## Duke Mathematical Journal

### Simple closed curves, finite covers of surfaces, and power subgroups of $\operatorname{Out}(F_{n})$

#### 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 $\operatorname{Aut}(F_{n})$-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 and Hensel, Kent, Looijenga, and Marché. We also show that the quotient of $\operatorname{Out}(F_{n})$ by the subgroup generated by $k^{th}$ powers of transvections often contains infinite-order elements, strengthening a result of Bridson and Vogtmann that it is often infinite. Finally, for any set $\mathcal{O}\subset F_{n}$ which is contained in the union of finitely many $\operatorname{Aut}(F_{n})$-orbits, we construct integral linear representations of free groups that have infinite image and that map all elements of $\mathcal{O}$ to torsion elements.

#### Article information

Source
Duke Math. J., Volume 168, Number 14 (2019), 2701-2726.

Dates
Revised: 12 January 2019
First available in Project Euclid: 10 September 2019

https://projecteuclid.org/euclid.dmj/1568081121

Digital Object Identifier
doi:10.1215/00127094-2019-0022

#### Citation

Malestein, Justin; Putman, Andrew. Simple closed curves, finite covers of surfaces, and power subgroups of $\operatorname{Out}(F_{n})$. Duke Math. J. 168 (2019), no. 14, 2701--2726. doi:10.1215/00127094-2019-0022. https://projecteuclid.org/euclid.dmj/1568081121

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