A celebrated theorem of Margulis characterizes arithmetic lattices in terms of density of their commensurators.
A question going back to Shalom asks the analogous question for thin subgroups. We shall report on work during the
last decade or so and conclude with a recent development. In recent work with Thomas Koberda, we were able to show
that for a large class of normal subgroups of rank one arithmetic lattices, the commensurator is discrete.