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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


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