Counting Reciprocal Hyperbolic Elements in Hecke Groups

2025/05/27 by Ara Basmajian, Basmajian, Ara, Blanca T Marmolejo +3
Mathematics · #20F69 #32G15 #57K20 #Analytic Number Theory Research #Combinatorics (math.CO) #FOS: Mathematics #Geometric Topology (math.GT) #Group Theory (math.GR) #History and Theory of Mathematics #Point processes and geometric inequalities

paper · pdf · doi:10.48550/arxiv.2505.21365

openalex publication_date 2025/05/27 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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.

Citations

• Counting cusp excursions of reciprocal geodesics
• Low-lying geodesics on the modular surface and necklaces
• Reciprocity in the Hecke Groups
• Counting and equidistribution of reciprocal geodesics and dihedral groups
• Combinatorial Growth in the Modular Group
• Counting curves, and the stable length of currents
• A remark on the word length in surface groups
• Counting Curves in Hyperbolic Surfaces

Related

• Spherical growth of reciprocal classes in the Hecke Groups
• Asymptotic growth of the number of Reciprocal Classes in the Hecke Groups
• Counting subgroups via Mirzakhani's curve counting
• Counting and equidistribution of reciprocal geodesics and dihedral groups
• Counting subgraphs in hyperbolic graphs with symmetry