University of Illinois Urbana-Champaign

Trees, dendrites, and the Cannon-Thurston map

Field, Elizabeth C

Content Files
FIELD-DISSERTATION-2020.pdf

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https://hdl.handle.net/2142/107885

Description

Title
Trees, dendrites, and the Cannon-Thurston map
Author(s)
Field, Elizabeth C
Issue Date
2020-04-16
Director of Research (if dissertation) or Advisor (if thesis)
Kapovich, Ilya
Doctoral Committee Chair(s)
Leininger, Christopher J
Committee Member(s)
  • Dunfield, Nathan
  • Schupp, Paul
Department of Study
Mathematics
Discipline
Mathematics
Degree Granting Institution
University of Illinois at Urbana-Champaign
Degree Name
Ph.D.
Degree Level
Dissertation
Date of Ingest
2020-08-26T21:54:20Z
Keyword(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.
Graduation Semester
2020-05
Type of Resource
Thesis
Permalink
http://hdl.handle.net/2142/107885
Copyright and License Information
Copyright 2020 Elizabeth Field

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