Hyperbolic extensions of free groups

Dowdall, Spencer; Taylor, Samuel

doi:10.2140/gt.2018.22.517

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Hyperbolic extensions of free groups

Spencer Dowdall and Samuel J Taylor

Geometry & Topology 22 (2018) 517–570

DOI: 10.2140/gt.2018.22.517

Abstract

Given a finitely generated subgroup $Γ \leq Out (F)$ of the outer automorphism group of the rank- $r$ free group $F = F_{r}$ , there is a corresponding free group extension $1 \to F \to E_{Γ} \to Γ \to 1$ . We give sufficient conditions for when the extension $E_{Γ}$ is hyperbolic. In particular, we show that if all infinite-order elements of $Γ$ are atoroidal and the action of $Γ$ on the free factor complex of $F$ has a quasi-isometric orbit map, then $E_{Γ}$ is hyperbolic. As an application, we produce examples of hyperbolic $F$ –extensions $E_{Γ}$ for which $Γ$ has torsion and is not virtually cyclic. The proof of our main theorem involves a detailed study of quasigeodesics in Outer space that make progress in the free factor complex. This may be of independent interest.

Keywords

hyperbolic group extensions, $\mathrm{Out}(\mathbb{F}_n)$, Outer space, free factor complex

Mathematical Subject Classification 2010

Primary: 20F28, 20F67

Secondary: 20E06, 57M07

References

Publication

Received: 9 March 2016

Revised: 20 October 2016

Accepted: 23 November 2016

Published: 31 October 2017

Proposed: Walter Neumann

Seconded: Benson Farb, Bruce Kleiner

Authors

Spencer Dowdall
Department of Mathematics Vanderbilt University Nashville, TN United States

Samuel J Taylor
Department of Mathematics Temple University Philadelphia, PA United States

Volume 22, issue 1 (2018)