Geometry, Rigidity and Group Actions (Chicago Lectures in Mathematics Series CLM) - Hardcover

Farb, Benson

Hardcover

ISBN 10: 0226237885 ISBN 13: 9780226237886

Publisher: University of Chicago Press, 2011

Synopsis

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, Gregory 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.

About this title

About the Author

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.

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Geometry, Rigidity, and Group Actions

THE UNIVERSITY OF CHICAGO PRESS

Copyright © 2011 The University of Chicago
All right reserved.
ISBN: 978-0-226-23788-6

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

AN EXTENSION CRITERION FOR LATTICE ACTIONS ON THE CIRCLE TO BOB ZIMMER

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 to the quotient by this cyclic group.

Our main result is:

THEOREM 1.2. Let G < G be a lattice in a locally compact, second countable group G and

? : G -> Homeo⁺ (S¹)

be a minimal unbounded action. Then the following are equivalent:

1) The real bounded Euler class [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] of ? is in the image of the restriction map

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

2) The strongly proximal factor ?sp of ? extends continuously to G.

The main ingredient in the proof of Theorem 1.2 is a result that, for any countable group G, characterizes the bounded classes in H²_b (G,R) obtained from minimal strongly proximal actions in terms of certain cocycles defined on an appropriate Poisson boundary of G; see Theorem 4.5 in Section 4, where the result is proven in the more general context of locally compact, second countable groups. This leads to results of independent interest concerning the extent to which an action is determined by its real bounded Euler class and to the question of the possible values of its norm. These results are summarized in the following theorem and corollaries.

THEOREM 1.3. Let G be a countable group.

1) For a homomorphism ? : G ? Homeo⁺ (S¹) we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

with equality if and only if ? is quasiconjugated to a minimal strongly proximal action.

2) If two minimal strongly proximal actions ?_{1, ?2 are not conjugated, then}

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

REMARK 1.4.

1) The first assertion in Theorem 1.3 echoes results obtained in concerning tight homomorphisms with values in a Lie group of Hermitian type.

2) Let ESP(G) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII](G,R) denote the subset consisting of the real bounded Euler classes of minimal strongly proximal G-actions and Z[ESP(G)] its Z-span. We will show (see Section 5) that the norm takes half-integral values on Z[EPS(G)]. In this context the following question arises, namely if Z[SP(G)] denotes the free abelian group on the set of conjugacy classes of minimal strongly proximal actions, can one determine the kernel of the homomorphism

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

and what is its significance for the dynamics of G-actions on S¹?

The following immediate corollary is another instance of the general principle that groups whose second bounded cohomology is finite-dimensional exhibit rigidity phenomena; compare, for example, with the case of actions by isometries on Hermitian symmetric spaces (see [4], [6]).

COROLLARY 1.5. Let G be a countable group and assume that H²_b (G,R) is finite-dimensional. Then there are, up to conjugation, only finitely many minimal strongly proximal G-actions on S¹. Together with the information concerning centralizers of minimal actions we deduce from Theorem 1.3,

COROLLARY 1.6.

1) For a homomorphism ? : G ? Homeo⁺ (S¹) we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

and this value equals (2k) ^-1 if and only if ? is quasiconjugated to a minimal unbounded action whose centralizer is of order k.

2) For minimal unbounded actions ?¹, ?² we have [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] if and only if, up to conjugation, [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where h is a homomorphism with values in the centralizer of ?¹(G).

The extension criterion in Theorem 1.2 leads to rigidity theorems when combined with the following two ingredients, namely vanishing theorems in bounded cohomology and the description of continuous homomorphisms from a locally compact group into Homeo⁺ (S¹).

Concerning the first ingredient, we know that the restriction map [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is a isomorphism in the following cases:

1) Products (see [8], [21]):G = G₁ × ··· × G_n is a Cartesian product of locally compact second countable groups and G has dense projection in every factor G_i.

2) Higher-rank Lie groups (see [8]): G = G(k) where G is a connected almost simple k-group, k is a local field and rank_{k G = 2.}

Since it is elementary to classify continuous homomorphisms from a semi-simple Lie group over a local field into Homeo⁺ (S¹) one obtains by combining (1) and (2),

COROLLARY 1.7. ([15], [27])

Let [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] be an irreducible lattice where ka are local fields, G_alpha] is a connected, simply connected, almost simple k_alpha]-group of positive rank. Assume that the sum of the k_alpha]-ranks of G_a is at least 2.

For a homomorphism ? : G ? Homeo⁺ (S¹) one of the following holds up to quasiconjugation:

1) ? has a finite orbit.

2) ? is minimal unbounded and its strongly proximal factor ?sp extends continuously to G factoring via the projection on a factor of the form G_alpha](k_a) = SL(2,R).

Concerning locally compact subgroups of Homeo⁺ (S¹), one has, owing to the solution of Hilbert's fifth problem, a wealth of information and in particular, those that are connected and minimal have a simple classification; they are up to conjugation, either Rot the subgroup of rotations or PSL(2,R)_k, the k-fold cyclic covering of PSL(2,R) (see [10], [16]). When studying continuous homomorphisms from a locally compact group G into Homeo⁺ (S¹) one has to deal with the fact that the image is not necessarily closed. In any case we have,

THEOREM 1.8. Let G be a locally compact group and p : G ? Homeo⁺ (S¹) a continuous and minimal action. Then one of the following holds:

1) p is conjugated into the group Rot of rotations and has a dense image in it. 2) p surjects onto PSL(2,R)_{k for some k = 1, up to conjugation. 3) Ker p is an open subgroup of G.}

From Theorem 1.8 and the vanishing result for products mentioned above we obtain,

COROLLARY 1.9. ([26], [2])

Let G = G₁ × ··· × G_n be a product of locally compact second countable groups and G < G a lattice with dense projections on each factor G_i. Assume that ? : G ? Homeo⁺ (S¹) is minimal unbounded. Then the strongly proximal quotient ?_sp extends continuously to G,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

and we have one of the following:

1) Ker(?_sp)^ext is open in G.

2) Up to conjugation (?_sp)^ext factors via a projection onto some factor G _i followed by a continuous surjection onto PSL(2,R) [subset] Homeo⁺ (S¹).

Finally, as shown by Bader, Furman, and Shaker, there is also in the context of actions on the circle a commensurator superrigidity theorem, which we state in a way that is somewhat different, but equivalent to their thm. C in [2].

THEOREM 1.10.

Let G be a locally compact second countable group, G < G a lattice, and ? < G a subgroup such that G [subset] ? [subset] Comm_G G and ? is dense in G. Let ? : ? ? Homeo⁺ (S¹) be a homomorphism such that ?(_) is minimal and unbounded. Then the strongly proximal quotient ?_sp extends continuously to G,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

and we have one of the following:

1) Ker(?_sp)^ext is open of infinite index in G.

2) (?_sp)^ext surjects onto PSL(2,R) [subset] Homeo⁺ (S¹).

Let us make the following comment about the hypothesis of Theorem 1.10. If ? : ? ? Homeo⁺ (S¹) is a homomorphism such that ?(G) is unbounded, then either ?(G) is minimal or there is an exceptional minimal set K [subset] S¹ (see Section 3) that is easily seen to be ?(?)-invariant. Thus ? is quasiconjugated to an action of ? for which G is minimal and unbounded.

The above result follows easily from the extension criterion (Theorem 1.2) and the following general fact concerning bounded cohomology:

THEOREM 1.11. ([20])

Let G < G and G < ? < Comm_G G be as in Theorem 1.10, in particular ? is dense in G. Then the image of the restriction map

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

coincides with the image of [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

We will leave to the interested reader the exercise of deducing Theorem 1.10 from Theorem 1.11 and refer to [20] for elementary proofs of the isomoranphism results for products and higher-rank groups mentioned above, as well as the proof of Theorem 1.11.

ACKNOWLEDGMENTS: Thanks to Luis Hernandez for his kind invitation to CIMAT where this work was completed. Thanks to Uri Bader and Alex Furman for sharing with me their result on boundary maps.

2. Boundary Maps

In this section we present a general existence and uniqueness result concerning measurable equivariant maps that is due to Bader-Furman and is of general interest in rigidity theory.

Here and in the sequel G is a second countable locally compact group, G × M ? M is a continuous action on a compact metrisable space M, µ [element of] M¹ (G) is a spread-out probability measure on G, and (B, v_B) is a standard Lebesgue G-space such that ?B is µ-stationary.

THEOREM 2.1.

Assume that

1) the G-action is minimal and strongly proximal.

2) For every sequence (g_n)n=1 in G there exists a subsequence (n_n)k=1 such that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

converges pointwise.

Then every measurable G-equivariant map

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

takes values in the subset of Dirac measures.

REMARK 2.2. When G is discrete countable and M = S¹, a result essentially equivalent to Theorem 2.1 has been obtained by Deroin, Kleptsyn, and Navas (see [9]).

We recall here that a G-action is strongly proximal if for every µ [element of] M¹ (M) the closure [bar Gµ] [subset] M¹ (M) contains a Dirac mass.

Proof. The main step consists in showing that the G-space M is µ-proximal in the sense of (2.6), def. 3 in VI: this means that for every µ-stationary measure v [element of] M¹ (M) the map

2.1 [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

with values in M¹(M), defined almost everywhere on the infinite product [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], takes values in the set of Dirac measures; this we proceed to show now. The existence of the limit (2.1) follows from the martingale convergence theorem (prop. 2.4 in [22] VI); in addition one has the following remarkable fact ([13], [17] and lemma 1.33 in [11]):

2.2 [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

(Continues...)

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Edition details

• Publisher: University of Chicago Press
• Publication date: 2011
• Language: English
• ISBN 10: 0226237885
• ISBN 13: 9780226237886
• Binding: Hardcover
• Number of pages: 552