Gradient forms and strong solidity of free quantum groups

Consider the free orthogonal quantum groups $O_N^{+}(F)$ and free unitary quantum groups $U_N^{+}(F)$ with $N\ge 3$. In the case $F={\mathrm{i}\mathrm{d}}_N$ it was proved both by Isono and Fima-Vergnioux that the associated finite von Neumann algebra $L_{\mathrm{\infty }}(O_N^{+})$ is strongly solid. Moreover, Isono obtains strong solidity also for $L_{\mathrm{\infty }}(U_N^{+})$ . In this paper we prove for general $F \in GL_N($\mathbb{C}$)$thatthevonNeumannalgebras$L_\infty(O_N^+(F))$and$L_\infty(U_N^+(F))$ are strongly solid. A crucial part in our proof is the study of coarse properties of gradient bimodules associated with Dirichlet forms on these algebras and constructions of derivations due to Cipriani–Sauvageot.