Amenability of Discrete Groups by Examples

180 Pages·2022·1.482 MB·English

by  Kate Juschenko

Mathematical
Surveys
and
Monographs
Volume 266
Amenability of Discrete
Groups by Examples
Kate Juschenko
Amenability of Discrete
Groups by Examples

Mathematical
Surveys
and
Monographs
Volume 266
Amenability of Discrete
Groups by Examples
Kate Juschenko
EDITORIAL COMMITTEE
Ana Caraiani Natasa Sesum
Michael A. Hill Constantin Teleman
Bryna Kra (chair) Anna-Karin Tornberg
2020 Mathematics Subject Classification. Primary 20-03, 20L99, 22-02.
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Contents
Preface ix
Chapter 1. Introduction 1
Chapter 2. The first germs of amenability: Paradoxical decompositions 5
1. Paradoxical actions 6
2. Hausdorff paradox 8
3. Banach-Tarski paradox 10
4. Pea to Sun paradox 11
5. First definitions: invariant mean, Følner condition 12
6. First examples 14
7. Operations that preserve amenability 15
8. Amenable actions: first definitions 17
9. Paradoxical decompositions and amenable actions 19
10. Faithful amenable actions of non-amenable groups 19
Chapter 3. Elementary amenable groups 25
1. Simplification of the class of elementary amenable groups 26
2. Elementary classes of amenable groups 28
3. Torsion groups 29
4. Dichotomy of the growth of elementary amenable groups 29
5. Groups of intermediate growth 32
6. Groups acting on rooted trees 33
7. Non-elemetary amenable groups acting on rooted trees 35
8. The first Grigorchuk’s group 36
9. Basilica group 37
10. Finitely generated branch groups 38
Chapter 4. The topological full group of a Cantor minimal system 41
1. The topological full group of Cantor minimal systems. Definition and
basic facts 42
2. Basic facts on dynamics on the Cantor space 45
3. Simplicity of the commutator subgroup 46
4. Finite generation of the commutator subgroup of a minimal subshift 50
5. Spatial isomorphism of topological full groups 53
6. Boyle’s flip conjugacy theorem 57
7. Examples of non-isomorphic topological full groups 58
Unique ergodicity 59
Chapter 5. Lamplighter actions and extensive amenability 63
1. Extensive amenability: basic definition and properties 64
v
vi CONTENTS
2. Recurrent actions: definition and basic properties 68
3. Recurrent actions are extensively amenable 70
4. Extensions of actions 73
5. Functorial properties of extensive amenability 74
6. Extensively amenable and amenable actions 76
7. Inverted orbits of random walks 77
Proof of “recurrent implies extensive amenability” using inverted orbits 79
Chapter 6. Amenability of topological full groups 81
1. Amenability of topological full groups of Cantor minimal system: first
proof 82
2. The interval exchange transformation group 85
3. Realization of finitely generated subgroups of IET as subgroups of the
full topological group of Zd 85
4. Amenability of subgroups of IET of low ranks 86
Chapter 7. Subgroups of topological full groups of intermediate growth 89
1. Multisections 89
2. Adding two multisections together 91
3. Bisections 92
4. Finitely generated actions 92
5. An introduction to periodicity and intermediate growth 95
6. Examples 109
Chapter 8. An amenability criterion via actions 115
1. Group theoretical version of the criteria 115
2. Actions by homeomorphisms. Preliminary definitions 117
3. Topological version of criterion 118
Chapter 9. Groups acting on Bratteli diagrams 121
1. Homeomorphisms of Bratteli of bounded type 122
2. Vershik transformations 122
3. Realization of Cantor minimal systems as homeomorphisms of
Bratteli-Vershik diagrams 123
4. One-dimensional tilings 123
Chapter 10. Groups acting on rooted trees 125
1. Finite automata of bounded activity 127
2. Groups of Neumann-Segal type 127
3. Iterated monodromy groups of polynomial iterations 128
4. Penrose tilings 129
5. Linearly and quadratically growing automata 131
Appendix A. Definitions of amenability and basic facts 135
1. Means and measures 135
2. Almost invariant vectors of the left regular representation 137
3. Weakening of the Følner condition 138
4. Kesten’s criterion 139
5. Hulanicki’s criterion 141
6. Weak containment of representations 142
CONTENTS vii
7. Day’s fixed point theorem 143
8. Tarski theorem and Tarski numbers 144
9. Gromov’s doubling condition. Grasshopper criteria 148
Appendix B. Related open problems 151
Is this group amenable? 152
Topological full groups 153
Bibliography 155
Subject Index 163
Notation Index 165

Preface
This book aims to treat all currently known examples of amenable groups and
techniques behind proving amenability. We do not elaborate on any examples of
non-amenablegroupsindetail(exceptfreegroups); rather,welistallknownexam-
ples of non-amenable groups which do not contain free groups in the Introduction.
The subject of amenability has its roots in measure theory. In 1904, Lebesgue
askedwhetherafinitelyadditivetranslation-invariantmeasuredefinedonallLebes-
gue measurable sets is necessarily the Lebesgue measure. This question was an-
swered negatively by Banach. The core of his proof is the Hahn-Banach theorem
and a connection between finitely additive measures and means. This essentially
led to a first example of an amenable group: the group of real numbers.
ThenextstepinthedevelopmentofamenabilitywastheHausdorffparadox. It
statesthattheunitspherecanbedecomposedintofinitelymanypiecesinawaythat
rotating these pieces around one can obtain two spheres. Hausdorff showed that
the non-abelian free group on two generators (which is now considered a canonical
example of a non-amenable group) is a subgroup of the group of rotations. It is
easy to make a paradoxical decomposition of the free group itself. Having this
decomposition in hand, Hausdorff pushed it to the action of the rotation group on
the sphere, which implied his famous paradox.
Consideringaunitballandprojectingeachpointoftheball(exceptthecenter)
onto the sphere, one obtains a paradoxical decomposition of the ball without a
center. Notably, the entire ball admits a paradoxical decomposition if one allows
use of rotations and translations in the rearranging of the sets. This counter-
intuitiveparadoxwasprovedbyBanachandTarskiin1929. Astrongversionofthe
Banach-Tarski paradox claims that any bounded set with non-empty interior can
be decomposed and rearranged to any other bounded set with non-empty interior.
This is striking, as it implies that a tiny pea that can be decomposed to the sun.
In 1929, von Neumann extracted an essential property of the group that for-
bids paradoxical decomposition: the existence of an invariant mean. He coined a
German name “meßbar”, which translates to English as “measurable”. Studying
the properties of meßbar groups, he showed that this class is closed under tak-
ing natural group operations: subgroups, extensions, quotients, and direct limits.
Moreover, it contains all finite and abelian groups. This immediately explained
why the Banach-Tarski paradox is impossible in the two dimensional case, where
the rotation group is solvable.
In 1957, Day gave an English term “amenable” as analog of von Neumann’s
“meßbar”groups. Hedefinedtheclassofelementaryamenablegroupsasthesmall-
estclassofgroupscontainingallfinitegroupsandabeliangroups, andclosedunder
taking subgroups, extensions, quotients, and direct limits. No substantial progress
in understanding this class was made until the ’80s, when Chou showed that all
ix