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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title = "Cohomology of hyperfinite Borel actions",

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",

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year = "2021",

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doi = "10.4171/GGD/633",

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AU - Sanadhya, Shrey

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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

UR - https://www.scopus.com/pages/publications/85121979052

U2 - 10.4171/GGD/633

DO - 10.4171/GGD/633

M3 - Article

AN - SCOPUS:85121979052

SN - 1661-7207

VL - 15

SP - 1363

EP - 1398

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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