SPEAKER 1: Our first talk this morning is Anna Wienhard, who's going to talk about geometric structures and representation varieties.
ANNA WIENHARD: Well, thank you very much. It's a great pleasure to be here. [INAUDIBLE]. So I had only one personal encounter with him in person, which happened in 2009 when I was giving a talk at the [INAUDIBLE]. And actually, it was an encounter I didn't-- and I only realized 10 minutes after that I had encountered Thurston.
So I was giving my talk. After my talk, someone came to me. And we had a very nice discussion, and he asked very interesting questions. But I didn't recognize Thurston, because he had short hair, and I knew these old pictures with him having long hair. And then, just 10 minutes later, in the coffee break, I said, oh, damn, I mean, that was Thurston.
So let me start by explaining why I chose this title of "Geometric Structure and Representation Varieties." Because this is, in a sense, the first mathematical encounter I had with Thurston's work, which was when I was an undergraduate student at Bonn. We had a seminar on hyperbolic structures, geometric structures, and in particular hyperbolic structures and manifolds, going through the Thurston-Gromov proof of Mostow rigidity for three-dimensional hyperbolic manifold. And in a sense of the first theorem I encountered which was attached with Thurston's name was the Ehresmann-Thurston principle.
So let me know if I write too small.
So what does the Ehresmann-Thurston principle say? So it says that, if you look at geometric structures on a manifold M-- so which means, just to remind you, that locally you model M on a geometry. So you have charts in M, which map open subsets of M into open subsets of some homogeneous space X. And such that, if you have overlapping charts, then over the intersection the coordinate transformation is given by an element in a group G, acting as group of transformations on your own your space X.
So looking at all geometric structures, you can build the deformation space of all such structures on a given topological manifold, with respect to a given geometry. And then you can, for such a structure, look at the holonomy of the structure. So one way to think about this is you take M, you lift the structure to the universe [INAUDIBLE] covering [? it ?] there. It's equivariant with respect to the action of the fundamental group of M. And the action-- since coordinate transformations are given by elements in G-- the action of the fundamental group is given by the action of G on X.
So if you look at the holonomy of such a geometric structure, you get some homomorphism from the fundamental group of M into your group of transformations G. And, since here you modeled by some equivalence class of geometric structure here. It actually goes into the space of homomorphism [? on ?] the action of G by conjugation. And what does the Ehresmann-Thurston principle say? The Ehresmann-Thurston says that this map is a local homeomorphism. So what does this mean? In other words, if I'm given representation from the fundamental group of M into G, of which I know that it is the holonomy of a geometric structure, if I wiggle it a little bit it's still the holonomy of a geometric structure on the manifold.
So what I would like to do in this talk, explore this relation between homomorphisms on one side and geometric structures on the manifold on the other, a bit further. But I will focus on individual homomorphisms. So what I want to describe, in this talk, and I will describe it in some examples, are not a complete generality.
So given some homomorphism from the fundamental group of your manifold into G, find-- or let me think differently-- realize it as a holonomy of a geometric structure. And so one approach-- and this is, in a sense, the approach I'm going to follow-- is, one approach is to try to just find nice, for this given homomorphism, you have the action of G on X. And you can try to find nice subsets of your homogeneous space X on which the action of the fundamental [INAUDIBLE] M via your representation is nice. And by "nice," I mean in particular property-discontinuous, so that you have a really nice quotient manifold-- or orbifold, but I will only look at quotient manifolds. And, the second property, it's very nice if the quotient is compact-- so if the action on the subset of X is not just property-discontinuous but also cocompact. Because then we really have a nice, compact quotient manifold.
So what I want to somehow illustrate is that, in order to do that, it's sometimes very useful to not think about the geometry given by G and X but thinking of a bigger geometry. So I want to illustrate that it's often useful to consider not just G but G as a subgroup of a bigger group G prime. And, in a sense, if we also take into account the homogeneous space X, to consider G,X as a subset geometry of a bigger geometry G prime, X prime.
OK. So clearly, I'm not going to do this for an arbitrary homomorphism, so I want to just define a class of homomorphisms or class of representations for which you can actually give a nice realization as holonomies of geometric structures. So the first thing I want to do is define this class of representations which goes under the name of "Anosov representations." And in this first part, I will also, for Anosov representations, give a very general construction which realizes the representation as a holonomy of a geometric structure in the way I described, by finding a nice, open subset of X on which the action is nice.
In the second part, I want to discuss mainly two applications of this general construction. And, in the third part, if I'm not cut-- so this will be very short. If I'm not cut out by the fire alarm, I want to say something about geometric transitions.
OK. So the first, now, I want to give you the definition of what an Anosov representation is. And I will not give you the most general definition, but I will restrict to a slightly simpler situation. So the class of Anosov representations have been introduced by Francois Labourie in the context of representation of surface groups into an arbitrary semisimple Lie group G. You can extend it to representation of an arbitrary hyperbolic group, into an arbitrary semisimple Lie group G, but I will stick with surfaces, for the moment, just to make things a bit simpler.
So I will look at fundamental groups of surfaces. And I will also not take the most generous semisimple Lie group, but I want to take one specific example. And actually, I'm going to take two examples here. So I will look at representations into the symplectic group Sp 2n. And I take 2, because I want to take K where K can be either the reals or the complex.
So given representation from the fundamental group of a closed surface into the symplectic group, I want to give you the definition of what an Anosov representation is. So this representation is Anosov. And now I don't give you the original definition, which involves more dynamical systems. I give you basically a characterization of Anosov representation which reflects the property I'm going to use later.
So such a representation is said to be Anosov if we have a nice map from the boundary of this group into some flag variety for that group. So if there exist a continuous rho equivariant map-- let's call it xi-- from the boundary of the surface group, which you can identify with the boundary of the hyperbolic plane-- so with S1-- into the projective space of dimension 2n minus 1, which carries an action of the symplectic group. And this boundary map has to have one additional condition. It has to be transverse. And what does this mean? So-- i.e.
So what does this mean? I write it here. For every two points t and t prime in S1 which are distinct, we get two lines. We have xi of t, and we get xi of t prime.
We want that the map is injective, but we want more than that. So these lines should not just be not equal but, if I take one line in a symplectic vector space, and can take its orthogonal complement with respect to the symplectic form, and I get a hyperplane. So I want that the line of one is in direct [INAUDIBLE] with the hyperplane of the other. So in that sense--
OK. So this is almost the right characterization. It's the right characterization-- or, I mean, it really is an Anosov representation if your representation here would be irreducible, then you don't need any additional assumption. If it's not irreducible, you need some dynamical property for sequences in your group going out to infinity. And, if you're interested in knowing precisely what this dynamical property, I'm happy to tell you later. Right now, I would like to leave it at this and give you some examples, because-- and this is really the property we're going to use later.
OK so what are examples?
So let's look at the first example-- so the easiest-- n equals 1 and K equals R. And then this group is Sp(2,R), which is the same as SL(2,R). And, in this situation, Anosov is the same as being a discrete embedding-- so-- and the same as having an embedding of the surface group as a discrete subgroup of SL(2,R). And so it's basically-- forget about difference between PSL(2,R) and SL(2,R). It's really basically the same as the holonomy of a hyperbolic structure.
That's the first example. The next example, n equals 1, K equals the complex numbers. So I have SL(2,C), there. Then, in this case-- and again, I neglect the difference between SL(2,C) and PSL(2,C). Anosov is the same as having a quasi Fuchsian representation.
And in this situation-- so you can see-- I mean, what is the space here? So the space here is RP1, if K is R. It's CP1 for SL(2,C). And being transverse in this situation just means being injective. So we have a continuous injective map from the circle into your group, which gives you the quasi [INAUDIBLE] the boundary.
OK, so now, let me continue with examples. So the next example is an example for arbitrary n and K being R. Then there is the so-called Hitchin component, which is a subset of the space of all homomorphisms from the fundamental group of the surface into Sp(2n,R), which is characterized as follows. It's all representations which you can deform to a particular embedding of SL(2,R) into Sp(2n,R). So it's the connected component of the representation rho 0, which takes the fundamental group of your surface into SL(2,R), via some discrete embedding. And then you take SL(2,R) into Sp2nR via-- and here you have to choose a specific embedding, namely the one you get from the 2n-dimensional [INAUDIBLE] representation.
OK so this is not the set of all Anosov representation but a subset of the set of Anosov representations of the fundamental of your surface into Sp(2n,R). So this Hitchin-- so I want to just shortly say something about the Hitchin component. And we'll hear more about that tomorrow morning, in Francois's talk, probably.
The Hitchin component was introduced by Hitchin, beginning of the '90s. And what he showed is that the Hitchin component-- he called it "Teichmuller component"-- is homeomorphic to a vector space, the way we know it from Teichmuller space. Which is the Hitchin component for Sp(2,R)-- so in the case where n is 1.
And there has been a lot of similar results about the Hitchin component in the last 20 years or so, showing for example that every representation in Hitchin component is discrete and faithful, which is a work by Francois, extending even more properties of Teichmuller space to the Hitchin component. So they're coordinates, like [INAUDIBLE] type coordinates, they're shear coordinates on Hitchin components. There is a Vi Peterson Riemannian metric on Hitchin components. And, in a sense, it's--
So one challenge to-- if you want to generalize everything you know about Thurston's work or other people's work on Teichmuller space involving the hyperbolic-- somehow the hyperbolic geometry point of view in Teichmuller space-- to the Hitchin component. And it's an even bigger challenge to-- and this is-- I mean, Francois will tell us something about this to develop classical Teichmuller theory-- so more complex, analytic tools-- to this Hitchin component.
But, for me, the Hitchin component itself will not play such a big role. I will come back to it shortly. Once again, for me, the bigger space of Anosov representations will play the bigger role. And so I want to make one remark, here, without writing things down.
So this is a simplified definition. There is actually, in a sense, a definition of being i Anosov, where i can run from 1 to n, where you take the same definition, but just that the map you take takes, here, it takes value from S1 into the space of one-dimensional subspaces of your vector space. In general, for i, it will take values in i-dimensional isotropic subspaces of your vector space.
So if you do this, then there are other-- I mean, you get more examples of i Anosov representations. One property of the Hitchin component is that it's i Anosov for every i. So not just for the lines but also for i-dimensional subspaces. .
OK. So now let me give you, for Anosov representations, a way of defining, describing some geometric structure.
So I want to describe an Anosov representation as holonomy of a geometric structure by, as I said, finding a domain of discontinuity in some homogeneous space. Finding a domain of discontinuity in some x-- which would be Sp(2n,K) [INAUDIBLE] something, which I will specify in a little bit.
So let me just draw a picture for quasi Fuchsian representations. So for quasi Fuchsian representations, so we have CP1. We have our curved psi. So I'm not good at drawing a fractal curve, so-- some kind of quasi circle. And there we know how to get a domain of discontinuity, even in CP1, namely, if we have a quasi Fuchsian representation, if this is the limit set, lambda, then, if we take the complement of lambda, this, the action of the fundamental group of the surface-- I'll just write it here-- i1 of sigma x via rho-- on this domain, properly discontinuous, and even cocompactly, we get two surfaces as our quotient.
So now we want to do something similar, but it turns out that it's not very useful to look at the domain of discontinuity in this space, in general. But, for the symplectic group, if n is bigger than 1, we have different flag varieties. We have lines, but we also have i-dimensional isotropic subspaces. And it turns out to be much more useful to mix one space where the map takes its value with another space where you define the domain of discontinuity.
So let me state it first as a theorem. So this is joint work with Olivier Guichard. So let rho be Anosov. Then there exists some open subset in the space of maximal isotropic subspaces-- so the space of Lagrangians of this symplectic vector space-- on which pi 1 of sigma via rho acts properly discontinuously. Actually, freely and cocompact.
OK, and I want to give you now-- so this is not just one, some domain of discontinuity, but we explicitly constructed, and the construction is, in a sense, similar to what you do for quasi Fuchsian representation, just that you have to play with lines and n-dimensional subspaces on the other side. So let me describe this here.
So we have our continuous equivariant map, from S1 into KP2n minus 1. And so now, for basically every point on the image of this map, we want to remove a certain subset of the space of Lagrangians. And the set we remove is all the Lagrangians which in a sense are not in general position with that line.
So what does this mean? So we define some set K xi which takes some-- take all Lagrangians K to n, such that there's just some t, such that this line in K to n is contained in the Lagrangian. So this is, in a sense, the bad set in the space of Lagrangians defined by some kind of limit set in the projective space. And this domain omega is precisely the complement of that.
AUDIENCE: Is this about the n?
ANNA WIENHARD: Hmm?
AUDIENCE: This doesn't look as if it's true for n equals 1 and K equals R.
ANNA WIENHARD: Yes, exactly. So it's still true, but if you put the word "nonempty," then it's not true for [INAUDIBLE]. So it's still true, but in the case where n is 1 and K is R, you would take projective space-- so RP1-- and you have a map from S1 to RP1 which hits everything. So in this situation, omega is empty. But that's the only situation. So for all the other cases, you get some nonempty domain of discontinuity.
And so this is another-- so just one remark. So this is, for example, one example where you see that it's useful, in this situation, to think of your Fuchsian representation or Anosov representation into SL(2,R) in a bigger group. Because now, if you embed SL(2,R) into SL(2,C), you have the limits that are just the equator, the RP1 and CP1, and the domain of discontinuity you would construct is precisely the upper hemisphere and lower hemisphere. And so you would get some nice, actually, nice domain of discontinuity and some nice structure on the surface.
OK, so this is the general construction. I want to make one remark about that. So you see, here we have an interplay between space of line, space of Lagrangians. You can reverse the roles. So if you have something which is n Anosov-- so where you have your map xi goes into the space of Lagrangians-- you can define a domain of discontinuity in the space of lines, by basically doing the same thing. So for every Lagrangian you pick up in your limit set, you remove all lines in this Lagrangian.
And, if you do that-- so for example, if you do that for the Hitchin component, this way you can construct for every Hitchin representation a rear-projective contact structure on some bundle over the surface and actually show that the Hitchin component parametrizes such rear-projective contact structures on certain bundles over the surface. Yeah?
AUDIENCE: [INAUDIBLE] does the topology of omega depend on which Anosov [INAUDIBLE] you choose-- for a fixed n [INAUDIBLE]?
ANNA WIENHARD: Well, the topology, yes, if you just look at all Anosov representations. If you look at a connected component, it doesn't.
ANNA WIENHARD: But, if you-- I mean, there you have basically, by Ehresmann-Thurston principle, you have-- but if you-- and the first case where you can see that is, if you-- the easiest case, perhaps, to-- so there-- two easiest cases. For the symplectic group, you can look at it for Sp(4,R). And so there you have the Hitchin component. You also have spaces of maximal representations, which are Anosov representations with respect to the Lagrangian.
When you do the construction in projective space, you get projective structures on S1 bundles, or more generally, O2 bundles over the surface. And which O2 bundle it is depends on the representation. So for example, whether the quotient is connected or not depends on the representation. But it's very useful, then, in the connected component, it doesn't, because that allows you, in some cases, to control actually the topology of the quotient manifold.
OK. So now I want to describe two applications of this construction where somehow you do this general construction and you get some more geometric information about your group action or about some quotient manifold. So while I wipe the board, just one-- so one other remark--
If you don't like the symplectic group, you can easily adapt what I just described for the automorphism group of some nondegenerate, bilinear form on a vector space. So you can just write down precisely the same. And actually what-- I mean, you can do everything more generally for an arbitrary semisimple Lie group and an arbitrary hyperbolic group. Also this construction, in a sense, works, but it's more involved, what you do. Yes?
AUDIENCE: Is there a corresponding result [? with the ?] [? other dimensions ?] [INAUDIBLE] one-dimensional, an n-dimensional, [INAUDIBLE] [? i-dimensional ?] [INAUDIBLE]?
ANNA WIENHARD: Yeah, so it's not-- so sometimes, not always. So you have to be a bit more-- so having somehow the extrema of the spectrum gives you a nicer description. So there's a very general construction that, whenever you have something which is Anosov with respect to all i-- so i Anosov with respect to all i-- then you pick a flag variety of the form K subspaces in n-minus-K dimensional subspaces. And I can give you a domain of discontinuity, but you can't go directly between K and n minus K.
So one point which plays a role here is the structure that-- and you see that very easily-- when you start with a map into the space of Lagrangians and then remove the projective-- I mean, the lines, in this Lagrangian, the map is transfer. So if you have two Lagrangians, they don't intersect except in zero. So if you remove line from one, and line from the other, the set you remove from projective space, these are disjoint sets.
So you actually control very nicely the topology of the set you remove. And if you do this with arbitrary K and n minus K, you don't. And so you can still get some proper discontinuity. But, in general, you don't know whether the quotient is compact or not.
So the first application I want to discuss gives you some answer or result about proper actions. And actually the second-- well also, for the second, I will focus on a different aspect.
So we all know, if we have a discrete subgroup of the isometry group of a Riemannian manifold, then this group acts properly discontinuously on the manifold. This is not true for semi-Riemannian spaces anymore. So a discrete subgroup of the isometry group of a semi-Riemannian manifold does not need to act properly.
So one example is-- and this is, in a sense, one of the easiest example of semi-Riemannian manifold-- is, if you take-- well, let's take SL(2,R), also. If you take SL(2,R), and you look at this as your manifold, with the Killing metric. So with the metric coming from the Killing form.
So this is a nondegenerate metric, by linear form of signature 1, 2. So this is not Riemannian, but it has a nice isometry group. So if you look at the group of isometries, it's SL(2,R) cross SL(2,R), acting from the right and from the left [? on ?] by right and left multiplication. And, if you don't like SL(2,R), you can just put any group G here with the Killing metric, and we have G cross t as the isometry group.
OK. So now, if I have a discrete subgroup of SL(2,R) cross SL(2,R), it does not have to act properly discontinuously on SL(2,R). And one example you can take is, just take your preferred Fuchsian representation and take your surface group embedded by your preferred Fuchsian representation into the first and into the second factor. This will not act properly on SL(2,R).
And this is actually-- I mean, this-- you can also do two different Fuchsian representations. And the fact that, whenever you take two different-- pick as you'd like-- Fuchsian representations of the fundamental group of your surface to embed your fundamental group of the surface into SL(2,R) cross SL(2,R), this will never act properly on SL(2,R). And this is actually related to Thurston's stretch metric on Teichmuller space. This is somehow the reason why you can define Thurston's stretch metric on Teichmuller space.
AUDIENCE: Could you say the representation again [INAUDIBLE]?
ANNA WIENHARD: Hmm? If you take two Fuchsians, two discrete embeddings, of the [? fundamental-group ?] surface-- say, rho 1, rho 2-- you embed your [? fundamental group ?] then diagonally, embedding gamma into rho 1 of gamma and rho 2 of gamma, this group will never act properly on SL(2,R). And it's related to the fact that you don't find two hyperbolic structures where, in one hyperbolic structure, all the curves are shorter than in the other, if you have a closed surface.
But we can ask-- I mean, we don't have to take two Fuchsian representations, so it does take two representations from pi 1 of your surface into SL(2,R). And say that this pair of representations is admissible, if the action, induced action of the fundamental group on SL(2,R) is properly discontinuous. OK. So there are such--
So for example, you can check that, if you take rho to be of your preferred Fuchsian representation J to be the identity, then-- so the trivia representation, then this is satisfied. And there are more examples. Actually, there are a lot of-- so I don't have time to discuss all the results here, but there has been recent work by Kassel, Guéritaud, Wolff, and by Thom Delzant and Deroin, who show that-- I mean, when you take J to be in any connected component, not the space of discrete [? phase-four ?] representations, you find a discrete [? phase-four ?] representation to make this admissible. So there's actually a huge set.
So what I would like to do is relate this admissibility question to the Anosov property. And there's the following theorem, which is joint work with Guéritaud, Francois Guéritaud, Olivier Guichard, Fanny Kassel, and myself. And I state it for SL(2,R).
So the following are equivalent. So one is, if we have such a pair for representation, this pair is admissible. And the second is, a certain representation is Anosov, but not this representation into SL(2,R) and cross SL(2,R). But, to see the admissibility or see the properness of the action, we actually have to take SL(2,R) cross SL(2,R) and embed it into a bigger group.
So let me put it here. The equivalent state--
--point, in this representation [INAUDIBLE], which which come from integral representations, so which are in the integral group. But to really have somehow monodromy representations or things which-- I mean, I don't know. [INAUDIBLE].
SPEAKER 1: Any questions? If not, let's thank Anna again.
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Anna Wienhard of the University of Heidelberg and Princeton University describes some developments in the study of locally homogeneous geometric structures and related subsets of representation varieties of hyperbolic groups into Lie groups of higher rank, June 25, 2014 at the Bill Thurston Legacy Conference.
The conference, "What's Next? The mathematical legacy of Bill Thurston," held at Cornell June 23-27, 2014, brought together mathematicians from a broad spectrum of areas to describe recent advances and explore future directions motivated by Thurston's transformative ideas.