Detecting Free Group Automorphisms via Virtual Homology Representations

TL;DR

This paper introduces a homological method to detect automorphisms of free groups and their actions on surface covers, establishing new characterizations and asymptotic properties of endomorphisms.

Contribution

It provides a novel homological characterization of free group automorphisms and shows that the endomorphism monoid is asymptotically linear, extending to surface homeomorphisms.

Findings

Automorphisms of free groups are detected via homology of finite covers.

End(F_n) is shown to be asymptotically linear.

Homology actions detect surface homeomorphisms in homotopy classes.

Abstract

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 $R_n$.