Trees, dendrites, and the Cannon-Thurston map

Abstract

When 1 -> H -> G -> Q -> 1 is a short exact sequence of three infinite, word-hyperbolic groups, Mahan Mitra (Mj) has shown that the inclusion map from H to G extends continuously to a map between the Gromov boundaries of H and G. This boundary map is known as the Cannon-Thurston map. In this context, Mitra associates to every point z in the Gromov boundary of Q an "ending lamination" on H which consists of pairs of distinct points in the boundary of H. We prove that for each such z, the quotient of the Gromov boundary of H by the equivalence relation generated by this ending lamination is a dendrite, that is, a tree-like topological space. This result generalizes the work of Kapovich-Lustig and Dowdall-Kapovich-Taylor, who prove that in the case where H is a free group and Q is a convex cocompact purely atoroidal subgroup of Out(F_N), one can identify the resultant quotient space with a certain $\mathbb{R}$-tree in the boundary of Culler-Vogtmann's Outer space.

arXiv comment27 pages. Added Convention 4.9, which alters the proofs of the remaining results in section 4, as well as other minor revisions. Version accepted to Algebraic and Geometric Topology

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Cited by in corpus (2)

• Pullbacks of metric bundles and Cannon-Thurston maps
• Limit pretrees for free group automorphisms: existence