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 which is contained in the union of finitely many -orbits, we construct finite-index normal subgroups of whose first rational homology is not spanned by powers of elements of . These examples answer questions of Farb and Hensel, Kent, Looijenga, and Marché. We also show that the quotient of by the subgroup generated by powers of transvections often contains infinite-order elements, strengthening a result of Bridson and Vogtmann that it is often infinite. Finally, for any set which is contained in the union of finitely many -orbits, we construct integral linear representations of free groups that have infinite image and that map all elements of to torsion elements.
"Simple closed curves, finite covers of surfaces, and power subgroups of ." Duke Math. J. 168 (14) 2701 - 2726, 1 October 2019. https://doi.org/10.1215/00127094-2019-0022