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

Published online by Cambridge University Press:  02 March 2009

A. H. DOOLEY  and
V. YA. GOLODETS
A. H. DOOLEY
Affiliation:
School of Mathematics, University of New South Wales, Sydney, NSW 2052, Australia (email: [email protected], [email protected])
V. YA. GOLODETS
Affiliation:
School of Mathematics, University of New South Wales, Sydney, NSW 2052, Australia (email: [email protected], [email protected])
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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 ℬ be the set of all finitely generated amenable groups all of whose subgroups are also finitely generated, and let 𝒜 be the subset of ℬ consisting of finite groups, torsion-free groups and their finite extensions. In this paper we study finite free products K of groups in 𝒜. 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}n∈ℕ,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)}s∈ℕ 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 s≠s′.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 29 , Issue 6 , December 2009 , pp. 1789 - 1814
Copyright
Copyright © Cambridge University Press 2009

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