Gromov boundaries of complexes associated to surfaces

Pho-on, Witsarut

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https://hdl.handle.net/2142/97398 Copy

Description

Title

Gromov boundaries of complexes associated to surfaces

Author(s)

Pho-on, Witsarut

Issue Date

2017-04-19

Director of Research (if dissertation) or Advisor (if thesis)

Leininger, Christopher

Doctoral Committee Chair(s)

Dunfield, Nathan

Committee Member(s)

• Kapovich, Ilya
• Bradlow, Steven

Department of Study

Mathematics

Discipline

Mathematics

Degree Granting Institution

University of Illinois at Urbana-Champaign

Degree Name

Ph.D.

Degree Level

Dissertation

Keyword(s)

• Gromov boundary
• Curve complex
• Arc complex
• Lamination
• Surface
• Unicorn curve
• Bicorn curve

Abstract

In 1996, Masur and Minsky showed that the curve graph is hyperbolic. Recently, Hensel, Przytycki, and Webb proved a stronger result which was the uniform hyperbolicity of the curve graph, and they also gave the first proof of the uniform hyperbolicity of the arc graph using unicorn arcs. For closed surfaces, their proof is indirect, but Przytycki and Sisto gave a more direct proof of hyperbolicity in that case using bicorn curves. In this dissertation, we extend the notion of unicorn arcs and bicorn curves between two arcs or curves to the case where we replace one arc or curve with a geodesic asymptotic to a lamination or a leaf of the lamination. Using these paths, we give new proofs of the results of Klarreich and Schleimer identifying the Gromov boundaries of the curve graph and the arc graph, respectively, as spaces of laminations.

Graduation Semester

2017-05

Type of Resource

text

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http://hdl.handle.net/2142/97398

Copyright and License Information

Copyright 2017 Witsarut Pho-on

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