Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Determining surfaces by short curves and applications

Cayo Dória, Nara Paiva

Abstract

The goal of this work is to give new quantitative results about the distribution of semi-arithmetic hyperbolic surfaces in the moduli space of closed hyperbolic surfaces. We show that two coverings of genus $g$ of a fixed arithmetic surface $S$ are $P(\frac{1}{g})$ apart from each other with respect to Teichmuller metric, where $P$ is a polynomial depending only on $S$ whose degree is universal. We also give a super-exponential upper bound for the number of semi-arithmetic hyperbolic surfaces with bounded genus, stretch and degree of the invariant trace field, generalizing for this class similar well known bounds for arithmetic hyperbolic surfaces. In order to get these results we establish, for any closed hyperbolic surface $S$ with injectivity radius at least $s$, a parametrization of the Teichmuller space by length functions whose values on $S$ are bounded by a linear function (with constants depending only on $s$) on the logarithm of the genus of $S.$

Determining surfaces by short curves and applications

Abstract

The goal of this work is to give new quantitative results about the distribution of semi-arithmetic hyperbolic surfaces in the moduli space of closed hyperbolic surfaces. We show that two coverings of genus of a fixed arithmetic surface are apart from each other with respect to Teichmuller metric, where is a polynomial depending only on whose degree is universal. We also give a super-exponential upper bound for the number of semi-arithmetic hyperbolic surfaces with bounded genus, stretch and degree of the invariant trace field, generalizing for this class similar well known bounds for arithmetic hyperbolic surfaces. In order to get these results we establish, for any closed hyperbolic surface with injectivity radius at least , a parametrization of the Teichmuller space by length functions whose values on are bounded by a linear function (with constants depending only on ) on the logarithm of the genus of
Paper Structure (9 sections, 15 theorems, 50 equations, 6 figures)

This paper contains 9 sections, 15 theorems, 50 equations, 6 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. The geometry of the moduli spaces
  4. Semi-Arithmetic hyperbolic surfaces
  5. Number theoretic facts about semi-arithmetic Fuchsian groups
  6. Length coordinates with logarithmic bound
  7. Applications
  8. Minimal distance between coverings of arithmetic hyperbolic surfaces
  9. Counting semi-arithmetic hyperbolic surfaces

Key Result

Theorem 1.1

Given $s>0$, for any marked closed hyperbolic surface $S \in \mathcal{T}_g$ with injectivity radius at least $s,$ there exist finitely many marked closed curves $\gamma_1, \ldots, \gamma_n$ in $S$ with $n \leq 15g-15$ such that:

Figures (6)

  • Figure : Trivalent vertex
  • Figure : Vertex with high degree
  • Figure : Step 1: Arcs joining a vertex with boundary
  • Figure : Step 2: Concatenation of arcs $c_i's$
  • Figure : Step 3: New arcs
  • ...and 1 more figures

Theorems & Definitions (25)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 2.1
  • Proposition 2.2
  • proof
  • Lemma 2.3
  • proof
  • Remark
  • Lemma 3.1
  • ...and 15 more