Hecke Triangle Groups and Dessins d'Enfants

Abstract

We study finite-index subgroups of the Hecke triangle group $H_q$ of signature $(2,q,\infty )$ through the finite-edge tessellation, special polygons, and the associated dessins. The quotient finite-edge graph is a connected bipartite ribbon graph in which even vertices have degree $1$ or $2$ and odd vertices may have any degree $d\mid q$; geometrically, degree $d$ records a $d$-cluster and an elliptic stabilizer of order $q/d$. We determine two finite ambiguities that arise when one passes from this quotient graph to a special polygon. First, all tree diagrams obtained from a fixed ribbon graph are classified by the possible spanning-tree cuts modulo graph automorphisms. Second, at a proper cluster one must record which cyclic gap contains the omitted branches of the universal $q$-star; every such choice develops to a special polygon. Combining the two gives an explicit finite description of all polygon orbits over a fixed dessin, together with a separate count of normalized realizations in the fixed tessellation. Concrete examples for $H_5$, $H_6$, and $H_7$ illustrate the geometry, and two arithmetic corollaries concern level-$2$ cusp labels and Galois orbits of principal congruence dessins.

arXiv comment20 pages, 4 figures