Counting Conjugacy Classes in $Out(F_N)$

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 $\varphi$ from an $R$-ball in the Cayley graph of $Out(F_N)$ with $\log \lambda (\varphi )$ on the order of $R$ grows exponentially in $R$.