Duke Mathematical Journal

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

Justin Malestein and Andrew Putman

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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 OFn which is contained in the union of finitely many Aut(Fn)-orbits, we construct finite-index normal subgroups of Fn whose first rational homology is not spanned by powers of elements of O. These examples answer questions of Farb and Hensel, Kent, Looijenga, and Marché. We also show that the quotient of Out(Fn) by the subgroup generated by kth powers of transvections often contains infinite-order elements, strengthening a result of Bridson and Vogtmann that it is often infinite. Finally, for any set OFn which is contained in the union of finitely many Aut(Fn)-orbits, we construct integral linear representations of free groups that have infinite image and that map all elements of O to torsion elements.

Article information

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

Received: 25 March 2018
Revised: 12 January 2019
First available in Project Euclid: 10 September 2019

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Digital Object Identifier

Primary: 57M10: Covering spaces
Secondary: 20F34: Fundamental groups and their automorphisms [See also 57M05, 57Sxx] 20C05: Group rings of finite groups and their modules [See also 16S34]

covering spaces surfaces restricted Lie algebras central series p-groups transvections linear representations mapping class group outer automorphism group of the free group representations of finite groups simple closed curves primitives


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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