Monodromy Groups of Dessins d'Enfant on Rational Triangular Billiards Surfaces

TL;DR

This paper classifies the monodromy groups of dessins d'enfant associated with rational triangular billiards surfaces, linking combinatorial graph structures to geometric billiard dynamics.

Contribution

It provides a classification of monodromy groups for dessins on rational triangular billiards surfaces, a novel connection between algebraic invariants and billiard geometry.

Findings

Monodromy groups are classified for dessins on these surfaces.

The work reveals algebraic structures underlying billiard trajectories.

New links between dessins d'enfant and rational billiard dynamics are established.

Abstract

A dessin d'enfant, or dessin, is a bicolored graph embedded into a Riemann surface, and the monodromy group is an algebraic invariant of the dessin generated by rotations of edges about black and white vertices. A rational billiards surface is a two dimensional surface that allows one to view the path of a billiards ball as a continuous path. In this paper, we classify the monodromy groups of dessins associated to rational triangular billiards surfaces.