2015/06/23 by Dowdall, Spencer, Kapovich, Ilya, Taylor, Samuel J.
#37B #37D #Dynamical Systems (math.DS) #FOS: Mathematics #Geometric Topology (math.GT) #Group Theory (math.GR) #Primary 20F65 #Secondary 57M
paper · doi:10.48550/arxiv.1506.06974
This paper gives a detailed analysis of the Cannon--Thurston maps associated to a general class of hyperbolic free group extensions. Let FN denote a free groups of finite rank N≥ 3 and consider a convex cocompact subgroup Γ≤ Out(FN), i.e. one for which the orbit map from Γ into the free factor complex of FN is a quasi-isometric embedding. The subgroup Γ determines an extension EΓ of FN, and the main theorem of Dowdall--Taylor \citeDT1 states that in this situation EΓ is hyperbolic if and only if Γ is purely atoroidal. Here, we give an explicit geometric description of the Cannon--Thurston maps ∂ FN→∂ EΓ 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 2N. This theorem generalizes the main result of Kapovich and Lustig \citeKapLusCT 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.