Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Counting cusp excursions of reciprocal geodesics

Ara Basmajian, Robert Suzzi Valli

Abstract

For a fixed cusp neighborhood (determined by depth D) of the modular surface, we investigate the class of reciprocal geodesics that enter this neighborhood (called a cusp excursion) a fixed number of times.

Counting cusp excursions of reciprocal geodesics

Abstract

For a fixed cusp neighborhood (determined by depth D) of the modular surface, we investigate the class of reciprocal geodesics that enter this neighborhood (called a cusp excursion) a fixed number of times.
Paper Structure (5 sections, 4 theorems, 20 equations, 3 figures, 1 table)

This paper contains 5 sections, 4 theorems, 20 equations, 3 figures, 1 table.

Table of Contents

  1. Introduction
  2. Some background and preliminaries
  3. Growth rates of reciprocal geodesics
  4. The case of $2n$ excursions of depth bigger than 1
  5. The case of two excursions of depth bigger than $D$

Key Result

Theorem 1.1

For any integers $n\geq 0$, For any integer $D\geq 2$, where $d_{D}=\frac{\alpha_D -1}{2+(D+1)(\alpha_D -2)}$, and $\alpha_D$ is the unique positive root of $z^D-z^{D-1}-\dots-1$.

Figures (3)

  • Figure 1: A cusp excursion of depth greater than $D$ on $S$
  • Figure 2: Correspondence between $N_{4t}$ and $C_{t}$
  • Figure 3: $2\leq k\leq t-r$

Theorems & Definitions (7)

  • Theorem 1.1
  • Theorem 3.1
  • proof
  • Lemma 3.2
  • proof
  • Theorem 3.3
  • proof