Topology of Manifolds with Higher Rank Lattice Actions
by
David Fisher
University of Chicago

Two groups A and B are quasi-isometric if there are commuting properly discontinuous, cocompact actions of A and B on a metric space M. This gives an action of A on the compact space M/B. If M is a simply connected manifold, it follows that A acting in this particular way on N = M/B is equivalent to \pi₁(N) being quasi-isometric to A.

For lattices in semisimple Lie groups, there is a complete quasi-isometry classification. This can be interpreted as restricting the possible actions of lattices in Lie groups on compact manifolds via the description of quasi-isometries given above. For example, any group quasi-isometric to SL₃(Z) is virtually a conjugate of it in SL₃(R).

My research involves placing different conditions on actions of lattices in higher rank Lie groups in order to obtain information about the action and the fundamental group of the manifold acted upon.

A volume preserving action is said to be ergodic if all invariant sets have full or null measure. An ergodic action is said to be engaging if any lift of the action to a finite cover is ergodic. Given any lattice Gamma in a higher rank semisimple Lie group G, my work shows that if Gamma has an engaging action on a compact manifold M, then \pi₍M) is an arithmetically constructed subgroup of an algebraic group, H. Furthermore, there are relations between G and H, that allow us to give a description of the action of Gamma, at least on a set of full measure.

Date received: February 11, 1999

Copyright © 1999 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # caca-36.