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Pseudo-Anosovs are exponentially generic in mapping class groups

Inhyeok Choi

Abstract

Given a finite generating set $S$, let us endow the mapping class group of a closed hyperbolic surface with the word metric for $S$. We discuss the following question: does the proportion of non-pseudo-Anosov mapping classes in the ball of radius $R$ decrease to 0 as $R$ increases? We show that any finite subset $S'$ of the mapping class group is contained in a finite generating set $S$ such that this proportion decreases exponentially. Our strategy applies to weakly hyperbolic groups and does not refer to the automatic structure of the group.

Pseudo-Anosovs are exponentially generic in mapping class groups

Abstract

Given a finite generating set , let us endow the mapping class group of a closed hyperbolic surface with the word metric for . We discuss the following question: does the proportion of non-pseudo-Anosov mapping classes in the ball of radius decrease to 0 as increases? We show that any finite subset of the mapping class group is contained in a finite generating set such that this proportion decreases exponentially. Our strategy applies to weakly hyperbolic groups and does not refer to the automatic structure of the group.

Paper Structure

This paper contains 9 sections, 1 theorem, 117 equations, 6 figures.

Table of Contents

  1. Introduction
  2. History and related problems
  3. Strategy for Theorem \ref{['thm:generic']}
  4. Witnessing and Schottky sets
  5. Pivoting in a random walk
  6. Pivoting and translation lengths
  7. Proof of Theorem \ref{['thm:generic']}
  8. The proof of Claim \ref{['claim:prop4.2']}
  9. Sketch of the proof of Proposition \ref{['prop:genericGekht']}

Key Result

Theorem A

Let $X$ be either a Gromov hyperbolic space or $\mathop{\mathrm{\mathcal{T}}}\nolimits(\Sigma)$. Let also $G$ be a finitely generated non-elementary subgroup of $\mathop{\mathrm{Isom}}\nolimits(X)$ and $S' \subseteq G$ be a finite subset. Then there exist $L, K>0$ and a finite generating set $S \sup holds for each $n$.

Figures (6)

  • Figure 1: Schematics for $[x, y]$ being $D$-marked with $([x_{i}, y_{i}])_{i=1}^{4}$, $([z_{i}, x_{i}])_{i=1}^{4}$.
  • Figure 2: Schematics for Fact \ref{['lem:concat']} and \ref{['lem:concatUlt']}.
  • Figure 3: Schematics for Fact \ref{['lem:farSegment']}, \ref{['lem:1segment']}.
  • Figure 4: Words $w_{j}$'s, $a_{j}$'s and $b_{j}$'s that arise from a trajectory.
  • Figure 5: Schematics for Criteria (A), (B) for the construction of $P_{k}$. The upper configuration describes the situation when $k$ is added in $P_{k}$. In the lower configuration, $\{i(1) < i(2) < i(3)\}$ satisfies items (i), (ii) in Criterion (B). Here, the shaded subsegments of the dashed lines fellow travel $\gamma_{1}$, $\eta_{2}$, $\gamma_{2}$ and $\eta_{3}$, from left to right, respectively. The newly chosen $z_{k}$ is highlighted by a circle.
  • ...and 1 more figures

Theorems & Definitions (21)

  • Theorem A: Translation length grows linearly
  • Definition 2.1: Witnessing in $\delta$-hyperbolic spaces
  • Definition 2.2: Witnessing in $\mathop{\mathrm{\mathcal{T}}}\nolimits(\Sigma)$
  • Definition 2.3
  • proof
  • Definition 2.10: Schottky set
  • Remark 2.11
  • proof
  • Claim 2.13
  • proof
  • ...and 11 more