## Discrete Subgroup of Lie Groups

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 Teichmüller theory). In recent years, many ideas from Teichmüller 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:

• Canary, "Informal lecture notes on Anosov representations"
• Helgason, "Differential Geometry, Lie groups, and Symmetric spaces"
• Knapp, "Lie groups beyond an introduction"
• Ginzburg, Guillemin, Karshon, "Moment maps, cobordisms and Hamiltonian group actions" (appendix on proper actions)

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.

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