Hyperbolic extensions of free groups

Abstract

Given a finitely generated subgroup Γ ≤ Out(F) of the outer automorphism group of the rank r free group F = F r , there is a corresponding free group extension 1 → F → E Γ → Γ → 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.

The co‐surface graph and the geometry of hyperbolic free group extensions

“…Fix a rose R ∈ X and a primitive conjugacy class α represented by a petal of R. Fix a finitely generated subgroup Γ Out(F) such that the orbit map Γ → F given by g → g · α is a quasi-isometric embedding. In [15], we show that this implies that the orbit Γ · R has strong quasiconvexity properties in X (for example, Theorem 2.6). For the application needed here, the following proposition from [13] is most convenient.…”

“…Theorem 2.6 (Dowdall-Taylor [15]). Let γ : I → X be a K-quasigeodesic whose projection π • γ : I → F is also a K-quasigeodesic.…”

“…The folding path comes equipped with a family of folding maps {φ st : G s → G t } s t satisfying φ 0L = φ, φ ss = Id Gs , and φ rt = φ st • φ rs with φ rs and φ rt inducing the same illegal turn structure on G r for all 0 r s t L. Our convention is to parameterize the folding path {G t } t∈[0,L] with respect to arc length in Outer space. With this convention, the assignment t → γ(t) = G t defines a directed geodesic γ : [15,Section 2] for the details of this construction; further properties will be recalled in § 6.2.…”

“…Following our convention from [15, § 6], the legal length of α|G t is defined to be the sum leg(α|G t ) of the lengths of those maximal legal segments that have length at least 3. The following basic fact appears as [15, Lemma 6.10] and follows directly from work in [3]: Lemma 6.2 [15]. For any folding path G t , t ∈ [a, b], every nontrivial conjugacy class α ∈ F satisfies…”

“…Theorem 1.1 [15]. Suppose that a finitely generated subgroup Γ Out(F) is purely atoroidal and qi-embeds into F. Then the free group extension E Γ in equation (1) is hyperbolic.…”