Abstract:
Lattices in simple groups of higher rank have property (T).
But lattices in semisimple groups of higher rank might fail to have it, due to the presence of a rank 1 factor. This is unfortunate, as otherwise these guys are rather rigid.
In my talk I will discuss a precise conjectural picture explaining this failure and relate it to several, known or conjectural, results.
Later, I will mention cases in which the conjectural picture is actually proven and in particular, I will discuss a recent joint work with Gelander and Levit in which we show that certain unitary representations of higher rank lattices have a spectral gap. I will draw some conclusions, extending the scope of Margulis Normal Subgroup Theorem.