Trees, dendrites, and the Cannon-Thurston map

Field, Elizabeth C

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

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

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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