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Trees, dendrites and the Cannon–Thurston map

Elizabeth Field

Algebraic & Geometric Topology 20 (2020) 3083–3126
DOI: 10.2140/agt.2020.20.3083

When 1 H G Q 1 is a short exact sequence of three word-hyperbolic groups, Mahan Mj (formerly Mitra) 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, Mj 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 and Lustig and Dowdall, Kapovich and Taylor, who prove that in the case where H is a free group and Q is a convex cocompact purely atoroidal subgroup of Out(FN), one can identify the resultant quotient space with a certain –tree in the boundary of Culler and Vogtmann’s Outer space.

Cannon–Thurston map, hyperbolic group, algebraic lamination, dendrite, Gromov boundary
Mathematical Subject Classification 2010
Primary: 20F65
Secondary: 20E07, 20F67, 57M07
Received: 25 July 2019
Revised: 22 January 2020
Accepted: 15 February 2020
Published: 8 December 2020
Elizabeth Field
Department of Mathematics
University of Illinois at Urbana-Champaign
Urbana, IL
United States