Presentation for 213A, Winter 2013Geometric group theory

Andreas Ns Aaserud

Department of Mathematics,UCLA

11 March 2013

Outline

Introduction

Fundamental observation

Gromovs criterion

Outline

Introduction

Fundamental observation

Gromovs criterion

Outline

Introduction

Fundamental observation

Gromovs criterion

Geometric group theory

We will give an overview of the area of group theory known asgeometric group theory, where one studies large scale orcoarse properties of finitely generated (f.g.) groups.

To make this precise, we need to view such a group as a metricspace and to introduce some notions from the theory of metricspaces. First things first:

DefinitionLet be a finitely generated (f.g.) group and fix a finite symmetricgenerating set S (i.e., s1 S whenever s S). Then we candefine a metric dS on by setting dS(g , h) = h1gS , where S denotes the word-length with respect to S , i.e., the numberof factors in a reduced decomposition.

Examples

Note: We may think of the metric space (, dS) as the Cayleygraph of (,S) equipped with the graph-metric. This helps usvisualize (, dS).

Examples:

I (Zn, {e1, . . . ,en});I (Z, {2,3});I (F2, {a1, b1});I The modular group: Z/2 Z/3;

(See pp. 78-81 of Pierre de la Harpes book Topics in GeometricGroups Theory for pretty pictures.)

Quasi-isometries

DefinitionLet (X , d) and (Y , d ) be metric spaces. Given constants 1and C > 0, a (,C )-quasi-isometric embedding of X into Y is amap f : X Y with the property that

1

d(x1, x2) C d (f (x1), f (x2)) d(x1, x2) + C

for all x1, x2 X . If moreover every y Y is within C of f (X ),then we call f a (,C )-quasi-isometry.

A quasi-isometric embedding (resp. quasi-isometry) is a mapf : X Y that is a (,C )-quasi-isometric embedding (resp. a(,C )-quasi-isometry) for some constants C > 0, 1.If there exists a quasi-isometry as above, then we say that (X , d)and (Y , d ) are quasi-isometric.

Groups as coarse spaces

We want to be able to say that a f.g. group is quasi-isometric toa metric space (and, in particular, to another f.g. group). For thisto make sense, we need the following proposition:

Proposition

Let S and S be two finite symmetric generating sets for a group .Then (, dS) and (, dS ) are quasi-isometric.

Proof.We claim that the identity map is a quasi-isometry. Indeed, let Cbe the largest S -word-length of an element in S , and C the largestS-word-length of an element in S . Then dS(, 1) C dS (, 1)and dS (, 1) CdS(, 1) for all . As dS and dS are invariantunder left-multiplication, this completes the proof.

First examples of (non-)quasi-isometric groups

Examples: 1. Any finite group is quasi-isometric to {1}.

2. We show that Z is not quasi-isometric to Z2. (The proof isanalogous to the proof that R is not homeomorphic to R2.)I Let f : Z2 Z be a (,C )-quasi-isometric embedding.I There is a constant k N such that

d(f (u1), f (u2)) < k whenever u1 and u2 are adjacent.

I Consider the finite set B = f 1({1, 2, . . . , k}).I Choose u, v Z2 such that f (u) < 1 and f (v) > k and let

u = u1, u2, . . . , un = v be a path in Z2 that avoids B.I Then f (uj) < 1 for all j , a contradiction.

Geometric properties

Now, we may define a geometric property of a f.g. group G to be aproperty (P) that only depends on the quasi-isometry-class of G .

Examples: The following properties are geometric.

(i) Growth rate (polynomial of degree d ; intermediate;exponential);

(ii) Hyperbolicity;

(iii) Number of ends;

(iv) Amenability;

(v) Being virtually free;

(vi) Being virtually free abelian;

(vii) Being virtually nilpotent (= having polynomial growth by adeep theorem due to M. Gromov);

(viii) Being virtually infinite cyclic;

(ix) Being finitely presented.

Non-geometric properties

Counterexamples: The following properties are not geometric.(Below, we say that two groups are commensurable if they haveisomorphic subgroups of finite index.)

(i) Being commensurable with a solvable group;

(ii) The sign of the Euler-Poincare characteristic;

(iii) Being commensurable with a simple group;

(iv) Being commensurable with a group which has a non-abelianfree quotient.

Open problems: It is not known whether the following propertiesare geometric.

(i) Property (T);

(ii) Being virtually polycyclic.

Fundamental observation

DefinitionLet be an action of the group G on the topological space X .

I is proper if {g G |gK K 6= } is finite for everycompact set K X ;

I is cocompact if the orbit space X/G is compact.

DefinitionLet X be a metric space. Then

I X is proper if all closed balls in X are compact;

I X is geodesic if, given x , y X , there exists a geodesicsegment from x to y , i.e., an isometry L : [0, d(x , y)] Xsuch that L(0) = x and L(d(x , y)) = y .

Note: In this context it is convenient to identify a f.g. group Gwith the metric space obtained by embedding [0, 1] isometricallyalong each edge of a Cayley graph of G .

The following result is an important tool for showing that variousf.g. groups and metric spaces are quasi-isometric.

Theorem (Fundamental observation of geometric grouptheory; Efremovich 53, Svarc 55, Milnor 68)

Let be a group with a proper cocompact left action on a propergeodesic metric space by isometries. Then is f.g. and the map 3 7 x0 X is a quasi-isometry for every x0 X .Proof.

I Define metric d on X/ by

d(x , y) = inf{d(x , y ) | x x , y y}.

It is positive-definite because X is proper.

I X/ is a compact metric space, hence bounded. PutR = diam(X/).

I Fix a base point x0 X and put B = {x X | d(x , x0) R}.As X is proper, {B} is a covering of X .

I Consider the finite set S = {s | s 6= 1, sB B 6= }.I Put r = inf{d(B, B) : / S {1}} > 0.

S generates :

I Let \ (S {1}). As d(x0, x0) R, there exists aunique positive integer k such that

R + (k 1)r d(x0, x0) < R + kr .

By choosing a geodesic path from x0 to x0, we find elementsx0, x1, . . . , xk+1 = x0 X such that d(x0, x1) < R andd(xi , xi+1) < r for i = 1, . . . , k .

I As X = B, there are 1 = 0, 1, . . . , k = such thatxi i1B, i = 1, . . . , k. Then = s1 sk , wheresi =

1i1i satisfies d(B, siB) < r , and S k .

I Define a map f : X by f () = x0.The map f is a quasi-isometry:

I Let x X so that x = x for some and x B. Thend(f (), x) R.

I Given 1, 2 and applying the previous slide with = 12 1, we have

d((1), (2)) = d(x0, x0) R + (k 1)r R + (S 1)r = (R r) + dS(1, 2)r .

I By induction on S , we get that

d((1), (2)) = d(x0, x0) dS(1, 2),

where = max{d(sx0, x0) | s S}.This completes the proof.

Examples: 1. Zn acts on Rn by translation. Hence these twometric spaces are quasi-isometric.

2. Let G be a f.g. group with a finite index subgroup H. Then Hacts on (the Cayley graph of) G by left-translation. Thus H is f.g.and is quasi-isometric to G . In particular, the groups Fn (for eachn = 2, 3, . . .), PSL2(Z) = Z/2 Z/3, and SL2(Z) = Z/4 Z/2 Z/6are all quasi-isometric.

3. Let 0 A B C 0 be a short exact sequence of groups,where A is finite and C is f.g. Then B acts on (the Cayley graphof) C by left-translation. As the pre-image of every c C is finite,this action is proper. Thus B is f.g. and is quasi-isometric to C .

4. Let be the fundamental group of a compact Riemannianmanifold Y . Then acts on the universal cover of Y . Hence isf.g. and is quasi-isometric to the universal cover of Y .

Gromovs criterion

We will next prove a topological criterion for quasi-isometry of f.g.groups due to M. Gromov.

LemmaLet be a f.g. group. Then is quasi-isometric to the oppositegroup op.

We leave the easy proof of the lemma to the viewer.

Theorem (Gromov 93)

Let and be f.g. groups. They are quasi-isometric if and only ifthere exist commuting proper cocompact (left or right) actions of and on some locally compact Hausdorff topological space Xby homeomorphisms.

Proof.1 only if: Assume first that there exists a (,C )-quasi-isometry and let X be the set consisting of such maps.I Equip X with the topology of pointwise convergence. Then X

is a locally compact Hausdorff topological space. Indeed, given0 X , 6= F finite and k > 0, the neighborhood

V = { X | x F : d(0(x), (x)) k}

of 0 is compact by the Arzela-Ascoli theorem.

I Define a left action of on X and a right action of on X by

()(x) = 1(1x).

I These actions are proper and the orbit spaces X/ and \X ,with the quotient topologies, are compact.

2 if: Assume that we have two commuting right actions, butwrite the action of on the left.

Given S = , construct suitable generating set for :I Choose a compact set K X such that K = X/ and

K = \X . (That is, each orbit intersects K .)I Let s S . As sK is compact, sK tTsKt for some finite

set Ts .I Put TI = sSTs and TII = { |K K 6= }. Then

T = TI TII generates by the choice of K .Define a quasi-isometry and an inverse map:

I Fix a point x0 K . Then, given , there is suchthat x0 K . Define : by setting () = 1 .

I Given 1, 2 , we have dT ((1), (2)) 2dS(1, 2).I Analogously, construct a finite generating set S for from T

and define : .I Then dS ((1), (2)) 2dT (1, 2), dT (, ()) 1 and

dS (, ()) 1. It follows that is a quasi-isometry.

Reference

Pierre de la Harpe.Topics in Geometric Group Theory.Chicago Lectures in Mathematics, Chicago and London, 2000.

IntroductionFundamental observationGromov's criterion