Reciprocity in the Hecke Groups

An element $g$ in a group $G$ is called \emph{reciprocal} if there exists $h\in G$ such that $g^{-1}=hg{h}^{-1}$. 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 ${\mathrm{\Gamma }}_p$ for $p\ge 3$. This generalizes a result by Sarnak for reciprocal elements in the modular group.