Detecting Free Group Automorphisms via Virtual Homology Representations

Let $F_n=F⟨x_1,\mathrm{.}\mathrm{.}\mathrm{.},x_n⟩$ denote the free group of rank $n\ge 2$ and let $\mathrm{E}\mathrm{n}\mathrm{d}(F_n)$ be the endomorphism monoid of $F_n$. We show that automorphisms of $F_n$ are detected via the $\mathrm{E}\mathrm{n}\mathrm{d}(F_n)$-action on the first integral homology of finite characteristic covers of the wedge of $n\ge 2$ circles $R_n$. This gives a homological characterization of homotopy equivalences of $R_n$ that we utilize to show that $\mathrm{E}\mathrm{n}\mathrm{d}(F_n)$ is asymptotically linear. We extend these results by showing that the $\mathrm{O}\mathrm{u}\mathrm{t}(F_n)$-action on the homology of iterated covers of a punctured surface ${\mathrm{\Sigma }}_g^b$ of the same homotopy type as $R_n$ detects homeomorphisms of ${\mathrm{\Sigma }}_g^b$ in homotopy classes of homotopy equivalences of $R_n$.