Geometric group theory, an introduction

264 Pages·2015·1.275 MB·English

by  Loh C

Clara L¨oh
Geometric group theory,
an introduction
January 30, 2015 – 22:12
Version for the course in 2014/15
Please send corrections and suggestions to [email protected]
Clara L¨oh
[email protected]
http://www.mathematik.uni-regensburg.de/loeh/
Fakult¨at fu¨r Mathematik
Universit¨at Regensburg
93040 Regensburg
Germany
Contents
1 Introduction 1
2 Generating groups 7
2.1 Review of the category of groups 8
2.1.1 Axiomatic description of groups 8
2.1.2 Concrete groups – automorphism groups 10
2.1.3 Normal subgroups and quotients 12
2.2 Groups via generators and relations 16
2.2.1 Generating sets of groups 16
2.2.2 Free groups 17
2.2.3 Generators and relations 23
2.3 New groups out of old 30
2.3.1 Products and extensions 30
2.3.2 Free products and free amalgamated products 33
3 Groups → geometry, I: Cayley graphs 39
3.1 Review of graph notation 40
3.2 Cayley graphs 43
3.3 Cayley graphs of free groups 46
3.3.1 Free groups and reduced words 47
3.3.2 Free groups → trees 50
3.3.3 Trees → free groups 52
iv Contents
4 Groups → geometry, II: Group actions 53
4.1 Review of group actions 54
4.1.1 Free actions 55
4.1.2 Orbits and stabilisers 59
4.1.3 Application: Counting via group actions 62
4.2 Free groups and actions on trees 64
4.2.1 Spanning trees 65
4.2.2 Completing the proof 67
4.3 Application: Subgroups of free groups are free 70
4.4 The ping-pong lemma 74
4.5 Application: Free subgroups of matrix groups 76
5 Groups → geometry, III: Quasi-isometry 79
5.1 Quasi-isometry types of metric spaces 80
5.2 Quasi-isometry types of groups 88
5.2.1 First examples 91
5.3 The Sˇvarc-Milnor lemma 94
5.3.1 Quasi-geodesics and quasi-geodesic spaces 94
5.3.2 The Sˇvarc-Milnor lemma 96
5.3.3 Applications of the Sˇvarc-Milnor lemma to group theory,
geometry and topology 102
5.4 The dynamic criterion for quasi-isometry 107
5.4.1 Applications of the dynamic criterion 113
5.5 Preview: Quasi-isometry invariants and geometric properties 116
5.5.1 Quasi-isometry invariants 116
5.5.2 Functorial quasi-isometry invariants 117
5.5.3 Geometric properties of groups and rigidity 122
6 Growth types of groups 125
6.1 Growth functions of finitely generated groups 126
6.2 Growth types of groups 129
6.2.1 Growth types 129
6.2.2 Growth types and quasi-isometry 131
6.2.3 Application: Volume growth of manifolds 136
6.3 Groups of polynomial growth 139
6.3.1 Nilpotent groups 140
Contents v
6.3.2 Growth of nilpotent groups 142
6.3.3 Groups of polynomial growth are virtually nilpotent 143
6.3.4 Application: Being virtually nilpotent is a geometric
property 145
6.3.5 Application: More on polynomial growth 146
6.3.6 Application: Quasi-isometry rigidity of free Abelian
groups 146
6.3.7 Application: Expanding maps of manifolds 147
7 Hyperbolic groups 151
7.1 Classical curvature, intuitively 152
7.1.1 Curvature of plane curves 152
7.1.2 Curvature of surfaces in R3 154
7.2 (Quasi-)Hyperbolic spaces 157
7.2.1 Hyperbolic spaces 157
7.2.2 Quasi-hyperbolic spaces 159
7.2.3 Quasi-geodesics in hyperbolic spaces 163
7.2.4 Hyperbolic graphs 170
7.3 Hyperbolic groups 171
7.4 Application: “Solving” the word problem 176
7.5 Elements of infinite order in hyperbolic groups 182
7.5.1 Existence of elements of infinite order in hyperbolic groups 182
7.5.2 Centralisers of elements of infinite order in hyperbolic
groups 189
7.5.3 Application: Products and negative curvature 200
7.5.4 Free subgroups of hyperbolic groups 201
8 Ends and boundaries 203
8.1 Geometry at infinity 204
8.2 Ends of groups 205
8.3 Boundary of hyperbolic groups 210
8.4 Application: Mostow rigidity 214
9 Amenable groups 217
9.1 Amenability via means 218
9.1.1 First examples of amenable groups 219
vi Contents
9.1.2 Inheritance properties 220
9.2 Further characterisations of amenability 223
9.2.1 Følner sequences 223
9.2.2 Paradoxical decompositions 227
9.2.3 Application: The Banach-Tarski paradox 228
9.2.4 (Co)Homological characterisations of amenability 230
9.3 Quasi-isometry invariance of amenability 231
9.4 Quasi-isometry and bilipschitz equivalence 233
9.4.1 Bilipschitz equivalence rigidity 233
A Appendix A.1
A.1 The fundamental group, a primer A.2
A.2 The (von) Neumann forest A.5
A.3 Geometric realisation of graphs A.7
1
Introduction
What is geometric group theory? Geometric group theory investigates the
interaction between algebraic and geometric properties of groups:
– Can groups be viewed as geometric objects and how are geometric
and algebraic properties of groups related?
– More generally: On which geometric objects can a given group act in
areasonableway,andhowaregeometricpropertiesofthesegeometric
objects/actions related to algebraic properties of the group?
How does geometric group theory work? Classically, group-valued in-
variants are associated with geometric objects, such as, e.g., the isometry
group or the fundamental group. It is one of the central insights leading
to geometric group theory that this process can be reversed to a certain
extent:
1. We associate a geometric object with the group in question; this
can be an “artificial” abstract construction or a very concrete model
space (such as the Euclidean plane or the hyperbolic plane) or action
from classical geometric theories.
2. Wetakegeometricinvariantsandapplythesetothegeometricobjects
obtained by the first step. This allows to translate geometric terms
such as geodesics, curvature, volumes, etc. into group theory.
Usually, in this step, in order to obtain good invariants, one restricts
2 1. Introduction
Z×Z Z Z∗Z
Figure 1.1.: Basic examples of Cayley graphs
attention to finitely generated groups and takes geometric invariants
from large scale geometry (as they blur the difference between differ-
ent finite generating sets of a given group).
3. We compare the behaviour of such geometric invariants of groups
with the algebraic behaviour, and we study what can be gained by
this symbiosis of geometry and algebra.
A key example of a geometric object associated with a group is the so-
called Cayley graph (with respect to a chosen generating set) together with
thecorrespondingwordmetric. Forinstance,fromthepointofviewoflarge
scale geometry, the Cayley graph of Z resembles the geometry of the real
line, the Cayley graph of Z×Z resembles the geometry of the Euclidean
plane, while the Cayley graph of the free group Z ∗ Z on two generators
has essential features of the geometry of the hyperbolic plane (Figure 1.1;
exact definitions of these concepts are introduced in later chapters).
More generally, in (large scale) geometric group theoretic terms, the
universe of (finitely generated) groups roughly unfolds as depicted in Fig-
ure 1.2. The boundaries are inhabited by amenable groups and non-
positively curved groups respectively – classes of groups that are (at least
partially) accessible. However, studying these boundary classes is only
the very beginning of understanding the universe of groups; in general,
knowledge about these two classes of groups is far from enough to draw
conclusions about groups at the inner regions of the universe:
“Hic abundant leones.” [15]
“A statement that holds for all finitely generated groups
has to be either trivial or wrong.” [attributed to M. Gromov]
3
elementary amenable
solvable
s
p polycyclic
u
o
r
g
e
bl nilpotent
a
n
e
m
a
s Abelian
p
u
o
gr 1 Z ?
e
t
i
n free groups
fi
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po hyperbolic groups
siti
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y CAT(0)-groups
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Figure 1.2.: Theuniverseofgroups(simplifiedversionofBridson’suniverse
of groups [15])
4 1. Introduction
Why study geometric group theory? On the one hand, geometric group
theoryisaninterestingtheorycombiningaspectsofdifferentfieldsofmath-
ematics in a cunning way. On the other hand, geometric group theory has
numerous applications to problems in classical fields such as group theory
and Riemannian geometry.
For example, so-called free groups (an a priori purely algebraic notion)
can be characterised geometrically via actions on trees; this leads to an
elegant proof of the (purely algebraic!) fact that subgroups of free groups
are free.
Further applications of geometric group theory to algebra and Rieman-
nian geometry include the following:
– Recognising that certain matrix groups are free groups; there is a geo-
metriccriterion,theso-calledping-pong-lemma,thatallowstodeduce
freeness of a group by looking at a suitable action (not necessarily
on a tree).
– Recognising that certain groups are finitely generated; this can be
done geometrically by exhibiting a good action on a suitable space.
– Establishingdecidabilityofthewordproblemforlargeclassesofgroups;
for example, Dehn used geometric ideas in his algorithm solving the
word problem in certain geometric classes of groups.
– Recognising that certain groups are virtually nilpotent; Gromovfound
a characterisation of finitely generated virtually nilpotent groups in
terms of geometric data, more precisely, in terms of the growth func-
tion.
– Proving non-existence of Riemannian metrics satisfying certain cur-
vature conditions on certain smooth manifolds; this is achieved by
translating these curvature conditions into group theory and looking
at groups associated with the given smooth manifold (e.g., the funda-
mental group). Moreover, a similar technique also yields (non-)split-
ting results for certain non-positively curved spaces.
– Rigidity results for certain classes of matrix groups and Riemannian
manifolds; here, the key is the study of an appropriate geometry at
infinity of the groups involved.
– Geometric group theory provides a layer of abstraction that helps to
understand and generalise classical geometry – in particular, in the
case of negative or non-positive curvature and the corresponding ge-
ometry at infinity.