Bimodule coefficients, Riesz transforms on Coxeter groups and strong solidity

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 $H_{\Gamma }^{\otimes n}$ for $n\ge \frac{p}{2}$ 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 ${\mathcal{S}}_p$ 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 Sinclair for discrete groups admitting a proper cocycle into a $p$-integrable representation.