Ergodic Theory and Dynamical Systems

Research Article

The geometric dimension of an equivalence relation and finite extensions of countable groups

A. H. DOOLEYa1 and V. YA. GOLODETSa1

a1 School of Mathematics, University of New South Wales, Sydney, NSW 2052, Australia (email: a.dooley@unsw.edu.au, v.golodets@unsw.edu.au)

Abstract

We say that the geometric dimension of a countable group G is equal to n if any free Borel action of G on a standard Borel probability space (X,μ), induces an equivalence relation of geometric dimension n on (X,μ) in the sense of Gaboriau. Let xs212C be the set of all finitely generated amenable groups all of whose subgroups are also finitely generated, and let xs1D49C be the subset of xs212C consisting of finite groups, torsion-free groups and their finite extensions. In this paper we study finite free products K of groups in xs1D49C. The geometric dimension of any such group K is one: we prove that also geom-dim(Gf(K))=1 for any finite extension Gf(K) of K, applying the results of Stallings on finite extensions of free product groups, together with the results of Gaboriau and others in orbit equivalence theory. Using results of Karrass, Pietrowski and Solitar we extend these results to finite extensions of free groups. We also give generalizations and applications of these results to groups with geometric dimension greater than one. We construct a family of finitely generated groups {Kn}nxs2208xs2115,n>1, such that geom-dim(Kn)=n and geom-dim(Gf(Kn))=n for any finite extension Gf(Kn) of Kn. In particular, this construction allows us to produce, for each integer n>1, a family of groups {K(s,n)}sxs2208xs2115 of geometric dimension n, such that any finite extension of K(s,n) also has geometric dimension n, but the finite extensions Gf(K(s,n)) are non-isomorphic, if ss′.

(Received September 26 2008)

(Revised October 14 2008)