Geodesic
Minimal topological actions do not determine the measurable orbit equivalence class
Tullio Ceccherini-Silberstein1 ; Gábor Elek2 orcid
1Università del Sannio, Benevento, Italy
2Hungarian Academy of Sciences, Budapest, Hungary
Groups, geometry, and dynamics, Tome 2 (2008) no. 2, pp. 139-163

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DOI

We construct an amenable action Φ of a non-amenable group Γ on a discrete space. This action extends to a minimal topological action Φ of Γ on a Cantor set C. We show that Φ is non-uniquely ergodic and furthermore there exist ergodic invariant measures μ1 and μ2 such that (Φ,C,μ1) and (Φ,C,μ2) are not orbit equivalent measurable equivalence relations. This also provides an instance of the failure of equivalence between the notions of “global” and “local” amenability for countable equivalence relations.
DOI : 10.4171/ggd/34
Classification : 37-XX, 43-XX, 00-XX
Mots-clés : Minimal actions, measurable equivalence relations, orbit equivalence, amenable actions, amenable graphs, hyperfiniteness

Tullio Ceccherini-Silberstein  1   ; Gábor Elek  2

1 Università del Sannio, Benevento, Italy
2 Hungarian Academy of Sciences, Budapest, Hungary 
Tullio Ceccherini-Silberstein; Gábor Elek. Minimal topological actions do not determine the measurable orbit equivalence class. Groups, geometry, and dynamics, Tome 2 (2008) no. 2, pp. 139-163. doi: 10.4171/ggd/34
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