Geometric group theory, an introduction

247 Pages·2016·1.206 MB·English

by  Loh C

Clara L¨oh
Geometric Group Theory
An Introduction
December 15, 2016 – 15:24
Book project
Incomplete draft version!
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
Part I Groups 7
2 Generating groups 9
2.1 Review of the category of groups 10
2.1.1 Abstractgroups:axioms 10
2.1.2 Concretegroups:automorphismgroups 12
2.1.3 Normalsubgroupsandquotients 16
2.2 Groups via generators and relations 19
2.2.1 Generatingsetsofgroups 19
2.2.2 Freegroups 20
2.2.3 Generatorsandrelations 25
2.2.4 Finitelypresentedgroups 29
2.3 New groups out of old 31
2.3.1 Productsandextensions 31
2.3.2 Freeproductsandfreeamalgamatedproducts 34
2.E Exercises 38
iv Contents
Part II Groups → Geometry 47
3 Cayley graphs 49
3.1 Review of graph notation 50
3.2 Cayley graphs 53
3.3 Cayley graphs of free groups 56
3.3.1 Freegroupsandreducedwords 57
3.3.2 Freegroups→trees 59
3.3.3 Trees→freegroups 61
3.E Exercises 62
4 Group actions 69
4.1 Review of group actions 70
4.1.1 Freeactions 71
4.1.2 Orbitsandstabilisers 74
4.1.3 Application:Countingviagroupactions 77
4.2 Free groups and actions on trees 78
4.2.1 Spanningtreesforgroupactions 79
4.2.2 Completingtheproof 80
4.3 Application: Subgroups of free groups are free 83
4.4 The ping-pong lemma 86
4.5 Application: Free subgroups of matrix groups 88
4.5.1 ThegroupSL(2,Z) 88
4.5.2 Regulargraphsoflargegirth 91
4.5.3 TheTitsalternative 93
4.E Exercises 96
5 Quasi-isometry 107
5.1 Quasi-isometry types of metric spaces 108
5.2 Quasi-isometry types of groups 114
5.2.1 Firstexamples 117
5.3 The Sˇvarc-Milnor lemma 119
5.3.1 Quasi-geodesicsandquasi-geodesicspaces 119
5.3.2 TheSˇvarc-Milnorlemma 121
5.3.3 ApplicationsoftheSˇvarc-Milnorlemmatogrouptheory,
geometry,andtopology 126
5.4 The dynamic criterion for quasi-isometry 130
5.4.1 Applicationsofthedynamiccriterion 135
5.5 Preview: Quasi-isometry invariants and geometric properties 137
5.5.1 Quasi-isometryinvariants 137
5.5.2 Functorialquasi-isometryinvariants 138
5.5.3 Geometricpropertiesofgroupsandrigidity 143
5.E Exercises 145
Contents v
Part III Geometry of groups 153
6 Growth types of groups 155
6.1 Growth functions of finitely generated groups 156
6.2 Growth types of groups 159
6.2.1 Growthtypes 159
6.2.2 Growthtypesandquasi-isometry 161
6.2.3 Application:Volumegrowthofmanifolds 164
6.3 Groups of polynomial growth 167
6.3.1 Nilpotentgroups 168
6.3.2 Growthofnilpotentgroups 169
6.3.3 Groupsofpolynomialgrowtharevirtuallynilpotent 170
6.3.4 Application:Virtualnilpotenceisageometricproperty 172
6.3.5 Moreonpolynomialgrowth 173
6.3.6 Quasi-isometryrigidityoffreeAbeliangroups 173
6.3.7 Application:Expandingmapsofmanifolds 174
6.4 Groups of uniform exponential growth 175
6.4.1 Uniformexponentialgrowth 176
6.4.2 Uniformuniformexponentialgrowth 177
6.4.3 TheuniformTitsalternative 178
6.4.4 Application:TheLehmerconjecture 179
6.E Exercises 181
7 Hyperbolic groups 187
8 Ends and boundaries 189
9 Amenable groups 191
9.1 Amenability via means 192
9.1.1 Firstexamplesofamenablegroups 192
9.1.2 Inheritanceproperties 194
9.2 Further characterisations of amenability 196
9.2.1 Følnersequences 197
9.2.2 Paradoxicaldecompositions 200
9.2.3 Application:TheBanach-Tarskiparadox 202
9.2.4 (Co)Homologicalcharacterisationsofamenability 204
9.3 Quasi-isometry invariance of amenability 205
9.4 Quasi-isometry vs. bilipschitz equivalence 207
9.E Exercises 211
vi Contents
Part IV Reference material 217
A Appendix 219
Bibliography 221
Index 231
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
a reasonable way, and how are geometric properties of these geometric
objects/actions related to algebraic properties of the group?
How does geometric group theory work? Classically, group-valued invari-
ants are associated with geometric objects, such as, e.g., the isometry group
orthefundamentalgroup.Itisoneofthecentralinsightsleadingtogeometric
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. We take geometric invariants and apply these to the geometric objects
obtainedbythefirststep.Thisallowstotranslategeometrictermssuch
as geodesics, curvature, volumes, etc. into group theory.
Usually, in this step, in order to obtain good invariants, one restricts
attention to finitely generated groups and takes geometric invariants
fromlargescalegeometry(astheyblurthedifferencebetweendifferent
finite generating sets of a given group).
3. We compare the behaviour of such geometric invariants of groups with
thealgebraicbehaviour,andwestudywhatcanbegainedbythissym-
biosis 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
2 1. Introduction
Z×Z Z Z∗Z
Figure 1.1.: Basic examples of Cayley graphs
the corresponding word metric. For instance, from the point of view of large
scalegeometry,theCayleygraphofZresemblesthegeometryoftherealline,
the Cayley graph of Z×Z resembles the geometry of the Euclidean plane,
whiletheCayleygraphofthefreegroupZ∗Zontwogeneratorshasessential
featuresofthegeometryofthehyperbolicplane(Figure1.1;exactdefinitions
of these concepts are introduced in later chapters).
More generally, in (large scale) geometric group theoretic terms, the uni-
verseof(finitelygenerated)groupsroughlyunfoldsasdepictedinFigure1.2.
Theboundariesareinhabitedbyamenablegroupsandnon-positivelycurved
groups respectively – classes of groups that are (at least partially) accessi-
ble. 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.” [22]
“A statement that holds for all finitely generated groups
has to be either trivial or wrong.” [attributed to M. Gromov]
Why study geometric group theory? On the one hand, geometric group
theory is an interesting theory combining aspects of different fields of math-
ematics in a cunning way. On the other hand, geometric group theory has
numerous applications to problems in classical fields such as group theory,
Riemannian geometry, topology, and number theory.
Forexample,so-calledfreegroups(anaprioripurelyalgebraicnotion)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.
FurtherapplicationsofgeometricgrouptheorytoalgebraandRiemannian
geometry include the following:
• Recognising that certain matrix groups are free groups; there is a geo-
metric criterion, the so-called ping-pong-lemma, that allows to deduce
freeness of a group by looking at a suitable action (not necessarily on
a tree).
3
elementary amenable
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Figure 1.2.: The universe of groups (simplified version of Bridson’s universe
of groups [22])
• Recognising that certain groups are finitely generated; this can be done
geometrically by exhibiting a good action on a suitable space.
• Establishing decidability of the word problem for large classes of groups;
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; Gromovfounda
characterisationoffinitelygeneratedvirtuallynilpotentgroupsinterms
of geometric data, more precisely, in terms of the growth function.
• Proving non-existence of Riemannian metrics satisfying certain cur-
vature conditions on certain smooth manifolds; this is achieved by
4 1. Introduction
translating these curvature conditions into group theory and looking
at groups associated with the given smooth manifold (e.g., the funda-
mentalgroup).Moreover,asimilartechniquealsoyields(non-)splitting
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.
• Group-theoretic reformulation of the Lehmer conjecture; bytheworkof
Breuillard et al., the Lehmer conjecture in algebraic number theory is
equivalent to a problem about growth of certain matrix groups.
• Geometric group theory provides a layer of abstraction that helps to
understandandgeneraliseclassicalgeometry –inparticular,inthecase
of negative or non-positive curvature and the corresponding geometry
at infinity.
• The Banach-Tarski paradox (a sphere can be divided into finitely many
piecesthatinturncanbepuzzledtogetherintotwospherescongruentto
the given one [this relies on the axiom of choice]); the Banach-Tarski
paradox corresponds to certain matrix groups not being “amenable”,
a notion related to both measure theoretic and geometric properties of
groups.
• A better understanding of many classical groups; this includes, for in-
stance, mapping class groups of surfaces and outer automorphisms of
free groups (and their behaviour similar to certain matrix groups).
Overviewofthebook.Asthemaincharactersingeometricgrouptheoryare
groups, we will start by reviewing some concepts and examples from group
theory, and by introducing constructions that allow to generate interesting
groups(Chapter2).Readersfamiliarwithgrouptheorycanhappilyskipthis
chapter.
Thenwewillintroduceoneofthemaincombinatorialobjectsingeometric
grouptheory,theso-calledCayleygraph,andreviewbasicnotionsconcerning
actionsofgroups(Chapter3–4).Afirsttasteofthepowerofgeometricgroup
theorywillthenbepresentedinthediscussionofgeometriccharacterisations
of free groups.
As next step, we will introduce a metric structure on groups via word
metrics on Cayley graphs, and we will study the large scale geometry of
groups with respect to this metric structure, in particular, the concept of
quasi-isometry (Chapter 5).
After that, invariants under quasi-isometry will be introduced – this in-
cludes, in particular, growth functions, curvature conditions, and the geom-
etry at infinity (Chapters 6–8).
Finally, we will have a look at the class of amenable groups and their
properties (Chapter 9).
Basics on fundamental groups and properties of the hyperbolic plane are
collected in the appendices (Appendix A).