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An Introduction to Riemann Surfaces and Algebraic Curves: Complex 1-Tori and Elliptic Curves

IIT Madras, , Prof. T.E. Venkata Balaji

Updated On 02 Feb, 19

Overview

Includes

Lecture 29: The Unimodular Group is Kleinian

4.1 ( 11 )


Lecture Details

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 httpwww.nptel.iitm.ac.insyllabus111106044Goals To see that a Kleinian subgroup of Moebius transformations is a discrete subspace of the space of all Moebius transformations and also that such a subgroup is either finite or countable as a set To define a subgroup of Moebius transformations to be Fuchsian if it maps a half-plane or a disc onto itself To see that a discrete Fuchsian subgroup is Kleinian. For example, the unimodular group is thus Kleinian To conclude using the results of the previous lecture that the quotient of the upper half-plane by the unimodular group is a Riemann surfaceKeywords Schwarzs Lemma, Riemann Mapping Theorem, properly discontinuous action, Kleinian subgroup of Moebius transformations, region of discontinuity of a subgroup of Moebius transformations, upper half-plane, unimodular group, projective special linear group, discrete subgroup of Moebius transformations, Fuchsian subgroup of Moebius transformations, holomorphic automorphisms, extended plane, stabilizer (or) isotropy subgroup, orbit map, second countable metric space, space of matrices, space of invertible matrices, space of determinant one matrices

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Sam

Excellent course helped me understand topic that i couldn't while attendinfg my college.

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Dembe

Great course. Thank you very much.

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