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The orbifold notation for surface groups. (English) Zbl 0835.20048

Liebeck, Martin (ed.) et al., Groups, combinatorics and geometry. Proceedings of the L.M.S. Durham symposium, held July 5-15, 1990 in Durham, UK. Cambridge: Cambridge University Press. Lond. Math. Soc. Lect. Note Ser. 165, 438-447 (1992).
The paper is an illustration of the idea of Bill Thurston that geometrical groups should be studied through their orbifolds. Given a group \(G\) of isometries of a surface and a discrete subgroup \(\Gamma\) acting on the surface manifold, an orbifold consists of the orbits of \(\Gamma\). The surface manifolds considered here are: 1. the sphere, 2. the Euclidean plane and 3. the hyperbolic plane.
An orbifold involves invariant topological properties of fundamental domains of \(\Gamma\) in \(G\). These can be formulated in terms of boundaries (given by mirror-invariant lines), by cone points (associated with rotations in the group with center not on a mirror line), handles, crosscaps, punched holes added on a sphere and local collapses. The author’s orbifold notation takes into account all these possibilities which follow from the classification of connected compact 2- manifolds.
It is also possible to generalize the Euler characteristic of a manifold \((V - E + F)\) to one of an orbifold. This generalized Euler characteristic can be computed directly from the orbifold notation. This notation is explicitly given for the finite spherical groups (case 1.), for the 17 two-dimensional space groups (case 2.) and for two examples of hyperbolic groups (in case 3.) illustrated by the Escher’s patterns known as Circle limit III (colored fishes) and Circle limit IV (black devils and white angels).
For the entire collection see [Zbl 0762.00006].

MSC:

20F65 Geometric group theory
57M50 General geometric structures on low-dimensional manifolds
20H15 Other geometric groups, including crystallographic groups
57M60 Group actions on manifolds and cell complexes in low dimensions
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