: a hyperfinite approachisaac/nsfungrouptalk.pdf · the problem end compactifications of finite...

⇡1(|�|): a hyperfinite approach

Isaac Goldbring(joint work with Alessandro Sisto)

University of Illinois at Chicago

Model theory and applications to geometryLisbon, Portugal

July 19, 2013

Isaac Goldbring (UIC) ⇡1(|�|): a hyperfinite approach Lisbon July 19, 2013 1 / 32

The problem

1 The problem

2 Nonstandard analysis

3 The Main Theorem

4 An application to homology

The problem

⇡1(X )

Suppose that X is a space and p 2 X .Recall that ⇡1(X ; p) is the set of (continuous) loops based at pmodulo the relation of two loops being homotopic.The operation of concatenating loops based at p induces a groupoperation on ⇡1(X ; p) (with identity being the homotopy class ofthe constant loop at p).If X is pathconnected, then this group is independent of p and isdenoted by ⇡1(X ), referred to as the fundamental group of X .The typical example is ⇡1(S1) ⇠= Z, where S1 is the unit circle in C.This construction is functorial: if f : X ! Y is continuous, thenthere is an induced map f⇤ : ⇡1(X )! ⇡1(Y ) given byf⇤([↵]) := [f � ↵].X is called simply connected if it is pathconnected and⇡1(X ) = {1}.

The problem

⇡1(�) when � is finite

TheoremSuppose that � is a connected, finite graph. Then ⇡1(�) is a finitelygenerated free group.

Proof.

Let T be a spanning tree of �.Let ~e1, . . . ,~en be oriented chords of T , that is, edges of � not in T ,given a fixed orientation.Given [↵] 2 ⇡1(�), let r↵ be the reduced word on {e±1

1 , . . . , e±1n }

obtained by recording which chords ↵ traverses fully and in whichdirection.The map [↵] 7! r↵ : ⇡1(�)! Fn is an isomorphism.

The problem

End compactifications of finite graphs

We now consider infinite, locally finite, connected graphs.Many results from finite graph theory are plain false for infinitegraphs.However, by compactifying an infinite graph by adding its “ends,”one can obtain topological analogues of theorems from finitegraph theory.

The problem

DefinitionLet X be a metric space and p 2 X .

1 For x , y 2 X , we write x /n y to indicate that x and y are in thesame path component of X \ B(p; n).

2 For r1, r2 : [0,1)! X proper rays with r1(0) = r2(0) = p, we sayend(r1) = end(r2) if and only if:

(8n 2 N)(9m0 2 N)(8m � m0)(r1(m) /n r2(m)).

3 Ends(X ) := {end(r) | r a proper ray starting at p}.4 |X | := X [ Ends(X ) is the end compactification of X , topologized

in such a way so that proper rays converge to their ends.

The problem

The main problem

Question (Diestel/Sprüssel)

Is there a nice combinatorial characterization of the fundamental groupof the end compactification of a locally finite, connected graph in thespirit of the result in the second slide?

The problem

An example: the infinite sideways ladder

Consider the loop ↵ beginning at v0, going along the bottom rungof the ladder to the end at +1, and then back again along thebottom rung of the ladder. ↵ is certainly nullhomotopic (i.e.homotopic to the constant loop at v0).If we consider the topological spanning tree T for � pictured belowin bold with oriented edges ~e1,~e2, . . ., then the “word” ↵ induces is(~e1~e2 · · · )_(· · · �e2

�e1).This word is of order type ! + !⇤ with no consecutiveappearances of �!ei and �ei . So we cannot combinatorially tell thatthis loop is nullhomotopic.

The problem

Diestel and Sprüssel’s Result

Undaunted by the previous example, Diestel and Sprüssel offeredthe following solution to their question.Let � be an infinite, locally finite, connected graph with endcompactification |�|. Let T be a topological spanning tree for �with oriented chords X = {~e1,~e2, . . .}.Diestel and Sprüssel consider words on X of arbitrary countableorder type (e.g. the order type of Q!) and define a non-wellorderednotion of reduction of words.If F (X ) denotes the group of reduced words (in the above sense),Diestel and Sprüssel show that the map [↵] 7! r↵ : ⇡1(|�|)! F (X )is a well-defined injective group homomorphism (although thistakes � 15 pages!). They also identify the image.By considering finite subwords, they construct an injective groupmorphism F (X )! lim �Fn into an inverse limit of finitely generatedfree groups, once again identifying the image. (Algebraic andeasy.)

The problem

Can nonstandard analysis help?

After seeing my nonstandard treatment on ends, Diestel asked me thefollowing question:

Question (Diestel)

Can nonstandard analysis make any of this simpler?

Answer (G., Sisto)

The problem

The infinite sideways ladder revisitedLet ⌫ be an infinite natural number. We can then consider the followinghyperfinite extension of �:

Our loop ↵ from before “clearly” induces the hyperfinite word�!e1�!e2 · · ·

�!e⌫ �e⌫ · · · �e2

�e1,

which “clearly” internally reduces to the empty word, exhibiting that ↵ isnullhomotopic.

In this way, we get an injective group morphism ⇡1(|�|) ,! ⇡1(�⌫),where ⇡1(�⌫) is the internal fundamental group of �⌫ , which is ahyperfinitely generated internally free group on ⌫ generators.

The problem

�!e⌫ �e⌫ · · · �e2

�e1,

The problem

�!e⌫ �e⌫ · · · �e2

�e1,

Nonstandard analysis

1 The problem

3 The Main Theorem

NSA in a nutshell

Every set X gets enlarged, in a functorial fashion, to a set X ⇤, thenonstandard extension of X .X ⇤ “logically behaves” like X (Transfer Principle), but contains new“ideal” elements, e.g. R⇤ contains infinitesimal and infinitenumbers.In a natural way, P(X )⇤ embeds into P(X ⇤). The subsets of P(X ⇤)that belong to P(X )⇤ are called the internal subsets of X ⇤;noninternal subsets of X ⇤ are called external.The similarity in logical behavior applies only to internal subsets ofX ⇤. For example, internal subsets of R⇤ that are bounded abovehave suprema; it follows that the set of infinitesimal numbers isexternal.

The ultraproduct approach

Suppose that U is a nonprincipal ultrafilter on N, that is, U is a{0, 1}-valued measure on P(N) such that finite sets get measure0.For f , g : N! X , write f ⇠U g to mean f = g a.e.Set XU := XN/ ⇠U , the ultrapower of X with respect to U .This construction is easily seen to be functorial and the fact thatXU behaves “logically” like X is known to model theorists as Łos’theorem.In this setting, A ✓ XU is internal if there are An ✓ X such thatA =

QU An := (

Qn An)/ ⇠U .

N := [(1, 2, 3, . . .)]U 2 N⇤ is a positive infinite number whosereciprocal 1

N = [(1, 12 , 1

3 , . . .] 2 R⇤ is a positive infinitesimal.If A :=

QU An with each An finite, then we say that A is hyperfinite

with internal cardinality [(|An|)] 2 N⇤.

Nonstandard metric spaces

If (X , d) is a metric space, then (X ⇤, d) is almost a metric spaceexcept for the fact that the metric takes values in R⇤ rather than inR.There are two important subsets of X ⇤ to consider:

Xns := {a 2 X ⇤ | there is b 2 X with d(a, b) infinitesimal}.Xfin := {a 2 X ⇤ | there is b 2 X with d(a, b) finite}.

Clearly Xns ✓ Xfin with equality holding if and only if X is a propermetric space, that is, closed balls are compact.If X is proper, then a ray r : [0,1)! X is proper if and only ifr(�) 2 Xinf for all infinite elements � of R⇤.

The nonstandard approach to ends

Suppose that (X , d) is a proper, geodesic metric space and p 2 X .For x , y 2 X ⇤, write x / y to mean there is ↵ 2 C([0, 1], X )⇤ (aninternal path in X ⇤) such that ↵(0) = x , ↵(1) = y , and↵(t) 2 Xinf := X ⇤ \ Xfin for all t 2 [0, 1]⇤.“x and y are in the same path component at infinity.”

Theorem (G.)

1 end(r1) = end(r2) if and only if for all (equiv. for some) �, ⌧ 2 R>0inf ,

r1(�) / r2(⌧).2 Set IPC(X ) := {[x ] | x 2 Xinf}, where [x ] denotes the equivalence

class of x with respect to /. Fix � 2 R>0inf . Then the map

end(r) 7! [r(�)] : Ends(X )! IPC(X ) is a bijection.

The nonstandard approach to ends

Suppose that (X , d) is a proper, geodesic metric space and p 2 X .For x , y 2 X ⇤, write x / y to mean there is ↵ 2 C([0, 1], X )⇤ (aninternal path in X ⇤) such that ↵(0) = x , ↵(1) = y , and↵(t) 2 Xinf := X ⇤ \ Xfin for all t 2 [0, 1]⇤.“x and y are in the same path component at infinity.”

Theorem (G.)

1 end(r1) = end(r2) if and only if for all (equiv. for some) �, ⌧ 2 R>0inf ,

r1(�) / r2(⌧).2 Set IPC(X ) := {[x ] | x 2 Xinf}, where [x ] denotes the equivalence

class of x with respect to /. Fix � 2 R>0inf . Then the map

end(r) 7! [r(�)] : Ends(X )! IPC(X ) is a bijection.

The Main Theorem

1 The problem

3 The Main Theorem

The Main Theorem

Collapsing graphs

From now on, � is an infinite, locally finite, connected graph.Let ✓n : �! �n be the map which collapses path components of� \ B(p; n) to points. Note �n is a finite graph.It is straightforward to check that ✓n extends continuously to✓n : |�|! �n.Set �hyp :=

QU �n, a hyperfinite graph.

�hyp arises from the internal map ✓ : �⇤ ! �hyp arising fromcollapsing internal path components of �⇤ \ B(p; N) to points,where N := [(1, 2, 3, . . .)] 2 N⇤ \ N.✓ extends to an internally continuous ✓ : |�⇤|! �hyp, where |�⇤|denotes the internal end compactification of �⇤.

The Main Theorem

Collapsing graphs

The Main Theorem

Collapsing graphs

The Main Theorem

Collapsing graphs

The Main Theorem

Collapsing graphs

The Main Theorem

Collapsing graphs

The Main Theorem

Collapsing graphs (cont’d)

✓ : |�⇤|! �hyp.Digesting the definitions, one sees that |�⇤| = |�|⇤.By the Transfer Principle applied to the functoriality of thefundamental group, we get an internal map⇥ : ⇡1(|�|⇤)! ⇡1(�hyp), where the ⇡1’s here denote internalfundamental groups.More digesting of notation reveals ⇡1(|�|⇤) = (⇡1(|�|))⇤, so ⇡1(|�|)is a subgroup of ⇡1(|�|⇤).

Theorem (G., Sisto)

⇥ � ⇡1(|�|) : ⇡1(|�|)! ⇡1(�hyp) is injective.

The Main Theorem

Theorem (G., Sisto)

⇥ � ⇡1(|�|) : ⇡1(|�|)! ⇡1(�hyp) is injective.

The Main Theorem

Theorem (G., Sisto)

⇥ � ⇡1(|�|) : ⇡1(|�|)! ⇡1(�hyp) is injective.

The Main Theorem

Theorem (G., Sisto)

⇥ � ⇡1(|�|) : ⇡1(|�|)! ⇡1(�hyp) is injective.

The Main Theorem

Theorem (G., Sisto)

⇥ � ⇡1(|�|) : ⇡1(|�|)! ⇡1(�hyp) is injective.

The Main Theorem

About the theorem

Theorem (G., Sisto)

⇥ � ⇡1(|�|) : ⇡1(|�|)! ⇡1(�hyp) is injective.

Remarks

1 ⇥ is generally not injective.2 The result does not imply that ⇡1(|�|) is free. (This happens if and

only if every end is contractible.) Indeed, internally free groups(e.g. ⇡1(�hyp)) need not be free. For example, Z⇤ is internally freeon one generator, while, for infinite M, N 2 N⇤ with M

N infinite, wehave the map (a, b) 7! aM + bN : Z2 ! Z⇤ is injective. If Z⇤ werefree, then Z2 would be free.

3 ⇡1(|�|) has the same universal theory as the theory of free groups.(If ⇡1(|�|) were finitely generated, we would say it is a limit group.)

The Main Theorem

About the theorem

Theorem (G., Sisto)

⇥ � ⇡1(|�|) : ⇡1(|�|)! ⇡1(�hyp) is injective.

Remarks

The Main Theorem

About the theorem

Theorem (G., Sisto)

⇥ � ⇡1(|�|) : ⇡1(|�|)! ⇡1(�hyp) is injective.

Remarks

The Main Theorem

Universal covers

Given a topological space X , a cover of X is a topological space Cand a surjective, continuous map p : C ! X such that, for everyx 2 X , there is an open neighborhood U of x such that p�1(U) isa disjoint union of open sets in C, each of which is mappedhomeomorphically onto U by p.A universal cover of X is a cover of X whose associatedtopological space is simply connected. “Nice” spaces haveuniversal covers.For example, the universal cover of S1 is R.If G is a finite graph, its universal cover is a tree.If p : C ! X is a cover of X and � is a path in X , then for everyc 2 p�1(�(0)), there is a unique path in C lying over � starting atc. If p is the universal cover of X and � is a loop, then this uniquepath in C is also a loop.

The Main Theorem

Universal covers

The Main Theorem

Universal covers

The Main Theorem

Universal covers

The Main Theorem

Universal covers

The Main Theorem

Universal covers

The Main Theorem

About the proof

Theorem (G., Sisto)

⇥ � ⇡1(|�|) : ⇡1(|�|)! ⇡1(�hyp) is injective.

Idea of Proof

Suppose ✓(↵) is internally nullhomotopic.

Since �hyp is hyperfinite, its internal universal cover g�hyp is aninternal tree. This passage to universal covering tree is notpossible in the standard approach!✓(↵) lifts to an internal loop in g�hyp.

Using nice geodesic paths in g�hyp, we can project back onto �hypto construct a homotopy witnessing that ↵ is nullhomotopic.We do not need to consider topological spanning trees in �, anadded bonus since their existence is nontrivial.

The Main Theorem

Connection with the standard result

With more effort, we can completely recover the Diestel-Sprüsselresult.However, we can easily recover the embedding of ⇡1(|�|) into aninverse limit of f.g. free groups.The maps ✓n : |�|! �n yield a homomorphism

: ⇡1(|�|)! lim �⇡1(�n).

Define � : lim �⇡1(�n)!QU ⇡1(�n) = ⇡1(�hyp) by �((xn)) := [(xn)].

Check that ⇥ � ⇡1(|�|) = � � . Since ⇥ � ⇡1(|�|) is injective, so is .As a result, we see that ⇡1(|�|) is !-residually free, a propertyknown to be equivalent to being a limit group for finitely generatedgroups.

An application to homology

1 The problem

3 The Main Theorem

The first homology group

Definition

1 If G is a group, its first homology group is the groupH1(G) := G/[G, G].

2 If X is a pathconnected space, its first singular homology group isH1(X ) := H1(⇡1(X )).

For finite graphs, the first singular homology group coincides witha familiar combinatorial object, the so-called cycle space C(�).For infinite graphs, Diestel and Sprüssel devised an ad hochomology theory for |�|, the topological cycle space Ctop(�).They wondered if the topological cycle space coincides with thefirst singular homology.

The first homology group

Definition

1 If G is a group, its first homology group is the groupH1(G) := G/[G, G].

2 If X is a pathconnected space, its first singular homology group isH1(X ) := H1(⇡1(X )).

For finite graphs, the first singular homology group coincides witha familiar combinatorial object, the so-called cycle space C(�).For infinite graphs, Diestel and Sprüssel devised an ad hochomology theory for |�|, the topological cycle space Ctop(�).They wondered if the topological cycle space coincides with thefirst singular homology.

The topological cycle space

Set ~E(�) := {' : Eor(�)! Z | '(~e) = �'( �e )}.If ('n)n2N ✓ ~E(�) is such that, for all edges e, 'n(e) 6= 0 for finitelymany n, then we may form the thin sum of ('n),

Pn 'n 2 ~E(�).

Given a circle ↵ in |�|, get '↵ 2 ~E(�) by setting '↵(~e) = 1 if ↵traverses ~e, �1 if it traverses �e , and 0 otherwise. Call '↵ anoriented circuit.Ctop(�) is the subgroup of ~E(�) obtained by taking thin sums oforiented circuits.

Theorem (Diestel-Sprüssel)

There is a surjective group homomorphism H1(|�|)! Ctop(�) suchthat:

the homomorphism is an isomorphism when � is finite;if ↵ is a loop in |�|, then the image of [↵] is 0 in Ctop(�) if and only if↵ traverse each edge the same number of times in each direction.

Pn 'n 2 ~E(�).

Two different homology theories

Theorem (Diestel/Sprüssel)

The loop ↵ depicted below is trivial in Ctop(�) but not in H1(|�|).

Why is ↵ not nullhomologous?

A finite version of ↵ would trace the word

�!e1 · · ·�!en �e1 · · ·

�en 2 [⇡1(�), ⇡1(�)],

whence the finite version of ↵ would be nullhomologous.Diestel and Sprüssel give a topological proof that the finite versionof the loop ↵ is nullhomologous. They then remark “But we cannotimitate this proof for ↵ and our infinite ladder, because homologyclasses in H1(|�|) are still finite chains: we cannot add infinitelymany boundaries to subdivide ↵ infinitely often.” (I.e. Wouldn’t itbe great if nonstandard analysis existed?)Instead, Diestel and Sprüssel use their analysis of thefundamental group to attach a complicated invariant to loopswhich vanish on nullhomologous loops. They then show that theinvariant for ↵ is nonzero.

�!e1 · · ·�!en �e1 · · ·

�en 2 [⇡1(�), ⇡1(�)],

�!e1 · · ·�!en �e1 · · ·

�en 2 [⇡1(�), ⇡1(�)],

�!e1 · · ·�!en �e1 · · ·

�en 2 [⇡1(�), ⇡1(�)],

A simple lemma

Recall our maps ✓ : �! �hyp and ⇥ : ⇡1(|�|)⇤ ! ⇡1(�hyp).

LemmaIf the loop ↵ is null-homologous, then ✓(↵) has finite commutatorlength as an element of ⇡1(�hyp).

Proof.Let g 2 ⇡1(|�|) be the element represented by ↵ and let f : G! H1(G)be the natural map. If f (g) = 0, then we can write g as the product of,say, n commutators. As ⇥ is a group homomorphism, we have that⇥(g) can be written as a product of n commutators as well.

A simple proof that ↵ is not nullhomologous

Fact (Goldstein-Turner; G.-Sisto)

If e1, . . . , en generate a free group, then e1 · · ·ene�11 · · ·e�1

n hascommutator length bn

By transfer, ✓(↵) induces the word e1 · · ·e⌫e�11 · · ·e�1

⌫ in �hyp.By transfer of the above fact, ✓(↵) has commutator length b⌫2c > N.By the previous lemma, ↵ is not nullhomologous.

Remarks about homology

We have seen that ✓(↵) internally nullhomotopic implies ↵nullhomotopic.The preceding example shows that ✓(↵) internally nullhomologousdoes not necessarily imply ↵ nullhomologous.In fact, one can check that ✓(↵) is internally nullhomologous if andonly if ↵ is trivial in Ctop(�). So, in some sense, their ad hochomology theory is really the internal version of the ordinaryhomology theory.

References

R. Diestel, Locally finite graphs with ends: a topological approach,I. Basic theory, Discrete Math. 311 (2011), 1423-1447.R. Diestel and P. Sprüssel, The fundamental group of a locallyfinite graph with ends, Adv. Math. 226 (2011), 2643-2675.I. Goldbring and A. Sisto, The fundamental group of a locally finitegraph with ends: a hyperfinite approach, submitted.I. Goldbring, Ends of groups: a nonstandard perspective, J. Log.Anal. 3 (2011), Paper 7, 1-28.

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