# Geometry, Rigidity, and Group Actions

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.

*Preface*

PART 1 ||

*Group Actions on Manifolds*

*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*

**Nicolas Monod, École Polytechnique Fédérale de Lausanne**

*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.”

**Physical Sciences: **
Experimental and Applied Physics

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