Strongly solid group factors which are not interpolated free group factors

Abstract

We give examples of non-amenable ICC groups $\Gamma$ with the Haagerup property, weakly amenable with constant $\Lambda_{\cb}(\Gamma) = 1$, for which we show that the associated ${\rm II_1}$ factors $L(\Gamma)$ are strongly solid, i.e. the normalizer of any diffuse amenable subalgebra $P \subset L(\Gamma)$ generates an amenable von Neumann algebra. Nevertheless, for these examples of groups $\Gamma$, $L(\Gamma)$ is not isomorphic to any interpolated free group factor $L(\F_t)$, for $1 < t \leq \infty$.