Margulis–Zimmer conjecture on commensurated subgroups of S-arithmetic lattices

Let $\mathbb{G}$ be a semisimple algebraic group defined over a number field $k$, and let $S$ be a finite set of valuations of $k$. Assume that $\mathbb{G}$ has higher $S$-rank, and that

$$\Lambda=\mathbb{G}(\mathcal{O}_S)$$

is an $S$-arithmetic lattice in $\mathbb{G}$. A subgroup $\Gamma\subseteq\Lambda$ is commensurated if $\Lambda\subseteq\operatorname{Comm}_{\mathbb{G}}(\Gamma)$.

Margulis–Zimmer conjecture. Every commensurated subgroup of $\Lambda$ is either finite or $S'$-arithmetic for some $S'\subseteq S$.

This conjecture seeks to classify commensurated subgroups of higher-rank lattices and was advertised by Margulis and Zimmer in the late 1970s. It is presented as a major motivation for the Greenberg–Shalom hypothesis; the supplied text does not state a resolution.