Spectral gap for irreducible subgroups and a strong version of Margulis normal subgroup theorem

Let G = G1 × G2 be a product of non-compact simple Lie groups. A subgroup ∆ is called irreducibly confined if it intersects trivially each factor and no conjugate limit is contained in one factor. We prove that such a group must be a lattice.

We deduce the following generalization of the NST: Let Γ be an irreducible lattice in a higher rank semisimple Lie group G. Let N < Γ be a confined subgroup. Then N is of finite index.

The case where G (or one of its factors) has Kazhdan’s property (T) was established in my joint work with Mikolaj Frakzyc. As in the original NST, without property (T) the problem is considerably harder. The main part is to prove a spectral gap for L₂(G/∆).

We also prove a variant in the context of products of general locally compact groups generalizing the Bader–Shalom normal subgroup theorem.

This is a joint work with Uri Bader and Arie Levit.