Counting conjugacy classes of fully irreducibles: double exponential growth

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.

updated version; to appear in Geometriae Dedicata

References in corpus (1)

• A train track directed random walk on $Out(F_r)$

Cited by in corpus (1)

• Taking the High-Edge Route Through Outer Space in Rank 3