Geometry, Rigidity, and Group Actions
edited by Benson Farb and David Fisher, by Robert J Zimmer
University of Chicago Press, 2011
Cloth: 978-0-226-23788-6 | Paper: 978-0-226-23789-3 | Electronic: 978-0-226-23790-9


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.


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.


“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, École Polytechnique Fédérale 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



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? / Dave Witte Morris

6. Some Remarks on Area-Preserving Actions on 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 CO Zimmer Program / Shmuel Weinberger

Part 2: Analytic, Ergodic, and Measurable Group Theory

9. Calculus on Nilpotent Lie Groups / Michael G. Cowling

10. A Survery of Measured Group Theory / Alex Furman

11. On Relative Property (T) / Alessandra Iozzi

12. Noncommutative Ergodic Theorems / Anders Karlsson and Francois 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, Francois 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 of Aut (Fn) Actions on Group Presentations and Representations / Alexander Lubotzky

List of Contributors