Cohomology of hyperfinite Borel actions

Sergey Bezuglyi; Shrey Sanadhya

doi:10.4171/GGD/633

Cohomology of hyperfinite Borel actions

Sergey Bezuglyi, Shrey Sanadhya

Research output: Contribution to journal › Article › peer-review

Abstract

We study cocycles of countable groups Γ of Borel automorphisms of a standard Borel space (X, B) taking values in a locally compact second countable group G. We prove that for a hyperfinite group Γ 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.

Original language	English
Pages (from-to)	1363-1398
Number of pages	36
Journal	Groups, Geometry, and Dynamics
Volume	15
Issue number	4
DOIs	https://doi.org/10.4171/GGD/633
State	Published - 1 Jan 2021
Externally published	Yes

Keywords

Borel automorphism
Coboundary
Cocycle
Hyperfinite countable Borel equivalence relation
Odometer

ASJC Scopus subject areas

Geometry and Topology
Discrete Mathematics and Combinatorics

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10.4171/GGD/633

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abstract = "We study cocycles of countable groups Γ of Borel automorphisms of a standard Borel space (X, B) taking values in a locally compact second countable group G. We prove that for a hyperfinite group Γ 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.",

keywords = "Borel automorphism, Coboundary, Cocycle, Hyperfinite countable Borel equivalence relation, Odometer",

author = "Sergey Bezuglyi and Shrey Sanadhya",

note = "Funding Information: The authors are very thankful to Palle Jorgensen and Paul Muhly for their interest to this work and useful discussions. We thank the referee for the careful reading and valuable suggestions. Publisher Copyright: {\textcopyright} 2021 European Mathematical Society Published by EMS Press.",

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N1 - Funding Information: The authors are very thankful to Palle Jorgensen and Paul Muhly for their interest to this work and useful discussions. We thank the referee for the careful reading and valuable suggestions. Publisher Copyright: © 2021 European Mathematical Society Published by EMS Press.

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N2 - We study cocycles of countable groups Γ of Borel automorphisms of a standard Borel space (X, B) taking values in a locally compact second countable group G. We prove that for a hyperfinite group Γ 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.

AB - We study cocycles of countable groups Γ of Borel automorphisms of a standard Borel space (X, B) taking values in a locally compact second countable group G. We prove that for a hyperfinite group Γ 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.

KW - Borel automorphism

KW - Coboundary

KW - Cocycle

KW - Hyperfinite countable Borel equivalence relation

KW - Odometer

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JO - Groups, Geometry, and Dynamics

JF - Groups, Geometry, and Dynamics

IS - 4

ER -

Cohomology of hyperfinite Borel actions

Abstract

Keywords

ASJC Scopus subject areas

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