Fundamental Groups for Ergodic Actions and Actions with Unit Fundamental Groups

Abstract

Fundamental groups of ergodic dynamical systems with invariant measure are considered. For a given countable subgroup $\Lambda \subset {\mathbf{R}}_{+}^{\ast }$, an ergodic dynamical system is constructed whose fundamental group is countable and contains $\Lambda$. G.A. Margulis and D. Sullivan revealed that the orthogonal group contains a dense countable subgroup with the proprety T. The fact is applied to construct ergodic actions with a unit fundamental group. The group of outer automorphisms of these actions has also been completely calculated. The proofs are based on the results of studies of the properties of action centralizers.

In the Supplement we introduce the concept of a fundamental group for ergodic actions of continuous groups. It will be shown by means of results of the R. Zimmer's rigidity theory that any finite measure-preserving action of the lattice of the simple Lie group, whose real rank is not less 2, has a unit fundamental group.