) → H

(2)

1 (L,H))

≤ 1√♯F (1)

· dimL(H) ker ∂(2)n |

C(2)1 ( H(S

(0)p K),H)

) ⊖H1.

where H1 is as in the lemma above with K replaced by S

(0)p K, considered as a

complex with an action of H .Now, if E(1) is a fundamental domain for the action of S2 × Cp × H on

(S(0)p K)(1) then by the proof of lemma 6.6 we have ♯E(1) ≤ ♯F (1), and contin-

uing the above calculation we find

1√♯F (1)

· dimL(H) ker ∂(2)n |

C(2)1 ( H (S

(0)p K),H)

) ⊖H1 ≤

≤ p√♯E(1)

· dimL(Cp×H) ker ∂(2)n |

C(2)1 ( Cp×H(S

(0)p K),Cp×H)

) ⊖H1.

Let the connected components of (S(0)p K)(1), considered as a graph, G on

(S(0)p K)(0), be (Gi)i and consider the random graph u = ×iuGi

where uGi

denotes the FUSF on Gi. Since the Cp ×H acts transitively on the vertices ofG the average degrees of any two vertices are equal, and in fact equal to theaverage degree of any vertex ρ ∈ V (G1) with respect to uG1 .

Choose such a ρ. Then ρ is a fundamental domain for the action of Cp×Hon (S

(0)p K)(0) and the set of edges l ∈ (S

(0)p K)(1) such that l+ = ρ constitutes

a fundamental domain E(1) for the action of S2 ×Cp ×H on (S(0)p K)(1). Thus

with fu(h) = u(F | ρ, h.ρ ∈ E(F)) we get

p√♯E(1)

· dimL(Cp×H) ker∂(2)n |

C(2)1 ( Cp×H(S

(0)p K),Cp×H)

) ⊖H1 ≤

≤ p√♯E(1)

∑

l∈E(1)

〈(1− PH1)l, l〉

=p

2√♯E(1)

∑

l∈E(1)

〈(1− Pℓ2(E(G1)))χl, χl〉

where the final equality uses the identification of C(2)1 with ℓ2alt and the fact that

the cycle-space of a graph is the direct sum of the cycle-spaces of its clusters.


Proceeding as in the proof of theorem 5.8 we then get by the commentsabove†

p

2√♯E(1)

∑

l∈E(1)

〈(1− Pℓ2(E(G1)))χl, χl〉 = pdeg(uG1)√♯E(1)

= pdeg(u)√♯E(1)

≤ p‖fu‖ℓ2(Cp×H)

≤ pM‖fu‖T1(Cp×H)

≤ 2pM.

Here the final inequality‡ uses lemma 5.9, noting that the assumption that G bea Cayley graph is superfluous as long as it has bounded degree and Cp ×H actsfreely and transitively on the vertices. Since we can arrange that m = ♯F (0) − 1and p ≤ 2m we are done. ////

Remark. • Of course, as a corollary we get theorem 5.2.

• Replacing K with its suspension in the statement of the theorem, the in-equality ♯E(1) ≤ ♯F (1) can be improved to ♯F (0) · ♯E(1) = ♯F (1) so that wecould replace ♯F (0) with

√♯F (0) in the statement also.

• It is perhaps interesting to note that the uniformity over subgroups followsfrom the particular case stating the theorem just for G, using inducedcomplexes.

†Actually, in order to write deg(uG1) we need to note that G1 has a transitive automorphism

group. But this is clear, taking for instance the subgroup of Cp ×H generated by elements h

with h.ρ ∈ V (G1).‡Note also that we do not actually have to use the equality deg(uG1

) = deg(u). It justseemed more in line with the picture in my own head to do so. (If we did not, of courseinstead of Cp × H the final inequalities would involve the subgroup of automorphisms of G1

mentioned in the previous footnote. In fact, this acts freely on the vertices of the connectedgraph G1, so that actually this is its Cayley graph.)

Chapter 7

An annotated list of

problems

And the seventh rule about fight club is, if this is your first nightat fight club, you have to fight.

– Fight Club by Chuck Palahniuk.key words:

Various open problems

7.1 Behavior under exact sequences. Does unitarisability behave wellwith respect to exact sequences? That is, if

0 // G // Γ // A // 0

is exact and G,A are unitarisable, is Γ then necessarily unitarisable? (see chap-ter 4.)

Also, if G1, G2 are unitarisable, is then even G1 ×G2 unitarisable? This isbasically a question of whether one can simultaneously “unitarize” two com-muting representations, and simultaneous similarity problems are typically hard.

7.2 The Dixmier problem. Dixmier’s question remains unanswered: Isevery (countable) discrete unitarisable group amenable?

To finish the proof that they are using theorem 5.2 one would have to showthat unitarisability implies the vanishing of all ℓ2-Betti numbers (even for groupsthat are not necessarily residually finite), and then show the existence of atreeable action - see proposition 6.10 in [Gab02].

There is a very recent paper of Monod and Ozawa in which the followingstrong result is proved.

( [MO09]) Theorem. For a countable group G the following are equivalent:

(i) G is amenable.

63

CHAPTER 7. AN ANNOTATED LIST OF PROBLEMS 64

(ii) For every abelian group A, A ≀G is unitarisable.∗

(iii) There is an infinite abelian group A such that A ≀G is unitarisable.

The authors use this to show that the Burnside groups B(p, q) are indeednot unitarisable (when they are infinite.) The proof uses bounded cohomology,and the important paper [GL07] by Gaboriau-Lyons.

In the light of this result it is perhaps not surprising that Pisier found ques-tion 7.1 hard. Indeed, resolving it in the positive would resolve Dixmier’s prob-lem in the positive as well by the above, using that the abelian group ⊕g∈GAis (amenable hence) unitarisable.

7.3 Transitive unimodular graphs. In percolation theory there is a ruleof thumb to the effect that most things that are true for Cayley graphs can beextended to transitive graphs, usually by imposing unimodularity instead.

In [Gab05], Gaboriau defines the first ℓ2-Betti number of a locally finite,transitive unimodular graph. That is, the graph G is locally finite and has atransitive, unimodular (closed, and not necessarily countable) group of auto-morphisms H ⊆ Aut(G).

It seems plausible that one could prove that if the first ℓ2-Betti number ofthe graph is non-zero, then H is not unitarisable.

Indeed, if Kρ is the isotropy group of a basepoint ρ ∈ V (G) then we should

have that β(2)1 (G) = [Kρ : H1] · β(2)

1 (H/H1) where H1 is the component of theidentity in H , which is compact and open. Note that the index is finite sinceKρ is also compact, open.

7.4 The fixed price question. A countable group G is said to havefixed price if cost(RG) is independent on the particular free measure-preservingaction of G. A well-known open problem is the so called fixed price question,asking whether any group at all fails to have fixed price.

In the original study by Gaboriau [Gab00], there is only a single canlculationestablishing the cost of a group without first showing that it has fixed price,namely corollary VI.30 of that paper: Any lattice in a connected semi-simpleLie group of real rank at least two has cost one.†

Settling the fixed price question would have ramifications for instance whendealing with the continuity of cost when taking increasing unions. At present,to conclude that an increasing union of cost one groups has cost one, one needsto show that they all have fixed price.

7.5 cost vs. ℓ2-Betti numbers. Does every countable group G satisfy

cost(G) − 1 = β(2)1 (G)?

More generally, one can ask whether every (SP1) equivalence relation Rsatisfies costµ(R) − 1 = β

(2)1 (R, µ). If this could be proved true, it would at

∗Recall that A ≀ G = (⊕g∈GA) ⋊ G.†There are other instances (or at least one - V I.19) but these do not concern explicit

groups.


once settle not just the first question but also the fixed price question.Recall from chapter 5 that we have

deg u = 2(β(2)1 (G) + 1) ≤ 2cost(G) ≤ deg m.

Thus, one can obtain bounds at least, by investigating the free uniform/minimalspanning forests on Cayley graphs of G.

This, in particular implies the fixed price question, 7.4 and is recognizedin the literature as a central and very important problem in measurable grouptheory.

7.6 Treeability vs. anti-treeabiility. Does there except a count-able group permitting a free measure-preserving action (on a standard Borelprobability space) which is treeable, as well as one which is not?

I do not expect this to be at all tractable, and I would guess no one hasany idea concerning how to go about it, but I mention it since I mentionedtreeability problems above.

7.7 Determinantal measures and a bound in dimension n. To provethe vanishing of higher ℓ2-Betti numbers along the lines of proof developed inchapters 5,6 one would presumably need to generalize the uniform spanningforrest from graphs to complexes. It is, of course, not the uniform-ness of itthat we care about, but more that the higher dimensional analogue of theorem3.7 hold. One way to accomplish this is to use the determinantal measures ofLyons, as exposed in [Lyo03,Lyo08].

The following analogue of theorem 3.7 summarizes the existence of a relevantgeneralization of the FUSF.

7.8 [Lyo08] Theorem. Let K be a simplicial complex with a free action of thecountable group G. Then in each dimension n such that the n-skeleton of K is

locally finite, there is a G-invariant probability measure uFn on 2K(n)

uo , supported

on the set of elements with T ⊆ K(n)uo such that no finitely supported c ∈ ker ∂n

has support contained in T , and satisfying for all s ∈ K(n)

uFn (s ∈ T ) =

1

(n+ 1)!〈(1− P)χs, χs〉.

In the statemeent above we identify C(2)n (K) with ℓ2alt(K

(n)), the subspace ofℓ2(K(n)) of functions such that (σ.f)(s) = signσ · f(s) with the obvious actionof Sn+1. Also, we define χs =

∑σ∈Sn+1

(signσ)δσ.s. Finally, P is of coursethe orthogonal projection onto the closed subspace generated by the finitelysupported s ∈ ker∂n.

One could hope that this could somehow give a strategy for proving thevanishing of higher ℓ2-Betti numbers by finding a suitable orientation trick as inthe proof of dimension 1 to get an analogue of theorem 6.10. This however, hasresisted all my efforts. In fact, in higher dimensions, one gets a bound for free,without using even unitarisability. This comes about from the fact that, whereas


in dimension 1 one can for given vertices u, v have lots and lots of lines of theform (u, g.v), the same is not true in higher dimensions, relating n-simplicesto n − 1-simplices. This gives a concrete bound on the number of orbits ofn-simplices as a function of n and the number of orbits of n− 1-simplices.

Initially, getting a bound for free might seem like a great thing, but I thinkthat maaybe it is not so - that maybe getting something for free means if youwant more you have to work hard.

The next couple of questions are related to these matters.

7.9 Determinantal measures and uniformity. The determinantalmeasure associated to an invariant subspace of a hilbert space with a (unitary)representation of some group G is not uniform. In fact, from an invarianceperspective, it is probably not even the natural choice of measure. Indeed, if Eis a finite matroid and we consider an invariant subspace H of ℓ2(E), one of thekey properties of the determinantal measure PH is that PH(e ∈ T ) = 〈PHe, e〉where PH is the orthogonal projection onto H and e ∈ E, with T a sample.

The existence of a measure with this property shows that the convex setof al measures satisfying it is non-empty, and then usually, from an invarianceperspective, it is natural to consider the unique measure with minimal ℓ2-norm.

However, I could not determine if this gives rise to a measure on infinitematroids, since there is neither the niceties of uniform-ness, nor the good prop-erties of the determinant to rely on. In particular, I could not determine theprobability that e1, . . . , ek are all in a sample.

If one could overcome these difficulties, perhaps by placing some weak-ishrestrictions on the matroids / subspaces under consideration, this might be aninteresting generalization of uniform spanning tree/complex measures to a moregeneral class of matroids/subspaces.

However, note that this would lose the stronger version of the equailityPH(e ∈ T ) = const. · 〈PHχe, χe〉, giving the probability that any given set ofedges is in T as a determinant (for the FUSF this property is known as thetransfer current theorem). Thus what one gains from an invariance perspective,one probably loses from a combinatorial perspective several times over.

7.10 A hands on proof of theorem 7.8. The development, in partic-ular the definition of the determinantal measures in [Lyo03,Lyo08] is perhapsnot very intuitive. They are simply defined by a formula for any basis neigh-bourhood, giving the probability in terms of determinants of certain matrices.

It seems like a good idea to try to generalize the proof of theorem 3.7 givenabove, but one runs into problems that are homological in nature. Basically,the point is that in graphs when one has a simple cycle one has exactly oneclass in homology and the coefficients on all edges in the cycle are equal. Thismakes calculating adjoints of the P,T in chapter 3, or doing the averaging kindof trick done there, painfree.

In higher dimensions however, this need not be the case. To realize thisconcretely, consider for example a double Moore space kind of construction,attaching two (appropriately triangulated) n-cells to the same n − 1-cell with


different degrees and gluing them together in one triangle, which one then re-moves. Choosing the degrees properly (for instance 5 and 2) one then has nocycles. When one re-atteches the triangle one gets a cycle where some trian-gles appear with coefficient 2, some with coefficient 5, and one triangle withcoefficient 3.

The point here is that there is torsion in the homology group in dimension1, namely we can hit the cycle with coefficients 2 on all edges but not the onewith coefficients 1 on all edges.

Thus to do a proof in higher dimensions one needs to keep track of thesethings, which would complicate calculations somewhat.

Bibliography

[Akh09] Azer Akhmedov. A new metric criterion for non-amenability III:Non-amenability of r.thompson’s group f. Preprint, April 2009.

[BLPS01] Itai Benjamini, Russel Lyons, Yuval Peres, and Oded Schramm. Uni-form spanning forests. Annals of Probability, 29(1):1–65, 2001.

[CG86] Jeff Cheeger and Mikhael Gromov. l2-cohomology and group coho-mology. Topology, 25(2):189–215, 1986.

[Dye59] Henry Abel Dye. On groups of measure preserving transformations.I. American Journal of Mathematics, 81:119–159, 1959.

[Dye63] Henry Abel Dye. On groups of measure preserving transformations.II. American Journal of Mathematics, 85:551–576s, 1963.

[FM77a] Jakob Feldman and Calvin Moore. Ergodic equivalence relations, co-homology, and von neumann algebras. I. Transactions of the Ameri-can Mathematical Society, 234(2):289–324, 1977.

[FM77b] Jakob Feldman and Calvin Moore. Ergodic equivalence relations, co-homology, and von neumann algebras. II. Transactions of the Amer-ican Mathematical Society, 234(3):325–359, 1977.

[Gab00] Damien Gaboriau. Cout des relations d’equivalence et des groupes.Inventiones mathematicae, 139(1):41–98, 2000.

[Gab02] Damien Gaboriau. Invariants l2 de relations d’equivalence et degroupes. Publications Mathematiques de l’Institut des Hautes EtudesScientifiques, 95(1):93–150, 2002.

[Gab05] Damien Gaboriau. Invariant percolation and harmonic dirichlet func-tions. Geometric and Functional Analysis, 15(5):1004–1051, 2005.

[GL07] Damien Gaboriau and Russel Lyons. A measurable-group-theoreticsolution to von neumann’s problem. preprint, 2007.

[KM04] Alexander Kechris and Benjamin Miller. Topics in Orbit Equivalence.Springer, 2004.

68

BIBLIOGRAPHY 69

[LP09] Russell Lyons and Yuval Peres. Probability on trees and networks.Book in progress (available at http://mypage.iu.edu/∼rdlyons/),February 2009.

[LPS06] Russel Lyons, Yuval Peres, and Oded Schramm. Minimal spanningforests. Annals of Probability, 34(5):1665–1692, 2006.

[Luc98] Wolfgang Luck. Dimension theory of arbitrary modules over finitevon neumann algebras and l2-betti numbers I: Foundations. Journalfur die reine und angewandte Mathematik, 495:135–162, 1998.

[Luc02] Wolfgang Luck. L2-invariants: Theory and Applications to Geometryand K-Theory, volume 44 of Ergebnisse der Mathematik und ihrerGrenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics.Springer, 2002.

[Lyo03] Russell Lyons. Determinantal probability measures. PublicationsMathematiques de l’Institut des Hautes Etudes Scientifiques, 98:167–212, 2003.

[Lyo08] Russell Lyons. Random complexes and ℓ2-betti numbers. preprint,November 2008.

[ME08] Nicolas Monod and Inessa Epstein. Non-unitarisable representationsand random forests. Preprint, december 8th 2008.

[MO09] Nicolas Monod and Narutaka Ozawa. The dixmier problem, lamp-lighters and burnside groups. preprint, february 26th 2009.

[NW99] M. Nagisa and Shuhei Wada. Simultaneous unitarizability and sim-ilarity prooblem. Scientiae Mathematicae, 2(3):255–261, 1999. (Injapanese).

[Pis01] Gilles Pisier. Similarity Problems and Completely Bounded Maps.Lecture Notes in Mathematics. Springer, second, expanded editionedition, 2001.

[Pis05] Gilles Pisier. Are unitarizable groups amenable. In Infinite Groups:Geometric, Combinatorial and Dynamical Aspects, volume 248 ofProgress in Mathematics, pages 323–362. Birkhauser, 2005.

[Sha09] E.T. Shavgulidze. About amenability of subgroups of the group ofdiffeomorphisms of the interval. Preprint, May 2009.

the dixmier unitarisability problem and ℓ -betti numbers · the second chapter deals with the...

Documents