Cocycle superrigidity for profinite actions of irreducible lattices

Daniel Drimbe; Adrian Ioana; Jesse Peterson

doi:10.4171/GGD/700

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Cocycle superrigidity for profinite actions of irreducible lattices

Journal article

Open access

Cocycle superrigidity for profinite actions of irreducible lattices

Daniel Drimbe, Adrian Ioana and Jesse Peterson

Groups, Geometry, and Dynamics, Vol.17(1), pp.315-329

03/01/2023

DOI: 10.4171/GGD/700

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Abstract

Let [GAMMA] be an irreducible lattice in a product of two locally compact groups and assume that [GAMMA] is densely embedded in a profinite group K. We give necessary conditions which imply that the left translation action [GAMMA] [??] K is "virtually" cocycle superrigid: any cocycle w: [GAMMA] x K [right arrow] [DELTA] with values in a countable group [DELTA] is cohomologous to a cocycle which factors through the map [GAMMA] x K [right arrow] [GAMMA] x [K.sub.0] for some finite quotient group [K.sub.0] of K. As a corollary, we deduce that any ergodic profinite action of [GAMMA] = [SL.sub.2]([??][[S.sup.-1]]) is virtually cocycle superrigid and virtually W*-superrigid for any finite nonempty set of primes S. 2020 Mathematics Subject Classification. Primary 37A20; Secondary 46L36. Keywords. Cocycle superrigidity, deformation/rigidity theory, orbit equivalence, profinite action, irreducible lattice, strong ergodicity.

Cocycle superrigidity

deformation

rigidity theory

orbit equivalence

profinite action

irreducible lattice

strong ergodicity

Details

Title: Subtitle: Cocycle superrigidity for profinite actions of irreducible lattices
Creators: Daniel Drimbe - KU Leuven
Adrian Ioana - University of California, San Diego
Jesse Peterson - Vanderbilt University
Resource Type: Journal article
Publication Details: Groups, Geometry, and Dynamics, Vol.17(1), pp.315-329
Publisher: European Mathematical Society Publishing House
DOI: 10.4171/GGD/700
ISSN: 1661-7207
ISSN: 1661-7207
eISSN: 1661-7215
Grant note: NSF Career Grant DMS: 1253402 PIMS fellowshipNSF FRG Grant: 1854074, 1801125 NSF Grant DMS: 1853989
Daniel Drimbe was partially supported by NSF Career Grant DMS #1253402 and by PIMS fellowship. Adrian Ioana was partially supported by NSF Career Grant DMS #1253402 and NSF FRG Grant #1854074. Jesse Peterson was supported in part by NSF Grant DMS #1801125 and NSF FRG Grant #1853989.
Language: English
Date published: 03/01/2023
Description audience: Academic
Academic Unit: Mathematics
Record Identifier: 9984795474202771

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