#### Volume 20, issue 6 (2020)

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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 $Out\left({F}_{N}\right)$, one can identify the resultant quotient space with a certain $ℝ$–tree in the boundary of Culler and Vogtmann’s Outer space.

##### Keywords
Cannon–Thurston map, hyperbolic group, algebraic lamination, dendrite, Gromov boundary
##### Mathematical Subject Classification 2010
Primary: 20F65
Secondary: 20E07, 20F67, 57M07