In this talk, I will introduce a family of property (T) wreath-like product groups $G$, denoted as $\mathcal C$, for which we can describe explicitly all $\ast$-embeddings between their von Neumann algebras, $\mathcal L(G)$. Empolying deep methods in geometric group theory we show that $\mathcal C$ contains groups for which all injective endomomorphisms are inner automorphisms. Combining these results, we obtain the first examples of group factors which admit only inner endomorphisms. Additionally, these results yield computations of the fundamental semigroup of these factors.

This is based on a new joint work with A. Ioana, D. Osin and B. Sun.