Reciprocity in the 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.