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Counting conjugacy classes of fully irreducibles: double exponential growth

Ilya Kapovich, Catherine Pfaff

Abstract

Inspired by results of Eskin and Mirzakhani counting closed geodesics of length $\le L$ in the moduli space of a fixed closed surface, we consider a similar question in the $Out(F_r)$ setting. The Eskin-Mirzakhani result can be equivalently stated in terms of counting the number of conjugacy classes (within the mapping class group) of pseudo-Anosovs whose dilitations have natural logarithm $\le L$. Let $\mathfrak N_r(L)$ denote the number of $Out(F_r)$-conjugacy classes of fully irreducibles satisfying that the natural logarithm of their dilatation is $\le L$. We prove for $r\ge 3$ that as $L\to\infty$, the number $\mathfrak N_r(L)$ has double exponential (in $L$) lower and upper bounds. These bounds reveal behavior not present in the surface setting or in classical hyperbolic dynamical systems.

Counting conjugacy classes of fully irreducibles: double exponential growth

Abstract

Inspired by results of Eskin and Mirzakhani counting closed geodesics of length in the moduli space of a fixed closed surface, we consider a similar question in the setting. The Eskin-Mirzakhani result can be equivalently stated in terms of counting the number of conjugacy classes (within the mapping class group) of pseudo-Anosovs whose dilitations have natural logarithm . Let denote the number of -conjugacy classes of fully irreducibles satisfying that the natural logarithm of their dilatation is . We prove for that as , the number has double exponential (in ) lower and upper bounds. These bounds reveal behavior not present in the surface setting or in classical hyperbolic dynamical systems.

Paper Structure

This paper contains 15 sections, 17 theorems, 47 equations, 1 table.

Table of Contents

  1. Introduction
  2. Implied complications to defining a Patterson-Sullivan measure on $\partial CV_r$
  3. Proof complications, ideas, and sketch
  4. Related results
  5. Definitions and Background
  6. Train track maps & (fully) irreducible outer automorphisms.
  7. Periodic Nielsen paths
  8. Whitehead graphs
  9. Full irreducibility criterion.
  10. Fold lines in Outer space
  11. Stallings fold decompositions
  12. Lone axis fully irreducible outer automorphism
  13. The Automorphisms
  14. Counting
  15. Questions

Key Result

Theorem 1.1

For each integer $r\ge 3$, there exist constants $a=a(r)>1,b=b(r)>1, c=c(r)>1$ so that: For $L\ge 1$, let $\mathfrak N_r(L)$ denote the number of $\textup{Out}(F_r)$-conjugacy classes of fully irreducibles $\varphi\in \textup{Out}(F_r)$ with $\log\lambda(\varphi)\le L$. Then there exists an $L_0\ge where $e$ is the base of the natural logarithm. Therefore, $c^{e^L}$ bounds from below the number o

Theorems & Definitions (48)

  • Theorem 1.1
  • Theorem 1.2
  • Definition 2.1: Graph maps & train track maps
  • Definition 2.2: Directions
  • Definition 2.3: Turns
  • Definition 2.4: Transition matrix $M(g)$, Perron-Frobenius matrix, Perron-Frobenius eigenvalue
  • Definition 2.5: Stretch factor of a fully irreducible
  • Definition 2.6: Nielsen paths & rotationless powers
  • Proposition 2.7
  • Definition 2.8: Ageometric
  • ...and 38 more