Counting Conjugacy Classes in $Out(F_N)$

TL;DR

This paper demonstrates exponential growth in the number of conjugacy classes of certain elements in groups with specific actions, with applications to automorphisms of free groups and their growth rates.

Contribution

It establishes exponential growth of conjugacy classes in groups with non-elementary WPD actions, applying this to $Out(F_N)$ for $N \\ge 3$ to analyze fully irreducible automorphisms.

Findings

Exponential growth of conjugacy classes in groups with WPD actions.

Application to $Out(F_N)$ shows exponential growth of conjugacy classes of fully irreducible automorphisms.

Growth rate of $\\log\lambda(\phi)$ is proportional to the radius in the Cayley graph.

Abstract

We show that if a f.g. group $G$ has a non-elementary WPD action on a hyperbolic metric space $X$, then the number of $G$-conjugacy classes of $X$-loxodromic elements of $G$ coming from a ball of radius $R$ in the Cayley graph of $G$ grows exponentially in $R$. As an application we prove that for $N\ge 3$ the number of distinct $Out(F_N)$-conjugacy classes of fully irreducibles $\phi$ from an $R$-ball in the Cayley graph of $Out(F_N)$ with $\log\lambda(\phi)$ on the order of $R$ grows exponentially in $R$.