Some problems conjectures and results on commensurators and commensurated subgroups of arithmetic groups

Yehuda Shalom
Tel Aviv University

We shall begin by discussing the following provocative speculation:
If the commensurator inside a simple Lie group $G$, of a Zariski dense discrete subgroup $H \subset G$, is dense, then $H$ is a lattice
(hence, by Margulis, arithmetic). Even for infinitely generated Fucshian subgroups this seems open and hard. Then we discuss the anaytic approach towards the well known Margulis-Zimmer conjecture on commensurated subgroups of $S$-arithmetic groups. We describe results covering "most" of the "property (T) side" of the approach,
and discuss the difficulties on the amenability side.

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