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Lickorish, A finite set of generators for the homeotopy group of a 2–manifold, Proc. Camb. Phil. Soc. 60 (1964), pp. 769–778 J. Montesinos, Classical tessellations and three–manifolds, Springer–Verlag (1987) ¨ T. Rado, Uber den Begriff der Riemannsche Fl¨ache, Acta Univ. Szeged 2 (1924–26), pp. 101–121 J. Ratcliffe, Foundations of hyperbolic manifolds, Springer–Verlag GTM 149 (1994) K. Sch¨utte and B. L. van der Waerden, Auf welcher Kugel haben 5, 6, 7, 8 oder 9 Punkte mit Mindestabstand Eins Platz, Math.

A Coxeter group G is an abstract group defined by a group presentation of the form ri |(ri rj )kij where • the indices vary over some index set I • the exponent kij = kji is either a positive integer or ∞ for each pair i, j • kii = 1 for each i • kij > 1 for each i = j If kij = ∞ for some i, j then the corresponding relation is meaningless and may be deleted from the presentation. 44. The Coxeter graph of the Coxeter group G is a labelled graph Γ with vertices corresponding to the index set I and edges (i, j) : kij > 2 labelled by kij .

44. The Coxeter graph of the Coxeter group G is a labelled graph Γ with vertices corresponding to the index set I and edges (i, j) : kij > 2 labelled by kij . For simplicity, edges with kij = 3 are usually left unlabelled. 45. Finite Coxeter groups can be realized as properly discontinuous spherical reflection groups. Notice that fundamental groups of triangle orbifolds are index 2 subgroups of reflection groups whose Coxeter graphs have three vertices. 4. “Bad” orbifolds. If Σ is a spherical orbifold with two cone points of order p, q > 1 q−1 where p = q, the orbifold Euler characteristic of Σ is 2 − p−1 p − q > 0, so the universal 2 cover of Σ should be S .

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