Nilpotent dessins: Decomposition theorem and classification of the abelian dessins

2015/08/19 by Hu, Kan, Nedela, Roman, Wang, Na-Er
#05C10 #20B25 #Combinatorics (math.CO) #FOS: Mathematics

paper · doi:10.48550/arxiv.1508.04523

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.

Related

• Totally symmetric dessins with nilpotent automorphism groups of class three
• Nilpotent groups of class two which underly a unique regular dessin
• Complete regular dessins of odd prime power order
• Regular dessins with a given automorphism group
• Complete regular dessins and skew-morphisms of cyclic groups