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Group Actions in Ergodic Theory, Geometry, and Topology

Selected Papers

Robert J. Zimmer is best known in mathematics for the highly influential conjectures and program that bear his name. Group Actions in Ergodic Theory, Geometry, and Topology: Selected Papers brings together some of the most significant writings by Zimmer, which lay out his program and contextualize his work over the course of his career. Zimmer’s body of work is remarkable in that it involves methods from a variety of mathematical disciplines, such as Lie theory, differential geometry, ergodic theory and dynamical systems, arithmetic groups, and topology, and at the same time offers a unifying perspective. After arriving at the University of Chicago in 1977, Zimmer extended his earlier research on ergodic group actions to prove his cocycle superrigidity theorem which proved to be a pivotal point in articulating and developing his program.  Zimmer’s ideas opened the door to many others, and they continue to be actively employed in many domains related to group actions in ergodic theory, geometry, and topology.

In addition to the selected papers themselves, this volume opens with a foreword by David Fisher, Alexander Lubotzky, and Gregory Margulis, as well as a substantial introductory essay by Zimmer recounting the course of his career in mathematics. The volume closes with an afterword by Fisher on the most recent developments around the Zimmer program.

672 pages | 7 x 10 | © 2019

Mathematics and Statistics


“Zimmer is one of the most influential contemporary American mathematicians. The corpus of Zimmer's contributions stands out by its coherence and its grand vision. Much more than being a strong problem-solver, more even than being a theory-builder, Zimmer is a mathematician with an overarching sense of the destination, with a domineering command of all relevant available techniques, with the inspiration to bring to life those techniques that were not available yet, with, in conclusion, and to use a trivial formulation, a program.”

Nicolas Monod, École Polytechnique Fédérale de Lausanne

“The classic works of Furstenberg, Mostow, and especially Margulis in the ’60s and ’70s led to a satisfactory understanding of the linear actions of lattices in high rank simple Lie groups. In the ’80s, Zimmer  conceived a profoundly ambitious plan for understanding non-linear actions, known today as the Zimmer program. The present book provides the blue prints of the plan whose resultant architecture has inspired an entire generation of mathematicians working on Lie groups, differential geometry, ergodic theory, dynamics, arithmetic groups and more. This publication comes out at a perfect time when a number of his disciples have recently succeeded in proving some of his primary conjectures.”

Alexander Lubotzky, Einstein Institute of Mathematics, Hebrew University

“This selection of Zimmer’s mathematical papers allows the reader to follow the natural evolution of his far-reaching results and ideas on groups actions in geometry and topology. The final two chapters (written by David Fisher) on the ‘Zimmer Program’ and its recent progress, highlight the broad and profound impact that Zimmer’s works continue to enjoy.”

Peter Sarnak, Institute for Advanced Study

Table of Contents

Foreword, by David Fisher, Alexander Lubotzky, and Gregory Margulis

Introduction, by Robert J. Zimmer

1        Spectra and Structure of Ergodic Actions

A. Extensions of Ergodic Group Actions, Illinois Journal of Mathematics (1976)
B. Ergodic Actions with Generalized Discrete Spectrum, Illinois Journal of Mathematics (1976)
C. Orbit Spaces of Unitary Representations, Ergodic Theory, and Simple Lie Groups, Annals of Mathematics (1977)

2        Amenable Actions, Equivalence Relations, and Foliations

A. Amenable Ergodic Group Actions and an Application to Poisson Boundaries of Random Walks, Journal of Functional Analysis (1978)
B. Induced and Amenable Ergodic Actions of Lie Groups, Annales scientifiques de l’École normale supérieure (1978)
C. Hyperfinite Factors and Amenable Ergodic Actions, Inventiones mathematicae (1977)
D. Curvature of Leaves in Amenable Foliations, American Journal of Mathematics (1983)
E. Amenable Actions and Dense Subgroups of Lie Groups, Journal of Functional Analysis (1987)

3        Orbit Equivalence and Strong Rigidity

A. Strong Rigidity for Ergodic Actions of Semisimple Lie Groups, Annals of Mathematics (1980)
B. Orbit Equivalence and Rigidity of Ergodic Actions of Lie Groups, Ergodic Theory and Dynamical Systems (1981)
C. Ergodic Actions of Semisimple Groups and Product Relations, Annals of Mathematics (1983)

4        Cocycle Superrigidity and the Program to Describe Lie Group and Lattice Actions on Manifolds

A. Volume Preserving Actions of Lattices in Semisimple Groups on Compact Manifolds, Institut des Hautes Études Scientifiques Publications Mathématiques (1984)
B. Kazhdan Groups Acting on Compact Manifolds, Inventiones mathematicae (1984)
C. Actions of Lattices in Semisimple Groups Preserving a G-Structure of Finite Type, Ergodic Theory and Dynamical Systems (1985)
D. Actions of Semisimple Groups and Discrete Subgroups, Proceedings of the International Congress of Mathematicians, August 3–11, 1986 (1987)
E. Split Rank and Semisimple Automorphism Groups of G-Structures, Journal of Differential Geometry (1987)
F. Manifolds with Infinitely Many Actions of an Arithmetic Group (with Richard K. Lashof), Illinois Journal of Mathematics (1990)
G. Spectrum, Entropy, and Geometric Structures for Smooth Actions of Kazhdan Groups, Israel Journal of Mathematics (1991)
H. Cocycle Superrigidity and Rigidity for Lattice Actions on Tori (with Anatole Katok and James Lewis), Topology (1996)

I. Volume-Preserving Actions of Simple Algebraic Q-Groups on Low-Dimensional Manifolds (with Dave Witte Morris), Journal of Topology and Analysis (2012)

5        Stabilizers of Semisimple Lie Group Actions: Invariant Random Subgroups

A. Stabilizers for Ergodic Actions of Higher Rank Semisimple Groups (with Garrett Stuck), Annals of Mathematics (1994)

6        Representations and Arithmetic Properties of Actions, Fundamental Groups, and Foliations

A. Arithmeticity of Holonomy Groups of Lie Foliations, Journal of the American Mathematical Society (1988)
B. Representations of Fundamental Groups of Manifolds with a Semisimple Transformation Group, Journal of the American Mathematical Society (1989)
C. Superrigidity, Ratner’s Theorem, and Fundamental Groups, Israel Journal of Mathematics (1991)
D. Fundamental Groups of Negatively Curved Manifolds and Actions of Semisimple Groups (with Ralf J. Spatzier), Topology (1991)
E. A Canonical Arithmetic Quotient for Simple Lie Group Actions (with Alexander Lubotzky), Proceedings of the International Colloquium on Lie Groups and Ergodic Theory: Mumbai, 1996 (1998)
F. Arithmetic Structure of Fundamental Groups and Actions of Semisimple Lie Groups (with Alexander Lubotzky), Topology (2001)
G. Entropy and Arithmetic Quotients for Simple Automorphism Groups of Geometric Manifolds, Geometriae Dedicata (2004)
H. Geometric Lattice Actions, Entropy and Fundamental Groups (with David Fisher), Commentarii Mathematici Helvetici (2002)

7        Geometric Structures: Automorphisms of Geometric Manifolds and Rigid Structures; Locally Homogeneous Manifolds

A. On the Automorphism Group of a Compact Lorentz Manifold and Other Geometric Manifolds, Inventiones mathematicae (1986)
B. Semisimple Automorphism Groups of G-Structures, Journal of Differential Geometry (1984)
C. Automorphism Groups and Fundamental Groups of Geometric Manifolds, Proceedings of Symposia in Pure Mathematics, vol. 54, pt. 3, Differential Geometry: Riemannian Geometry (1993)
D. Discrete Groups and Non-Riemannian Homogeneous Spaces, Journal of the American Mathematical Society (1994)
E. On Manifolds Locally Modelled on Non-Riemannian Homogeneous Spaces (with François Labourie and Shahar Mozes), Geometric and Functional Analysis (1995)
F. On the Non-existence of Cocompact Lattices for SL(n)/SL(m) (with François Labourie), Mathematical Research Letters (1995)

8        Stationary Measures and Structure Theorems for Lie Group Actions

A. A Structure Theorem for Actions of Semisimple Lie Groups (with Amos Nevo), Annals of Mathematics (2002)
B. Entropy of Stationary Measures and Bounded Tangential de-Rham Cohomology of Semisimple Lie Group Actions (with Amos Nevo), Geometriae Dedicata (2005)
C. Invariant Rigid Geometric Structures and Smooth Projective Factors (with Amos Nevo), Geometric and Functional Analysis (2009)

Group Actions on Manifolds: Around the Zimmer Program, by David Fisher

Afterword: Recent Progress in the Zimmer Program, by David Fisher


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