Cantor systems and quasi-isometry of groups

Abstract

The purpose of this note is twofold. In the first part we observe that two finitely generated non-amenable groups are quasi-isometric if and only if they admit topologically orbit equivalent Cantor minimal actions. In particular, free groups if different rank admit topologically orbit equivalent Cantor minimal actions unlike in the measurable setting. In the second part we introduce the measured orbit equivalence category of a Cantor minimal system and construct (in certain cases) a representation of this category on the category of finite-dimensional vector spaces. This gives rise to novel fundamental invariants of the orbit equivalence relation together with an ergodic invariant probability measure.

incorporated referee's suggestions and corrections

References in corpus (1)

• Vanishing of ${\ell }^2$-Betti numbers of locally compact groups as an invariant of coarse equivalence

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