# Trees, dendrites, and the Cannon-Thurston map

## TL;DR

This paper proves that for certain hyperbolic group extensions, the boundary quotient associated with ending laminations forms a dendrite, generalizing previous results about trees in Outer space.

## Contribution

It establishes that the boundary quotient space is a dendrite for a broad class of hyperbolic group extensions, extending prior specific cases.

## Key findings

- The boundary quotient is a dendrite for hyperbolic group extensions.
- Generalizes previous results from free groups to broader hyperbolic groups.
- Connects boundary laminations with dendritic topologies in hyperbolic group theory.

## 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.