Counting conjugacy classes of fully irreducibles: double exponential growth

TL;DR

This paper establishes double exponential growth bounds for counting conjugacy classes of fully irreducible automorphisms in free groups, revealing complex behavior distinct from surface or hyperbolic systems.

Contribution

It proves double exponential bounds for the number of conjugacy classes of fully irreducibles in Out(F_r), a novel result in the context of free group automorphisms.

Findings

Double exponential growth bounds for conjugacy classes

Behavior differs from surface and hyperbolic systems

Provides new insights into Out(F_r) dynamics

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.