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Geometry, Rigidity, and Group Actions

THE UNIVERSITY OF CHICAGO PRESS

Contents

Preface.............................................................................................................................................................ix1. An Extension Criterion for Lattice Actions on the Circle Marc Burger............................................................................................32. Meromorphic Almost Rigid Geometric Structures Sorin Dumitrescu..................................................................................................323. Harmonic Functions over Group Actions Renato Feres and Emily Ronshausen.........................................................................................594. Groups Acting on Manifolds: Around the Zimmer Program David Fisher..............................................................................................725. Can Lattices in SL (n, R) Act on the Circle? Dave Witte Morris..................................................................................................1586. Some Remarks on Area-Preserving Actions of Lattices Pierre Py...................................................................................................2087. Isometric Actions of Simple Groups and Transverse Structures: The Integrable Normal Case Raul Quiroga-Barranco..................................................2298. Some Remarks Inspired by the C0 Zimmer Program Shmuel Weinberger................................................................................................2629. Calculus on Nilpotent Lie Groups Michael G. Cowling.............................................................................................................28510. A Survey of Measured Group Theory Alex Furman..................................................................................................................29611. On Relative Property (T) Alessandra Iozzi......................................................................................................................37512. Noncommutative Ergodic Theorems Anders Karlsson and François Ledrappier...................................................................................39613. Cocycle and Orbit Superrigidity for Lattices in SL (n, R) Acting on Homogeneous Spaces Sorin Popa and Stefaan Vaes.............................................41914. Heights on SL2 and Free Subgroups Emmanuel Breuillard...............................................................................................45515. Displacing Representations and Orbit Maps Thomas Delzant, Olivier Guichard, François Labourie, and Shahar Mozes...........................................49416. Problems on Automorphism Groups of Nonpositively Curved Polyhedral Complexes and Their Lattices Benson Farb, Chris Hruska, and Anne Thomas.....................51517. The Geometry of Twisted Conjugacy Classes in Wreath Products Jennifer Taback and Peter Wong....................................................................56118. Ergodicity of Mapping Class Group Actions on SU(2)-Character Varieties William M. Goldman and Eugene Z. Xia....................................................59119. Dynamics of Aut (Fn) Actions on Group Presentations and Representations Alexander Lubotzky..........................................................609List of Contributors................................................................................................................................................645

Chapter One

1. Introduction

Let G < G be a lattice in a locally compact second countable group G. The aim of this paper is to establish a necessary and sufficient condition for a G-action by homeomorphisms of the circle to extend continuously to G. This condition will be expressed in terms of the real bounded Euler class of this action. Combined with classical vanishing theorems in bounded cohomology, one recovers rigidity results of Ghys, Witte-Zimmer, Navas, and Bader-Furman-Shaker in a unified manner. For a survey of various approaches to the problem of classifying lattice actions on the circle we refer to the paper by David Witte Morris in this volume.

Let Homeo⁺ (S¹) be the group of orientation-preserving homeomorphisms of the circle and e [element of] H² (Homeo⁺ (S¹), Z) the Euler class; recall that e corresponds to the central extension defined by the universal covering of Homeo⁺ (S¹). The Euler class admits a representing cocycle that is bounded and this defines a bounded class [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]) called the bounded Euler class. The relevance of bounded cohomology to the study of group actions on the circle comes from a result of Ghys, namely that the bounded Euler class [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] of an action ?: G -> Homeo⁺ (S¹) determines ? up to quasiconjugation; a quasiconjugation is a self-map of the circle that is weakly cyclic order preserving and in particular not necessarily continuous; see Section 3 for details. If e^b_R denotes then the bounded class obtained by considering the bounded cocycle defining e^b as real valued, we call the invariant [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] the real bounded Euler class of ?. From this point of view we have the following dichotomy (see Proposition 3.2):

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]: in this case, ? is quasiconjugated to an action of G by rotations; as far as the extension problem is concerned, it reduces to the properties of the restriction map

Hom_c (G,R/Z) -> Hom(G, R/Z). [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] in this case, ? is quasiconjugated to a minimal unbounded action; that is, every orbit is dense and the group of homeomorphisms ?(G) is not equicontinuous.

In the first case (E) we call ? elementary and in the second (NE) nonelementary; nonelementary actions are our main object of study in this paper.

Concerning the extension problem, an issue that has to be taken care of is the existence of a nontrivial centralizer of the action under consideration. This is illustrated by the following:

EXAMPLE 1.1. Let G < PSL(2,R) be a lattice that is nonuniform and torsion free. Since G is a free group we can lift the identity to a homomorphism ?_k: G -> PSL(2,R) _k [subset] Homeo⁺ (S¹) into the k-fold cyclic covering of PSL(2,R), and this for every k = 1. In this way we get an action that is minimal, unbounded, but for k = 2 does not extend continuously to PSL(2,R). If G is torsion-free cocompact, this construction applies provided k divides the Euler characteristic of G, which is always the case for k = 2.

Thus given a minimal unbounded action one is led to consider its topological S¹-factors; those are easily classified and in particular there is, up to conjugation, a unique factor

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

that is strongly proximal (Proposition 3.7).

This relies on arguments of Ghys that establish that the centralizer of ?(G) is a finite cyclic group; the strongly proximal quotient is then obtained by passing...