Reciprocity in the Hecke Groups

TL;DR

This paper classifies and parametrizes reciprocal elements in Hecke groups, extending Sarnak's results from the modular group to a broader class of groups, thereby advancing understanding of their algebraic structure.

Contribution

It provides a complete classification and parametrization of reciprocal elements in Hecke groups, generalizing previous work on the modular group.

Findings

Reciprocal elements are classified in Hecke groups for all p ≥ 3.

A parametrization of reciprocal classes in these groups is established.

The results extend known classifications from the modular group to Hecke groups.

Abstract

An element $g$ in a group $G$ is called \emph{reciprocal} if there exists $h \in G$ such that $g^{-1}=hgh^{-1}$. The reciprocal elements are also known as `real elements' or `reversible elements' in the literature. We classify the reciprocal elements and parametrize the reciprocal classes in the Hecke groups $\Gamma_p$ for $p\geq 3$. This generalizes a result by Sarnak for reciprocal elements in the modular group.