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

TL;DR

The paper addresses the rigidity of II_1 factors arising from ICC groups with property (T) and proves that for a wide class of such groups, Out($L(G)$) ≅ Char($G$) ⋊ Out($G$), confirming Jones' conjecture in this setting. It develops a robust group-theoretic framework based on wreath-like products and Cohen-Lyndon subgroups, augmented by Dehn filling techniques in relatively hyperbolic groups to construct groups with prescribed outer automorphism structures and no nontrivial characters. By combining these group-theoretic tools with Popa-style deformation/rigidity methods, the authors obtain W*-superrigidity results and explicit computations of Out($L(G)$) for wreath-like group factors, including the realization that any countable group $Q$ can occur as Out($L(G)$) for some ICC group $G$ with property (T). The work thus provides both a complete realization of Jones' conjecture in a broad setting and a flexible construction method for II_1 factors with predetermined symmetry groups, with potential applications to C*-algebras and embeddings of factors. The results significantly advance understanding of how algebraic structures at the group level transfer to symmetries of associated von Neumann algebras and reinforce the deep connections between geometric group theory and operator algebras.

Abstract

Given a countable group $G$, let ${\rm L}(G)$ 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 $Out(\text{L}(G))\cong Char (G)\rtimes Out(G)$. As an application, we show that, for every countable group $Q$, there exists an ICC group $G$ with property (T) such that $Out(\text{L}(G))\cong Q$.

Theorems & Definitions (141)

• Example 1.1
• Theorem 1.2: Connes, Co80
• Conjecture 1.4: Jones
• Theorem 1.5
• Definition 2.1
• Example 2.2
• Theorem 2.3: Chifan--Ioana--Osin--Sun, CIOS1
• Example 2.4
• Definition 2.5
• Proposition 2.6