A Normal Subgroup Theorem for Commensurators

We present a Normal Subgroup Theorem for (dense) commensurators of lattices in arbitrary locally compact groups (not necessarily Lie).
In particular, any normal subgroup of a (dense) commensurator of an
(integrable) lattice in a simple topological group necessarily contains (up to finite index) the lattice.

The approach, as in Margulis’ Normal Subgroup Theorem for lattices, involves, on the one hand, using cohomology and rigidity theory to prove a certain group has property (T), and on the other hand, Furstenberg’s Boundary Theory to prove this group is also amenable.

This is joint work with Yehuda Shalom.