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1 October 2019 Simple closed curves, finite covers of surfaces, and power subgroups of Out(Fn)
Justin Malestein, Andrew Putman
Duke Math. J. 168(14): 2701-2726 (1 October 2019). DOI: 10.1215/00127094-2019-0022

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 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.

Citation

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Justin Malestein. Andrew Putman. "Simple closed curves, finite covers of surfaces, and power subgroups of Out(Fn)." Duke Math. J. 168 (14) 2701 - 2726, 1 October 2019. https://doi.org/10.1215/00127094-2019-0022

Information

Received: 25 March 2018; Revised: 12 January 2019; Published: 1 October 2019
First available in Project Euclid: 10 September 2019

zbMATH: 07131297
MathSciNet: MR4012346
Digital Object Identifier: 10.1215/00127094-2019-0022

Subjects:
Primary: 57M10
Secondary: 20C05, 20F34

Rights: Copyright © 2019 Duke University Press

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Vol.168 • No. 14 • 1 October 2019
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