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Monodromy groups of dessins d'enfant on rational triangular billiards surfaces

Madison Mabe, Richard A. Moy, Jason Schmurr and Japheth Varlack

Vol. 16 (2023), No. 1, 49–58
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. We classify the monodromy groups of dessins associated to rational triangular billiards surfaces.

Keywords
dessins d'enfant, monodromy groups, triangular billiards surface
Mathematical Subject Classification
Primary: 11G32, 14H57
Secondary: 37C83
Milestones
Received: 14 July 2021
Revised: 5 March 2022
Accepted: 23 March 2022
Published: 14 April 2023

Communicated by Nathan Kaplan
Authors
Madison Mabe
Department of Mathematical Sciences
Lee University
Cleveland, TN
United States
Richard A. Moy
Department of Mathematical Sciences
Lee University
Cleveland, TN
United States
Jason Schmurr
Department of Mathematical Sciences
Lee University
Cleveland, TN
United States
Japheth Varlack
Department of Mathematical Sciences
Lee University
Cleveland, TN
United States