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Spectral gap for products and a strong normal subgroup theorem

Uri Bader, Tsachik Gelander, Arie Levit

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.

Spectral gap for products and a strong normal subgroup theorem

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.

Paper Structure

This paper contains 10 sections, 68 theorems, 125 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Unitary representations of product groups
  4. Koopman representations over actions with stabilizers
  5. Standard semisimple groups and irreducible lattices
  6. Discrete subgroups of standard semisimple groups
  7. Confined and strongly confined subgroups
  8. Confined subgroups of irreducible lattices
  9. Margulis functions on factors
  10. Strongly confined subgroups of semisimple groups

Key Result

Theorem 1.1

Let $G$ be a connected semisimple Lie group of real rank at least two and with trivial center. Let $\Gamma$ be an irreducible lattice in $G$. Then any confined subgroup of $\Gamma$ has finite index.

Theorems & Definitions (156)

  • Theorem 1.1
  • Theorem 1.1: Reformulation
  • Corollary 1.2
  • Definition 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Theorem 1.6: Spectral gap for actions of products
  • Corollary 1.7: A weak version of the Stuck--Zimmer conjecture
  • Definition 2.1
  • Lemma 2.2
  • ...and 146 more