Existential Inclusions of Bi-exact Groups are Conjugacy Representation Rigid

Abstract

If $\Lambda$ is a non-amenable bi-exact group and $\Lambda \hookrightarrow \Gamma$ is an existential embedding, then each of the intersections $\Lambda \cap g \Lambda g^{-1}$ for $g$ a member of $\Gamma \backslash \Lambda$ is amenable. This in conjunction with work of Bekka and Kalantar demonstrates that in this situation, the weak equivalence class of the quasi-regular representation $\lambda_{\Gamma/\Lambda}$ determines $\Lambda$ up to conjugacy among the self-commensurating subgroups of $\Gamma$.