A remark on thickness of free-by-cyclic groups

Abstract

Let $F$ be a free group of positive, finite rank and let $\Phi \in Aut\left(F\right)$ be a polynomial-growth automorphism. Then $F{⋊}_{\Phi }\mathbb{Z}$ is strongly thick of order $\eta$, where $\eta$ is the rate of polynomial growth of $\varphi$. This fact is implicit in work of Macura, whose results predate the notion of thickness. Therefore, in this note, we make the relationship between polynomial growth of 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.