# Publications of the Research Institute for Mathematical Sciences

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**Volume 24, Issue 6, 1988, pp. 821–847**

**DOI: 10.2977/prims/1195173929**

Fundamental Groups for Ergodic Actions and Actions with Unit Fundamental Groups

Sergey L. Gefter^{[1]}and Valentin Ya. Golodets

^{[2]}(1) Department of Mechanics and Mathematics, Kharkov State University, Dzerzhinsky square 4, 310077, KHARKOV, UKRAINE

(2) Institute for Low Temperature Physics, Ukrainian Academy of Sciences, 47 Lenin avenue, KHARKOV, UKRAINE

Fundamental groups of ergodic dynamical systems with invariant measure are considered. For a given countable subgroup *A*⊂** R***, an ergodic dynamical system is constructed whose
fundamental group is countable and contains

*A*. 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.

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Gefter Sergey, Golodets Valentin: Fundamental Groups for Ergodic Actions and Actions with Unit Fundamental Groups. *Publ. Res. Inst. Math. Sci.* 24 (1988), 821-847. doi: 10.2977/prims/1195173929