Generic phenomena in groups: some answers and many questions

Summary

We give a survey of some known results and of the many open questions in the study of generic phenomena in geometrically interesting groups.

In this paper we will discuss a number of loosely related questions, which emanate from Thurston’s geometrization program for three-dimensional manifolds, and the general Thurston “yoga” that most everything is hyperbolic. We venture quite far afield from three-dimensional geometry and topology—to the geometry of higher-rank symmetric spaces, to number theory, and probability theory, and to the theory of finite groups. In Section 2 we describe the underpinnings from the theory of three-dimensional manifolds as envisaged by W. Thurston. In Section 3 we will describe one natural approach to describing randomness in groups. In Section 5 we describe an approach to actually producing random matrices in lattices in semisimple Lie groups using the philosophy in Section 3. In Section 6 we describe a different approach to randomness, and the questions it raises.