Integrable measure equivalence for groups of polynomial growth

Tim Austin, Lewis Bowen

Research output: Contribution to journalArticle

Abstract

Bader, Furman and Sauer have recently introduced the notion of integrable measure equivalence for finitely-generated groups. This is the sub-equivalence relation of measure equivalence obtained by insisting that the relevant cocycles satisfy an integrability condition. They have used it to prove new classification results for hyperbolic groups. The present work shows that groups of polynomial growth are also quite rigid under integrable measure equivalence, in that if two such groups are equivalent then they must have bi-Lipschitz asymptotic cones. This will follow by proving that the cocycles arising from an integrable measure equivalence converge under re-scaling, albeit in a very weak sense, to bi-Lipschitz maps of asymptotic cones.

Original languageEnglish (US)
Pages (from-to)117-154
Number of pages38
JournalGroups, Geometry, and Dynamics
Volume10
Issue number1
DOIs
StatePublished - 2016

Fingerprint

Polynomial Growth
Asymptotic Cone
Equivalence
Cocycle
Lipschitz Map
Hyperbolic Groups
Finitely Generated Group
Rescaling
Equivalence relation
Integrability
Lipschitz
Converge

Keywords

  • Asymptotic cones
  • Cocycle ergodic theorems
  • Groups of polynomial growth
  • Integrable measure equivalence
  • Nilpotent groups

ASJC Scopus subject areas

  • Discrete Mathematics and Combinatorics
  • Geometry and Topology

Cite this

Integrable measure equivalence for groups of polynomial growth. / Austin, Tim; Bowen, Lewis.

In: Groups, Geometry, and Dynamics, Vol. 10, No. 1, 2016, p. 117-154.

Research output: Contribution to journalArticle

Austin, Tim ; Bowen, Lewis. / Integrable measure equivalence for groups of polynomial growth. In: Groups, Geometry, and Dynamics. 2016 ; Vol. 10, No. 1. pp. 117-154.
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