Existential Inclusions of Bi-exact Groups are Conjugacy Representation Rigid

If $\mathrm{\Lambda }$ is a non-amenable bi-exact group and $\mathrm{\Lambda }\hookrightarrow \mathrm{\Gamma }$ is an existential embedding, then each of the intersections $\mathrm{\Lambda }\cap g\mathrm{\Lambda }{g}^{-1}$ for $g$ a member of $\mathrm{\Gamma }\mathrm{\backslash }\mathrm{\Lambda }$ 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 ${\lambda }_{\mathrm{\Gamma }\mathrm{/}\mathrm{\Lambda }}$ determines $\mathrm{\Lambda }$ up to conjugacy among the self-commensurating subgroups of $\mathrm{\Gamma }$.