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Operations on hypermaps, and outer automorphisms. (English) Zbl 0698.05028

Let M be a map on a surface. There is a well-known set of five other maps closely associated with M. These are formed by various combinations of the operations of duality and the replacement of face boundaries with left-right paths (the Petrie polygons) of M. These related maps have the same number of flags and the same automorphism group as M. A common generalization of a graph embedding (or map) is a hypergraph embedding (or hypermap). These are commonly represented as an embedding of a cubic graph together with a proper 3-coloring of the faces. The three color classes correspond to the hypervertices, hyperedges, and hyperfaces of the embedding.
In this paper the author extends the above operations on maps to operations on hypermaps. In this extension there is some added flexibility in the operations forming related maps. The resulting group of operations is isomorphic to PG(2,\({\mathbb{Z}})\). These operations are induced by the group of automorphisms of a certain group G whose transitive permutation representations correspond to hypermaps.
Reviewer: D.S.Archdeacon

MSC:

05C10 Planar graphs; geometric and topological aspects of graph theory
20B27 Infinite automorphism groups
05C25 Graphs and abstract algebra (groups, rings, fields, etc.)
20F65 Geometric group theory
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