# A remark on thickness of free-by-cyclic groups

## TL;DR

This paper establishes a direct link between polynomial growth automorphisms of free groups and the thickness property of the resulting free-by-cyclic groups, clarifying their geometric structure.

## Contribution

It explicitly relates polynomial growth rates of automorphisms to the thickness order of free-by-cyclic groups, expanding understanding of their geometric and algebraic properties.

## Key findings

- Free-by-cyclic groups are strongly thick of order equal to the polynomial growth rate.
- The relationship between polynomial growth and thickness is made explicit.
- Free-by-cyclic groups admit relatively hyperbolic structures with thick peripheral subgroups.

## Abstract

Let $F$ be a free group of positive, finite rank and let $\Phi\in Aut(F)$ be a polynomial-growth automorphism. Then $F\rtimes_\Phi\mathbb Z$ is strongly thick of order $\eta$, where $\eta$ is the rate of polynomial growth of $\phi$. This fact is implicit in work of Macura, but her work predates the notion of thickness. Therefore, in this note, we make the relationship between polynomial growth and thickness explicit. Our result combines with a result independently due to Dahmani-Li, Gautero-Lustig, and Ghosh to show that free-by-cyclic groups admit relatively hyperbolic structures with thick peripheral subgroups.