Nilpotent groups of class two which underly a unique regular dessin

TL;DR

This paper classifies nilpotent groups of class two that underlie a unique regular dessin, advancing understanding of symmetries in graph embeddings on surfaces.

Contribution

It provides a classification of certain nilpotent groups of class two associated with unique regular dessins, using algebraic and group-theoretical methods.

Findings

Classification of nilpotent groups of class two with unique regular dessins

Identification of groups with highest external symmetry

Application of algebraic and group-theoretical techniques

Abstract

A dessin is an embedding of connected bipartite graph into an oriented closed surface. A dessin is regular if its group of colour- and orientation-preserving automorphisms acts transitively on the edges. In the present paper regular dessins with a nilpotent automorphism group are investigated, and attention are paid on those with the highest level of external symmetry. Depending on the algebraic theory of dessins and using group-theoretical methods, we present a classification of nilpotent groups of class two which underly a unique regular dessin.