Regular maps with nilpotent automorphism groups

Authors:
Aleksander Malnič;Roman Nedela;Martin ŠKoviera
Affiliations:
Pedagoška Fakulteta, Univerza v Ljubljani, Kardeljeva pl. 16, 1000 Ljubljana, Slovenia;Department of Mathematics, Matej Bel University, SK-975 49 Banská Bystrica, Slovakia;Department of Computer Science, Comenius University, SK-842 48Bratislava, Slovakia
Venue:
European Journal of Combinatorics
Year:
2012

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Abstract

We study regular maps with nilpotent automorphism groups in detail. We prove that every nilpotent regular map decomposes into a direct product of maps HxK, where Aut(H) is a 2-group and K is a map with a single vertex and an odd number of semiedges. Many important properties of nilpotent maps follow from this canonical decomposition, including restrictions on the valency, covalency, and the number of edges. We also show that, apart from two well-defined classes of maps on at most two vertices and their duals, every nilpotent regular map has both its valency and covalency divisible by 4. Finally, we give a complete classification of nilpotent regular maps of nilpotency class 2.