The Unimodular Group is Kleinian

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 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 surface


Keywords: Schwarz's 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