Simple closed curves, finite covers of surfaces, and power subgroups of Out(F_n)

TL;DR

This paper constructs specific examples of finite covers of surfaces and subgroups of free groups where homology and automorphism properties defy previous expectations, providing new insights into the structure of Out(F_n).

Contribution

It introduces novel constructions of finite covers and subgroups with unusual homological and automorphic properties, answering several open questions.

Findings

Finite covers where homology isn't spanned by lifts of simple closed curves.

Subgroups of F_n with homology not generated by powers of certain elements.

The quotient of Out(F_n) by powers of transvections often contains infinite order elements.

Abstract

We construct examples of finite covers of punctured surfaces where the first rational homology is not spanned by lifts of simple closed curves. More generally, for any set $\mathcal{O} \subset F_n$ which is contained in the union of finitely many $Aut(F_n)$-orbits, we construct finite-index normal subgroups of $F_n$ whose first rational homology is not spanned by powers of elements of $\mathcal{O}$. These examples answer questions of Farb-Hensel, Kent, Looijenga, and Marche. We also show that the quotient of $Out(F_n)$ by the subgroup generated by kth powers of transvections often contains infinite order elements, strengthening a result of Bridson-Vogtmann saying that it is often infinite. Finally, for any set $\mathcal{O} \subset F_n$ which is contained in the union of finitely many $Aut(F_n)$-orbits, we construct integral linear representations of free groups that have infinite image and map all elements of $\mathcal{O}$ to torsion elements.