Simple closed curves, finite covers of surfaces, and power subgroups of 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 $Aut(F_n)$-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.

19 pages; minor revision; to appear in Duke Math. J

References in corpus (5)

• Quotients of surface groups and homology of finite covers via quantum representations
• Finite covers of graphs, their primitive homology, and representation theory
• Every Infinite order mapping class has an infinite order action on the homology of some finite cover
• Homological eigenvalues of lifts of pseudo-Anosov mapping classes to finite covers
• Virtual homological spectral radii for automorphisms of surfaces

Cited by in corpus (8)

• Curves with prescribed symmetry and associated representations of mapping class groups
• Subrepresentations in the homology of finite covers of graphs
• A remark on the homology of finite coverings of a surface
• Mapping class groups of surfaces of genus $\ge 3$ do not virtually surject to $\mathbb{Z}$
• Linear representations of hyperelliptic mapping class groups
• Graph coverings and (im)primitive homology: some new examples of exceptionally low degree
• The Extrinsic Primitive Torsion Problem
• Simple closed curves, non-kernel homology and Magnus embedding