Hecke triangle Groups and Dessin d'enfant

TL;DR

This paper constructs bipartite graphs called Dessin d'enfant for subgroups of Hecke triangle groups and explores their correspondence with polygons and tree diagrams, enhancing understanding of their geometric and algebraic structures.

Contribution

It introduces a method to associate Dessin d'enfant with finite index subgroups of Hecke triangle groups and establishes their correspondence with polygons and tree diagrams.

Findings

Bipartite graphs are constructed for subgroups of Hecke triangle groups.

Correspondences among polygons, graphs, and trees are demonstrated.

The work deepens the understanding of the geometric structures of Hecke triangle groups.

Abstract

In this work we will construct bipartite graphs, famously known as Dessin d'enfant, corresponding to finite index subgroups of Hecke triangle groups $(2, q, \infty )$. Then using a results of \cite{ll} we shall show the correspondences among the special polygons, the bi-partite graph, and the tree diagram for a finite index subgroup of the Hecke triangle groups $(2, q, \infty )$.