X-Sender: dwitte@littlewood.math.okstate.edu Mime-Version: 1.0 Date: Wed, 4 Feb 1998 03:32:40 -0600 To: adams@math.umn.edu From: Dave Witte Subject: descr.html Content-Type: text/plain; charset="us-ascii" Content-Length: 15116 Ergodic Theory, Groups, and Geometry Conference Description

NSF/CBMS Regional Conference in the Mathematical Sciences

Ergodic Theory, Groups, and Geometry

University of Minnesota, Minneapolis

June 22 ­ 26, 1998

Conference Description

1. Principal lecturer: Robert J. Zimmer (U of Chicago)

Professor Zimmer is a distinguished mathematician who has made tremendous contributions to the study of group actions on manifolds by ergodic-theoretic methods, and to related areas of research. He has given numerous invited lectures both domestically and internationally, and is also experienced and effective in written exposition [14], [15], [16], [18], [20].

2. Host institution: University of Minnesota

The University of Minnesota has suitable housing for conference attendees, and the immediate vicinity offers a variety of interesting and inexpensive places to eat. The campus also has appropriate lecture halls and lounges for the scheduled activities of the conference. Furthermore, computer facilities and the mathematics library will be available for the participants' use.

The conference organizers, Scot Adams and Dave Witte, both have active research programs in areas related to the topic of the conference. The university's Department of Professional Development and Conference Services will assist with publicity and local arrangements, and is providing other guidance.

Minneapolis has a major airport, from which the campus is easily accessible by taxi. The weather is pleasant in late June, and numerous recreational activities and tourist attractions are available in the Twin Cities area for interested participants.

3. Participants

The topic and principal lecturer of the conference (and the setting) can be expected to appeal to numerous mathematicians in the midwest region and beyond. This is a very active field of research, with direct connections to ergodic theory, differential geometry, number theory, and group representation theory. The organizers will make a particular effort to identify and attract appropriate beginning researchers and members of underrepresented groups.

4. Subject of the conference

The aim of the lectures will be to discuss the theory of simple Lie transformation groups, particularly the ergodic theoretic structure of such actions and applications to differential geometry. More precisely, let G be a non-compact simple Lie group. We will discuss the recent progress in the general program of understanding the structure of the actions of G and the geometric, topological, and analytic properties of the spaces on which it acts.

The study of a general action of a Lie group G is greatly facilitated by the study of actions on vector bundles and principal bundles. For example, if G acts smoothly on a smooth manifold, it acts on the tangent bundle and the frame bundle, as well as higher order versions of these bundles, namely various jet bundles and higher order frame bundles. In addition, it acts on all of the associated bundles to these principal bundles. If there is a geometric structure on the manifold that is preserved, this often leads to other principal or vector bundles on which the group acts. The study of the action on these bundles is a natural and powerful tool in the study of smooth transformation groups.

In another direction, the (universal covering of the) group will also act on the universal covering of the manifold. This of course holds in the context of much more general spaces (i.e. spaces which are not necessarily manifolds). Thus, associated to each representation of the fundamental group over any local field, there is a corresponding action of the group on the associated vector and principal bundles. The analysis of the G action on these bundles contributes to the understanding of the representation theory of the fundamental group, and hence to the group itself and various geometric properties of the space related to the fundamental group.

Another useful approach to understanding group actions, dual to the first, is understanding the realization of these spaces themselves as bundles over simpler spaces. In this way one can hope to obtain an inductive understanding of the original action.

These lectures will discuss systematic approaches to these issues for actions of simple Lie groups, and consequent results about the structure of actions and the geometry and topology of the underlying spaces. There will be four central themes, which we discuss below:

  1. Geometric Borel density theorem, generalizations, and geometric consequences.
  2. Superrigidity and geometric consequences.
  3. Arithmetic structure of actions and fundamental groups.
  4. Homogeneous projective quotients and stationary measures.

The Borel density theorem was originally a result about the Zariski density of certain discrete subgroups of semisimple Lie groups. From the perspective of transformation groups, it can be succinctly stated as asserting that every finite invariant measure for an algebraic action of a noncompact simple real algebraic group on an algebraic variety is supported on fixed points. For most smooth actions on manifolds, there is of course no algebraic structure for which the action will be algebraic. Nevertheless, by systematically using the dynamics of an action on a compact manifold, for example a finite invariant measure, and contrasting this with the algebraic nature of the fibers in the natural bundles over a manifold, one can utilize Borel's theorem in this form to prove what we call the geometric Borel density theorem, which can yield basic geometric information. For example, this was used in [17] to study the possible geometric structures on a manifold invariant under a non-compact simple Lie group. One consequence is that a non-compact simple Lie automorphism group of a compact Lorentz manifold must be locally isomorphic to SL(2,R). There have been some basic further developments in this direction since [17] appeared. The dynamics of a finite invariant measure were replaced by a weaker condition in [5], where automorphisms of non-compact Lorentz manifolds are successfully analyzed. In [1], a complete classification of those groups which can be the full connected automorphism group of a compact Lorentz manifold was obtained.

We shall also discuss in this context the question of the stabilizers of points in actions preserving a finite measure. Once again, the first results in [17], asserting essential local freeness of all (non-trivial) finite measure preserving actions, follow from arguments involving the Borel density theorem. Two significant developments regarding the almost everywhere triviality of stabilizers since then are the essential freeness in higher rank [11], and freeness results in the case of connection preserving volume preserving actions [12].

The work of Gromov [3] uses the geometric Borel density theorem to analyze the actions on bundles of higher order for actions preserving a rigid geometric structure. This leads to the construction of the Gromov representation of the fundamental group of a rigid geometric manifold which admits a simple noncompact automorphism group, which is a key ingredient in the analysis of this situation. Gromov's work also leads to an understanding of the local geometry of the structure.

When the group G is not only simple but of higher rank, we discuss the superrigidity theorem for actions on principal bundles. This was first proven in [13], generalizing Margulis' theorem for linear representations of discrete subgroups. Superrigidity for actions on bundles gives a complete description at the measure theoretic level of the structure of a G-action on a bundle with algebraic group as structure group. This has basic implications for many issues. For example, we discuss applications to the classification of low dimensional actions of discrete groups [19]. In addition, we discuss applications to the classical question in differential geometry of determining which homogeneous spaces admit a compact form. This question has been attacked by a variety of techniques, and superrigidity has provided the basis for showing non-existence in a variety of important and natural test cases, such as SL(n,R)/SL(m,R) for suitable n and m [6].

The nature of the topology of a manifold supporting an action of a given group is a basic one in the theory of transformation groups. For a higher rank Lie group G, we discuss the question of the arithmetic structure of the fundamental group of the space. The basic results are contained in the recent work [7], where arithmeticity is established under very general conditions. Basic ergodic theoretic ingredients in this work are superrigidity and Ratner's work on invariant measures [10]; in addition, there is significant arithmetic argument required. The work of [7] represents a significant extension of Margulis' work on arithmeticity [8]. In this context, however, there are examples of actions not satisfying the hypotheses in [7]. We discuss these as well. They appear and are analyzed in [2], and are an outgrowth of earlier constructions in [4]. We also discuss the relationship of [7] to rigid geometric structures and the Gromov representation.

Most of the above requires the existence of a finite invariant measure. However, this is a significant assumption, and the recent work of [9] begins a systematic approach to the non-finite invariant measure case through the study of stationary measures (which always exist on a compact space.) The main results deal with the question of when a general action can be expressed as one induced from a finite measure preserving action of a parabolic subgroup. This is established under very general conditions in [9] for actions of higher rank G. We also discuss counterexamples in the case of rank one groups.

The realization as an action induced from a parabolic implies that the action can be realized as a bundle over a homogeneous projective variety. Results of [7] on arithmeticity can be used to show that in the volume preserving case, one can under very general assumptions realize actions as bundles over arithmetic manifolds.

5. Outline of lectures

References cited