Asymptotic growth of the number of Reciprocal Classes in the Hecke Groups

TL;DR

This paper investigates how the number of reciprocal conjugacy classes in Hecke groups grows asymptotically, using probabilistic tools and the groups' free product structure to provide new insights.

Contribution

It introduces a novel approach employing probability theory to estimate the asymptotic growth of reciprocal classes in Hecke groups, differing from previous methods.

Findings

Derived asymptotic growth estimates for reciprocal classes

Utilized free product structure and word lengths in analysis

Provided a new probabilistic framework for group analysis

Abstract

We estimate the asymptotic growth of reciprocal conjugacy classes in Hecke groups using their free product structure and word lengths of reciprocal elements. Our approach is different from other works in this direction and uses tools from basic probability theory.