Cohomology of hyperfinite Borel actions

We study cocycles of countable groups \Gamma of Borel automorphisms of a standard Borel space (X, \mathcal{B}) taking values in a locally compact second countable group G . We prove that for a hyperfinite group \Gamma the subgroup of coboundaries is dense in the group of cocycles. We describe all Bo...

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Bibliographic Details
Published inGroups, geometry and dynamics Vol. 15; no. 4; pp. 1363 - 1398
Main Authors Bezuglyi, Sergey I., Sanadhya, Shrey
Format Journal Article
LanguageEnglish
Published European Mathematical Society Publishing House 01.01.2021
Subjects
Cohomology theory
Finite groups
Isomorphisms (Mathematics)
Mathematical research
United States
Online AccessGet full text
ISSN1661-7207
1661-7215
DOI10.4171/ggd/633

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Summary:We study cocycles of countable groups \Gamma of Borel automorphisms of a standard Borel space (X, \mathcal{B}) taking values in a locally compact second countable group G . We prove that for a hyperfinite group \Gamma the subgroup of coboundaries is dense in the group of cocycles. We describe all Borel cocycles of the 2 -odometer and show that any such cocycle is cohomologous to a cocycle with values in a countable dense subgroup H of G . We also provide a Borel version of Gottschalk–Hedlund theorem.
ISSN:1661-7207
1661-7215
DOI:10.4171/ggd/633