random walks and hamronic functions in hyperbolic geometric group theory albert garreta

7/31/2019 Random Walks and Hamronic Functions in Hyperbolic Geometric Group Theory Albert Garreta

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Random walks and harmonic functions inhyperbolic geometric group theory.

Albert Garreta Fontelles

May 28, 2012

Abstract

We provide a small survey about the basics of hyperbolic geometry and hyper-

bolic groups, focused on analytical aspects of these disciplines such as random walks

and harmonic functions on discrete groups. It is intended for non-group theorists and

non-geometers who wish to get a somewhat intuitive idea of these subjects.

1 Introduction.

The whole text constitutes an elementary introduction to some analytical aspects of hyper-

bolic geometric group theory, such as random walks and harmonic functions on a discrete

group. The main goal is to give enough information to get acquainted with some of itsmost basic results and ideas. Proofs are rarely provided, although they can all be found in

the appropiate references.

About the organization of the paper; the second section is devoted to provide basic def-

initions and examples in the areas of geometric group theory and hyperbolic geometry.

The third section deals with boundaries of hyperbolic spaces. These boundaries are very

well-behaved when they correspond to hyperbolic groups, and proof to be useful later in

the study of harmonic functions defined over those groups. In section four the notion of

a random walk on a discrete group, along with that of an harmonic function, is discussed,

and, afterwards, it is stated -but not proved-, that, a) in hyperbolic groups almost every

random walk converges to a point in the boundary; b) that this fact allows one to define

a measure on it, and c) that, using this measure, harmonic functions defined on the group

look the same as bounded functions on the boundary, which is a variation of some stan-

dard results in harmonic analysis. The main goal of this survey is to give enough concepts

so as to properly understand these three results. Finally we introduce a notion of conver-

gence of metric spaces and, as a simple example, we prove that an infinite grid can be

shown to converge to the plane endowed with the taxicab metric, and also that harmonic

functions on the grid converge -in a sense we will precise- to harmonic functions on the

plane -in the usual sense-.

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2 Hyperbolic geometry and groups.

2.1 General definitions.

First, the notion of a geodesic metric space, along with some examples and non-examples,

is provided. Following it comes the definition of hyperbolic spaces in the sense of Gro-

mov, and as a key example the Poincares disk is presented. Next we define the Cayley

graph with respect to a presentation of a group and characterize hyperbolic groups as

those groups which posses an hyperbolic such graph. Next the notion of quasi-isometry

plays a crucial role in showing that hyperbolicity does not depend on the presentation of

the group.

DEFINITION 1. An isometric embedding of an interval [a,b] R into a metric space(X,d) is a continuous injective application : [a,b] X such that |r s| = d((r),(s))

for all r, s [a,b].DEFINITION 2. A metric space (X,d) is said to be geodesic if, for every pair of points x,

y X, there exists an isometric embedding of an interval I = [a,b] R, : I X, suchthat (a) = x and (b) = y.

Thus, in a geodesic space any two points can be connected by a path whose lenght is the

distance between these two points. We provide some examples and non-examples.

1. The usual euclidean space (Rn,dE) is indeed a geodesic space, its geodesics beingstraight segments.

2. (Rn 0,dE) is not geodesic, for any path connecting (1,0,...,0) with (1,0,...,0)must jump the point 0 and therefore have lenght strictly greater than 2.

3. A locally finite graph (a graph in which all its vertices have a finite number of

neighbours) naturally induces a geodesic metric space: transform each of its edges

into isometric copies of the unit interval. This yields a continuous set of points, c.

Now define the distance d between two points x, y c as the minimum lenghtamong the paths connecting x to y (this lenght being induced by the distance on the

unit interval edges). This minimum is attained because is locally finite. Moreover

(c,d) is clearly a geodesic metric space.

4. The same continuous set c induced by a locally finite graph , embedded into aneuclidean space, and furnished with the induced euclidean metric, is not geodesic.

DEFINITION 3. Given a geodesic space (X,d) and three points x, y, z, consider geodesicsconnecting x to y, y to z, and z to x. The set formed by these geodesics is called a geodesic

triangle with vertices x, y, and z.

Note that, in general, geodesics are not unique, and therefore in most cases three points

in a geodesic space determine several triangles.

DEFINITION 4. A geodesic triangle with geodesics x, y, z is said to be - thin if, for

any point on any side, say x x, the following holds: in f{d(x,y)|y y or y z} .(figure 1).

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The following is a celebrated definition due to Gromov.

DEFINITION 5. A geodesic space (X,d) is said to be -hyperbolic if all its geodesictriangles are -thin. (X,d) is said to be hyperbolic if it is -hyperbolic for some .

Again we provide some examples and non-examples. The most illustrative of them, the

Poincare disk, is posposed to the following subsection due to its general importance.

1. The usual euclidean space (Rn,dE) is not hyperbolic, for one can easily find fam-ilies of triangles and points on its sides such that the distance to its other sides is

arbitrarily large.

2. A graph with no cycles, eg, a tree, induces a 0-hyperbolic space (c,d). (Seefigure 1).

2.2 Poincares Disk.

This section follows closely section 5 in Bowditchs introductory notes [2].

Let D = {z C||z| < 1} be the open unit disk on the complex plane. Consider the follow-ing application:

: D (0,)

z 2

1 |z|2

Now let : ID be a smooth path. Well say that its hyperbolic speed at time t is

vH(t) := |(t)|((t))

Predictably enough, the hyperbolic length of is

lH() :=

IvH(t)dt

Then, the hyperbolic distance between two pointsx andy D, dH(x,y), is the minimum ofthe hyperbolic lengths of smooth paths from x to y (the minimum is, indeed, attained). A

path attaining such a minimum is of course a geodesic from x to y. By the way has beendefined, small euclideally looking things near the boundary of D are in fact extremely

large and, moreover, one has to travel an infinite distance in order to touch that boundary.

It can be shown that, in the hyperbolic diskD, geodesics are contained in arcs of euclidean

circles orthogonal to D. This gives us an euclidean way of picturing geodesic triangles.FIGURE 13 (geodesics and geodesic triangles in D).

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As one would expect, the hyperbolic disk is indeed a -hyperbolic space, and the mini-mum such can be cheked to be 1

2log3.

We conclude the section with a picture of a tilling of D due to the artist M.C Escher. All

the birds in the picture have the same hyperbolic area, although their euclidean area variesdrastically as one approaches the boundary. One can also appreciate geodesics drawn by

the birds backbone.

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Figure 1: Eschers tessellation of the Poincare disk.

2.3 Hyperbolic groups and quasi-isometries.

One can associate to a finitely generated group G a locally finite graph ; the so called

Cayley graph ofG. The study of the induced geodesic metric space (,d) can illustrateseveral important algebraic properties of G. This study is often made by looking at the

graph from the infinity. All these techniques and concepts were introduced by Gro-mov around 1980 and used to prove some remarkable results, such as his famous growth

theorem.

DEFINITION 6. Let G be a group generated by a finite set of elements S = {s1,...,st}. TheCayley graph associated to G and S, (G,S), is defined by taking as vertices all elementsin G, and connecting two of them, x, y, if either xy1 S or x1y S (more cases?).

The Cayley graph (G,S) is clearly locally finite. And algorithmic and descriptive way oflooking at it consists in making the following construction: start with the identity element

e, and connect to it everey generator si. Next, from each generator si add edges to the

elements sis1, ..., sist, s1si , ..., stsi. Now, identify two elements if they are equal in G.Repeating this process infinitely many times one obtains the Cayley graph (G,S).

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Some examples are provided, but first we recall the definition of some of the groups

involved:

1. The free group of rank two, F2

, is probably, along with the infinite cyclic group Z,

the most basic infinite group. It consists on two generators a,b, and no relations. Its

elements are words in the alphabet {a,a1,b,b1} such that aa1, a1a, bb1, andb1b do not ocur as subwords.

F2 =< a,b| >

2. Given two groups A and B, generated by the sets S and T, and with relations RSand RT, respectively, its free product A B is the group generated by S T, withrelations RS RT. For example, F2 = Z Z. It is analogous to the direct cartesianproduct, with the difference that in the latter one adds relations so the elements in

A commute with the elements in B.

A B =< S T|RS RT >

3. The Baumslar-Solitar groups, BS(n,m) are groups given by the following presen-tation:

BS(n,m) =< a,b|abna1 = bm >

They posess many interesting properties and have served as counterexamples to

important conjectures. Further work: investigate why the faces of the Cayley

graph for BS(1,2) look like an hyperbolic semiplane.

4. The Discrete Heisenberg group H3 is given by the presentation

H3 =< a,b,c|ac = ca,bc = cb,ab = cba >

It is isomorphic to a subgroup of the group of invertible matrices with integer coef-

ficients. The continuous Heisenberg has a strong relation with hyperbolic geometry.

Further work: investigate if similar relations hold for the discrete version.

Next we present the Cayley graph of some groups and their presentations.

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Figure 2: Cayley graph of the finite cyclic group Z3().

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Figure 3: Cayley graph ofZZ.

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Figure 4: Cayley graph of the free group of rank two (define this group too) F2.

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Figure 5: Cayley graph of the free product of two finite cyclic groups, Z6 Z3.

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Figure 6: A more complicated Cayley graph: Baumslag-Solitar group BS(1,2).

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Figure 7: Two cut offs of the previous Cayley graph.

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Figure 8: Discrete Heisenberg group.

DEFINITION

7. A finitely generated group is said to be word-hyperbolic if, for some setof generators S, (G,S) is an hyperbolic space.

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Next some examples and non-examples are provided:

1. One of the most important examples of hyperbolic group is the free group F2 =, in particular, F2 is 0-hyperbolic.2. The fundamental group of a surface with negative Euler characteristic is an hyper-

bolic group. This includes, for example, compact orientable surfaces with genus

greater or equal to 2, eg, torus with at least one additional hole.

3. The fundamental groups of compact Riemannian surfaces with strictly negative cur-

vature are hyperbolic.

4. Any group containig Z2 is not hyperbolic, for Z2 itself can be seen to contain trian-

gles which are not -thin for any .

5. Baumslag-Solitar groups, BS(n,m), are not hyperbolic.

The dependence of the Cayley graph on the set of generators is crucial, and it is, in fact,

what motivates the notion of quasi-isometry in a group theoretic enviornment.

DEFINITION 8. Given two metric spaces (X,dX), (Y,dY), an application f : X Y issaid to be a quasi-isometry if there exist constants C and K such that

1

CdX(x,y) K dY(f(x), f(y)) CdX(x,y) + K

In this case, X and Y are said to be quasi-isometric.

Informally speaking, a quasi-isometry preserves large distances, while it has no restriction

on small ones. This is clear: if dX(x,y) is extremely larger than Cand K, dY(f(x), f(y)) isvirtually equal to dX(x,y), while if it is radically smaller than Cand K, dY(f(x), f(y)) canvary widely before exceeding the required inequalities. Two quasi-isometric spaces tend

to look as the same space as one walks away from them. For example, a finite space tends

to look like a point as we walk away from it; therefore it is quasi-isometric to a point. In

the Cayley graph ofZ2, the squares become smaller as we distantiate from them; this

is because this graph is quasi-isometric to an infinite plane.

It is an easy but fundamental observation that, for two finite sets S and T of generators for

a group, (G,S) and (G,T) are quasi-isometric. Therefore the study of quasi-isometricinvariants of the Cayley graph of a group does not depend on the choice of generators.The alert reader is surely to be expecting the following result:

THEOREM 1. The hyperbolicity of a geodesic metric space is a quasi-isometric invariant.

COROLLARY 1. A finitley generated group G is a word-hyperbolic group if and only if

(G,S) is an hyperbolic space for any finite set of generators S.

(Note, that, however, -hyperbolicity is not invariant.)

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3 Boundaries of hyperbolic spaces and groups.

In this section we first define the geodesic boundary for arbitrary hyperbolic spaces, which

consists of certain equivalence classes of geodesic rays. Afterwards we introduce thenotion of Gromov product, which in a manner of speaking measures for how long two

geodesics travel close together. The understanding of this product allows us to define

the sequential boundary, or simply the boundary, of the hyperbolic space, which is made

of equivalence classes of sequences converging to infinity. We next see how these two

boundaries can be topologized. Finally we see that when the space is nice enough the

two boundaries coincide, that the boundary can be viewed as a compactification of the

space, and that it is invariant under quasi-isometries. This section follows closely the

corresponding section in [7].

DEFINITION 9. A geodesic ray in a geodesic metric space (X,d) is an isometric embed-

ding : [0,) X. Two such rays, 1 and 2, are said to be equivalent if there exist K> 0such that d(1(t),2(t)) K for any t. In this case it will be written 1 2.

is, indeed, an equivalence relation, which yields to the following definition;

DEFINITION 10. Given an hyperbolic metric space (X,d), its geodesic boundary is theset of equivalence classes of geodesic rays:

g(X) := {[]| G(X)}

where G(X) is the set of geodesic rays in X. Its geodesic boundary with basepoint x X

is the set of equivalence classes of geodesic rays starting at x;

xg := {[]| G(X,x)}

where G(X,x) is the set of geodesic rays such that(0) = x.

Figure 9: Geodesic bi-infinite rays on the Poincare disk. a and b are two equivalent rays.

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Let (X,d) be a -hyperbolic metric space and take any three points x, y and z in it. Con-sider geodesic rays xy and xz starting at x and containing y and z, respectively. Intuitively,there are points y xy and z

xz such that the geodesic segments [x,y] and [x,z] satisfy

the following:

1. [x,y] and [x,z] are 2- close (in the Hausdorff sense).

2. The lenght of[x,y] and [x,z] are the same, say l.

3. Any two similar geodesic segments with lenght l are not 2 -close.

There is, in fact, a way of calculating l in terms ofx, y and z:

DEFINITION 11. For thre points x, y and z in an arbitrary metric space (X,d) the Gromovproductof them is defined as;

(y,z)x =1

2(d(x,y) + d(x,z) d(y,z))

(y,z)x is the lenght l to which we were refering above, and therefore the Gromov productmeasures how long two geodesics travel closely together.

Since the next definition is rather technically complicated, we warn the casual reader who

is just looking to get a general picture of the subject, that a simple and practical notion of

is is given later.

DEFINITION 12. Take any x in a hyperbolic metric space (X,d), and let (xn) be a se-

quence of points. (xn) is said to converge to infinity if

liminfi,j(xi,xj)x =

If (yn) is another sequence converging to infinity, we say (xn) and (yn) are equivalent,(xn) (yn), if

liminfi,j(xi,yj)x =

So a sequence of points converges to infinity if one can always find two of its terms whose

geodesics from x travel 2-close together for an arbitrary large length. This amountsto saying that the geodesics from x to xn tend to be 2-close and therefore equivalent

geodesic rays, so, in fact, looking at this sequence from an arbitrarily far distance reultsin seeing a sequence lying on a geodesic ray. Similarly, the equivalence between two

sequences converging at infinity can be thought as the fact that the two geodesic rays on

which these sequences lie when looked from infinity are equivalent.

Again, is an equivalence relation, and its classes define a new kind of boundary:

DEFINITION 13. Given an hyperbolic metric space (X,d), its sequantial boundary, orsimply the boundary, is the set of equivalence classes of sequences converging to infinity:

(X) := {[(xn)]|(xn) S(X)}

where S(X) denotes the set of sequences in X converging to infinity.

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All this boundaries can be furnished with a topological structure:

DEFINITION 14. Given a -hyperbolic metric space (X,d), take any p gx (X), t (X), and, for r 0, define

V(p, r) = {q xg|p = [1],q = [2], liminft(1(t),2(t))x r}

U(t,r) = {q (X)|t = [(xn)],q = [(yn)], liminfi,j(xi,yj)x r}

Then xg(X) (resp. (X)) is topologized by taking as neighborhood basis {V(p,r)|p

xg, rR+} (resp. {U(r,p)}).

Next, as promised, we provide the simple but practical explanation of the above concepts.

A metric space (X,d) is called proper if it is locally compact and complete. For properhyperbolic spaces, the class of a sequence converging to infinity can be represented by a

sequence all of whose points lie on a geodesic ray in an ordered way. Considering thisand the interpretation given to the Gromov product, neighborhoods of p x

g(X) andt (X) are determined by representative geodesic rays (resp. by sequences on geodesicrays) which stay 2- close (in the Hausdorff sense) at least for a lenght r.These informal considerations are even more useful once the following result is obtained:

THEOREM 2. For two proper hyperbolic spaces X and Y , the following holds:

1. xg(X) is homeomorphic to (X) for all x X.

2. (X) and X(X) are compact spaces.

3. Two quasi-isometric proper hyperbolic spaces have homeomorphic boundaries.

DEFINITION 15. For a word-hyperbolic group G, its boundary is defined as the boundary

of any of its Cayley graphs, eg,

G := (G,S)

Where (G,S) is the boundary of the Cayley graph as defined above.

By theorem 2, this definition does not depend on the generating set S.

Next we define the notion of geometric action of a group G on a metric space X and

provide some classic and fundamental results about hyperbolic groups. These notions and

theorems will not be used further in the text. The casual reader may skip the definition

and think about a geometric action just as a really nice action, or simply skip the whole

remaining of the section.

DEFINITION 16. Given a group G acting on a metric space X, the action is called geo-

metric if it is properly discontinous (the set {g G|gK K = /0} is finite for all compactsubsets K X), cocompact (G/X is a compact space with the quotient topology), andisometric (d(gx,gy) = d(x,y) for all x, y X and g G).

THEOREM 3. A group G is word-hyperbolic if and only if it acts geometrically on a

proper hyperbolic space (X,d). In this case G andX are homeomorphic.

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Note that in the above theorem necessity is quite easy since it is immediate to check that

any group G acts geometrically on its Cayley graphs. Suficiency is the profound part of the

theorem: a group acting geometrically on any hyperbolic space, for example on Poincares

disk, has hyperbolic Cayley graphs, and the boundary of them are homeomorphic to theboundary of the space.

The theorem can be proved using the so-called Schwarz-Milnor lemma.

LEMMA 1. (Schwarz-Milnors lemma.)

If G is a group acting geometrically on a geodesic space X , then it is finitely generated,

X is proper, and X is quasi-isometric to (G,S) for any finite generating set of G.

The following result shows that the boundary of an hyperbolic group has a rather restricted

structure.

THEOREM 4. Let G be an hyperbolic group. Then one and only one of the following

holds:

1. G is finite and G is trivial.

2. G contains a cyclic subgroup of finite index and G consists of two points.

3. G contains an isomorphic copy ofF2, the free group of rank two, and G is home-omorphic to a Cantor set.

DEFINITION 17. The Cantor set is the set

P(N) = {(xn)nN|xn {0,1}} =

{0,1}

endowed with the product topology when considering the discrete topology on each copy

of{0,1}.

The Cantor set can be seen to posess some remarcable properties. Namely, it is a totally

disconnected, uncountable, compact, complete, Hausdorff space, in which every point is

never an interior point and always an accumulation point. Moreover, any compact metric

space is a continuous image of the Cantor set.

Note: Usually the Cantor set is defined in a somewhat different way as a subset of the

unit interval which is obtained by appropiately applying infinite iterations of the actionremove one third of a segment. The following picture, we hope, perfectly illustrates

this process.

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Set.jpg

Figure 10: Each line is the result of an iteration. The Cantor set is obtained by applying

infinite iterations.

4 Browinian movement, harmonic functions, and lapla-

cians on discrete groups.

First of all we point out a way of modelling card shuffling via random walks on groups,

which we hope will serve as a motivation for the study of abstract random walks. Basic

definitions are given afterwards, followed by an introduction to the notion of harmonic

functions defined over groups, which implicitly uses what we call a discrete laplacian.

Next we see that, for the vast majority of hyperbolic groups, harmonic functions are in an

isometric correspondance with bounded functions on the boundary, which is a variaton ofseveral well-known results in harmonic analysis.

4.1 Motivation: shuffling cards.

A single card shuffle of a deck with 52 cards can be viewed as a randomly chosen ele-

ment of the symmetric group S52. Suppose you have your deck perfectly ordered, and then

apply to it a random permutation from S52. Since a single permutation is clearly not suf-

ficient, you decide to apply a new randomly chosen permutation to your now disordered

deck. The process of applying n permutations to the deck can be modelled as a randaom

walk in Sn

. If you decide to make the shuffling in a more sofisticated way, say, for in-

stance, by repeteadly transposing two randomly chosen cards, then what you are doing

can be viewed as random walks on S52 assuming that the transition probability between

two permutations which do not differ by a transposition is 0 (precise definitions will be

given later). In fact, it was proved by Diaconis and Shahshahani, using this random walk

model that, for general Sn, the distribution is essentially the uniform distribution after ap-

plying 12

nlogn randomly chosen transpositions. This means that, after applying about 100

random transpositions on a deck with 52 cards, the probability that the card in position x

is the card y is essentially 152

. Using similar techniques, Bayer and Diaconis proved that

only seven riffle shuffles are enough to completely shuffle a deck. The interested reader

on these topics can check the introductory paper by Saloff-Coste [4].

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4.2 Definition of random walks on discrete groups.

Suppose G is a countably infinite discrete group endowed with a probability measure ,eg, an application : P(G) R which is countably additive and such that (G) = 1.The support of is the set of elements g in G such that (g) = 0;

supp() = {g G|(g) = 0}

is said to be nondegenrate if the semigroup generated by supp() equals G; equiva-lently, is nondegenarte if every element in g can be written as a product of elements insupp() without allowing inverses of those elements to ocurr in the product (unless thoseinverses are already in supp()). Unless otherwise specified will always be nondegen-erate.

Recall that a Markov chain is a sequence of random variables X1, X2, ... such that

Pr(Xn+1 = x|X1 = x1,...,Xn = n) = Pr(Xn+1 = x|Xn = xn)

The possible values of the variables Xi form the state space of the Markov chain and

the probabilities Pr(Xn+1 = x|Xn = xn) := p(xn|x) are called the transition probabilities.Informally, a Markov chain can be pictured as a labelled directed graph, where the set

of vertices is the state space, and any ordered pair of vertices (x,y), (x = y is allowed) isconnected by a directed edge xy with label

p(x|y)

In practice the random variables are usually ignored.

DEFINITION 18. Given a group G with probability measure , its (right) random walk(G,) is the Markov chain whith state space G and transition probabilities p(g|h) givenby

p(g|h) = (h1g)

Note that p(g|h) is invariant under left translations, that is, p(g|h) =(h1g) =(h1a1ag) =p(ah|ag) for all a G. So, in fact,

p(g|h) = p(1|g1h)

Markov chains can be viewed as models of stochastic processes that ocurr during time.

Informally, if we leave a monkey on a vertex v of the graph associated to the Markovchain, the labells p(v|x) indicate the probability of finding the monkey in the vertex xafter a minute has passed (this probability may not be uniform, for the monkey is more

likely to be in a vertex full of bananas than in an empty one). More generally, given a

path of vertices v, x1, ..., xn, the number p(v|x1)p(x1|x2)...p(xn1|xn) indicates how likelyis the monkey to walk that precise path during the first n minutes. Of course, if p(x,n)denotes the probability of finding the monkey at vertex x after n minutes, one has

p(x,n) =y

p(y,n 1)p(y|x)

What we are really saying is that Markov chains are a good tool for modelling discreteBrownian movement. Note that we have started the process with a single monkey on a

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fixed vertex, while several of them could have been placed in different random vertices

following a distribution law. Formally speaking, the distribution ruling the way the pro-

cess starts is called the initial distribution of the Markov chain.

Let us now go back to G and its random walk(G,). If the initial distribution is a singleelement g in G, then one can define an associated Borel measure Pg on the trajectory

space G = G G ... (a Borel measure on a topological space is a measure defined onthe minimal -algebra containing the open subsets -here the topology is the product ofdiscrete topologies), given by

G R

(x1,x2,...) (x1)p(g|gx1)p(gx1|gx1x2)...

where (x1) = 1 if x1 = g and (x1) = 0 otherwise. One can obviously assure P is 0outside the set G,g

:= {(y1,y2,...) G|y1 = 1,(yn+1

1yn) = 0}. For an arbitraryinitial distribution , one can also define an associated Borel measure P in terms of Pgfor g G:

P =g

(g)Pg

where (g) denotes the probability of one of the starting vertices being g.

DEFINITION 19. The measure space (G,P), which is a Lebesgue space, will be calledthe random trajectory space.

Note that the above is the definition one expects to encounter when reading stuff about

random walks; that is, randomly chosen paths.

4.3 Harmonic functions on discrete groups.

DEFINITION 20. For G endowed with a probability measure , a function f : G C iscalled -harmonic if

f(g) = xG

f(gx)(x)

The space of bounded -harmonic functions on G is denoted by H(G), or just H(G).

It is a Banach space once we endow it with the supremum norm.

DEFINITION 21. Suppose G is endowed with a probability measure . Then the discrete

laplacian for G with respect to is an operator on the space of funtions on G definedas

f(g) = xG

(x)f(gx) f(g)

It is immediate to check that a function f is -harmonic if and only iff(g) = 0 for all gin G. (This is a variation of the classical mean-value theorem.)

From now on we will consider the most natural case for . Namely, we assume that, ifS = {s1,...,st} is a set of generators for G, then (g) = 0 if g / S and (si) =

1t

for all

i. The corresponding random walk on G is the most natural one could think of: at each

vertex v the probability of stepping into a nonadjacent vertex is 0, while the probability

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of going into an adjacent one is 1t

(note that the Cayley graph of a group with respect to a

generating set S is always |S|- regular). In this case, for a function f : G C one has

f(g) = sS

1

|S| f(gs) f(g)

This probability measure will be denoted G or just .

4.4 Boundaries, random walks, and harmonic functions.

THEOREM 5. Let G be an hyperbolic group with a non-degenerate probability measure

such that neither G nor< supp() > contain a finite-index subgroup isomorphic copyofZ. Then the trajectory of almost every -random walk, eg, almost every sequence(xn) (G

,P) converges with respect to the topology on G G to some point in G.

From now on until the end of this subsection G will be assumed to be as in the hypotheses

of the previous theorem. In this case, G can be equiped with a measure , which will becalled the harmonic measure ofG. Namely, for a Borel subset K G, (K) is definedas the probability that an infinite -random path originating at 1 converges to some pointin K, with respect to the topology of G G.Note that G acts on its boundary G. Namely, if [(xn)] is a point in G, then an elementin g transforms [(xn)] into [(gxn)]. This action is well defined. If y G, the action ofg G over y is denoted by gy.Now consider the following map from the space of bounded functions defined on G,L(G) to the set of-harmonic functions on G, H(G):

: L(G) H(G)

(f)(g) =G

f(y)d(gy)

Now, for a probability measure one can define its logarithmic moment:

LM() := gG

(g)log|g|

and its entropy:

H() := gG

(g)log(g)

It turns out to be that if these two values are finite then is a really nice map:

THEOREM 6. Let G andbe as above and such that LM() < and H()


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map can be defined from the set of continuous (and thus bounded) functions on the surface

S of a ball B in Rn to the set of harmonic functions inside the ball by;

L(u)(x) =

S u()P(,x)d()

where u() is a continuous function on S, and P(,x) is the so-called Poisson kernel forthe ball B.

5 Ultralimits, asymptotic cones, convergence of metric

spaces, and convergence of harmonic functions.

In this section we provide a new tool which allows one to look at metric spaces from

infinity: this tool uses ultrafilters and ultralimits in order to stablish a formal definitionof convergence of metric spaces. The asymptotic cone of a space will be shown to be

the ultralimit of a special sequence of metric spaces. Finally, all these concepts are used

to show how Brownian movement on the plane can be modelled as random walks on an

infinite grid and then taking the asymptotic cone of it. Also, discrete harmonic functions

on the grid are extended to harmonic functions (in the classical sense) on the plane.

We first give an intuitive definition of ultrafilter and ultralimit and afterwards the for-

mal definitions are provided. The uninterested reader can ceirtanly stick to the intuitive

version.

Given a bounded sequence of points on a metric space (xn) X, the ultralimit (withrespect to an ultrafilter w) of it is an accumulation point of the set {xn|n N}. It alwaysexists, it is unique, and is denoted by limwxn. It is a well known theorem that a sequence

of bounded points on a metric space have at least one convergent subsequence. What the

ultralimit does is in fact selecting one of these subsequences and then takes its limit. The

chosen subsequence depends on the ultrafilter w. An ultralimit is always linear (whenever

it is possible to talk about linearity), and, provided the ultrafilter has got an additional

special property and that the sequence converges, it coincides with the usual notion of

limit.

DEFINITION 22. Let X be a set, an application w :P(X) {1,2} is called an ultrafilteron X if the following holds:

1. w(X) = 1.

2. IfA B X, and w(A) = 1, then w(B) = 1.

3. IfA, B P(X) and w(A) = w(B) = 1, then w(A B) = 1.

4. w(XA) = 1 w(A).

Note that, in particular, w is a measure on X.

If X = N and, for every finite subset S N, one has w(S) = 0, then w is called a non-principal ultrafilter.

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PROPOSITION 1. Let w be a non-principal ultrafilter (on N). Then the ultralimit of a

bounded sequence of points (xn) X in a metric space (X,d) is an accumulation point xof{xn|n N} such that, for every > 0,

w{n N|d(xn,x) < } = 1

This point x exists, it is unique and, if(xn) converges in the usual way to y, then x = y.

DEFINITION 23. Given a metric space (X,d), consider the sequence Xn = (X,dn) wheredn(x,y) =

1n

d(x,y) is a reescaled metric. Fix a point o X. Now define XbN to be the set

of sequences (xn) in X such that (dn(o,xn)) is a bounded set of real numbers. Take now anon-principal ultrafilter on N, say w. For any two sequences (xn) and (yn) in Xb

N, define

its distance with respect to w to be

dw((xn),(yn)) = limwdn(xn,yn)

This is well defined because (dn(xn,yn)) is a bounded sequence of real numbers. We saythat (xn) (yn) ifdw((xn),(yn)) = 0. This can be seen to be an equivalence relation. Thenthe space (Xb

N/ ,dw) is a metric space (where dw is taken over any representative of thecorresponding equivalence classes). This metric space is called the asymptotic cone of X

(with respect to w). It is often denoted

Conew(X)

The vast majority of known examples of asymptotic cones are independent of the choice

of the ultrafilter w. Nevertheless, the Cayley graph of some infinitely presented groups

have been shown to posses non-homeomorphic asymptotic cones for different ultrafilters,

see [12]. Whether this can be the case or not for finitely presented groups is an open

problem.

EXAMPLE 1. In the next example it is shown in a rough way that discrete random infi-

nite paths and -harmonic functions on the discrete infinite grid can be seen to converge,in an appropiate sense, to continuous random infinite paths and the usual harmonic func-

tions on the euclidean plane. The infinite grid can be conveniently made to converge to

the plane endowed with a special metric. It is not always the case, though, that Cayley

graphs converge to nice manifolds: this is the reason we stick to the infinite grid. One

would like to see an example of an hyperbolic group converging to the Poincares disk.Unfortunately, the usual machinery used for this kind of convergence, namely ultralimits

and asymptotic cones, is not valid, for the asymptotic cone of any -hyperbolic space is0-hyperbolic. More precisely, such a limit is either the real line or an infinite space in

which any two points are connected by a single path which is also a geodesic ( this kind

of metric space is called a real tree) and such that there are infinitely many geodesics rays

starting at any point [11].

Let G = Z2 be generated by a and b. Recall that the Cayley graph in this case is aninfinite grid, which will be denoted by , and its vertices by (v1,v2). Now, on the planeR2 consider the metric dL(z1,z2) = |x(z2) x(z1)| + |y(z2) y(z1)|, where y(z) and x(z)

denote the x-th and y-th coordinate of z R2

, respectively. With this metric, the balls in(R2,dL) look like squares rotated by 45 degrees.

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It is not difficult to see that the space (,d), where d is the word-metric for , can beisometrically embedded into (R2,dL).The next proposition shows that the asymptotic cone ofis, in fact, (R2,dL).

PROPOSITION 2. Following the above notation, one has

Conew() = (R2,dL)

Where = means isometric equivalence. Moreover, while the isometry depends on theultrafilter w, the property of being iometric is independent of it.

Proof. The next figure illustrates a sequence of points (xn) in bounded by the unit ballon each n. It is also interesting to observe how the balls in n resemble the balls in

(R2; dL).

Suppose n is isometrically embedded in (R2,dL) and take a sequence (zn) = (xn,yn).

Then (dn(o,zn)) = xn2 +yn

2 is bounded. Now define:

: Conew() (C,dL)

[(xn,yn)] (limwxn, limwyn)

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where [(zn)] denotes the equivalence class of(zn) with respect to . is well defined. Indeed: suppose (zn) = (xn,yn) (vn,un) = (sn). Then, by definition,

limw1

n d(zn,sn) = limw (|xn vn| + |yn un|) = 0

This is equivalent to limwxn = limwvn and limwyn = limwun, since limw behaves mostlylike the ordinary limit (recall that limw can be viewed as choosing a convergent subse-

quence and then taking the usual limit of it).

Injectivity and exhaustivity of follow from similar arguments and elementary propertiesof the plane and the classical notion of convergence.

Finally, is an isometry; for, writing x = limwxn, y = limwyn, v = limwvn, and u = limwun,we have:

dL((x,y),(v,u)) = |x v| + |y u| = limw (|xn vn| + |yn un|) = dw([xn,yn], [vn,un])

The reader can try to define a similar map from Conew() to the euclidean plane. It willimmediately be found out that the arguments above do not work anymore.

Now take a bounded -harmonic function f H(). One can naturally define -harmonic functions fn on n, induced by f. Namely, fn(g) = f(ng). Next consider themap;

fw : Conew() C

[pn] limwfn(pn)

Assume that fw is well defined (this is work in progress). Then, for every [pn] take arepresentative sequence which is convergent in the usual sense. It then follows:

fn(pn) =1

4

fn(pn +

1

ne1) + fn(pn +

1

ne2) + fn(pn +

1

ne3) + fn(pn +

1

ne4)

Where e1 = (1,0), e2 = (0,1), e3 = (1,0), and e4 = (0,1).What we have here is maybe a familiar situation: if we now make appropiate manipula-

tions and take the limit we obtain that fw is an harmonic function on R2 in the usual sense.

Indeed, we have, once we leave a 0 on the left side and multiply by 4n2

;

0 =1

n2

fn(pn +

1

ne1) + fn(pn +

1

ne3) 2

1

nfn(pn)

+

1

n2

fn(pn +

1

ne2) + fn(pn +

1

ne4) 2

1

nfn(pn)

We set:

A(h) :=1

n2

fn(pn +

1

ne1) + fn(pn +

1

ne3) 2

1

nfn(pn)

B(h) := 1n2

fn(pn + 1n

e2) + fn(pn + 1n

e4) 2 1n

fn(pn)

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Now, using Taylor expansions on A(h) and B(h), along with calculus arguments, one canfinally show that fw is harmonic in the usual sense by taking the limit when n tends to

infinity.

Remark: Note that, since all metrics in R2

are equivalent, all these limits can be takenwith respect to the L1 metric or with respect to the standard euclidean metric without

changing any of them.

References

[1] Cornelia Drutu and Michael Kapovich, Lectures on geometric group theory.

[2] Brian H.Bowdich, A course on Geometric Group Theory, MSJ Memoirs, Mathemat-

ical Society of Japan, (2006).

[3] V.A. Kaimanovich, A.M. Vershick, Random Walks on Discrete Groups: Boundary

and Entropy, The Annals of Probability, Vol.11, No.3, 457-490, (1983).

[4] Laurent Saloff-Coste, Probability on Groups: Random Walks and Invariant Diffu-

sions, Notices of the AMS, 48 - 9, (2001).

[5] Georgios K. Alexopoulos, Random Walks on Discrete Groups of Polynomial Volume

Growth, The Annals of Probability, Vol. 30, No. 2, 723-801, (2002).

[6] Wolfgang Woess, Random Walks on Infinite Graphs and Groups - a Survey on Se-

lected Topics, Bull. London Math. Soc. 26 (1994) 1-60

[7] Ilya Kapovich and Nadia Benakli, Boundaries of hyperbolic groups, Contemporary

Mathematics, American Mathematical Society (2002).

[8] I. Mundet, Gromov i la geometria: un aperitiu, Butlleti de la SCM, Vol. 24 (2009),

Num. 1, 13-22.

[9] M. Gromov, Metric structures for Riemannian and non-Riemannian spaces,

Birkhuser (1999), ISBN 0-8176-3898-9.

[10] James W. Anderson, Hyperbolic Geometry, second edition, Springer, (2005).

[11] Robert Young, Notes on asymptotic cones, (2008).

[12] Cornelia Drutu, Quasi-isometric invariants and asymptotic cones.

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