Hyperbolic geometry, discrete groups, and manifolds 1000-1M26HG

The following plan is tentative; each item is meant to correspond to roughly one week (i.e. one lecture and one exercise session).

I. The hyperbolic plane
I.1) Historical background & definition of the Poincaré half-plane
I.2) Background on Riemannian geometry
I.3) The Poincaré disc, Klein disc, and hyperboloid models

II. PSL(2,R) and hyperbolic geometry
II.1) Projective geometry, Möbius transformations, & the cross-ratio
II.2) Geodesics in the hyperbolic plane
II.3) Action of PSL(2,R) and classification of isometries

III. Fuchsian groups
III.1) Discrete groups and their action
III.2) Fundamental domains
III.3) Poincaré’s polygon theorem

IV. Surfaces and their geometric structures
IV.1) Background on fundamental groups and covering spaces
IV.2) Geometric structures on surfaces
IV.3) Teichmüller space and the Fenchel–Nielsen parameters

V. Higher-dimensional hyperbolic manifolds
V.1) The hyperbolic space in higher dimensions and its isometries
V.2) Elementary groups, Kazhdan–Margulis, & thick-thin decompositions
V.3) Outlook: hyperbolisation & Mostow rigidity