Gromov boundaries of complexes associated to surfaces

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.

Name thesis:degree_name

Ph.D.

Level thesis:degree_level

Dissertation

Discipline thesis:degree_discipline

Mathematics

Grantor

University of Illinois at Urbana-Champaign

Year dc:date

2017

Author dc:creator

• Pho-on, Witsarut

Contributors dc:contributor

• Leininger, Christopher
• Dunfield, Nathan
• Kapovich, Ilya
• Bradlow, Steven

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

Statement dc:rights

• Copyright 2017 Witsarut Pho-on

Language dc:language

en

Handle dc:identifier

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

OAI identifier oai:identifier

oai:www.ideals.illinois.edu:2142/97398