Hyperbolic Surfaces and Their Teichmüller Spaces - Lecture 2

Explore hyperbolic surfaces and their Teichmüller spaces in this advanced lecture from the Geometry, Groups and Dynamics program at the International Centre for Theoretical Sciences. Delve into key concepts like the collar lemma, surface group representations, and proper discontinuous actions. Examine the relationship between hyperbolic metrics and geometric structures modeled on PSL2(R) and H2. Investigate developing maps, holonomy representations, and their properties. Gain insights into the continuous nature of these mathematical structures through detailed proofs and examples presented over the course of 71 minutes.

Geometry, Groups and Dynamics GGD -2017
Hyperbolic surfaces and their Teichmuller spaces Lecture - 02
Last time
Teichmuller space
One more geometric facts
Collar lemma
Collar function
Picture of the hyperbolic surface
Corollary "Margules Lemma"
Today: Surface group representation
Recall
Fact
Proper discontinuous action
Consequence - Corollary
Examples
Two Remarks
Proposition
A hyperbolic metric on S is a geometric structure modelled on PSL2IR, H2
Global description
Developing map
Holonamy representation
Note
Corollary
Proof
Proposition
Why continuous?
Fact 1
Fact 2