Parallel products in groups and maps
type: Item
instance of: scholarly article
title: Parallel products in groups and maps
zbMATH Open document ID: 0836.20032
published in: Journal of Algebra
publication date: 1996-04-16T00:00:00Z
review text: Given a bijective correspondence between generators \(g_i\) and \(f_i\) of two groups \(G_1\) and \(G_2\), the parallel product \(G_1 |G_2\) is defined as the subgroup of \(G_1 \times G_2\) generated by \((g_i, f_i)\), \(i \in I\). If \(G_i = F/H_i\), for a free group \(F\) with generators \(x_i\), then \(G_1 |G_2 = F/H_1 \cap H_2\). This construction can be considered as a special case of the parallel product of two actions of a group on two sets, considering the action on an orbit of the product of the two sets. The construction is applied to the theory of maps on surfaces which are embeddings of graphs in surfaces such that each complementary region is simply connected; in a purely combinatorial way, each map can be considered as an action of a certain universal group by permutations on the set of flags of the map. Thus the parallel product of two (rooted) maps is defined which is again a map and a covering of the two original ones. In this context, various results concerning coverings of maps, regular (rotary, reflexible, chiral), orientable maps and symmetry groups of maps are discussed. For example, the parallel product of two reflexible maps is again reflexible, and its symmetry group is the parallel product of the symmetry groups of the individual maps. As another application, any chiral map has a finite smooth cover which is reflexible.
Mathematics Subject Classification ID: 20E22
Mathematics Subject Classification ID: 20F65
Mathematics Subject Classification ID: 57M07
zbMATH DE Number: 638505
zbMATH Keywords: regular maps
zbMATH Keywords: generators
zbMATH Keywords: parallel product
zbMATH Keywords: maps on surfaces
zbMATH Keywords: embeddings of graphs in surfaces
zbMATH Keywords: orientable maps
zbMATH Keywords: symmetry groups
zbMATH Keywords: reflexible maps
author: Steve Wilson
reviewed by: Bruno Zimmermann
MaRDI profile type: Publication
full work available at URL: jabr.1994.1200
OpenAlex ID: W2065872561
DOI: 10.1006/JABR.1994.1200
description: scientific article; zbMATH DE number 638505