The arithmetic and geometry of Bianchi groups.
Item
-
Title
-
The arithmetic and geometry of Bianchi groups.
-
Identifier
-
AAI9732988
-
identifier
-
9732988
-
Creator
-
Yao, Wei-Chen.
-
Contributor
-
Adviser: Carlos J. Moreno
-
Date
-
1997
-
Language
-
English
-
Publisher
-
City University of New York.
-
Subject
-
Mathematics
-
Abstract
-
In this thesis, we discuss the arithmetic and geometric properties of Bianchi groups. Let {dollar}K = Q(\sqrt{lcub}-d{rcub}){dollar} be an imaginary quadratic field, {dollar}{lcub}\cal O{rcub}{dollar} be its ring of integers. The Bianchi group is defined by {dollar}PSL\sb2({lcub}\cal O{rcub}){dollar}.;The following are the contents of the thesis. In chapter 1, we apply the ideas of Bianchi, Humbert, and Swan to construct a fundamental domain of the Bianchi group in H and give a sharper bound for Swan's algorithm. This fundamental domain is bounded by four perpendicular planes and by a finite number of geodesic hemi-spheres centered on the complex plane. The principal result of this thesis is the following.;Theorem A. If a geodesic hemi-sphere S meets the fundamental domain {dollar}{lcub}\cal D{rcub}{dollar}, then the radius of S is greater than {dollar}Cd\sp{lcub}-3.5{rcub}{dollar} where C is an absolute constant.;In this thesis we make applications of the above estimate to the following areas.;(1) Reduction theory of binary Hermitian forms.;Theorem B. For any fixed discriminant D, there are at most a finite number of non-equivalent binary Hermitian forms. In fact, the number of non-equivalent binary Hermitian forms is less than {dollar}Cd\sp{lcub}10.5{rcub}{dollar} where C is a constant depending only on D.;(2) Reduction theory of binary quadratic forms.;Theorem C. For any fixed discriminant D, there are at most a finite number of non-equivalent binary quadratic forms over the ring of integers of K. In fact, the number of non-equivalent binary quadratic forms is less than {dollar}C\sp1d\sp{lcub}10.5{rcub}{dollar} where {dollar}C\sp\prime{dollar} is a constant depending only on D.;The complexity of determining reduced sets of Binary Hermitian forms and Binary quadratic forms is decided by the fundamental domain of Bianchi groups. Conversely, the determination of reduced Hermitian forms implies the determination of fundamental domain of Bianchi group.;(3) Dilogarithm and Bloch groups.;Following Thurston's idea, we can triangulate the fundamental domain as a union of ideal tetrahedra parametrized by complex numbers {dollar}z\sb1{dollar}, ...,{dollar}z\sb{lcub}k{rcub}{dollar} each count certain multiplicity {dollar}n\sb{lcub}i{rcub}{dollar}. The volume of the fundamental domain of Bianchi group PSL{dollar}\sb2{dollar}({dollar}{lcub}\cal O{rcub}){dollar} is then shown to be equal to {dollar}\sum n\sb{lcub}i{rcub}D\sb2(z\sb{lcub}i{rcub}){dollar} where D{dollar}\sb2{dollar} is the Bloch-Wigner Dilogarithm function. From this representation, we get an element in the pre-Bloch group. The determination of the fundamental domain allows us to calculate this element effectively.
-
Type
-
dissertation
-
Source
-
PQT Legacy CUNY.xlsx
-
degree
-
Ph.D.
