Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Arithmetic modular links

José Andrés Rodríguez Migueles, Tali Pinsky, Jessica S. Purcell

Abstract

We construct a sequence of geodesics on the modular surface such that the complement of the canonical lifts to the unit tangent bundle are arithmetic 3-manifolds.

Arithmetic modular links

Abstract

We construct a sequence of geodesics on the modular surface such that the complement of the canonical lifts to the unit tangent bundle are arithmetic 3-manifolds.
Paper Structure (13 sections, 15 theorems, 17 equations, 9 figures, 2 algorithms)

This paper contains 13 sections, 15 theorems, 17 equations, 9 figures, 2 algorithms.

Table of Contents

  1. Introduction
  2. Acknowledgements
  3. Surfaces and unit tangent bundles
  4. The modular surface
  5. The once-punctured torus
  6. Curves on the once-punctured torus and the Farey tessellation
  7. Arithmetic Kleinian groups
  8. Regular octahedra for neighboring slopes
  9. Projecting and lifting on the modular surface
  10. Cutting sequences
  11. Modular links
  12. Volume versus hyperbolic length
  13. Further questions

Key Result

Theorem 1.1

There exists a sequence $\{\gamma_n\}_{n\in{\mathbb{N}}}$ of distinct closed geodesics on the modular surface such that for each $n$, the union of the first $n$ canonical lifts $\bigcup_{j=1}^n \widehat{\gamma_j}$ has complement in the unit tangent bundle $\mathop{\mathrm{UT}}\nolimits({\Sigma_{\rm{

Figures (9)

  • Figure 1: The branched surface inside the complement of the trefoil, with the direction of the semiflow indicated, pointing downwards from the branchline.
  • Figure 2: Taking the quotient of ${\mathbb{R}}^2{\smallsetminus}\Lambda$ by translations gives $\Sigma_{1,1}$. Quotient further by $2\pi/3$ rotations about centres of triangles, and $\pi$ rotations about centres of edges to obtain ${\Sigma_{\rm{Mod}}}$.
  • Figure 3: Starting on the left with $\Sigma_{1,1}\times [0,1]$ with $\alpha=1/0$ drilled from $\Sigma_{1,1}\times\{0\}$ and $\beta=0/1$ drilled from $\Sigma_{1,1}\times\{1\}$, obtain a regular ideal octahedron on the right.
  • Figure 4: A rotation by $2\pi/3$ about the centre of each equilateral triangle takes the curve $p/q$ to the curve $\overline{(p-q)/p}$, and a further rotation takes it to $\overline{q/(q-p)}$. Shown is the case $q>p>0$. Similar pictures give other cases.
  • Figure 5: The Farey graph of rational slopes.
  • ...and 4 more figures

Theorems & Definitions (32)

  • Theorem 1.1
  • Corollary 1.2
  • Corollary 1.3
  • Lemma 2.2
  • proof
  • Remark 2.3
  • Definition 2.4
  • Definition 3.1
  • Lemma 3.2
  • Lemma 4.1
  • ...and 22 more