Highly symmetric maps on surfaces with boundary
type: Item
instance of: scholarly article
title: Highly symmetric maps on surfaces with boundary
zbMATH Open document ID: 1395.57004
published in: Journal of Mathematical Sciences (New York)
publication date: 2018-01-04T00:00:00Z
review text: A map on a surface is called \textit{regular} if its automorphism group has a single orbit on flags (incident vertex-edge-face triples), or sometimes if the surface is orientable and the group of orientation-preserving automorphisms has a single orbit on arcs (incident vertex-edge pairs). In recent years such maps have attracted a lot of attention, but almost entirely on surfaces without boundary. In this paper the author considers fully regular maps on surfaces with non-empty boundary. He shows that every flag of such a map must meet the boundary, and hence that the automorphism group is always cyclic or dihedral, and then uses these facts to give a complete classification. In fact it turns out that all such maps lie on the closed disk, and so are quotients of regular maps on the sphere. In particular, there are two mutually dual infinite families, and six sporadic examples. Also he extends this classification to regular hypermaps (which are relevant to the study of algebraic curves over real algebraic number fields), and to arc-transitive maps, and observes that a similar classification of edge-transitive maps on surfaces with non-empty boundary would be unrealistic because of the wide variety of examples that occur in that case.
Mathematics Subject Classification ID: 57M15
Mathematics Subject Classification ID: 05C10
Mathematics Subject Classification ID: 05E18
Mathematics Subject Classification ID: 20B25
Mathematics Subject Classification ID: 57M60
zbMATH DE Number: 6822073
zbMATH Keywords: regular map
zbMATH Keywords: surface with boundary
zbMATH Keywords: arc-transitive
zbMATH Keywords: flag-transitive
author: Gareth Jones
reviewed by: Marston D. E. Conder
MaRDI profile type: Publication
full work available at URL: s10958-017-3553-7
OpenAlex ID: W2756279969
cites work: FOUNDATIONS OF THE THEORY OF MAPS ON SURFACES WITH BOUNDARY
cites work: Regular maps and hypermaps of Euler characteristic \(-1\) to \(-200\)
cites work: Q4071773
cites work: Q3855356
cites work: Q3110201
cites work: Locally finite, planar, edge-transitive graphs
cites work: Q4364755
cites work: Operations on maps, and outer automorphisms
cites work: REAL BELYI THEORY
cites work: Graphs on surfaces and their applications. Appendix by Don B. Zagier
cites work: Regular maps whose groups do not act faithfully on vertices, edges, or faces
cites work: Q4145882
cites work: Generation of classical groups
cites work: Some Remarks on Infinite Groups
cites work: Q3424899
cites work: Q3656228
cites work: Realizing finite edge-transitive orientable maps
cites work: Hypermaps versus bipartite maps
DOI: 10.1007/S10958-017-3553-7
description: scientific article; zbMATH DE number 6822073