Detecting Free Group Automorphisms via Virtual Homology Representations

Let $F_n= F\langle x_1,...,x_n\rangle$ denote the free group of rank $n\ge 2$ and let $\mathrm{End}(F_n)$ be the endomorphism monoid of $F_n$. We show that automorphisms of $F_n$ are detected via the $\mathrm{End}(F_n)$-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 $\mathrm{End}(F_n)$ is asymptotically linear. We extend these results by showing that the $\mathrm{Out}(F_n)$-action on the homology of iterated covers of a punctured surface $\Sigma_g^b$ of the same homotopy type as $R_n$ detects homeomorphisms of $\Sigma_g^b$ in homotopy classes of homotopy equivalences of $R_n$.