Journal article
Measure equivalence rigidity via s-malleable deformations
Compositio mathematica, Vol.159(10), pp.2023-2050
10/01/2023
DOI: 10.1112/S0010437X2300739X
Abstract
We single out a large class of groups M for which the following unique prime factorization result holds: if Gamma(1),..., Gamma(n) is an element of M and Gamma(1) x center dot center dot center dot x Gamma(n) is measure equivalent to a product Lambda(1) x center dot center dot center dot x Lambda(m) of infinite icc groups, then n >= m, and if n = m, then, after permutation of the indices, Gamma(i) is measure equivalent to Lambda(i), for all 1 <= i <= n. This provides an analogue of Monod and Shalom's theorem [Orbit equivalence rigidity and bounded cohomology, Ann. of Math. 164 (2006), 825-878] for groups that belong to M. Class M is constructed using groups whose von Neumann algebras admit an s-malleable deformation in the sense of Sorin Popa and it contains all icc non-amenable groups Gamma for which either (i) Gamma is an arbitrary wreath product group with amenable base or (ii) Gamma admits an unbounded 1-cocycle into its left regular representation. Consequently, we derive several orbit equivalence rigidity results for actions of product groups that belong toM. Finally, for groups G satisfying condition (ii), we show that all embeddings of group von Neumann algebras of non-amenable inner amenable groups into L(G) are 'rigid'. In particular, we provide an alternative solution to a question of Popa that was recently answered by Ding, Kunnawalkam Elayavalli, and Peterson [Properly Proximal von Neumann Algebras, Preprint (2022), arXiv:2204.00517].
Details
- Title: Subtitle
- Measure equivalence rigidity via s-malleable deformations
- Creators
- Daniel Drimbe - KU Leuven
- Resource Type
- Journal article
- Publication Details
- Compositio mathematica, Vol.159(10), pp.2023-2050
- Publisher
- Cambridge Univ Press
- DOI
- 10.1112/S0010437X2300739X
- ISSN
- 0010-437X
- eISSN
- 1570-5846
- Number of pages
- 29
- Language
- English
- Date published
- 10/01/2023
- Academic Unit
- Mathematics
- Record Identifier
- 9984696656102771
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