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.

Given a free-by-cyclic group $G = F_N \rtimes_\varphi \mathbb{Z}$ determined by any outer automorphism $\varphi \in \mathrm{Out}(F_N)$ which is represented by an expanding irreducible train-track map $f$, we construct a $K(G,1)$ $2$-complex $X$ called the folded mapping torus of $f$, and equip it with a semiflow. We show that $X$ enjoys many similar properties to those proven by Thurston and Fried for the mapping torus of a pseudo-Anosov homeomorphism. In particular, we construct an open, convex cone $\mathcal{A} \subset H^1(X;\mathbb{R}) = \mathrm{Hom}(G;\mathbb{R})$ containing the homomorphism $u_0 \colon G \to \mathbb{Z}$ having $\mathrm{ker}(u_0) = F_N$, a homology class $\epsilon \in H_1(X;\mathbb{R})$, and a continuous, convex, homogeneous of degree $-1$ function $\mathfrak H\colon\mathcal{A} \to \mathbb{R}$ with the following properties. Given any primitive integral class $u \in \mathcal{A}$ there is a graph $\Theta_u \subset X$ such that: (1) the inclusion $\Theta_u \to X$ is $\pi_1$-injective and $\pi_1(\Theta_u) = \mathrm{ker}(u)$, (2) $u(\epsilon) = \chi(\Theta_u)$, (3) $\Theta_u \subset X$ is a section of the semiflow and the first return map to $\Theta_u$ is an expanding irreducible train track map representing $\varphi_u \in \mathrm{Out}(\mathrm{ker}(u))$ such that $G = \mathrm{ker}(u) \rtimes_{\varphi_u} \mathbb{Z}$, (4) the logarithm of the stretch factor of $\varphi_u$ is precisely $\mathfrak H(u)$, (5) if $\varphi$ was further assumed to be hyperbolic and fully irreducible then for every primitive integral $u\in \mathcal{A}$ the automorphism $\varphi_u$ of $\mathrm{ker}(u)$ is also hyperbolic and fully irreducible.Comment: v7: Minor organizational and stylistic changes incorporating referee's suggestions. Notably, section 6.3 in v6 has been moved to section 4.5 in v7. 67 pages, 13 figures. Final version; accepted for publication in Geometry & Topolog

We show that any isomorphism between mapping class groups of orientable infinite-type surfaces is induced by a homeomorphism between the surfaces. Our argument additionally applies to automorphisms between finite-index subgroups of these 'big' mapping class groups and shows that each finite-index subgroup has finite outer automorphism group. As a key ingredient, we prove that all simplicial automorphisms between curve complexes of infinite-type orientable surfaces are induced by homeomorphisms.

Abstract. Consider a group G and an epimorphism u 0 : G → Z inducing a splitting of G as a semidirect product ker(u 0 ) ϕ Z with ker(u 0 ) a finitely generated free group and ϕ ∈ Out(ker(u 0 )) representable by an expanding irreducible train track map. Building on our earlier work [DKL], in which we realized G as π 1 (X) for an Eilenberg-Maclane 2-complex X equipped with a semiflow ψ, and inspired by McMullen's Teichmüller polynomial for fibered hyperbolic 3-manifolds, we construct a polynomial invariant m ∈ Z[H 1 (G; Z)/torsion] for (X, ψ) and investigate its properties.Specifically, m determines a convex polyhedral cone C X ⊂ H 1 (G; R), a convex, real-analytic function H : C X → R, and specializes to give an integral Laurent polynomial mu(ζ) for each integral u ∈ C X . We show that C X is equal to the "cone of sections" of (X, ψ) (the convex hull of all cohomology classes dual to sections of of ψ), and that for each (compatible) cross section Θu ⊂ X with first return map fu : Θu → Θu, the specialization mu(ζ) encodes the characteristic polynomial of the transition matrix of fu. More generally, for every class u ∈ C X there exists a geodesic metric du and a codimension-1 foliation Ωu of X defined by a "closed 1-form" representing u transverse to ψ so that after reparametrizing the flow ψ u s maps leaves of Ωu to leaves via a local e sH(u) -homothety.Among other things, we additionally prove that C X is equal to (the cone over) the component of the BNSinvariant Σ(G) containing u 0 and, consequently, that each primitive integral u ∈ C X induces a splitting of G as an ascending HNN-extension G = Qu * φu with Qu a finite-rank free group and φu : Qu → Qu injective. For any such splitting, we show that the stretch factor of φu is exactly given by e H(u) . In particular, we see that C X and H depend only on the group G and epimorphism u 0 .

This paper gives a detailed analysis of the Cannon-Thurston maps associated to a general class of hyperbolic free group extensions. Let F denote a free groups of finite rank at least 3 and consider a convex cocompact subgroup Γ ≤ Out(F), i.e. one for which the orbit map from Γ into the free factor complex of F is a quasi-isometric embedding. The subgroup Γ determines an extension E Γ of F, and the main theorem of Dowdall-Taylor [DT1] states that in this situation E Γ is hyperbolic if and only if Γ is purely atoroidal.Here, we give an explicit geometric description of the Cannon-Thurston maps ∂F → ∂E Γ for these hyperbolic free group extensions, the existence of which follows from a general result of Mitra. In particular, we obtain a uniform bound on the multiplicity of the Cannon-Thurston map, showing that this map has multiplicity at most 2 rank(F). This theorem generalizes the main result of Kapovich and Lustig [KL5] which treats the special case where Γ is infinite cyclic. We also answer a question of Mahan Mitra by producing an explicit example of a hyperbolic free group extension for which the natural map from the boundary of Γ to the space of laminations of the free group (with the Chabauty topology) is not continuous.

Abstract. In this paper we explore the idea that Teichmüller space is hyperbolic "on average." Our approach focuses on studying the geometry of geodesics which spend a definite proportion of time in some thick part of Teichmüller space. We consider several different measures on Teichmüller space and find that this behavior for geodesics is indeed typical. With respect to each of these measures, we show that the average distance between points in a ball of radius r is asymptotic to 2r, which is as large as possible. Our techniques also lead to a statement quantifying the expected thinness of random triangles in Teichmüller space, showing that "most triangles are mostly thin."

We introduce the co-surface graph CS of a finitely generated free group F and use it to study the geometry of hyperbolic group extensions of F. Among other things, we show that the Gromov boundary of the co-surface graph is equivariantly homeomorphic to the space of free arational F-trees and use this to prove that a finitely generated subgroup of Out(F) quasi-isometrically embeds into the co-surface graph if and only if it is purely atoroidal and quasi-isometrically embeds into the free factor complex. This answers a question of I. Kapovich. Our earlier work [S. Dowdall and S. J. Taylor, 'Hyperbolic extensions of free groups', to appear in Geom. Topol.] shows that every such group gives rise to a hyperbolic extension of F, and here we prove a converse to this result that characterizes the hyperbolic extensions of F arising in this manner. As an application of our techniques, we additionally obtain a Scott-Swarup type theorem for this class of extensions.

No abstract