Bimodule coefficients, Riesz transforms on Coxeter groups and strong solidity

TL;DR

This paper investigates the properties of bimodules over Coxeter groups, linking Schatten class coefficients to strong solidity of associated von Neumann algebras, and introduces new characterizations and results in deformation-rigidity theory.

Contribution

It provides new characterizations of gradient bimodules with Schatten class coefficients for Coxeter groups and extends strong solidity results beyond right-angled Coxeter groups.

Findings

Gradient bimodules with Schatten class coefficients are quasi-contained in the coarse bimodule for large tensor powers.

Equivalence between gradient-$ ext{S}_p$ property, smallness at infinity, and parity paths in Coxeter groups.

Extended strong solidity results for Coxeter group von Neumann algebras beyond previously known cases.

Abstract

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 $\mathbb{C}[\Gamma]$, with $\Gamma$ 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 $\mathcal{S}_p$ class $p \in [2, \infty)$ then the $n$-fold tensor power $H^{\otimes n}_\Gamma$ for $n \geq p/2$ is quasi-contained in the coarse bimodule. We apply this to gradient bimodules associated with the carr\'e du champ of a symmetric quantum Markov semi-group. For Coxeter groups we give a number of characterizations of having coefficients in $\mathcal{S}_p$ for the gradient bimodule constructed from the word length function. We get equivalence of: (1) the gradient-$\mathcal{S}_p$ 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 T. Sinclair for discrete groups admitting a proper cocycle into a $p$-integrable representation.