Hyperbolic Geometry, the Modular Group and Diophantine Approximation - Lecture 1

Explore hyperbolic geometry, the modular group, and Diophantine approximation in this comprehensive lecture from the Geometry, Groups and Dynamics program at the International Centre for Theoretical Sciences. Delve into key concepts such as the hyperbolic plane, its boundary, subgroups of SL(2,R), and the unit tangent bundle. Examine the geodesic flow, fundamental domains, and the Dirichlet fundamental domain. Learn about important propositions and proofs related to these topics. This 1-hour 14-minute lecture serves as an in-depth introduction to these advanced mathematical concepts, suitable for graduate students and researchers in geometry, dynamical systems, and group theory.

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Hyperbolic geometry, the modular group and Diophantine approximation Lecture - 01
H Hyperbolic plane
Boundary of H
Subgroups of SL2,R
Observation
Let S'H be the unit tangent bundle over H
Observation:
Hence
Geodesic flow
Observation
Note
Recall
Example
Fundamental domains
Dirichlet fundamental domain
Proposition
Proof
Claim
Imaginary part