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Foundations of the theory of maps on surfaces with boundary. (English) Zbl 0565.05026
A map is an embedded graph in a surface (2-manifold) S with some special features (a collection of ”darts” resp. ”blades”). Associated to a map is a group G of permutations (of the darts resp. blades). If S is orientable without boundary G is a factor group of a triangle group \((2,m,n)=<X,Y|\) \(X^ 2=Y^ m=(XY)^ n=1>\) [see G. A. Jones and D. Singerman, Proc. Lond. Math. Soc., III. Ser. 37, 273-307 (1978; Zbl 0391.05024), for this case]. In the present paper maps on surfaces which are not necessarily orientable and may have boundary are discussed. In this case G is a quotient of an extended triangle group \([2,m,n]=<A,B,C|\) \(A^ 2=B^ 2=C^ 2=(AB)^ 2=(BC)^ m=(AC)^ n=1>\) (isomorphic to the group generated by the reflections in the sides of a triangle). On the other hand there is a definition of an algebraic map as a permutation group G with certain special properties. One of the main results of the paper asserts that every algebraic map comes from a topological map on some surface. Various other concepts arising from maps are discussed. The paper is carefully written, many examples are given.
Reviewer: B.Zimmermann

MSC:
05C10 Planar graphs; geometric and topological aspects of graph theory
05C25 Graphs and abstract algebra (groups, rings, fields, etc.)
57N05 Topology of the Euclidean \(2\)-space, \(2\)-manifolds (MSC2010)
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