**Proper actions on symplectic groups and their Lie algebras- Jean-Philippe Burelle (Sherbrooke)**

Abstract: Danciger-Guéritaud-Kassel developed a theory of proper actions on PSL(2,R) (anti-de Sitter space) and its Lie algebra sl(2,R) (Minkowski space) using length contraction/expansion properties. They applied this machinery to obtain several results on the structure of anti-de Sitter and flat Lorentzian manifolds in dimension 3, including proofs of the tameness of Margulis space times and of the crooked plane conjecture. I will show how part of this theory generalizes to symplectic groups Sp(2n,R) and their Lie algebras sp(2n,R). This is joint work with Fanny Kassel.

**Growth of quadratic forms under Anosov subgroups- León Carvajales (Universidad de la República & Sorbonne)**

Abstract: For positive integers p and q we define a counting problem in the (pseudo-Riemannian symmetric) space of quadratic forms of signature (p,q) on R^{p+q}. This is done by associating to each quadratic form o a totally geodesic copy of the Riemannian symmetric space of PSO(o) inside the Riemannian symmetric space of PSL_{p+q}(R), and by looking at the orbit of this geodesic copy under the action of a discrete subgroup of PSL_{p+q}(R). We then present some contributions to the study of this counting problem for Anosov subgroups of PSL_{p+q}(R).

**Topological dynamics of the Weyl chamber flow- Nguyen-Thi Dang (Heidelberg)**

Abstract: Let G be a connected, real linear, semi-simple Lie group without compact factors and D be a discrete subgroup. Let K be a maximal compact subgroup, A a Cartan subgroup for which Cartan decomposition holds. Consider an action D\G on the right by a one-parameter subgroup of A. As a family of topological dynamical systems, when do all orbits diverge ? What are the interesting sets for the dynamics ? Are there dense orbits in those sets ? Is there topological mixing ? When D is a cocompact lattice, there are no diverging orbits and every flow is exponentially mixing by a result of Howe-Moore. Assume now that D is only Zariski dense in G. For SO(n,1)⁰, the corresponding flow is the frame flow which factors over the geodesic flow of a hyperbolic manifold. Mixing properties of such flows in convex-cocompact hyperbolic manifolds were obtained by Winter in 2016 and improved recently by Winter-Sarkar. In 2017, Maucourant-Schapira proved topological mixing when D is Zariski dense. In this talk, I'll focus in the case where G is a higher rank split simple Lie group and D a Zariski dense subgroup. The existence of Weyl chambers in A allows to define regular Weyl chamber flows as those parametrised by an element of the interior a Weyl chamber. The Benoist cone, which contains all the information about the spectrum of D will then give us a necessary condition for the existence of non-diverging orbits. Finally I'll explain a topological mixing criteria for regular flows.

## 8 replies on “Geometric representation theory: Dynamics of a representation”

[…] Dynamics of a representation: […]

Hi everyone, I’m J-P. I started a position last year as an assistant professor at the University of Sherbrooke, in Canada.

This is my first “virtual conference” but I really like the format and I hope it will work well. I’ll be happy to answer any questions about my talk in this comment thread or in the Zoom session for graduate students on Thursday.

Hi J-P, thanks for the talk!

One question for you (not sure how well-formed this is … ): if I were to start taking limits of sequences of Schottky groups of the sort you describe in your talk, will every limit be a Schottky group (except possibly with reflexive convex sets which “touch” / have non-disjoint closures), or can you get other limits?

Hi Feng, thanks for the question 🙂

One thing that you could do is shrink some convex sets to points, in which case the corresponding generators would blow up (more precisely, they would converge to a projection to that point).

Some reflexive convex sets have stabilizers which are unbounded, so you could even keep the convex sets fixed and degenerate in some directions.

If you only care about convergent sequences, another thing is that the convex sets we use have a stronger property than being disjoint : they are transverse, in the sense that every point in one convex set is transverse to every point in another. One way of degenerating would be to go to a representation where this fails for some boundary points.

Hi everyone, I’m Thi and I started a post-doc in the differential geometry team in Heidelberg last October, just after my PhD at IRMAR in the University of Rennes 1, which is a city in the center of Brittany in France.

This is also the first “virtual conference” for me, the first time I ever made a video so my skills in this regard, well let’s say they have plenty of room for improvement.

I happily welcome you all to ask questions in this section or in the zoom office hour on Friday. Let’s engage into fruitful and lively discussions !

Finally, I wish you all to find a good way to balance your lives with the confinement, social distancing, upcoming changes and keep in good health !

Hi everyone, I’m León. I’m finishing my PhD at Universidad de la República (Uruguay) and Sorbonne Université (France). I’m interested in geometry and dynamics of discrete subgroups of Lie groups. I’ll be happy to discuss about my talk or any related subject in the next days!

Hey León, I liked your talk!

I would like to understand the proof a bit better, can you help me? Which Benoist results are you using to estimate 1/2 lambda(…) and what is the “cross ratio” involved there? Also, can you say a bit more about this flow you are looking at? Which space does it live on? I assume you somehow construct it out of the flow associated to an Anosov representation, is that right?

Hey Florian, thanks for your comment! Yes, I was a bit sloppy in the last part of the talk 😉

1) You can find the estimate of Benoist in “Propriétés asymptotiques…” (this is a series of 2 papers, the second one contains the estimate). The definition of the cross-ratio is also in that paper, but you can check it also in my preprint “Growth of quadratic…” in arxiv. The cross-ratio assigns a vector in \mathfrak{b} to each 4-tuple of full flags satisfying some transversality condition and naturally generalizes the cross-ratio between 4 lines in R^2.

2) Concerning the flow, yes: it is a Hölder reparametrization of the geodesic flow of the representation. There is a procedure for giving “Hopf coordinates” to this flow: it depends on the choice of a cocycle over the boundary of the group. There is an equivalence relation between cocycles (called Livsic cohomology) with the property that, for cohomologous cocycles, the associated flows are conjugate. However, the result that we want to prove is not purely dynamical, so the difficulty to obtain the counting theorem relies on the fact that we are interested in a “geometric cocycle” (a specific representative in the cohomology class). This geometric cocycle in this case is constructed using the Busemann function of the Riemannian symmetric space: given a full flag \xi and an element \gamma in your group, you compute the Busemann function based at \xi and with respect to the orthogonal projection of \xi to S^o and the orthogonal projection of \xi to \rho\gamma\cdot S^o.

Once you have your Busemann cocycle you can define a “Gromov product”, assigning a vector to each pair of transverse flags. It turns out that if you evaluate this Gromov product in ((\rho\gamma)_-,(\rho\gamma)_+) you obtain the cross-ratio you want…

Does this makes any sense? I realize that maybe is too sloppy still, so do not hesitate to ask again!

Best,

León