Unbounded asymmetry of stretch factors

TL;DR

This paper demonstrates that the bound relating stretch factors of automorphisms of free groups and their inverses depends on the rank of the group, showing the bound cannot be uniform across all ranks.

Contribution

It proves that the constant bounding the ratio of logarithms of stretch factors varies with the free group's rank, refuting the possibility of a universal bound.

Findings

The constant C_N depends on N and cannot be chosen independently.

The ratio of logarithms of stretch factors is unbounded as N varies.

The result refines understanding of automorphism dynamics in free groups.

Abstract

A result of Handel-Mosher guarantees that the ratio of logarithms of stretch factors of any fully irreducible automorphism of the free group $F_N$ and its inverse is bounded by a constant $C_N$. In this short note we show that this constant $C_N$ cannot be chosen independent of $N$.