PolExp growth for automorphisms of toral relatively hyperbolic groups
Let $G$ be a toral relatively hyperbolic group, and let $\varphi\in\mathrm{Aut}(G)$. We prove that, under iteration of $\varphi$, the conjugacy length $||\varphi^n(g)||$ of every element $g\in G$ grows like $n^d\lambda^n$ for some $d\in\mathbb{N}$ and some algebraic integer $\lambda\geq 1$. For a given $\varphi$, only finitely many values of $d$ and $\lambda$ occur as $g$ varies in $G$. The same statements hold for the growth of the word length $|\varphi^n(g)|$. For $G$ hyperbolic, we generalize polynomial subgroups: we show that, for a given growth type $n^d\lambda^n$ other than $1$, there is a malnormal family of quasiconvex subgroups $K_1,\dots,K_p$ such that a conjugacy class $[g]$ grows at most like $n^d\lambda^n$ if and only if $g$ is conjugate into one of the subgroups $K_i$.