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

Justin Malestein; Andrew Putman

doi:10.1215/00127094-2019-0022

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arXiv · 1708.06486

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

Justin Malestein·Andrew Putman

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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.

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BibTeX RIS

Justin Malestein, Andrew Putman. 2019-03-20. Simple closed curves, finite covers of surfaces, and power subgroups of Out(F_n). https://doi.org/10.1215/00127094-2019-0022

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