Wreath-like products of groups and their von Neumann algebras III: Embeddings

TL;DR

The paper advances the study of von Neumann algebra embeddings for wreath-like product groups with property $(T)$ by proving that any embedding $L(G)\hookrightarrow L(H)^t$ between ICC property $(T)$ wreath-like product II$_1$ factors is, up to unitary conjugacy, induced by a group embedding $\delta:G\hookrightarrow H$ and a character $\rho:G\to\mathbb{T}$, with $t\in\mathbb{N}$; it extends previous isomorphism rigidity to the embedding setting and gives an explicit decomposition into induced pieces. It develops a robust rigidity toolkit—intertwining-by-bimodules, cocycle superrigidity, orbit equivalence rigidity, and second-cohomology control—to analyze embeddings and classify them in terms of group-theoretic data. The results yield a continuum of ICC property $(T)$ groups whose II$_1$ factors are pairwise non-embeddable, and produce groups whose von Neumann algebras have only inner endomorphisms, while explicitly computing several invariants such as End$(M)$, Out$(M)$, and fundamental/embedding semigroups. Together, the findings reinforce Connes-type rigidity for a broad class of non-amenable groups and provide concrete, highly structured examples of II$_1$ factors with prescribed endomorphism and embedding behavior.

Abstract

For a class of wreath-like product groups with property (T), we describe explicitly all the embeddings between their von Neumann algebras. This allows us to provide a continuum of ICC groups with property (T) whose von Neumann algebras are pairwise non (stably) embeddable. We also give a construction of groups in this class only having inner injective homomorphisms. As an application, we obtain examples of group von Neumann algebras which admit only inner endomorphisms.

Theorems & Definitions (53)

• Definition 1.1
• Remark 1.2
• Theorem 1.3
• Example 1.4
• Corollary 1.5
• Theorem 1.6
• Lemma 2.1: Bowditch
• Lemma 2.2: Ols93
• Theorem 2.3: Osi07SunDGOCIOS1CIOS3
• Lemma 2.4