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Rigidity of the dynamics of ${\rm Aut}({\mathsf{F}}_n)$ on representations into a compact group

Serge Cantat, Christophe Dupont, Florestan Martin-Baillon

Abstract

Let $G$ be a compact Lie group. Let ${\mathsf{F}}_n$ be the free group of rank $n$. We describe the orbits of ${\mathsf{Aut}}(\mathsf{F}_n)$ on ${\mathsf{Hom}}(\mathsf{F}_n;G)$ when $n$ is sufficiently large. The dynamics stabilizes: orbit closures and invariant probability measures are algebraic, as in Ratner's theorems.

Rigidity of the dynamics of ${\rm Aut}({\mathsf{F}}_n)$ on representations into a compact group

Abstract

Let be a compact Lie group. Let be the free group of rank . We describe the orbits of on when is sufficiently large. The dynamics stabilizes: orbit closures and invariant probability measures are algebraic, as in Ratner's theorems.
Paper Structure (35 sections, 30 theorems, 54 equations)

This paper contains 35 sections, 30 theorems, 54 equations.

Table of Contents

  1. Introduction
  2. Nielsen moves
  3. Stabilization of the dynamics
  4. Redundancy
  5. Finite groups
  6. Rank, compressibility, and chains
  7. Transitivity
  8. Jordan families
  9. Compact Lie groups
  10. Peter-Weyl theorem
  11. Jordan theorem
  12. Generators of finite groups
  13. Generators of compact groups
  14. A redundancy Theorem
  15. Redundancy
  16. ...and 20 more sections

Key Result

Theorem A

Let $G$ be a compact Lie group. If $n$ is large enough, then

Theorems & Definitions (61)

  • Theorem A
  • Theorem B
  • Example 2.1
  • Example 2.2
  • Theorem 2.3
  • proof : Proof of (1)
  • Remark 2.4
  • Corollary 2.5
  • proof : Proof of Corollary \ref{['cor:lubotzky']} (2)
  • Example 2.6
  • ...and 51 more