NDLI: Regular maps with nilpotent automorphism group

Content Provider	Springer Nature Link
Author	Škoviera, Martin Du, Shaofei Conder, Marston Nedela, Roman
Copyright Year	2016
Abstract	A 2-cell decomposition of a closed orientable surface is called a regular map if its automorphism group acts transitively on the set of all its darts (or arcs). It is well known that the group $$G = \mathrm{Aut}^+(\mathcal {M})$$ of all orientation-preserving automorphisms of such a map $$\mathcal {M}$$ is a finite quotient of the free product $$\Gamma = \mathbb {Z}* C_2$$ . In this paper we investigate the situation where G is nilpotent and the underlying graph of the map is simple (with no multiple edges). By applying a theorem of Labute (Proc Amer Math Soc 66:197–201, 1977) on the ranks of the factors of the lower central series of $$\Gamma $$ (via the associated Lie algebra), we prove that the number of vertices of any such map is bounded by a function of the nilpotency class of the group G. Moreover, we show that for a fixed nilpotency class c there is exactly one such simple regular map $$\mathcal {M}_c$$ attaining the bound, and that this map is universal, in the sense that every simple regular map $$\mathcal {M}$$ for which $$\mathrm{Aut}^+(\mathcal {M})$$ is nilpotent of class at most c is a quotient of $$\mathcal {M}_c$$ . In particular, there are finitely many such quotients for any given value of c, and every regular map $$\mathcal {M}$$ , whether simple or non-simple, for which $$\mathrm{Aut}^+(\mathcal {M})$$ is nilpotent of class at most c, is a cyclic cover of exactly one of them.
Ending Page	874
Page Count	12
Starting Page	863
File Format	PDF
ISSN	09259899
e-ISSN	15729192
Journal	Journal of Algebraic Combinatorics
Issue Number	4
Volume Number	44
Language	English
Publisher	Springer US
Publisher Date	2016-06-10
Publisher Place	New York
Access Restriction	One Nation One Subscription (ONOS)
Subject Keyword	Planar graphs; geometric and topological aspects of graph theory Regular map Group actions in low dimensions Convex and Discrete Geometry Nilpotent group Computer Science Graph embedding Group actions on combinatorial structures Order, Lattices, Ordered Algebraic Structures Group Theory and Generalizations Combinatorics Relations with graph theory
Content Type	Text
Resource Type	Article
Subject	Discrete Mathematics and Combinatorics Algebra and Number Theory

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