Bimodule coefficients, Riesz transforms on Coxeter groups and strong solidity

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 carre 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 three strong solidity results: two are known, one is new. The first result is a concise proof of a result by T. Sinclair for discrete groups admitting a proper cocycle into a $p$-integrable representation. The second result is strong solidity for hyperbolic right-angled Coxeter groups. The final -- and new -- result extends current strong solidity results for right-angled Hecke von Neumann algebras beyond right-angled Coxeter groups that are small at infinity.