Cantor systems and quasi‐isometry of groups. Issue 4 (8th June 2017)

• Record Type: Journal Article
• Authors: Medynets, Kostya; Sauer, Roman; Thom, Andreas
• Is Part Of: Bulletin of the London Mathematical Society. Volume 49:Issue 4(2017)
• Page Start: 709
• Page End: 724
• Publication Date: 2017-06-08
• Subjects: 37B99 (primary) -- 20F65 (secondary); Mathematics -- Periodicals; 510
• DOI: 10.1112/blms.12059
• Languages: English
• ISSNs: 0024-6093

Abstract

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 of 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.