Fundamental domains of modular subgroups using isometric circles.
Item
-
Title
-
Fundamental domains of modular subgroups using isometric circles.
-
Identifier
-
AAI9108105
-
identifier
-
9108105
-
Creator
-
Fung, Terry Y.
-
Contributor
-
Adviser: Harvey Cohn
-
Date
-
1990
-
Language
-
English
-
Publisher
-
City University of New York.
-
Subject
-
Mathematics
-
Abstract
-
Let {dollar}\Gamma{dollar} denote the inhomogeneous modular group acting on the upper half plane {dollar}{lcub}\cal H{rcub}{dollar} in this way:{dollar}{dollar}z\mapsto{lcub}az+b\over cz+d{rcub},\ ad-bc=1,\ a,b,c,d\in{lcub}\cal Z{rcub}{dollar}{dollar}Sometimes we denote the element of {dollar}\Gamma{dollar} by its matrix form {dollar}A=\pm\left(\matrix{lcub}a&b\cr c&d\cr{rcub}\right){dollar}.;We consider the subgroup of {dollar}\Gamma{dollar} such as{dollar}{dollar}\Gamma\sp\circ (N) = \left\{lcub}\left(\matrix{lcub}a&b\cr c&d\cr{rcub}\right) \vert b \equiv 0\quad ({lcub}\rm mod{rcub}\ N)\right\{rcub}{dollar}{dollar}We construct the fundamental domain for {dollar}\Gamma\sp\circ (N){dollar} using isometric circles in a manner based upon the divisibility of prime factor(s) of {dollar}N{dollar}. The use of isometric circles was Poincare's original constructional and existential method for fundamental domains.;In this paper, we construct a fundamental domain of {dollar}\Gamma\sp\circ (N){dollar} for {dollar}N{dollar} which has at most four distinct prime factors. The fixed circle of {dollar}\Gamma\sp\circ (N){dollar} is the real axis and the cusps are zero and the multiples of the prime factors. We construct it by using isometric circles with the relation {dollar}(z - {lcub}L\over Q{rcub})(z\prime - {lcub}M\over Q{rcub}) = -{lcub}1\over Q\sp2{rcub}{dollar} provided that {dollar}LM \equiv -1{dollar} (mod {dollar}QN{dollar}) where {dollar}{lcub}L\over Q{rcub}, {lcub}M\over Q{rcub}{dollar} are centers of circles with radius {dollar}r = {lcub}1\over Q{rcub}{dollar}. The idea of the method is to find the matching circles by checking {dollar}LM \equiv -1{dollar} (mod {dollar}QN{dollar}). We found that for {dollar}N{dollar} with one prime factor, the isometric circles will have radius {dollar}r = 1{dollar}. If {dollar}N{dollar} has two distinct prime factors, then the isometric circles have radius {dollar}r \geq {lcub}1\over 2{rcub}{dollar}. If {dollar}N{dollar} has three distinct prime factors then the isometric circles have radius {dollar}r \geq {lcub}1\over 3{rcub}{dollar} if {dollar}N{dollar} is odd, and {dollar}r \geq {lcub}1\over 4{rcub}{dollar} if {dollar}N{dollar} is even. If {dollar}N{dollar} has four distinct prime factors, the isometric circles will have radius {dollar}r \geq {lcub}1\over 4{rcub}{dollar} provided all the prime factors are greater than three, but if the prime factors are less than or equal to three, various special cases will occur leading to circles of radius {dollar}{lcub}1\over 7{rcub}{dollar} as we shall demonstrate.;This work generalizes a result of Cohn (3) for two prime factors. The method will be shown to be applicable to any number of prime numbers.
-
Type
-
dissertation
-
Source
-
PQT Legacy CUNY.xlsx
-
degree
-
Ph.D.
