Greenberg (1974) in special cases and
Shalom (roughly 2010) asked the following:
if G is a simple Lie group and \Gamma <G is
an infinite subgroup with dense commensurator,
is \Gamma a lattice in G?
I will discuss recent progress on this question
as well as connections to a variety of seemingly
unrelated questions in geometric group theory.
If time permits I will also discuss why the question
seems equivalent to certain questions about
ergodicity of geodesic flows and spectral
gaps. | 
