**University Lecture Series**

Volume: 40;
2006;
192 pp;
Softcover

MSC: Primary 05; 22; 43; 46;
Secondary 20; 28; 37; 51; 54; 60

Print ISBN: 978-0-8218-4137-2

Product Code: ULECT/40

List Price: $42.00

Individual Member Price: $33.60

**Electronic ISBN: 978-1-4704-2184-7
Product Code: ULECT/40.E**

List Price: $42.00

Individual Member Price: $33.60

#### Supplemental Materials

# Dynamics of Infinite-dimensional Groups: The Ramsey–Dvoretzky–Milman Phenomenon

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*Vladimir Pestov*

The “infinite-dimensional groups” in the
title refer to unitary groups of Hilbert spaces, the infinite symmetric group,
groups of homeomorphisms of manifolds, groups of transformations of measure
spaces, etc. The book presents an approach to the study of such groups based on
ideas from geometric functional analysis and from exploring the interplay
between dynamical properties of those groups, combinatorial Ramsey-type
theorems, and the phenomenon of concentration of measure.

The dynamics of infinite-dimensional groups is very much unlike that of
locally compact groups. For instance, every locally compact group acts freely
on a suitable compact space (Veech). By contrast, a 1983 result by Gromov and
Milman states that whenever the unitary group of a separable Hilbert space
continuously acts on a compact space, it has a common fixed point.

In the book, this new fast-growing theory is built strictly from
well-understood examples up. The book has no close counterpart and is based
on recent research articles. At the same time, it is organized so as to be
reasonably self-contained. The topic is essentially interdisciplinary and
will be of interest to mathematicians working in geometric functional
analysis, topological and ergodic dynamics, Ramsey theory, logic and
descriptive set theory, representation theory, topological groups, and
operator algebras.

#### Readership

Graduate students and research mathematicians interested in representation theory, dynamical systems, geometric functional analysis, Ramsey theory, and descriptive set theory.

#### Reviews & Endorsements

This is a very well-written and lively exposition, with a number of basic examples worked out in detail. In comparison to the original version, while the set of topics treated is essentially the same, some chapters have been reorganized, updated and largely expanded.

-- Zentralblatt MATH

#### Table of Contents

# Table of Contents

## Dynamics of Infinite-dimensional Groups: The Ramsey-Dvoretzky-Milman Phenomenon

- Cover Cover11 free
- Title i2 free
- Copyright ii3 free
- Contents vii8
- Preface v6 free
- Preface to the original IMPA edition ix10 free
- Introduction 112 free
- Chapter 1. The Ramsey–Dvoretzky–Milman phenomenon 1122 free
- Chapter 2. The fixed point on compact a property 3950
- Chapter 3. The concentration property 4960
- Chapter 4. Lévy groups 7586
- Chapter 5. Urysohn metric space and its group of isometries 103114
- Chapter 6. Minimal flows 125136
- 6.1. Universal minimal flow 125136
- 6.2. Group of circle homeomorphisms 127138
- 6.3. Infinite symmetric group 130141
- 6.4. Maximal chains and Uspenskij's theorem 132143
- 6.5. FraÏsse theory 135146
- 6.6. Extremely amenable subgroups of S[sub(∞)] 138149
- 6.7. Minimal flows of automorphism groups 142153
- 6.8. Fixed point on metric compacta property 147158

- Chapter 7. Further aspects of concentration 149160
- Chapter 8. Oscillation stability and distortion 171182
- Bibliography 183194
- Index 191202
- Back Cover Back Cover1204