Sieve methods in group theory \Rmnum{3}: $\aut(F_n)$

TL;DR

This paper investigates the structure of automorphisms of free groups, showing that non-iwip and non-hyperbolic elements form an exponentially small subset within a specific kernel subgroup.

Contribution

It provides a new exponential smallness result for certain automorphisms in the kernel of a natural epimorphism in group theory.

Findings

Non-iwip and non-hyperbolic elements are exponentially rare in the kernel subgroup.

The paper advances understanding of the distribution of automorphisms in free group automorphism groups.

Abstract

Let $\pi:\aut(F_n)\rightarrow \aut(\Z^n)$ be the epimorphism induced by the isomorphism $\Z^n \cong F_n/F_n'$ and define $\mathcal{T}_n:=\ker\pi$. We prove that the subset of $\mathcal{T}_n$ consists of all non-iwip and all non-hyperbolic elements is exponentially small.