Nilpotent dessins: Decomposition theorem and classification of the abelian dessins

TL;DR

This paper investigates regular dessins with nilpotent and abelian automorphism groups, providing a decomposition theorem and a complete classification and enumeration of symmetric cases.

Contribution

It introduces a decomposition theorem for nilpotent dessins and classifies all regular and symmetric dessins with abelian automorphism groups.

Findings

Each nilpotent dessin decomposes into a parallel product of Sylow subgroup dessins.

Complete classification of regular dessins with abelian automorphism groups.

Enumeration results for symmetric dessins with abelian automorphism groups.

Abstract

A map is a 2-cell decomposition of an orientable closed surface. A dessin is a bipartite map with a fixed colouring of vertices. A dessin is regular if its group of colour- and orientation-preserving automorphisms acts transitively on the edges, and a regular dessin is symmetric if it admits an additional external symmetry transposing the vertex colours. Regular dessins with nilpotent automorphism groups are investigated. We show that each such dessin is a parallel product of regular dessins whose automorphism groups are the Sylow subgroups. Regular and symmetric dessins with abelian automorphism groups are classified and enumerated.