By T. E. Venkata Balaji

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**Extra resources for An Introduction to Families, Deformations and Moduli**

**Example text**

Then this covering determines a subgroup of the fundamental group of M as above and this subgroup is found to be conjugate to G. Replacing G by any of its conjugates in π1 (M ) would lead to a covering isomorphic to p : M −→ M . Further, in the case when G is normal, this covering p : M −→ M turns out to be Galois and we have DeckM (M , p) = DeckM (M , p)/G. Of course the case when G is the trivial subgroup of the fundamental group of M gives the universal covering p : M −→ M itself! Thus there is a bijection between the set of conjugacy classes of subgroups of π1 (M ) and the set of isomorphism classes of coverings of M .

No torsion. 4 Corollary. The covering group of a Riemann surface cannot contain elliptic elements. 5 Corollary. The covering group of a Riemann surface with universal covering U contains only parabolic and hyperbolic elements. 3. Uniformization of Riemann Surfaces 13 We wish to describe Riemann surfaces with abelian fundamental group and universal covering U. To do so we will need the following result. 6 Lemma. Any Riemann surface with nonzero abelian π1 and universal covering U has cyclic covering group.

5 Any Riemann surface can be uniformized in the sense that it is the quotient of a simply connected domain in the extended complex plane by a group of M¨ obius transformations. The quotient is a universal covering with deck transformation group isomorphic to the fundamental group of the given Riemann surface. 3 alongwith a description of those M¨obius transformations that give rise to deck transformations in the universal covering. 3 is to develop the complete structure of universal coverings of annuli and elliptic curves and in the process we describe the universal coverings of those Riemann surfaces whose fundamental groups are abelian.