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Title:Gromov boundaries of complexes associated to surfaces
Author(s):Pho-on, Witsarut
Director of Research:Leininger, Christopher
Doctoral Committee Chair(s):Dunfield, Nathan
Doctoral Committee Member(s):Kapovich, Ilya; Bradlow, Steven
Department / Program:Mathematics
Degree Granting Institution:University of Illinois at Urbana-Champaign
Subject(s):Gromov boundary
Curve complex
Arc complex
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.
Issue Date:2017-04-19
Rights Information:Copyright 2017 Witsarut Pho-on
Date Available in IDEALS:2017-08-10
Date Deposited:2017-05

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