Bimodule coefficients, Riesz transforms on Coxeter groups and strong solidity

Matthijs Borst¹  ; Martijn Caspers¹  ; Mateusz Wasilewski² 

¹Delft University of Technology (TU Delft), The Netherlands
²KU Leuven, Belgium

Groups, geometry, and dynamics, Tome 18 (2024) no. 2, pp. 501-549

Résumé

In deformation-rigidity theory, it is often important to know whether certain bimodules are weakly contained in the coarse bimodule. Consider a bimodule H over the group algebra C[Γ] with Γ a discrete group. The starting point of this paper is that if a dense set of the so-called coefficients of H is contained in the Schatten Sp​ class p∈[2,∞), then the n-fold tensor power HΓ⊗n​ for n≥2p​ is quasi-contained in the coarse bimodule. We apply this to gradient bimodules associated with the carré du champ of a symmetric quantum Markov semi-group. For Coxeter groups, we give a number of characterizations of having coefficients in Sp​ for the gradient bimodule constructed from the word length function. We get equivalence of: (1) the gradient-Sp​ property introduced by the second named author, (2) smallness at infinity of a natural compactification of the Coxeter group, and for a large class of Coxeter groups, (3) walks in the Coxeter diagram called parity paths. We derive several strong solidity results. In particular, we extend current strong solidity results for right-angled Hecke von Neumann algebras beyond right-angled Coxeter groups that are small at infinity. Our general methods also yield a concise proof of a result by Sinclair for discrete groups admitting a proper cocycle into a p-integrable representation.

Détail

DOI : 10.4171/ggd/752

Classification : 46L10, 20F55
Mots-clés : Strong solidity, quantum Markov semi-groups, Coxeter groups, Akemann–Ostrand property, Riesz transform, Hecke von Neumann algebras, bimodule coefficients