Spectral gap for products and a strong normal subgroup theorem

TL;DR

This paper develops a spectral gap theorem for product group actions, leading to new results on the structure of subgroups in higher rank Lie groups, including strengthened versions of classical theorems without relying on property (T).

Contribution

It introduces a spectral gap theorem for product groups that generalizes Kazhdan's property (T) and applies it to subgroup classification in higher rank Lie groups, removing previous assumptions.

Findings

Confined subgroups of irreducible lattices are of finite index.

Confined discrete subgroups satisfying irreducibility are irreducible lattices.

Strengthens the normal subgroup theorem of Margulis without property (T).

Abstract

We establish a general spectral gap theorem for actions of products of groups which may replace Kazhdan's property (T) in various situations. As a main application, we prove that a confined subgroup of an irreducible lattice in a higher rank semisimple Lie group is of finite index. This significantly strengthens the classical normal subgroup theorem of Margulis and removes the property (T) assumption from the recent counterpart result of Fraczyk and Gelander. We further show that any confined discrete subgroup of a higher rank semisimple Lie group satisfying a certain irreducibility condition is an irreducible lattice. This implies a variant of the Stuck-Zimmer conjecture under a strong irreducibility assumption of the action.