By H.S.M. and W.O.J. Moser Coxeter
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Additional resources for Generators and Relations for Discrete Groups
We have seen that the only finite fundamental groups are those of the sphere and the projective plane. This is the topological counterpart of the geometrical fact that the sphere and the elliptic plane have finite area, whereas the Euclidean and hyperbolic planes have infinite area. (For a remarkable drawing of the case p = 2, see HILBERT and COHN-VOSSEN 1932, p. 228, Abb. ) Fig. 5c Along with the map of 2m-gons on the universal covering surface, it is convenient to consider the dual map (on the same surface) which has a vertex in each face of the original map, an edge crossing each edge, and a face surrounding each vertex.
3c. ·······-----... \. / -. . / / ·b/ the grou~ of order 16 generated by the permutatiOns R = (2 4) (3 6), S = (1 2) (3 5) (4 7) (6 8), T=(13)(28)(45)(67), we have an octagon joined to an octagram (Fig. 3c). 4 Planar diagrams the Cayley diagram is the Thomsen graph (Fig. 3d). The cyclic group [ 6 , generated by R = (1 2 3 4 5 6), S = (1 4) (2 5) (3 6) , is represented by the same graph with six edges directed as in Fig. 3e. In each of the last two cases the generatorS is redundant, and by deleting all the S-edges we obtain a simpler diagram for the same group with fewer generators.
Thus p g g is (oo, oo I 2, 2) in the notation of CoxETER (1939, pp. 74, 81). 5 The groups c m m and p 4 46 There is a half-turn T that interchanges the pairs of opposite reflections R1 , R 3 and R 2 , R 4 which generate p m m. 509) = R~ = T 2 = (R1 R 2) 2 = (R1 T R 2 T) 2 =E. It is of special interest as being the symmetry group of the "anomalous" uniform tessellation (COXETER 1940a, p. 405; STEINHAUS 1950, p. 65, Fig. 65) in which strips of squares alternate with strips of triangles. Ri Fig. 5 k.
Generators and Relations for Discrete Groups by H.S.M. and W.O.J. Moser Coxeter