Cannon–Thurston maps for hyperbolic free group extensions

Abstract

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.

Algebraic ending laminations and quasiconvexity

“…We now turn to Case 2 and indicate briefly how the arguments of [DKT16] go through in this case to prove the analogous statement Proposition 5.8 below. Theorem 4.1 and Lemma 4.12 of [DT14] establish stability of F n −progressing quasigeodesics.…”

“…Theorem 4.1 and Lemma 4.12 of [DT14] establish stability of F n −progressing quasigeodesics. While Lemma 5.5 of [DKT16] is necessary to prove flaring for Case 1 above, flaring in Case 2 follows from hyperbolicity. (In fact it is shown in [MS12, Section 5.3, Proposition 5.8] that flaring is equivalent to hyperbolicity of X z ).…”

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

“…For the application needed here, the following proposition from [13] is most convenient. Proposition 4.9 (Folding rays to infinity [13,Proposition 5.6]). Suppose that Γ Out(F) is purely atoroidal and qi-embeds into F. For any k 0 there is a K 0 such that if (g i ) i 0 is a k-quasigeodesic ray in Γ, then there is an infinite length folding ray γ : I → X parameterized at unit speed with the following properties.…”

“…These properties turn out to characterize convex cocompact subgroups of Out(F) among the class of subgroups inducing hyperbolic extensions of F. Suppose henceforth that 1 → F → E p → Q → 1 is a hyperbolic extension of F. This short exact sequence induces an outer action of Q on F given by the homomorphism Q → Out(F) sending q ∈ Q to the class of the automorphism that conjugates F E by any liftq ∈ E of q. We then have the commutative diagram (13) where Γ is the image of Q → Out(F). Fixing finite generating sets for E and Q, for each element a ∈ F we continue to write a * for a geodesic in E joining a −∞ ∈ ∂E to a ∞ ∈ ∂E.…”

“…We note that Mj and Rafi [39] have recently, and independently, proven Theorem 7.9 by very different methods. Their approach uses structural results on convex cocompact subgroups proven in [15] as well as a characterization of the Cannon-Thurston map for (1) that we obtained with Kapovich in [13] and which builds on earlier work of Mj [36]. Our proof is more direct and proceeds as follows.…”