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
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