Gradient forms and strong solidity of free quantum groups

Consider the free orthogonal quantum groups ON+(F) and free unitary quantum groups UN+(F) with N≥ 3. In the case F= id _N it was proved both by Isono and Fima-Vergnioux that the associated finite von Neumann algebra L∞(ON+) is strongly solid. Moreover, Isono obtains strong solidity also for L∞(UN+). In this paper we prove for general F∈ GL_N(C) that the von Neumann algebras L∞(ON+(F)) and L∞(UN+(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.