Non-simple closed geodesics on 2-orbifolds
Item
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Title
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Non-simple closed geodesics on 2-orbifolds
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Identifier
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d_2009_2013:f696762a2be9:12035
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identifier
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12687
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Creator
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Suzzi Valli, Robert,
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Contributor
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Ara Basmajian
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Date
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2013
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Language
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English
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Publisher
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City University of New York.
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Subject
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Mathematics | Applied mathematics
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Abstract
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Given a Fuchsian group Gamma, that is, a discrete subgroup of the group of orientation-preserving isometries of the hyperbolic plane H, the quotient H/Gamma is a 2-orbifold. If {dollar}Gamma{dollar} contains torsion then the resulting 2-orbifold contains cone points corresponding to the elliptic fixed points. In this thesis we focus on minimal length non-simple closed geodesics on 2-orbifolds. Nakanishi, Pommerenke and Purzitsky discovered the shortest non-simple closed geodesic on a 2-orbifold, which passes through a cone point of the orbifold. This raises questions about minimal length non-simple closed geodesics disjoint from the cone points. We explore once self-intersecting closed geodesics disjoint from the cone points of the orbifold, called figure eight geodesics. Using fundamental domains and basic hyperbolic trigonometry we identify and classify all figure eight geodesics on triangle group orbifolds. This classification allows us to find the shortest figure eight geodesic on a triangle group orbifold. We then generalize to find the shortest figure eight geodesic on a 2-orbifold without cone points of order two.
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Type
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dissertation
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Source
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2009_2013.csv
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degree
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Ph.D.
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Program
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Mathematics
