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Geometric representation theory: Limits of subgroups

Chabauty Limits of the Diagonal Subgroup in SL(n,Q_p)- Arielle Leitner (Weizmann)

Abstract: A conjugacy limit group is a limit of a sequence of conjugates of the positive diagonal subgroup C in SL(n) in the Chabauty topology. In low dimensions, there are finitely many limits up to conjugacy, and we explain why there are more limits over Q_p than over R. In higher dimensions there are infinitely many limits up to conjugacy. We can understand limits of C by understand how to go to infinity in the building (you won't need to know what a building is for this talk, we'll explain the geometry with low dimensional examples). We use the geometry of the building to classify limits of C. This is joint work with C. Ciobotaru and A. Valette. The hidden agenda of this talk is to convince you that Q_p is friendly, and things that we do over R and C can work over Q_p as well.


Sequences of Hitchin representations of Tree-Type- Giuseppe Martone (Michigan)

Abstract: Let S be an oriented surface of genus greater than 1. The Teichmuller space of S can be described as a connected component of the space of representations of the fundamental group of S into the Lie group PSL(2,R). The Hitchin component generalizes this classical picture to the Lie group PSL(d,R). Hitchin representations are a prominent subject of study in the field of Higher Teichmuller theory. Motivated by classical work of Thurston, one wishes to understand the asymptotic behavior of sequences of Hitchin representations. In this talk we describe non-trivial sufficient conditions on a diverging sequence of Hitchin representations so that its limit can be described as an action on a tree. In other words, we single out sequences whose asymptotic behavior is similar to diverging sequences in the Teichmuller space. Our non-trivial conditions are given in terms of Fock-Goncharov coordinates on moduli spaces of positive tuples of flags.

5 replies on “Geometric representation theory: Limits of subgroups”

I’ll try to make the conference feel more interactive by asking the first question here.

Giuseppe, I’d like some clarifications if possible on the shear parameters for closed leafs (which you describe at ~7:20). The data fixed at the beginning is all topological : a pants decomposition, an ideal triangulation of each pair of pants, and small transverse arcs to each closed curve in the pants decomposition. Then, to get parameters for a given hyperbolic metric, I assume you “straighten” the pants decomposition and the triangulation with respect to that metric, so that you have closed geodesics and infinite geodesic arcs. I guess there is no reasonable way to straighten the small arcs, so they just stay a topological arcs when you do this? Are the triangles on each side of the closed leaf determined by the endpoints of the arc before straightening or after straightening? What happens if the endpoints land on one of the open leaves?

Thanks!

Hi JP,

Thanks for the comment. Indeed, the choice of the small transverse arcs k needs some care.

Let me give additional details and then a precise reference.

Although the topological data is…topological…when carefully describing this parametrization, it is convenient choose an auxiliary hyperbolic metric m0 on S. Then, leaves and small transverse arcs can be required to be geodesic with respect to m0. Furthermore, we assume the endpoints of the small transverse arcs are disjoint from the leaves, and every transverse arc intersects a unique closed leaf.

Choose a lift K of the transverse arc k to the universal cover of the surface. Then, by construction the endpoints of K lie in the interior of two distinct geodesic ideal triangles. The rest of the steps necessary to define the shear invariants are as described in the talk.

There is a natural correspondence between m0-geodesics and geodesics with respect to another hyperbolic metric m, via the straightening procedure you hinted at.

A more detailed explanation on how to pick the transverse arcs is available in Bonahon-Dreyer “Parametrizing Hitchin components”. Specifically, on page 2943 (of the published version).

I hope this answers your questions. Otherwise, let me know and feel free to ask more!

Hi Arielle, thanks for your talk, I enjoyed it a lot. This blow up at dimension 5/6/7 is so surprising (to me at least)! Is there a way to understand why this happens? Can you actually write down an infinite family of limits in dimension 7?

Thanks Florian! Yes, the combinatorial explosion is rather surprising. Here is an example of an infinite family in dimension 7:

1 0 0 0 0 a 0
0 1 0 0 0 b b
0 0 1 0 0 c 2c
0 0 0 1 0 d xd
0 0 0 0 1 e f
0 0 0 0 0 1 0
0 0 0 0 0 0 1

Here a,b,c,d,e,f are any real (or p-adic numbers) which gives us a subgroup of SL(7) as we range over all reals. Let x be some fixed real (or p-adic) number. I claim that for x =/= y the groups are not conjugate.

[ You also need to check that these are limits of the diagonal, this is just a computation and you can find it in my paper].

First, define the orbit closure= closure of the orbit of a group in projective space. An orbit closure is a projective subspace. If two groups are conjugate, then there is a projective transformation between the orbit closures of the two groups. If we can show that there is no projective equivalence between the orbit closures, then the groups are not conjugate. This is what we will do.

We need to understand the orbit closures of the group acting on RP^6. Let RP^6 = , the projectivization of the 7 standard basis vectors.

Notice the group acts on as the identity, so the orbit of any point in this subspace has dimension 0 (projectively).

Now the rest of the orbit closures form a pencil of 5 dimensional subspaces around , and these have the form .

But there are 4 special subspaces that break down farther into 4 dimensional orbit closures, this happens when t \in {0,1,2,x}.

If we project this pencil of 5 dimensional subspaces onto a projective line, you will see 4 special points (from the 4 special orbit closures). The cross ratio is a projective invariant of 4 points on a projective line.

In case this sketch was too fast we can either talk about it more, or you can look at my paper. You can generalize it for n \geq 7 by using an unordered generalized cross ratio.

I really have no idea whether n=6 is finite or infinite, and I’d really love to find out!

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