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Geometry, Rigidity, and Group Actions

Edited by Benson Farb and David Fisher

Geometry, Rigidity, and Group Actions

Edited by Benson Farb and David Fisher

600 pages | 1 halftone, 15 line drawings | 6 x 9 | © 2011
Cloth $81.00 ISBN: 9780226237886 Published April 2011
E-book $10.00 to $80.99 About E-books ISBN: 9780226237909 Published April 2011

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.


PART 1 || Group Actions on Manifolds
1. An Extension Criterion for Lattice Actions on the Circle
Marc Burger
2. Meromorphic Almost Rigid Geometric Structures
Sorin Dumitrescu
3. Harmonic Functions over Group Actions
Renato Feres and Emily Ronshausen
4. Groups Acting on Manifolds: Around the Zimmer Program
David Fisher
5. Can Lattices in SL (n, R) Act on the Circle?
David Witte Morris
6. Some Remarks on Area-Preserving Actions of Lattices
Pierre Py
7. Isometric Actions of Simple Groups and Transverse Structures: The Integrable Normal Case
Raul Quiroga-Barranco
8. Some Remarks Inspired by the C0 Zimmer Program
Shmuel Weinberger

PART 2 || Analytic, Ergodic, and Measurable Group Theory

9. Calculus on Nilpotent Lie Groups
Michael G. Cowling
10. A Survey of Measured Group Theory
Alex Furman
11. On Relative Property (T)
Alessandra Iozzi
12. Noncommutative Ergodic Theorems
Anders Karlsson and François Ledrappier
13. Cocycle and Orbit Superrigidity for Lattices in SL (n, R) Acting on Homogeneous Spaces
Sorin Popa and Stefaan Vaes

PART 3 || Geometric Group Theory

14. Heights on SL2 and Free Subgroups
Emmanuel Breuillard
15. Displacing Representations and Orbit Maps
Thomas Delzant, Olivier Guichard, François Labourie, and Shahar Mozes
16. Problems on Automorphism Groups of Nonpositively Curved Polyhedral Complexes and Their Lattices
Benson Farb, Chris Hruska, and Anne Thomas
17. The Geometry of Twisted Conjugacy Classes in Wreath Products
Jennifer Taback and Peter Wong

PART 4 || Group Actions on Representations Varieties

18. Ergodicity of Mapping Class Group Actions on SU(2)-Character Varieties
William M. Goldman and Eugene Z. Xia
19. Dynamics and Aut (Fn) Actions on Group Presentations and Representations
Alexander Lubotzky

List of Contributors
Review Quotes
Nicolas Monod, École Polytechnique Fédérale de Lausanne

“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.”
Scot Adams, University of Minnesota

“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.”

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