Trees, dendrites and the Cannon–Thurston map

Field, Elizabeth

doi:10.2140/agt.2020.20.3083

Download this article
	For screen
	For printing

Recent Issues

The Journal

About the Journal

Editorial Board

Editorial Interests

Subscriptions

Submission Guidelines

Submission Page

Policies for Authors

Ethics Statement

ISSN (electronic): 1472-2739

ISSN (print): 1472-2747

Author Index

To Appear

Other MSP Journals

This article is available for purchase or by subscription. See below.

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.

PDF Access Denied

We have not been able to recognize your IP address 44.197.111.121 as that of a subscriber to this journal.
Online access to the content of recent issues is by subscription, or purchase of single articles.

Please contact your institution's librarian suggesting a subscription, for example by using our journal-recommendation form. Or, visit our subscription page for instructions on purchasing a subscription.

You may also contact us at contact@msp.org
or by using our contact form.

Or, you may purchase this single article for USD 40.00:

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)