Greenberg-Shalom's Commensurator Hypothesis and Applications

We discuss many surprising implications of a positive answer to a question raised in some cases by Greenberg in the $\mathrm{‘}70$s and more generally by Shalom in the early $2000$s. We refer to this positive answer as the Greenberg-Shalom hypothesis. This hypothesis then says that any infinite discrete subgroup of a semisimple Lie group with dense commensurator is a lattice in a product of some factors. For some applications it is natural to extend the hypothesis to cover semisimple algebraic groups over other fields as well.