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A Primer on Riemann Surfaces by A. F. Beardon

By A. F. Beardon

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As a torus is obtained by rotating Sxs a circle around an axis, it can be parametrised by so G/G is topologically a torus. We shall see later that every torus with a conformal structure can be obtained in this way. 6. 1. Let G - D {o} and let G be the cyclic group generated by g(z) = 2x + iy/ 2. Show that g is a homeomorphism of accumulates in and q(i) D. Let D q : D -* D/G are distinct points of onto itself and that no orbit be the natural projection so D/G. q(l) Show that q(l + 2_ni) = q( 2_n + i) and by letting n -► +°°, 2.

For each z in D select a compact disc finitely many images we may assume that G. g(K) K The restriction of K q to centred at z and lying in D. K g(K) are disjoint for each non-trivial and We choose one such K meet for each K2 z and denote its interior by is denoted by and so is a homeomorphism of Only and so by decreasing the radius of qz : this is onto an open subset 1-1 qz (Kz ) on of K g in Kz . Kz D/G. These open subsets cover D/G and we take as our inverse charts the functions % : Kz - W * The transition maps are of the form (q^) Select any in the domain of this map so 5 = (qw)"1qz (c) = g(c) say, for some g in G.

W QUOTIENT SURFACES We are going to construct Riemann surfaces as quotient spaces D/G derived from a group G of homeomorphisms of a plane domain D onto itself. As every Riemann surface arises in this way (we cannot prove this yet), this construction is evidently of great importance. Recall that a Mobius transformation , N az+b g(z) = . 4 and the Exercises. Our construction depends on finding certain groups of Mobius transformations. 1. Let D be a domain in and let G be a group of Mobius transformations with the properties (1) g(D) = D for every (2) if and g€G g g*I, in G?

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