Moduli spaces of compact Riemann surfaces admitting automorphisms.
Item
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Title
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Moduli spaces of compact Riemann surfaces admitting automorphisms.
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Identifier
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AAI9732985
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identifier
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9732985
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Creator
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Weaver, Anthony.
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Contributor
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Adviser: Ravi S. Kulkarni
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Date
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1997
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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
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Abstract
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By focussing on hyperelliptic surfaces, a geometric picture of a large subvariety of the moduli space of compact Riemann surfaces in arbitrary genus {dollar}g \ge 2{dollar} is obtained. Since the hyperelliptic involution is central in any automorphism group containing it, and since the quotient modulo the action of the group generated by the hyperelliptic involution is a sphere, any automorphism group of a hyperelliptic surface which contains the hyperelliptic involution is a central extension of {dollar}Z\sb2{dollar} by one of the finite automorphism groups of the Riemann sphere, namely, {dollar}Z\sb{lcub}n{rcub}{dollar}, {dollar}D\sb{lcub}2n{rcub}{dollar}, {dollar}A\sb4{dollar}, {dollar}S\sb4{dollar}, or {dollar}A\sb5{dollar}. There are eight infinite families of such groups, plus another eight individual groups. The possible branching data for the actions of these groups are determined from a knowledge of the orbits of the corresponding spherical automorphism group on the sphere. The actions of these groups are classified, up to topological equivalence, using finite group theory, and some of the theory of Fuchsian groups. A geometric picture of the moduli space of hyperelliptic Riemann surfaces is obtained, using the subgroup lattices of the groups. The full picture is obtained in genus 3. A generalization of the notion of hyperelliptic surface leads to clear lines of future research.
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Type
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dissertation
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Source
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PQT Legacy CUNY.xlsx
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degree
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Ph.D.
