Discrete Subgroup of Lie Groups

Website: https://web.ma.utexas.edu/users/stecker/disc/

Place & Time: Fall 2021, Tuesday & Thursday 14:00 - 15:30 in PMA 11.176

My office and email: Florian Stecker, PMA 10.132, ut@florianstecker.net

Short description: One of the fundamental facts about a Lie group G is that we can study its connected Lie subgroups by just looking at the Lie algebra Lie(G). The other extreme in the spectrum of subgroups are discrete ones, closed subgroups whose identity component is trivial. It is hard to say anything about them in general. But if G is for example the isometry group of hyperbolic plane, the theory of its discrete subgroups is equivalent to that of hyperbolic surfaces (sometimes called Teichm├╝ller theory). In recent years, many ideas from Teichm├╝ller theory have been extended to other semisimple Lie groups G, and I want to discuss some of these results. The class will be roughly split in three parts (of not necessarily equal in size): 1) discrete subgroups of SL(2,R), hyperbolic groups, convex cocompact representations 2) discrete subgroups of SL(n,R) for n > 2, limit sets, eigenvalues and singular values, Anosov representations 3) general semisimple Lie groups G.

Sources and references: Here is a list of books, lecture notes and papers that I'm drawing from, and that might be useful to read things up:

Homework: I publish exercise sheets every few weeks. Don't worry, you don't have to do them. But if are serious about learning the material, you probably should. Also, I try to choose exercises which are fun and interesting.
#1 hyperbolic trigonometry, orbit type stratification, triples at infinity

Notes: Here are some notes. The fact that they are written in TeX should not make you believe they are carefully written, or even correct. Most parts are just copied from other sources (see above), and probably the originals do a better job explaining it. I post these anyway in case someone missed a class etc. and finds them helpful.

Schedule:
Jan 18overview
Jan 20models of hyperbolic spacenotes
Jan 25proper actionsnotes
Jan 27slice theorem and consequencesnotes
Feb 1discrete subgroups of hyperbolic isometriesnotes
Feb 8Quasi-isometries, hyperbolic spaces, Milnor-Svarc lemma, Morse lemma
Feb 10proof of the Morse lemma, boundary of hyperbolic spaces
Feb 15hyperbolic groups, north-south dynamics on the boundary
Feb 17minimal action on the boundary, proof that axis is quasi-geodesic, definition of convex cocompactness
Feb 22properties of convex cocompactness, definition via convex sets
Feb 24examples of convex cocompact representations
Mar 1openness of convex cocompactness
Mar 3basics about SL(n,R), matrix decompositions, flag manifolds
Mar 8exterior power representations, proximality, eigenvalue estimates
Mar 22Zariski closed subgroup, existence of proximal elements
Mar 24Limit set and properties
Mar 29Building proximal elements with almost prescribed fixed points and eigenvalues
Mar 31Proof of the last Theorem, convexity of the limit cone