Title: Cannon-Thurston maps for hyperbolic free group extensions

 

Abstract: Let FN be a free group of finite rank N ≥ 3 and let Γ ≤ Out(FN) be a finitely generated "convex cocompact" subgroup, that is, such that the orbit map from Γ to the free factor complex of FN is a quasi-isometric embedding. Assume also that Γ is purely atoroidal. In this case Γ determines an extension group EΓ of Γ with the quotient EΓ/FN=Γ, and it is known by a result of Dowdall and Taylor that the group EΓ is then word-hyperbolic. By a general result of Mitra the inclusion of FN in EΓ extends to a continuous surjective FN-equivariant map between their hyperbolic boundaries j: ∂ FN → ∂ EΓ, called the Cannon-Thurston map. We analyze the structure of this map and prove that the map is finite-to-one, with multiplicity at most 2N. The talk is based on a joint paper with Spencer Dowdall and Sam Taylor.