All generating sets of all property T von Neumann algebras have free entropy dimension $\leq 1$

Abstract

Suppose $N$ is a diffuse, property T von Neumann algebra and X is an arbitrary finite generating set of selfadjoint elements for N. By using rigidity/deformation arguments applied to representations of N in full matrix algebras, we deduce that the microstate spaces of X are asymptotically discrete up to unitary conjugacy. We use this description to show that the free entropy dimension of X, $\delta_0(X)$, is less than or equal to 1. It follows that when N embeds into the ultraproduct of the hyperfinite $\mathrm{II}_1$-factor, then $\delta_0(X)=1$ and otherwise, $\delta_0(X)=-\infinity$. This generalizes the earlier results of Voiculescu, and Ge, Shen pertaining to $SL_n(\mathbb Z)$ as well as the results of Connes, Shlyakhtenko pertaining to group generators of arbitrary property T algebras.