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