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Commensurators of thin normal subgroups and abelian quotients

Thomas Koberda, Mahan Mj

Abstract

We give an affirmative answer to many cases of a question due to Shalom, which asks if the commensurator of a thin subgroup of a Lie group is discrete. In this paper, let $K<Γ<G$ be an infinite normal subgroup of an arithmetic lattice $Γ$ in a rank one simple Lie group $G$, such that the quotient $Q=Γ/K$ is infinite. We show that the commensurator of $K$ in $G$ is discrete, provided that $Q$ admits a surjective homomorphism to $\mathbb{Z}$. In this case, we also show that the commensurator of $K$ contains the normalizer of $K$ with finite index. We thus vastly generalize a result of the authors, which showed that many natural normal subgroups of $\mathrm{PSL}_2(\mathbb{Z})$ have discrete commensurator in $\mathrm{PSL}_2(\mathbb{R})$.

Commensurators of thin normal subgroups and abelian quotients

Abstract

We give an affirmative answer to many cases of a question due to Shalom, which asks if the commensurator of a thin subgroup of a Lie group is discrete. In this paper, let be an infinite normal subgroup of an arithmetic lattice in a rank one simple Lie group , such that the quotient is infinite. We show that the commensurator of in is discrete, provided that admits a surjective homomorphism to . In this case, we also show that the commensurator of contains the normalizer of with finite index. We thus vastly generalize a result of the authors, which showed that many natural normal subgroups of have discrete commensurator in .

Paper Structure

This paper contains 15 sections, 21 theorems, 36 equations.

Table of Contents

  1. Introduction
  2. Main result
  3. Tools and Techniques
  4. Structure of the paper
  5. Generalities on discrete subgroups of Lie groups
  6. Zariski dense subgroups and commensurators
  7. Preservation of lines with holes
  8. Homological lines with holes and Hodge theory
  9. Preserving homological lines with holes and cohomological consequences
  10. Hodge theory
  11. The commensurator of a form
  12. Abelian quotients and harmonic 1-forms
  13. Proof of Theorem \ref{['thm:main-general']}
  14. Applications
  15. Discrete Patterns of horoballs

Key Result

Theorem 1.1

Let $G$ be a semi-simple Lie group with no compact factors and let $\Gamma$ be an irreducible lattice in $G$. Then $\Gamma$ is arithmetic if and only if $\operatorname{Comm}_G(\Gamma)$ is dense in $G$.

Theorems & Definitions (40)

  • Theorem 1.1: Margulis
  • Theorem 1.3
  • Theorem 1.4
  • Lemma 2.1
  • Lemma 2.2
  • Lemma 2.3
  • proof
  • Proposition 2.4
  • Proposition 2.5
  • proof
  • ...and 30 more