Counting Reciprocal Hyperbolic Elements in Hecke Groups

TL;DR

This paper investigates the asymptotic growth and distribution of primitive reciprocal hyperbolic elements in Hecke groups, linking geometric geodesics to algebraic group elements.

Contribution

It provides the first asymptotic formulas for counting primitive reciprocal hyperbolic elements in Hecke groups, connecting geometric and algebraic properties.

Findings

Established the asymptotic growth rate of reciprocal hyperbolic elements.

Derived the limiting constant in terms of word length.

Linked geometric geodesics to algebraic conjugacy classes.

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.