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Counting Reciprocal Hyperbolic Elements in Hecke Groups

Ara Basmajian, Blanca Marmolejo, Robert Suzzi Valli

TL;DR

This work analyzes reciprocal geodesics on (2,k,∞) Hecke surfaces by translating geometric questions into conjugacy classes of reciprocal hyperbolic elements in Γ_k and deriving precise word-length growth rates. The authors develop a framework based on normal forms in the free product Z_2 * Z_k, establish recurrences for word counts, and identify dominant roots of two polynomial families P_{2m+1} and Q_{2m} that govern growth, with explicit constants obtained from a linear system. A key insight is that B-type words, which occur when k is even, are negligible, enabling clean asymptotics for both base points and their primitive counterparts, including the infinite-k limit. The paper also provides a unified computational method, illustrated by the k=4 example, to extract leading growth constants such as d_4 ≈ 0.44721 from the associated linear system.

Abstract

A reciprocal geodesic on a (2,k, $\infty$) Hecke surface is a geodesic loop based at an even order cone point p traversing its path an even number of times. Associated to each reciprocal geodesic is the conjugacy class of a hyperbolic element in the (2,k,$\infty$) Hecke group whose axis passes through a cone point that projects to p. Such an element is called a reciprocal hyperbolic element based at p. In this paper, we determine the asymptotic growth rate and limiting constant (in terms of word length) of the number of primitive conjugacy classes of reciprocal hyperbolic elements in a Hecke group.

Counting Reciprocal Hyperbolic Elements in Hecke Groups

TL;DR

This work analyzes reciprocal geodesics on (2,k,∞) Hecke surfaces by translating geometric questions into conjugacy classes of reciprocal hyperbolic elements in Γ_k and deriving precise word-length growth rates. The authors develop a framework based on normal forms in the free product Z_2 * Z_k, establish recurrences for word counts, and identify dominant roots of two polynomial families P_{2m+1} and Q_{2m} that govern growth, with explicit constants obtained from a linear system. A key insight is that B-type words, which occur when k is even, are negligible, enabling clean asymptotics for both base points and their primitive counterparts, including the infinite-k limit. The paper also provides a unified computational method, illustrated by the k=4 example, to extract leading growth constants such as d_4 ≈ 0.44721 from the associated linear system.

Abstract

A reciprocal geodesic on a (2,k, ) Hecke surface is a geodesic loop based at an even order cone point p traversing its path an even number of times. Associated to each reciprocal geodesic is the conjugacy class of a hyperbolic element in the (2,k,) Hecke group whose axis passes through a cone point that projects to p. Such an element is called a reciprocal hyperbolic element based at p. In this paper, we determine the asymptotic growth rate and limiting constant (in terms of word length) of the number of primitive conjugacy classes of reciprocal hyperbolic elements in a Hecke group.

Paper Structure

This paper contains 11 sections, 21 theorems, 82 equations, 1 figure, 2 tables.

Table of Contents

  1. Introduction
  2. Basics and notation
  3. Outline of proof
  4. Normal forms
  5. Reciprocal Growth based at the order 2 cone point
  6. Reciprocal growth based at the order two cone point, $k$ odd
  7. Reciprocal growth based at the order 2 cone point, $k$ even
  8. Reciprocal Growth based at the order $k$ cone point in $X_k$ for $k$ even.
  9. Growth in $\mathbb{Z}_2\ast \mathbb{Z}$
  10. Properties of the polynomial families
  11. The linear system associated to a recurrence relation (solving for coefficients)

Key Result

Lemma 2.1

Let $w \in \mathcal{W}$ where $w$ is not conjugate to $a$ or a power of $b$. Then

Figures (1)

  • Figure 1: A reciprocal geodesic $\gamma$ based at order 2 on $X_k$

Theorems & Definitions (41)

  • Lemma 2.1
  • Lemma 4.1: Normal forms
  • Lemma 4.2
  • proof : Proof of Lemma \ref{['lem: passing through both cone points']}
  • Lemma 4.3
  • proof
  • Proposition 4.4: Based at order 2
  • proof
  • Proposition 4.5: Based at order $k$
  • Remark 4.6
  • ...and 31 more