Wreath-like products of groups and their von Neumann algebras II: Outer automorphisms

Abstract

Given a countable group $G$, let $\mathrm{L}(G)$ denote its von Neumann algebra. For a wide class of ICC groups with Kazhdan's property (T), we confirm a conjecture of V.F.R. Jones asserting that $Out(\text{L}(G))\cong Char(G)⋊Out(G)$. As an application, we show that, for every countable group $Q$, there exists an ICC group $G$ with property (T) such that $Out(\text{L}(G))\cong Q$.

Some results of this paper were originally included in the first version of arXiv:2111.04708, which is now subdivided into three separate papers. Compared to arXiv:2111.04708v1, the main result (Theorem 1.5) is strengthened and its proof is based on a different idea. v3: Final version, to appear in Duke Math. J