By T. E. Venkata Balaji

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We next check that the map φ restricted to U (w) induces a homeomorphism onto an open subset of W/G which we denote by U (w). Then we verify that the inverse image of U (w) under φ splits into the disjoint union of translates of U (w) by various elements of G, and further that each of these translates is mapped homeomorphically by φ onto U (w). In this way φ : W −→ W/G becomes a topological covering with deck transformation group G. 3 Remark. We note that the statement of the above theorem and its proof can be easily adapted to any of the following categories: 1.

There exist discrete subgroups of PSL(2, C) that are For example, the Picard group defined by A(z) = az + b cz + d ad − bc = 1; a, b, c, d Gaussian Integers is a discrete but not Kleinian subgroup of SL(2, C) and the same are true of its image in PSL(2, C). However, the notions of discreteness and being Kleinian coincide for a group of conformal automorphisms of a disc (or half-plane). In this case the group is therefore also Fuchsian. Further the preserved disc (or half-plane) is entirely contained in the region of discontinuity.

If G is normal, W/G naturally becomes a complex Lie group and the map φ a morphism of Lie groups. 1 Example (Tori as Quotients). The complex vector space Cn is a complex commutative Lie group with vector addition as group operation. Since it is 2n-dimensional as a real vector space, we may fix a basis 22 1. Classification of Annuli and Elliptic Curves ω := w1 , . . , w2n . , a Z-submodule of maximal rank) and moreover, a discrete subgroup of the Lie group Cn . G has for its fundamental domain the 2n-dimensional real closed “parallelopiped” given by    2n  F = (tj wj ) ; 0 ≤ tj ≤ 1 .

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