Complex iso-length-spectrality in arithmetic hyperbolic 3-manifolds Pubblico

Abstract

Abstract
Complex iso-length-spectrality in arithmetic hyperbolic 3-manifolds
The (real-)length spectrum of a compact hyperbolic 3-manifold is the set of lengths
of all closed geodesics along with the multiplicity of each length. A closed geodesic
also has an imaginary part that represents the twist encountered by traveling once
around the closed geodesic. So, the complex length of a closed geodesic is +it, where
is the length of the closed geodesic and t is the twist with 0 ≤ t < 2π. The complex
length spectrum of a compact hyperbolic 3-manifold is the set of complex lengths of
all closed geodesics along with the multiplicity of each length. Two compact hyper-
bolic 3-manifolds are called iso-length-spectral if their length spectra are the same.
Also, two compact hyperbolic 3-manifolds are called complex iso-length-spectral if
their complex length spectra are the same. The aim of this paper is to investigate if
iso-length-spectral arithmetic hyperbolic 3-manifolds are complex iso-length-spectral.
Arithmetic hyperbolic 3-manifolds are a class of hyperbolic 3-manifolds where arith-
metic data about the manifolds tells us a great deal of information about the mani-
folds.

Table of Contents

Contents
1 Introduction 1
2 Geometric Preliminaries 8
2.1 Hyperbolic 3-Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
2.2 Hyperbolic 3-Manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . 11
3 Algebraic Preliminaries 18
3.1 Dedekind Domains . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
3.2 Number Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
3.3 A Bridge and a Brief Aside . . . . . . . . . . . . . . . . . . . . . . . . 25
3.4 Factorization of Prime Ideals . . . . . . . . . . . . . . . . . . . . . . . 26
3.5 Non-Archimedean Local Fields . . . . . . . . . . . . . . . . . . . . . . 27
3.6 Chebotarev Density Theorem . . . . . . . . . . . . . . . . . . . . . . 28
3.7 Quaternion Algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
4 Combining Geometric and Algebraic Knowledge 34
4.1 Arithmetic Hyperbolic 3-Manifolds . . . . . . . . . . . . . . . . . . . 34
4.2 Why Arithmetic Hyperbolic 3-Manifolds? . . . . . . . . . . . . . . . . 39
5 Previous Results 42
6 New Results 48
6.1 Angles of Loxodromic Eigenvalues . . . . . . . . . . . . . . . . . . . . 48
6.2 Trace Fields of Degree 3 . . . . . . . . . . . . . . . . . . . . . . . . . 53
6.3 Roots of Unity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
6.4 Back to Degree 3 and Onward to Degree p . . . . . . . . . . . . . . . 68
6.5 Isospectrality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
6.6 Salem Numbers and the Short Geodesic Conjecture . . . . . . . . . . 83
7 Appendix 90
7.1 More on loxodromic eigenvalues . . . . . . . . . . . . . . . . . . . . . 90
7.2 Salem numbers and possible angles . . . . . . . . . . . . . . . . . . . 91

About this Dissertation

Rights statement

• Permission granted by the author to include this thesis or dissertation in this repository. All rights reserved by the author. Please contact the author for information regarding the reproduction and use of this thesis or dissertation.

School	Laney Graduate School
Department	Math and Computer Science
Degree	PhD
Submission	Dissertation
Language	English
Research Field	Mathematics
Parola chiave	arithmetic hyperbolic 3-manifolds, complex length spectrum
Committee Chair / Thesis Advisor	Hamilton, Emily, Emory University
Committee Members	Brussel, Eric, Emory University Abrams, Aaron, Emory University

Ultima modifica

Primary PDF

Thumbnail	Title	Date Uploaded	Actions
	Complex iso-length-spectrality in arithmetic hyperbolic 3-manifolds ()	2018-08-28 16:10:15 -0400	Press to Select an action Download

Supplemental Files

Thumbnail	Title	Date Uploaded	Actions