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An effective closing lemma for unipotent flows

Elon Lindenstrauss, Gregory Margulis, Amir Mohammadi, Nimish Shah, Andreas Wieser

Abstract

We prove an effective closing lemma for unipotent flows on quotients of perfect real groups. This is largely motivated by recent developments in effective unipotent dynamics.

An effective closing lemma for unipotent flows

Abstract

We prove an effective closing lemma for unipotent flows on quotients of perfect real groups. This is largely motivated by recent developments in effective unipotent dynamics.

Paper Structure

This paper contains 13 sections, 22 theorems, 110 equations.

Table of Contents

  1. Introduction
  2. Heights of groups generated by lattice elements
  3. Chevalley representations for groups of class $\mathcal{H}$
  4. A bound on the number of the connected components
  5. $\mathbb{Q}$-groups generated by lattice elements
  6. Almost invariant Lie algebras
  7. Proof of Theorem \ref{['thm:main']}
  8. Visits to cusp neighborhoods
  9. A large set of non-Diophantine points
  10. A set of good density points
  11. Constructing a group of class $\mathcal{H}$
  12. Completion of the proof of Proposition \ref{['prop:mainstep']}
  13. Proof of Theorem \ref{['thm:main']}

Key Result

Theorem 1.1

Suppose that $\mathbf{G}$ is perfect. There exist constants ${\color{red}{A_{1}}},{\color{red}{A_{2}}}>1$ depending only on $N$, and $E>0$ depending on $N,G,\Gamma$ with the following property. Let $\mathfrak{h} < \mathfrak{g}$ be a (proper) subalgebra so that $\mathfrak{h} +\operatorname{rad}(\math Then one of the following is true:

Theorems & Definitions (39)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 2.1
  • Proposition 2.2: Chevalley representations and heights
  • Lemma 2.3: Degree bound
  • proof
  • proof : Proof of Proposition \ref{['prop:chevalley']}
  • Lemma 2.4
  • Lemma 2.5
  • proof
  • ...and 29 more