Geometry, Rigidity, and Group Actions

By: Benson Farb (Editor), David Fisher (Editor)

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Hardcover | 30 April 2011 | Edition Number 1

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600 Pages

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2.35 x 1.56 x 0.41

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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.

Industry 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, ?cole Polytechnique F?d?rale de Lausanne

"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, University of Minnesota

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You Can Find This Book In

Non-Fiction Mathematics Geometry Differential & Riemannian Geometry Algebra Topology Miscellaneous Items

Preface	p. ix
Group Actions of Manifolds
An Extension Criterion for Lattice Actions on the Circle	p. 3
Meromorphic Almost Rigid Geometric Structures	p. 32
Harmonic Functions over group Actions	p. 59
Groups Acting on Manifolds: Around the Zimmer Program	p. 72
Can Lattices in SL)nR) Act on the Circle?	p. 158
Some Remarks on Area Preserving Actions of Lattices	p. 208
Isometric Actions of Simple Groups and Transverse Structures: The Integrable Normal Case	p. 229
SOme Remarks Inspired by the C0 Zimemr Program	p. 262
Analytic Ergodic and Measurable Group Theory
Calculus on Nilpotent Lie Groups	p. 285
A Survey of Measured Group Theory	p. 296
On Relative Property (T)	p. 375
Noncommutative Ergodic Theorems	p. 396
Cocycle and Orbit Superrigidity for Lattices in SL (n, R) Acting on Homogeneous Spaces	p. 419
Geometric Group Theory
Heights on SL₂ and Free Subgroups	p. 455
Displacing Representations and Orbit Maps	p. 494
Problems on Automorphism Groups on Nonpositively Curved Polyhedral Complexes and Their Lattices	p. 515
The Geometry of Twisted Conjugacy Classes in Wreath Products	p. 561
Group Actions on Representation Varieties
Ergodicity of Mapping Class Group Actions on SU(2) Character Varieties	p. 591
Dynamics of Aut (F_n) Actions on Group Presentations and Representation	p. 609
List of Contributors	p. 645
Table of Contents provided by Ingram. All Rights Reserved.

Geometry, Rigidity, and Group Actions

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