Geometry, Rigidity, and Group Actions

by Benson Farb, David Fisher

Overview

The study of group actions is more than a hundred years old but remains to this day a vibrant and widely studied topic in a variety of mathematic fields. A central development in the last fifty years is the phenomenon of rigidity, whereby one can classify actions of certain groups, such as lattices in semi-simple Lie groups. This provides a way to classify all possible symmetries of important spaces and all spaces admitting given symmetries. Paradigmatic results can be found in the seminal work of George Mostow, Gergory Margulis, and Robert J. Zimmer, among others.

The papers in Geometry, Rigidity, and Group Actions explore the role of group actions and rigidity in several areas of mathematics, including ergodic theory, dynamics, geometry, topology, and the algebraic properties of representation varieties. In some cases, the dynamics of the possible group actions are the principal focus of inquiry. In other cases, the dynamics of group actions are a tool for proving theorems about algebra, geometry, or topology. This volume contains surveys of some of the main directions in the field, as well as research articles on topics of current interest.

Product Details

ISBN-13:	9780226237886
Publisher:	University of Chicago Press
Publication date:	04/30/2011
Series:	Chicago Lectures in Mathematics Series
Pages:	600
Product dimensions:	6.10(w) x 9.10(h) x 1.60(d)

About the Author

Benson Farb is professor of mathematics at the University of Chicago. He is the author of Problems on Mapping Class Groups and Related Topics and coauthor of Noncommutative Algebra. David Fisher is professor of mathematics at Indiana University.

Table of Contents

Preface ix

Part 1 Group Actions of Manifolds

1 An Extension Criterion for Lattice Actions on the Circle Marc Burger 3

2 Meromorphic Almost Rigid Geometric Structures Sorin Dumitrescu 32

3 Harmonic Functions over group Actions Renato Feres Emily Ronshausen 59

14 Groups Acting on Manifolds: Around the Zimmer Program David Fisher 72

5 Can Lattices in SL)nR) Act on the Circle? Dave Witte Morris 158

6 Some Remarks on Area Preserving Actions of Lattices Pierre Py 208

7 Isometric Actions of Simple Groups and Transverse Structures: The Integrable Normal Case Raul Quiroga Barranco 229

8 SOme Remarks Inspired by the C⁰ Zimemr Program Shmuel Weinberger 262

Part 2 Analytic Ergodic and Measurable Group Theory

9 Calculus on Nilpotent Lie Groups Michael G. Cowling 285

10 A Survey of Measured Group Theory Alex Furman 296

11 On Relative Property (T) Alessandra Iozzi 375

12 Noncommutative Ergodic Theorems Anders Karlsson Francois Ledrappier 396

13 Cocycle and Orbit Superrigidity for Lattices in SL (n, R) Acting on Homogeneous Spaces Sorin Popa Stefaan Vaes 419

Part3 Geometric Group Theory

14 Heights on SL₂ and Free Subgroups Emmanuel Breuillard 455

15 Displacing Representations and Orbit Maps Thomas Delzant Olivier Guichard François Labourie Shahar Mozes 494

16 Problems on Automorphism Groups on Nonpositively Curved Polyhedral Complexes and Their Lattices Benson Farb Chris Hruska Anne Thomas 515

17 The Geometry of Twisted Conjugacy Classes in Wreath Products Jennifer Taback Peter Wong 561

Part 4 Group Actions on Representation Varieties

18 Ergodicity of Mapping Class Group Actions on SU(2) Character Varieties William M. Goldman Eugene Z. Xia 591

19 Dynamics of Aut (F_n) Actions on Group Presentations and Representation Alexander Lubotzky 609

List of Contributors 645

Editorial Reviews

“There is no better way to pay tribute to Robert J. Zimmer’s deep and broad impact on mathematics than to assemble a selection of gems written by top experts from the fields that he influenced over the years and to this day. The texts collected by Farb and Fisher range from short masterpieces to authoritative surveys that will surely become classical references. Geometry, Rigidity, and Group Actions will appeal to a wide range of mathematicians.

Nicolas Monod

“For those interested in learning about the subject of large group actions, sometimes described as ‘Zimmer’s program,’ this is a key book to own, and it fits well alongside the earlier rigidity books by Zimmer, Margulis, Feres, and Witte Morris. This is an extensive area of mathematics, with many subareas of research, and for those already familiar with parts of the program, this book will also prove invaluable as a guide to many of the latest developments.”

Scot Adams

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