Quotients by Kleinian Subgroups give rise to Riemann Surfaces

An Introduction to Riemann Surfaces and Algebraic Curves: Complex 1-Tori and Elliptic Curves by Dr. T.E. Venkata Balaji, Department of Mathematics, IIT Madras. For more details on NPTEL visit http://www.nptel.iitm.ac.in/syllabus/111106044/

Goals: * To see how the quotient of the region of discontinuity by a Kleinian subgroup of Moebius transformations is a union of Riemann surfaces

  • To see how the quotients above are ramified (or) branched coverings of Riemann surfaces, with ramifications at the points with nontrivial isotropies (stabilizers)

  • To see in detail how to get a complex coordinate chart at the image point of a point of ramification

Keywords: Upper half-plane, unimodular group, fixed point, projective special linear group, quotient by a subgroup of Moebius transformations, holomorphic automorphisms, extended plane, properly discontinuous action, stabilizer (or) isotropy subgroup, region of discontinuity of a subgroup of Moebius transformations, limit set of a subgroup of Moebius transformations, elliptic Moebius transformations, isolated point, discrete subset, Kleinian subgroup of Moebius transformations, quotient topology, ramification (or) branch points, ramified (or) branched covering, unramified (or) unbranched covering, branch cut, slit disc

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