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Discrete Linear Groups containing Arithmetic Groups

Indira Chatterji, T. N. Venkataramana

TL;DR

The paper investigates conditions under which a Zariski dense discrete subgroup $\Gamma$ of a real simple Lie group $G$, containing a higher-rank lattice from a subgroup $H$, must be a lattice in $G$. It develops a super-rigidity framework using measurable equivariant maps, a mild generalization of Howe-Moore, and Margulis' super-rigidity to derive rigidity results in archimedean and non-archimedean settings. The authors obtain concrete arithmeticity-type consequences, notably that a $\Gamma<SL_n(\mathbb{R})$ containing $SL_3(\mathbb{Z})$ in the top-left corner and meeting $SL_3(\mathbb{Z})$ in finite index is commensurate to $SL_n(\mathbb{Z})$, with analogous results for symplectic groups and linear embeddings. They also demonstrate a rank-one counterpoint showing the general question cannot hold without additional hypotheses, clarifying the scope of the positive Nori-type results and highlighting the role of rank and isotropy in such rigidity phenomena.

Abstract

We prove in a large number of cases, that a Zariski dense discrete subgroup of a simple real algebraic group $G$ which contains a higher rank lattice is a lattice in the group $G$. For example, we show that a Zariski dense subgroup of $SL_n({\mathbb R})$ which contains $SL_3({\mathbb Z})$ in the top left hand corner, is conjugate to $SL_n({\mathbb Z})$ .

Discrete Linear Groups containing Arithmetic Groups

TL;DR

The paper investigates conditions under which a Zariski dense discrete subgroup of a real simple Lie group , containing a higher-rank lattice from a subgroup , must be a lattice in . It develops a super-rigidity framework using measurable equivariant maps, a mild generalization of Howe-Moore, and Margulis' super-rigidity to derive rigidity results in archimedean and non-archimedean settings. The authors obtain concrete arithmeticity-type consequences, notably that a containing in the top-left corner and meeting in finite index is commensurate to , with analogous results for symplectic groups and linear embeddings. They also demonstrate a rank-one counterpoint showing the general question cannot hold without additional hypotheses, clarifying the scope of the positive Nori-type results and highlighting the role of rank and isotropy in such rigidity phenomena.

Abstract

We prove in a large number of cases, that a Zariski dense discrete subgroup of a simple real algebraic group which contains a higher rank lattice is a lattice in the group . For example, we show that a Zariski dense subgroup of which contains in the top left hand corner, is conjugate to .

Paper Structure

This paper contains 16 sections, 31 theorems, 22 equations.

Table of Contents

  1. Introduction
  2. Preliminaries on measurable maps
  3. A Mild Generalisation of the Howe-Moore Theorem.
  4. The Margulis Super-Rigidity Theorem.
  5. Some Measure Theoretic Constructions
  6. Some Lie Theoretic Results
  7. Invariance properties of a map associated to $\phi$.
  8. An Equivariance Property of a Map Associated with $\phi$.
  9. Proof of the super-rigidity result (Theorem \ref{['theo:main']})
  10. The Non-Archimedean Case.
  11. The Archimedean Case
  12. Preliminaries on the $G'$-action on $G'/J$.
  13. A Lemma on Real Varieties.
  14. Applications (Proof of Corollary \ref{['SLn']})
  15. Proof of Theorem \ref{['spllinear']}
  16. ...and 1 more sections

Key Result

Theorem 1

Suppose that ${\rm SL}_3$ is embedded in ${\rm SL}_n$ (in the "top left hand corner") as above. Suppose that $\Gamma$ is a Zariski dense discrete subgroup of ${\rm SL}_n(\mathbb{R})$ whose intersection with ${\rm SL}_3(\mathbb{R})$ is a subgroup of ${\rm SL}_3(\mathbb{Z})$ of finite index. Then, $\G

Theorems & Definitions (52)

  • Theorem 1
  • Theorem 2
  • Theorem 3: Main result
  • Theorem 4
  • Corollary 5
  • Corollary 6
  • Corollary 7
  • Corollary 8
  • Lemma 1
  • proof
  • ...and 42 more