Maps, hypermaps and triangle groups. (English) Zbl 0833.20045

Schneps, Leila (ed.), The Grothendieck theory of dessins d’enfants. Cambridge: Cambridge University Press. Lond. Math. Soc. Lect. Note Ser. 200, 115-145 (1994).
The authors continue their work on the maps and hypermaps that they started earlier [Proc. Lond. Math. Soc., III. Ser. 37, 273-307 (1978; Zbl 0391.05024)]. It is shown that every map on a surface (which may be orientable, nonorientable or with boundary) gives rise to a permutation representation of \(C_2\), which contains Grothendieck’s cartographic group with index 2, and transitive iff the map is connected. Taking the hypercartographic group, (\(Cor Co Md\)), instead of the cartographic group extends this theory to hypermaps. At the end, the \(n\)-dimensional case is discussed on the fundamentals given elsewhere. The work on maps on orientable surfaces is extended to maps on arbitrary surfaces. It is also shown how the choice of a Belyi function determines hypermaps. Finally the functors induced by homomorphisms are studied and some examples are given.
For the entire collection see [Zbl 0798.00001].


20F65 Geometric group theory
30F10 Compact Riemann surfaces and uniformization
32Q99 Complex manifolds
32A45 Hyperfunctions
57N05 Topology of the Euclidean \(2\)-space, \(2\)-manifolds (MSC2010)
57M15 Relations of low-dimensional topology with graph theory


Zbl 0391.05024