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Hausdorff dimension, diverging Schottky representations and the infinite dimensional hyperbolic space

Antonin Guilloux, Gilles Courtois

Abstract

One of our main goals in this paper is to understand the behavior of limit sets of a diverging sequence of Schottky groups in the group of isometries of the N-dimensional hyperbolic space. This leads us to a generalization of a classical theorem of Bowen on variations of Hausdorff dimension of limit sets; and to a method of transforming a diverging sequence of Schottky groups into an almost converging sequence in the group of isometries of the infinite dimensional hyperbolic space. Our results apply in particular to an example of McMullen and generalize a previous work by Mehmeti and Dang.

Hausdorff dimension, diverging Schottky representations and the infinite dimensional hyperbolic space

Abstract

One of our main goals in this paper is to understand the behavior of limit sets of a diverging sequence of Schottky groups in the group of isometries of the N-dimensional hyperbolic space. This leads us to a generalization of a classical theorem of Bowen on variations of Hausdorff dimension of limit sets; and to a method of transforming a diverging sequence of Schottky groups into an almost converging sequence in the group of isometries of the infinite dimensional hyperbolic space. Our results apply in particular to an example of McMullen and generalize a previous work by Mehmeti and Dang.

Paper Structure

This paper contains 36 sections, 35 theorems, 150 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Revisiting an example of McMullen
  3. Bowen Theorem for non locally compact $\mathop{\mathrm{CAT}}\nolimits(-1)$-spaces
  4. Turning diverging sequences of representations into convergent ones
  5. Acknowledgments
  6. CAT(-1)-geometry
  7. $\mathop{\mathrm{CAT}}\nolimits(-1)$-spaces
  8. Visual metric and shadows
  9. Quasi-isometric representations
  10. Convergence of Hausdorff dimensions for Schottky actions in CAT(-1)-spaces
  11. Gibbs measures and coding of the limit set
  12. Shift and cylinders in $\partial \mathbf F_r$
  13. Schottky actions and coding of the limit set
  14. Gibbs measures for the shift
  15. Balls and cylinders in limit sets
  16. ...and 21 more sections

Key Result

Theorem 1.1

Let $\rho _k$, $k\in \mathbf N$, and $\rho$ be Schottky representations in the space $\mathop{\mathrm{Rep}}\nolimits^{\mathrm{Schottky}}_{\mathbf H^N} (\mathbf F_{r})$. We assume that $\lim_k \rho _k (g) =\rho (g)$ for every $g\in \mathbf F _r$. Then,

Figures (2)

  • Figure 1: McMullen example: Schottky action of $\rho_\infty$ on its minimal tree (left); quotient of this action (right).
  • Figure 2: A tripod in $\mathbf F_r$ (left); its image in $X$ by $\tau_\rho$ (right). In the right hand side picture, the distance between $p_n$ and each of the geodesic is at most $C_K$.

Theorems & Definitions (77)

  • Theorem 1.1: Bowen-IHES
  • Example 1.2: McMullen example
  • Theorem 1.3
  • Remark 1.4
  • Remark 1.5
  • Example 1.6: McMullen example, continued
  • Theorem 1.7: Bowen theorem for $\mathop{\mathrm{CAT}}\nolimits(-1)$-spaces
  • Corollary 1.8
  • Proposition 1.9
  • proof
  • ...and 67 more