Subgroups of Hecke groups and Hecke polygons.
Item
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Title
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Subgroups of Hecke groups and Hecke polygons.
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Identifier
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AAI9707107
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identifier
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9707107
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Creator
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Huang, Shuechin.
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Contributor
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Adviser: Ravi Kulkarni
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Date
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1996
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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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In this thesis, we study certain Fuchsian groups {dollar}{lcub}\cal H{rcub}(p\sb1,\...,p\sb{lcub}n{rcub}),{dollar} called Hecke groups. These groups are isomorphic to {dollar}\Pi\sbsp{lcub}j=1{rcub}{lcub}*n{rcub}.{dollar} Let {dollar}\Gamma{dollar} be a subgroup of finite index in {dollar}{lcub}\cal H{rcub}(p\sb1,\...,p\sb{lcub}n{rcub}).{dollar} By Kurosh's theorem, {dollar}\Gamma{dollar} is isomorphic to {dollar}F\sb{lcub}r{rcub}\*\Pi\sbsp{lcub}i=1{rcub}{lcub}*k{rcub} Z\sb{lcub}m\sb{lcub}i{rcub}{rcub},{dollar} where {dollar}F\sb{lcub}r{rcub}{dollar} is a free group of rank r, and each {dollar}m\sb{lcub}i{rcub}{dollar} divides some {dollar}p\sb{lcub}j{rcub}.{dollar} Moreover, H{dollar}\sp2/\Gamma{dollar} is Riemann surface. The numbers {dollar}m\sb{lcub}1{rcub},\...,m\sb{lcub}k{rcub}{dollar} are branching numbers of the branch points on H{dollar}\sp2/\Gamma{dollar}. Let g and t be the genus and the number of cusps of H{dollar}\sp2/\Gamma{dollar} respectively. The signature of {dollar}\Gamma{dollar} is ({dollar}g;m\sb1,\...,m\sb{lcub}k{rcub};t).{dollar}.;A purpose of this thesis is to consider two problems. First, determine the necessary and sufficient conditions for the existence of a subgroup of finite index of a given type in {dollar}{lcub}\cal H{rcub}(p\sb1,\...,p\sb{lcub}n{rcub}).{dollar} We also extend this work to extended Hecke groups {dollar}{lcub}\cal H{rcub}*(p\sb1,\...,p\sb{lcub}n{rcub}){dollar} which are isomorphic to D{dollar}p\sb1\ \*\sb{lcub}z\sb2{rcub}\cdots\*\sb{lcub}z\sb2{rcub}{dollar} D{dollar}\sb{lcub}p\sb{lcub}n{rcub}{rcub}{dollar} (amalgamated over {dollar}Z\sb2{dollar}'s generated by reflections), where each D{dollar}\sb{lcub}p\sb{lcub}j{rcub}{rcub}{dollar} is a dihedral group of order 2{dollar}\sb{lcub}p\sb{lcub}j{rcub}{rcub}{dollar}.;The second problem is the realizability problem for the existence of a subgroup with a given signature in {dollar}{lcub}\cal H{rcub}(p\sb1,\...,p\sb{lcub}n{rcub}).{dollar} This is a special case of the Hurwitz problem about the realizability of branched covers. Special cases of this work were also studied by Millington, Singerman, Hoare, Edmonds, Ewing and Kulkami. Our approach is based on constructing special Poincare polygons which are the same as fundamental domains for {dollar}{lcub}\cal H{rcub}(p\sb1,\...,p\sb{lcub}n{rcub}),\ {lcub}\cal H{rcub}*(p\sb1,\...,p\sb{lcub}n{rcub}){dollar} and their subgroups.
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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.
