## Files in this item

FilesDescriptionFormat

application/pdf

FIELD-DISSERTATION-2020.pdf (526kB)
(no description provided)PDF

## Description

 Title: Trees, dendrites, and the Cannon-Thurston map Author(s): Field, Elizabeth C Director of Research: Kapovich, Ilya Doctoral Committee Chair(s): Leininger, Christopher J Doctoral Committee Member(s): Dunfield, Nathan; Schupp, Paul Department / Program: Mathematics Discipline: Mathematics Degree Granting Institution: University of Illinois at Urbana-Champaign Degree: Ph.D. Genre: Dissertation Subject(s): Cannon-Thurston map hyperbolic group algebraic lamination dendrite Gromov boundary Abstract: When $1\to H\to G\to Q\to 1$ is a short exact sequence of three 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 $\mathrm{Out}(F_N)$, one can identify the resultant quotient space with a certain $\mathbb{R}$-tree in the boundary of Culler-Vogtmann's Outer space. Issue Date: 2020-04-16 Type: Thesis URI: http://hdl.handle.net/2142/107885 Rights Information: Copyright 2020 Elizabeth Field Date Available in IDEALS: 2020-08-26 Date Deposited: 2020-05
﻿