Geometry, Rigidity, and Group Actions chicago lectures in mathematics series Editors: Spencer J. Bloch, Peter Constantin, Benson Farb, Norman R. Lebovitz, Carlos Kenig, and J. P. May Other Chicago Lectures in Mathematics titles available from the University of Chicago Press Simplical Objects in Algebraic Topology, by J. Peter May (1967, 1993) Fields and Rings, Second Edition, by Irving Kaplansky (1969, 1972) Lie Algebras and Locally Compact Groups, by Irving Kaplansky (1971) Several Complex Variables, by Raghavan Narasimhan (1971) Torsion-Free Modules, by Eben Matlis (1973) Stable Homotopy and Generalised Homology, by J. F. Adams (1974) Rings with Involution, by I. N. Herstein (1976) Theory of Unitary Group Representation, by George V. Mackey (1976) Infinite-Dimensional Optimization and Convexity, by Ivar Ekeland and Thomas Turnbull (1983) Commutative Semigroup Rings, by Robert Gilmer (1984) Navier-Stokes Equations, by Peter Constantin and Ciprian Foias (1988) Essential Results of Functional Analysis, by Robert J. Zimmer (1990) Fuchsian Groups, by Svetlana Katok (1992) Topological Classification of Stratified Spaces, by Shmuel Weinberger (1994) Unstable Modules over the Steenrod Algebra and Sullivan’s Fixed Point Set Conjecture, by Lionel Schwartz (1994) Geometry of Nonpositively Curved Manifolds, by Patrick B. Eberlein (1996) Lectures on Exceptional Lie Groups, by J. F. Adams (1996) Dimension Theory in Dynamical Systems: Contemporary Views and Applications, by Yakov B. Pesin (1997) A Concise Course in Algebraic Topology, by J. P. May (1999) Harmonic Analysis and Partial Differential Equations: Essays in Honor of Alberto Calderon´ , edited by Michael Christ, Carlos Kenig, and Cora Sadosky (1999) Topics in Geometric Group Theory, by Pierre de la Harpe (2000) Exterior Differential Systems and Euler-Lagrange Partial Differential Equations, by Robert Bryant, Phillip Griffiths, and Daniel Grossman (2003) Ratner’s Theorems on Unipotent Flows, by Dave Witte Morris (2005) Geometry, Rigidity, and Group Actions Edited by Benson Farb and David Fisher the university of chicago press • chicago and london 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. The University of Chicago Press, Chicago 60637 The University of Chicago Press, Ltd., London © 2011 by The University of Chicago All rights reserved. Published 2011 Printed in the United States of America 20191817161514131211 12345 ISBN-13: 978-0-226-23788-6 (cloth) ISBN-10: 0-226-23788-5 (cloth) Library of Congress Cataloging-in-Publication Data Geometry, rigidity, and group actions/edited by Benson Farb and David Fisher. p. cm.—(Chicago lectures in mathematics series) Festschrift for Robert Zimmer on the occasion of his 60th birthday. ISBN-13: 978-0-226-23788-6 (cloth : alk. paper) ISBN-10: 0-226-23788-5 (cloth : alk. paper) 1. Rigidity (Geometry) 2. Group actions (Mathematics) 3. Mani- folds (Mathematics) I. Zimmer, Robert J., 1947– II. Farb, Benson. III. Fisher, David. IV. Series: Chicago lectures in mathematics series. QA613.G465 2011 516.11—dc22 2010026631 ∞ The paper used in this publication meets the minimum requirements of the American National Standard for Information Sciences—Permanence of Paper for Printed Library Materials, ANSI Z39.48-1992. To Bob, mentor and friend Contents Preface ix PART 1 || Group Actions on Manifolds 1. An Extension Criterion for Lattice Actions on the Circle 3 Marc Burger 2. Meromorphic Almost Rigid Geometric Structures 32 Sorin Dumitrescu 3. Harmonic Functions over Group Actions 59 Renato Feres and Emily Ronshausen 4. Groups Acting on Manifolds: Around the Zimmer Program 72 David Fisher 5. Can Lattices in SL (n, R) Act on the Circle? 158 Dave Witte Morris 6. Some Remarks on Area-Preserving Actions of Lattices 208 Pierre Py 7. Isometric Actions of Simple Groups and Transverse Structures: The Integrable Normal Case 229 Raul Quiroga-Barranco 8. Some Remarks Inspired by the C0 Zimmer Program 262 Shmuel Weinberger PART 2 || Analytic, Ergodic, and Measurable Group Theory 9. Calculus on Nilpotent Lie Groups 285 Michael G. Cowling viii / contents 10. A Survey of Measured Group Theory 296 Alex Furman 11. On Relative Property (T) 375 Alessandra Iozzi 12. Noncommutative Ergodic Theorems 396 Anders Karlsson and François Ledrappier 13. Cocycle and Orbit Superrigidity for Lattices in SL (n, R) Acting on Homogeneous Spaces 419 Sorin Popa and Stefaan Vaes PART 3 || Geometric Group Theory 14. Heights on SL2 and Free Subgroups 455 Emmanuel Breuillard 15. Displacing Representations and Orbit Maps 494 Thomas Delzant, Olivier Guichard, François Labourie, and Shahar Mozes 16. Problems on Automorphism Groups of Nonpositively Curved Polyhedral Complexes and Their Lattices 515 Benson Farb, Chris Hruska, and Anne Thomas 17. The Geometry of Twisted Conjugacy Classes in Wreath Products 561 Jennifer Taback and Peter Wong PART 4 || Group Actions on Representations Varieties 18. Ergodicity of Mapping Class Group Actions on SU(2)-Character Varieties 591 William M. Goldman and Eugene Z. Xia 19. Dynamics of Aut (Fn) Actions on Group Presentations and Representations 609 Alexander Lubotzky List of Contributors 645 Preface In September 2007, a conference was held at the University of Chicago in honor of the sixtieth birthday of Robert J. Zimmer. The conference was a testament to and a celebration of Zimmer’s continuing and lasting influence on the fields of geometry, rigidity, and group actions. The wide variety of pa- pers submitted to this volume are just one indication of that influence. This volume contains both survey papers and research papers. The com- mon theme throughout concerns group actions. The general setup is that we have a group G, which is either a Lie group or a discrete group, acting on some space X. The examples are quite varied, and include, in (informal) order of decreasing “rigidity”: 1. algebraic actions on varieties 2. isometric actions on metric spaces 3. smooth actions on manifolds 4. measurable actions on measure spaces In each instance the goal is to relate algebraic/combinatorial properties of G to geometric/topological/measure-theoretic properties of X and the G action on X. In many cases, properties of G impose surprising restrictions on the possible choices of X and the possible actions on X. The papers by Goldman-Xia and Lubotzky concentrate on (1). The varieties here are representation and character varieties of certain finitely generated groups , with the automorphism group G = Aut() acting by algebraic auto- morphisms. This setup can be viewed as a dynamical system, where one can study its dynamical properties (as in Goldman-Xia’s paper), or as an algebraic system (as in Lubotzky’s paper), where information about G can be gleaned from properties of the action. A number of the papers in this volume are concerned with (2). The paper by Farb-Hruska-Thomas considers the case where X is a nonpositively curved x / preface simplicial complex and the group G is a lattice in the simplicial automorphism group of X; here the goal is to extend the classical theory of lattices in alge- braic groups over fields with a discrete valuation to this more general context. Breuillard’s paper provides a strengthening of the famous Tits alternative, and gives further applications to spectral gaps, group growth, and more. Here the G is an arbitrary, finitely generated subgroup of GL(2, C), and X is a projec- tive space over various fields. The paper by Delzant-Guichard-Labourie-Mozes considers the general setup of groups acting isometrically on metric spaces. They relate the notion of a “displacing action” to the geometric property of an orbit being quasi-isometrically embedded. This setup is widely applicable, al- though the case of S-arithmetic groups is emphasized here. Iozzi considers isometric actions of semidirect products G of groups H with (normal) abelian groups A, acting unitarily on a Hilbert space X. The main goal is to give neces- sary and sufficient conditions for the pair (G, A) to have relative property (T). Taback-Wong use the large-scale geometry of wreath products to study twisted conjugacy classes in these groups. Cowling’s paper develops some calculus on nilpotent groups with a view toward generalizing Mostow’s rigidity theorem concerning groups acting on symmetric spaces. The general study of smooth (usually volume-preserving) actions of (non- compact) Lie groups and their lattices on smooth manifolds goes by the name of the Zimmer program. The paper by Fisher gives a broad survey of this area, while the paper by Morris surveys the already rich case of actions on the cir- cle. Burger’s paper also concerns actions of various large groups on the circle and provides a unified approach to many of the major results in this area. The Feres-Ronshausen paper studies general properties of groups acting on manifolds without an invariant volume by studying a certain type of harmonic function related to these actions. The primary applications in this paper also concern groups of diffeomorphisms of the circle. Moving up one dimen- sion, Py’s paper provides a brief introduction to an interesting approach to studying volume-preserving diffeomorphisms of compact surfaces using quasi-morphisms. A major theme in the Zimmer program is the study of group actions that preserve geometric structures, a topic represented here in the pa- pers by Dumitrescu and Quiroga-Barranco. While Zimmer’s