introduction to geometric group theory - bgubarakw/probseminar/nagorko/slides.pdf · introduction...

Introduction to Geometric Group Theory

Andrzej Nagorko

Warsaw University

June 24th, 2007

An introductory talk about basic notions of Geometric Group Theory,with emphasis on asymptotic invariants of groups.

A. Nagorko (Warsaw) Introduction to Geometric Group Theory June 24th, 2007 1 / 22

Introduction The grand plan

Outline

I plan to talk about the following asymptotic invariants of grups:

1 Volume growth.

2 Gromov hyperbolicity.

3 Asymptotic dimension.

I’ll start by introducing basic notions of the theory.


Introduction Dictionary definition of Geometric Group Theory

What is Geometric Group Theory?

A simple definition of Geometric group theory is that it is the study of groupsas geometric objects.

Thinking about groups this way was popularized by Gromov who revolutionizedthe subject of infinite groups. This is a new field, so there are many fundamen-tal theorems waiting to be discovered, and it is a rich field, lying at a juncturebetween algebra and topology, where a great variety of methods from otherbranches of mathematics can be utilized. Geometric group theory draws upontechniques from, and solves problems in the theory of 3-manifolds, hyperbolicgeometry, combinatorial group theory, Lie groups...

Dani Wise


Cayley Graphs Word length metric

Word length metric

Let G be a finitely generated group.

Let S be a finite generator set of G and let

S−1 ={

g ∈ G : g−1 ∈ S}

.

For an element g of G let

|g| = min{

n : ∃{s1,s2,...,sn}⊂S∪S−1 g = s1s2 · · · sn}

.

be the minimum length of a word in alphabet S ∪ S−1 that represents g in G.

A map d : G ×G→ [0,∞) defined by the formula

d(g, h) = |g−1h|

is a metric on G. We call it the word length metric.


Cayley Graphs Definition and basic properties

A Cayley graph of 〈G, S〉Recall from the previous slide:

d(g, h) = |g−1h|, where |g| = min˘

n : ∃{s1,s2,...,sn}⊂S∪S−1 g = s1s2 · · · sn¯

.

Consider an undirected graph with a vertex set G and a set of edges

E ={{g1, g2} ⊂ G : g−1

1 g2 ∈ S ∪ S−1 and g1 6= g2

}.

Realize it as a simplicial complex with uniform geodesic metric, i.e. with themaximal metric in which every edge is isometric to the unit interval [0, 1]. Wecall it a Cayley graph of G.

The distance from a vertex g to a vertex h in the Cayley graph of G is equal to

min{n : there exists a path from g to h with n + 1 vertices}.

It coincides with the word length metric on G, as every path in the graph corre-sponds to a word in alphabet S ∪ S−1.


Cayley Graphs Examples

A Cayley graph of Z2

With a standard generator set {a = (1, 0), b = (0, 1)}.



A Cayley graph of F2 (a free group on two generators)

With a standard generator set {a, b}.



A Cayley graph of F3

With a standard generator set {a, b, c}.



A Cayley graph of Z3 ∗ Z6

Let G and H be groups.

We say that a wordg1h1g2h2 · · · gnhn

in alphabet G t H is reduced, if

gi ∈ G, hi ∈ H, gi 6= e for i > 1 and hi 6= e for i < n.

A free product G∗H of G and H is a group whose elements are reduced wordsin alphabet GtH, with multiplication defined to be a concatenation followed bycancellation that transforms the concatenated word to a reduced form.

If S is a generator set of G and T is a generator set of H, then S · e t e · T is agenerator set of G ∗ H.



A Cayley graph of Z3 ∗ Z6

With a generator set {a, b}, where a generates Z3 and b generates Z6.



A Cayley graph of Baumslag-Solitar group BS(2, 1)

The Baumslag-Solitar group B(m, n) is given by the group presentation⟨a, b | an = bamb−1⟩ .

A Cayley graph of B(2, 1) with a generator set{

a, b, a−1, b−1}

:

(in fact, this is a picture of a discrete model of a hyperbolic plane)


Cayley Graphs Gromov-Hausdorff convergence

Gromov-Hausdorff convergence

A Gromov-Hausdorff distance between metric spaces X and Y is defined tobe the infimum of the Hausdorff distances between isometric embeddings of Xand Y into other metric spaces.

A metric space (X , d) seen from the infinity is the Gromov-Hausdorff limit of asequence (X , d/n) of metric spaces.

Examples:1 A bounded space see from the infinity is a point.2 Z2 seen from the infinity is R2.


Cayley Graphs Issues with the geometry

Dependence on a generator set

A Cayley graph of Z with a generator set {1}:

A Cayley graph of Z with a generator set {2, 3}:



Quasi-isometries

A map f from a metric space X into a metric space Y is a quasi-isometricembedding if there exist positive real numbers C and D such that for everytwo points x1 and x2 in X we have

1C

d(x1, x2)− D ≤ d(f (x1), f (x2)) ≤ Cd(x1, x2) + D.

We say that f is a quasi-isometry, if additionally its image is R-dense in Y forsome R <∞, i.e. if

∃R<∞∀y∈Y∃x∈X d(f (x), y) < R.

The Cayley graphs from the previous slide are quasi-isometric.



Examples and non-examples of quasi-isometriesExamples of quasi-isometries:

1 An inclusion Z2 ⊂ R2 (both spaces with euclidean metrics)2 A function f : R→ Z given by the formula f (x) = bxc.3 A constant map X → {0}, for any bounded space X .

Examples of maps that are not quasi-isometries:1 n→ n2.2 n2 → n3.

Examples of non-quasi-isometric metric spaces:1 Euclidean and hyperbolic plane.

Examples of quasi-isometric groups:1 Any two finite groups.2 Z and Z2 ∗ Z2.3 Any two ”virtually G” groups.



Independence of a generator set

Lemma. Cayley graphs corresponding to different generator sets of a groupare quasi-isometric.

Notation:

S, T - two finite generator sets of a group G.

|g|S and |g|T - minimal word lengths for S and T .

dS(g, h) = |g−1h|S and dT (g, h) = |g−1h|T - word length metrics on G.


Cayley Graphs Svarc-Milnor lemma

Svarc-Milnor lemma

A geometric action of a group G on a metric space X is a group action suchthat:

1 G acts on X by isometries,2 the action is properly discontinuous, i.e. for every compact set K ⊂ X

the set {g ∈ G : g(K ) ∩ K 6= ∅} is finite,3 the quotient X/G is compact in the quotient topology.

Svarc-Milnor Lemma. If a group G acts geometrically on a proper geodesicspace X, then it is finitely generated and quasi-isometric to X.


Asymptotic invariants

Asymptotic invariants:

Volume growth


Asymptotic invariants Volume growth

Volume growth

Let X be a metric space. We let VolX (t) be the supremum of cardinalities ofclosed t-balls in X . We say that the map VolX : R+ → N ∪ {∞} is the volumegrowth of X .

We say that X is of bounded geometry if VolX (t) <∞ for each t .

Let f , g : R+ → N. We write thata) f � g, if f (r) ≤ C1 (g(C2r + 1) + 1) for some constants C1, C2 and all r ;b) f ∼ g, if f � g and g � f ;c) f ≺ g, if f � g but it is not true that f ∼ g.

Proposition. If X and Y are quasi-isometric and of bounded geometry, thenVolX ∼ VolY .



Volume growth of Zn

We haveVolZn(t) ∼ tn.

If n 6= m, then tn 6∼ tm, hence Zn is not quasi-isometric to Zm for n 6= m.



Volume growth of BS(2, 1)

We haveVolBS(2,1)(t) ∼ 2t .

Note that for each a, b > 1 we have at ∼ bt and ta 6∼ bt .

There are groups with intermediate volume growth; so called Grigorchukgroups.


Asymptotic invariants: Volume growth Gromov’s polynomial growth theorem

Gromov’s polynomial growth theorem

Gromov theorem. A group G has polynomial volume growth if and only if it isvirtually nilpotent.

A group G has polynomial volume growth, if VolG(t) � tn for some n.

A group G is virtually nilpotent, if it contains a nilpotent group of finite index.

A group G is nilpotent, if its lower central series terminate in a trivial group.

Lower central series of a group G is a sequence G1 = G and Gn+1 =[Gn, G] =

{g−1h−1gh : g ∈ G1, h ∈ G

}.


introduction to geometric group theory - bgubarakw/probseminar/nagorko/slides.pdf · introduction...

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