Totally symmetric dessins with nilpotent automorphism groups of class three

TL;DR

This paper classifies totally symmetric dessins, which are highly symmetric graph embeddings on surfaces, specifically focusing on those with nilpotent automorphism groups of class three, expanding understanding of their structure.

Contribution

It provides a classification of totally symmetric dessins with nilpotent automorphism groups of class three, a previously unexplored category.

Findings

Classification of totally symmetric dessins with nilpotent automorphism groups of class three

Identification of structural properties of these dessins

Extension of symmetry theory in graph embeddings

Abstract

A dessin is a 2-cell embedding of a connected bipartite graph into an orientable closed surface. An automorphism of a dessin is a permutation of the edges of the underlying graph which preserves the colouring of the vertices and extends to an orientation-preserving self-homeomorphism of the supporting surface. A dessin is regular if its automorphism group is transitive on the edges, and a regular dessin is totally symmetric if it is invariant under all dessin operations. Thus totally symmetric dessins possesses the highest level of external symmetry. In this paper we present a classification of totally symmetric dessins with a nilpotent automorphism group of class three