Abstract:

A subgroup H < G is called confined if there is a compact subset K of G such that every conjugate of H intersects K at some point other than identity. We prove that every confined subgroup of an irreducible lattice in a higher rank semisimple Lie group has finite index. Since a non-trivial normal subgroup is confined, our result extends the Margulis normal subgroup theorem. We do not rely on Kazhdan’s property (T), and instead obtain spectral gap from a product structure. More generally, we show that any confined discrete subgroup of a higher rank semisimple Lie group satisfying a certain irreducibility condition is a lattice. This extends the recent work of Fraczyk and Gelander, removing the property (T) assumption. Joint work with Uri Bader and Tsachik Gelander.