Parallel products in groups and maps. (English) Zbl 0836.20032

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.


20E22 Extensions, wreath products, and other compositions of groups
20F65 Geometric group theory
57M07 Topological methods in group theory
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