|Introduction to large scale geometry and hyperbolic manifolds|
|Time：||Tue/Thu 10:10-12:00, 07-23~08-08|
|Instructor：||Feng Luo [Rutgers University]|
|Place：||Conference Room 1, Floor 1, Jin Chun Yuan West Building|
This is an introduction course for undergraduate and graduate students who are interested on low-dimensional geometry and topology. Our main goal is to prove the famous Mostow rigidity theorem for closed hyperbolic manifolds. We will follow Gromov’s proof of the Mostow theorem by first introducing the Gromov norm of manifolds and then show that the Gromov norm is related to the hyperbolic volume. If time permits, we will also use the large scale geometry to show Dehn-Nielsen’s theorem that the automorphisms of the fundamental group of a closed surface are induced by a self-homeomorphisms.
The topics to be covered are:
1. Hyperbolic spaces
2. Quasi isometry and large scale geometry
3. Gromov’shyperbolicity of metric spaces
4. Gromov norm and Gromov-Thurston’s theorem relating Gromov norm to volume
5. Mostow rigidity for hyperbolic metrics on 3-manifolds
Geometry of the mapping class groups of surfaces and Dehn-Nielsen theorem
Students should know some basic algebraic topology (fundamental groups, covering spaces, homology) and some basic Riemannian geometry.
 Riccardo Benedetti, Carlo Petronio, Lectures on Hyperbolic Geometry, Springer, 2003