21.125 (2026)

(M. Bridson). Let $F_m$ be a free group of rank $m$ and let $\phi \in \text{Aut}(F_m)$ be a polynomially growing automorphism of maximal degree $m - 1$, which means that for some (equivalently, any) free basis $\{x_1, \dots, x_m\}$ of $F_m$, the sequence $\max_i |\phi^n(x_i)|$ grows at the rate of $n^{m-1}$, where $|g|$ denotes the minimal length of $g$ in the $x_i$ and their inverses.
$\qquad$ a) Is the free-by-cyclic group $F_m \rtimes_\phi \mathbb{Z}$ virtually special?
$\qquad$ b) In particular, are the Hydra groups
$$G_m = F_m \rtimes \mathbb{Z} = \langle a_1, \dots, a_m, t \mid t^{-1}a_1t = a_1, t^{-1}a_it = a_i a_{i-1} \text{ for all } i > 1 \rangle$$ virtually special?