Trees, dendrites and the Cannon–Thurston map

Field, Elizabeth

doi:10.2140/agt.2020.20.3083

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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

Abstract

When $1 \to H \to G \to Q \to 1$ 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 $Out (F_{N})$ , one can identify the resultant quotient space with a certain $ℝ$ –tree in the boundary of Culler and Vogtmann’s Outer space.

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Keywords

Cannon–Thurston map, hyperbolic group, algebraic lamination, dendrite, Gromov boundary

Mathematical Subject Classification 2010

Primary: 20F65

Secondary: 20E07, 20F67, 57M07

References

Publication

Received: 25 July 2019

Revised: 22 January 2020

Accepted: 15 February 2020

Published: 8 December 2020

Authors

Elizabeth Field
Department of Mathematics University of Illinois at Urbana-Champaign Urbana, IL United States

Volume 20, issue 6 (2020)