 We are now having a problem session and the success of this problem session is going to be determined by how much everybody participates. We have heard lots of courses, we have heard lots of current research, but mathematics is not a spectator sport and we all learn it much better when we do problems. Mae'r hwyl wedi bod ni wedi boddi'n cael ei ddweud o'r ffordd gyda'r unrhyw i ddiwylliant a'i ddweud i gyd-fyrdd cymryddiol â'r hyn sy'n mynd, ac mae'n gwneud i chi'n bryd o'r holl o'r ffordd gyda'r hynny o'r hyn. Yn y gallu'r problem-sesion, is that some of us who have got a lot of experience, or some of you even who haven't got much experience, have got lots of maths problems that we want to think about, and there are lots of young people here looking for interesting problems to work on, and if this works well, then we have some problems that different people are going to propose. Some of you are going to be saying, I think I have an idea of how I might be able to solve that, and then we will talk to each other, and in the future there will be collaborations going on between people in different parts of the world, and then we all get to know each other better and the mathematics community gets strengthened, so that's my ideal for how it's going to work. So some people are going to be proposing problems, and I've asked a couple of people to take notes of these problems, and these are going to be typed up and then put on to the web page of the conference, and then together with the contact details that are also on the web page of the conference, if you're interested in this problem, you can then email the person who proposed it, maybe with some ideas or some questions, and perhaps in the future then you will have a project together, and so when we come back in five years' time, we will hear about many of these things. So that's the idea of where we're going. Now, I know some people have some problems. One of them, who's Bertrand, has to leave very soon, so I'm going to ask him if you'll come first and present his problems and then we'll move along to other people so that he can shoot off to the bus and have a good trip back. Thank you. So the problem I wanted to ask is a problem concerning stability theory. So we heard a lot about Kleinian groups acting on three-dimensional hyperbolic space or on CP1, and there is a very great theorem by Sullivan, so maybe let me state really briefly what is the theorem. So this is a theorem about stability of the action of the group on CP1. So it's a kind of so-called structural stability theorem, which says that you might take gamma in PGL2C, a Kleinian group, and then we have seen a lot of examples where you can deform the group gamma in a family of groups gamma t, t depending on some parameters, let's say 0, 1, such that gamma naught is equal to gamma. So Sullivan proposed the following definition of structural stability. He said gamma is structurally stable if for any such family there exists an epsilon, such that the action of gamma epsilon on the Riemann sphere looks exactly the same as the action of gamma on the Riemann sphere up to a homomorphism. Namely, structural stability means that there exists epsilon such that for any t there exists, so less than epsilon, there exists a homomorphism from the Riemann sphere to itself, such that the action, so the gamma action, let's briefly say that, the gamma naught action on CP1 is mapped under this homomorphism to the gamma t action here. So what Sullivan proved is that in fact the group which has structural stability is either the rigid group, so those groups that you cannot deform inside PSL2C, so for instance co-compact lattice in PSL2C we know that it is rigid by most rigidity theorem, or if it is not like that, in fact it is a convex co-compact group. So the stable group are the rigid group or the convex co-compact group. We know for a convex co-compact group we have seen a lot of examples for instance quasi-fucsian groups. They are structurally stable, namely if you deform, so you have a limit set which is a quasi-circle and we have seen a lot of talks here where the limit sets were moving continuously, for instance, my hand stalks this one. So my problem, yes, it is a homomorphism, so it satisfies the following. So if you take any gamma, so then it will satisfy this equation. So if you just follow the element in the deformation of your group, so you get from any gamma you get a gamma t for any t. And so the homomorphism should satisfy such an equation for any gamma. This is what I meant by phi is conjugating the gamma naught action on Cp1 to the gamma t action on Cp1. It's continuous, yes, it's continuous, continuous family. So maybe you could reformulate exactly the same thing as looking at the whole deformation space of the group. No, not just for all. So the Sylvain theorem says that if for all families, possible families, you have this structural ability, then in fact either you are a rigid group or convex co-compact group. So the problem is not quite a question. So the problem I would like to address, I think it's difficult, but still... In fact, I don't think there is any theorem like that for representation in higher dimensions. We have seen a lot of such representations, for instance in complex hyperbolic groups or in, let's say, SNR or SNC for n bigger than or equal to 3. And I think that would be really great to develop a kind of stability theory to see if such theorems could be true for instance. So the question is it true that if you have stability in this sense, structural stability in this sense, then in fact your group is anozoff in the sense of labourie, et cetera. I don't know, but it's possible. It would be really interesting to have, to answer these questions, even in particular cases I think, even in SL3 CEO or something. Okay, yes. So I think this is part of the problem to define the thing, but you can look at some action on projective space or flag varieties. I don't know. For another representation, I think you have stability on all these flag varieties probably. It is stable if the converse is true. So Bishop told you that the converse should be true in this context. But I think that, so the question that Nicolae was addressing is on which, so first question was on which space do you see, I mean, will you detect stability? So here it was P1C. But if you take a group acting, a subgroup of SL3C for instance, you could look at P2C or flag varieties. It's not clear what is a good space to look at. So I agree this is part of the problem. But then Nicolae said that even if you take for instance another representation, so we expect it is stable because another system, dynamical system, are always structurally stable. So we expect some structural stability properties of another representation, but Nicolae said that it is not even clear that this is true. So maybe of course we should look at, in fact I have not thought about that. But I'm sure that there are some stability properties for another representation because I'm sure of that. For instance you can restrict, you can make a weaker definition of stability only on the limit set. And for another representation I think this is quite clear that this would be true. I don't know. I mean this is more a problem than a question, a precise question I think, this is really interesting I think to think about these kind of things. Okay, thank you. Are there any more questions about Bertrand's problem? Well I've got Todd waiting in the wings. Some of these are actually problems, some of them are more directions to go in as it were. All right, I'll put my initials by them. Okay, I almost gave Sir a heart attack because I can't tell the difference between a torus, and a one-hold sphere, and a one-hold torus. I don't know why. But on the one-hold torus, with a cusp, you can look at McShane's identity. Sir told me to do this. Oh yeah, I remember this. Okay it's sum of 1 over e to the length of a simple closed geodesic, plus 1 is equal to 1, right? And in a recent paper by, it's on the archives, by Charette and Goldman, they differentiate this, and if you remember that, for those of you who weren't here, but for the students, if you remember what I was talking about, when you take the derivative, when you're expanding, what are alphas, those alphas, the Margulis invariants, right? They're derivatives of lengths, right? You're expanding lengths. So you can differentiate this thing, and you get some new sum is equal to some alpha because you're opening up the cusp, right? And so you get another identity here. Okay, so the question is, you go over and you see Sir 10, and you say, how many different identities do you have? There's not just one, there's like bushels of them. He has them growing all over his house. And you say, I'll take another one of those identities and do the same thing, differentiate it, see what you get. So it's a nice, I think this is an easy problem to do. It's a doable problem. See what happens. So try Sharet Goldman on another McShane type identity. So that's the first problem. I think it's kind of a nice easy problem. I think it's, I don't know if it's a thesis problem, but it's a nice thing to write up and see what you get. Maybe you have a nice organized way of looking at it like Sir does on all these McShane's identities too. All right, so we'll do another one. Well, you're going to look at, so differentiate means look at, differentiate means look at an affine deformation. Affine deformation, okay? So you take this torus, right? You have a surface and then you look at an affine deformation and that is differentiating the, that's going to take the derivative of the links. And whenever you take the derivative of the links, you get these alphas, these Margules invariants. So this sum is going to have, I mean I could almost write it, but I don't, I probably screwed up, but it's going to have lengths of closed geosics and for the affine deformation that you have, it's going to have an alpha for it too. So it's going to have, this is going to have lengths, it's going to be a function of lengths and alpha of gamma. So for a particular affine deformation. McShane's, I don't know the, Sharet Goldman? Look at Sir, do you have some references? No, I mean, but like other identities. Yeah, you're going to see a bunch of them. He's going to do a bunch of them. He just, they grow. I don't know how. They just push up. I'm sorry, I'm sorry about that. I got ahead of myself. We should have done this afterwards. All right, there's another, I'm just, Virginie Schill today. A long time ago, I don't know, a long time ago, she looked at, just a year ago, she looked at ideal affine deformations of ideal triangle groups. So in this case, you have three geodesics, which meet an infinity. You now draw their affine, you draw their associated closed geodesics, and you say, now you have three lines, you put it into R21, and you have three lines in general position. That's maybe, I don't know if they cross or whatever. And now you look at reflections in each one of those lines. Reflections on each one of those lines are going to, they're going to preserve a crooked plane. They're going to preserve a bunch of crooked planes. The vertex, this is going to be the spine. There's that unique space-like line on the crooked plane. They're going to define a crooked plane, but you can put the vertex different places. For your vertices. But it's rather rigid, because you can only move in those three lines. Okay? So there are examples of charats where, well, I don't know of examples, but she shows that there are ideal affine deformations, affine depths of ideal triangle groups, which properly on the affine space, it's E21, and I should say it's E21 up there. It doesn't matter. Okay? But do not crooked fundamental domain. Okay? And the idea here would be to try to find a nice-weighted domain. Kind of like when John Parker and Bill Goldman were talking about triangle groups, you had a conjecture of how far you could push this. Can you dimple, like he dimple, Rich Schwartz came along and dimpled the spinal spheres. So can you dimple the crooked planes in a nice way to create a nice fundamental domain? In general, I talked to Virginie about this, and she said, well, you should just redo that whole paper, just because we have new techniques and everything. So we're expanding the techniques. So this is to find nice domains, okay? They're going to have possibly pieces of these crooked fundamental domains, but maybe it's going to be dimpled. Maybe there's going to be another way to think about them. But it's much more rigid than the usual case because you have to have your crooked planes on the vertices on these lines. So you can't move them off these lines. All right? I'm not done. None of them are good, but we're good. No, this one's good. This is in the same vein. If you remember, again, people who weren't here, but hopefully you've heard enough of these things that you'll get used to this. Okay, in R21, or E21 if you want to call it, okay? The crooked plane conjecture holds, okay? And that means that every free, properly discontinuous action on E21, you can find a domain bounded by how many crooked planes you need, okay? Yeah. Sorry. Yeah? Oh, this is any... Oh, I haven't gotten to the next question, right? But see, this one was about free... This is about free... This is for free actions. These are... Oh, this is the problem. Because these are not free. These fix... You're going to have a lot of fixed points here. Okay? But you do have a finite cover, but you could say, well, what are those going to look like? So, you know, is there a nice one? Yeah, in some ways, it's not so shocking, but in other ways, it'd be nice to see these and see what rigidity you have on this, or to see what confirmation you have. But in this one... So, this is from the notorious DGK, since Fannie left, I can say that. Anybody heard of the note? Oh, that's right. Darn! They're notorious. They're terrible. But anyway, so... But also by the notorious DGK that in ADS 3, the crooked plane conjecture fails. They show examples, but all of their examples are about doing these lines and moving them on the hyperbolic surface. So they create... The last thing I did in my lecture was I took a line in this and I built a crooked plane. They look at how lines are moving on the surface and they relate that to the crooked planes. So the question is, again, can you find a nice fundamental domain for these things where the crooked plane conjecture doesn't hold? Again, same. Find a nice way to... nice domain. Probably related to crooked planes but slightly different. All right. So that's not... I'm not quite done. All right. There was a throwaway line that I told you that I said in my course. So... Yeah, okay. I'm not going to do that one because I'm not sure about that one. The next one, we're going to erase Bertrands because it's too crazy for words. Yeah, he's gone. So I can say what I want about him about Bertrands. He's too nice. So the natural objects that appear... Well, the natural things that appear in R21 don't appear in R31, right? Because in R31 you don't have one as an eigenvalue. So you don't get free proper actions, right? So you might look for the next place where this occurs. And strangely enough, the next place where this kind of Margulis spacetime might occur, or a Margulis spacetime like thing might occur is not in R32 because of some weird thing with the alphas, but in R43. So look at R43. Of course, now we can go crazy. R43, R65, et cetera, et cetera, et cetera. So in these spaces, don't ask me what this means, but they're real split. A generic element will look like generic element SO43 will be conjugate to something like 1, lambda, lambda inverse, lambda 1 inverse, lambda 2, lambda 2 inverse. These are all real numbers. It's ident, and you have these coming in pairs. Kind of like these inausal representations, but they're a little bit more general than the inausal ones that François gets, I think. All right. And so, in all of this stuff, you can define alpha, right? You can define alpha. So, and there exists free proper actions, okay? So, so, what you might want to say is is there a geometric version of this? Because we went a lot further than just the algebraic methods of Margulis and stuff. And Margulis and Seifer and Abel have done a lot of this. But there is a geometric version by, and I'm going to please try to get my, get this name right for me, Smylga? Is that the way we, is that the way, recent student of Yve Benoit, and now he's at Strasberg? Is he, yeah he's, what? Oh, he's at Yale, he's at Yale. Okay, that's right. Okay, he created these, a kind of crooked plane in 4-3 space. And, but the crooked plane that he created was a little flabby, so you can't calculate with it. So, the question is, can you find something a little bit more rigid, or change Smylga's approach so that you can get a little bit more rigid to get these things in R4-3 to figure out what the parameterization, the Margulis space looks like, okay? So, you can, you alter Smylga's approach to get some, to understand the deformation space. I've thought about this on and off, and I, you know, it hurts my head, because you know, I only think on every, it only works on every other Thursday, and then I can only do in Dimension 4, and I can't get up to Dimension 7. It's very confusing. Okay, there's another related one to this, and this will be my last one, thank God, as you all are saying. Get me back to some hyperbolic geometry, real hyperbolic geometry crap, or complex hyperbolic geometry stuff. And the last one is Al Sander's conjecture. Fanny talked about this. Al Sander's conjecture states that if M is affinely flat and compact, I'm writing it out because that's the most important part, all of our examples have been non-compact examples, then Pi 1 of M, by the way, you need complete 2, I'm sorry, we'll put complete here. The physicists are actually interested in non-complete things, but we like complete things. So then the question is, then Pi 1 is, I think it's virtually, virtually solvable. I think that's the... So this is true. It's proven in dimension less than or equal to 6. And there's a problem, and where's the problem? One of the problems is in, what about 4-3? The 4-3 signature. Okay? What about 4-3? This signature right here, these manifolds and higher manifolds. And is there an idea of how to figure out if you're forced to have compactness, can you get... Basically, if you're forced to get compactness, can you get rid of free groups? And that's the question. There's reasons for that. I forget all the reasons. Again, a lot of it, a big deal is Eigenvalue 1. But that doesn't throw away all of them, but it throws away a lot of them. That you want something who's... Because there's a bunch... The references to less than or equal to 6, I think it looked at... I'll probably be off on this, but look at the papers by Abel's, Margulis and Soyfer. I think they discuss it in detail and they go through it. Also, it's home and up, might have some of this too. So, you can look at... I don't know how much he has, but Tomino has some of these. And he's using very algebraic approaches here. Tomino does, but they all are using... All of this approach, the approach that they're trying to do with R43, is really back to the original Margulis paper, which is really a lot of estimates and a lot of angles that are just wacky. More questions? Okay. Well, Peppy tells me he has something up his very chic sleeve. I can't say that. Thanks. It's just one very basic problem, elementary. Consider Gaman, this kids group in PU21. So it acts CP2, preserving hyperbolic to space, which is a ball. Gaman is a lattice. It acts minimally on the boundary of hyperbolic space, the ideal boundary, which is this three-sphere. So in the three-sphere, every orbit is dense. And then it's an exercise, very easy, using practice geometry, to see that in price it acts minimally on CP2 minus the complex hyperbolic space. Okay? So now the question is, what if it is not a lattice? Okay? Suppose now that you have an action here, if Gaman is not a lattice, then you will have a fundamental domain hitting the interior of the ball, hitting the sphere at infinity with a fundamental domain for the action on this sphere. And then outside, you don't know what happens. The regional of, say, equicontinuity can have infinitely many components. You don't know what happens. You don't have to control in general. But let me say, let us assume that, let omega not be the connected, omega not equal connected component of the equicontinuity region that contains C. Okay? So this action is equicontinuity in the interior of the ball. Because of this, that's extending to some set. Around? So the action is, sorry, the problem is, study omega not and the quotient omega not divided by the action. So for example, can we describe a fundamental domain? Yes. On the limit set in the boundary. Yeah, yeah. The number of connected components? Yeah. Yes, yes. Yeah, there are. We have examples, there are examples in which you can have one, two, three, four or infinitely many. I don't know if that's always the case or that this quotient is compact. It should be. So it's a compact manifold that contains a sub-manifold or an open sub-manifold which is complex hyperbolic and then you have a projective structure on the compact one. Can you have compact quotients? In the city? We know the answer but I don't know. Yeah. Si ti'n es i'r context o'r cyflantie? Pwedeis ta'n erw'n coosint i'n compacto? No, so sorry. You cannot have a compact quotient. We have the classification of the compact quotients that you have and then looking at that classification you can see which kind of open sets you can have and corresponding groups and so on. No idea. The dimension three is wild. Thank you. As the other three organisers have all presented problems I'm going to present some problems and then I'm going to open it to everybody else for more problems. So the broad general idea would be if I've got s a surface finite type I've got pi1 of s fundamental group then study the representations of pi1 of s into pu21 and the geometry. So that's a very broad theme. There's various things already known. I've got a survey paper a few years ago with Platys where we gathered together many things but there are still many things that are unknown. So one thing that you might like to try to do so problem one would be see how far you can go in the direction that Mahon was talking about about Canon Thurston maps convergence of limit sets and so forth. So complex hyperbolic version I'm just going to say let's talk. So this would be limit sets Canon Thurston etc. So I mean yes that's going to be my next question. Right? So I mean I didn't mean the sophisticated part of his talk I meant the baby part of his talk. Okay? So let's just think about can you get a just this basic thing of do you have this Canon Thurston map and does that exist and so forth and if you have a sequence of groups how do things behave? So a second question could be what happens as you go to the boundary? My belief which I have tentatively stated as a conjecture is that you always get pinching i.e. curve going to a cusp or something so curve going to a cusp or I don't know what rank the cusp would be If that's true which curves can you pinch how many can you pinch at the same time and which ones can you pinch at the same time? Sorry? Yeah yeah? Oh it's going to be yeah yeah yeah yeah So then three Who or how many can you pinch? So Pierre and I have a one parameter family of three punctured spheres where you have seven curves pinched but we don't know whether these are discreet or not we just that we have seven conjugacy classes of parabolic elements Can you do better? Can you get eight? It's in the Almora proceedings Yeah So my belief would be that somehow your boundary of your quasi-fugtion space has got some sort of polyhedral structure where you would have Codimension 1 facets where one curve is pinched and these intersect along Codimension 2 curves where two are pinched Yeah So the boundary of Mae'n gweithio'r clywed dynol yn cyfnodol. Mae'r gweithio'r dynol o'r bwysig yng nghymru, ydy'r bwysig iddyn nhw'n ddweud beth rwy'n ddweud hynny'n ddweud. Mae rhai gyrfa mhag o ddweud yn S-U-2-1 yn ei ddweud a'u codiwn dynol 1, wrth gwrs, yn S-L-2-C yn ei ddweud a'u codiwn dynol 2, ac ydych chi'n gweithio'r cyfnod o'r cyfnodau, sy'n gweithio'r cyfnod, sy'n gweithio'r cyfnod 8g-8g, ac mae'n gweithio'r cyfnod o'r locus, oedd y ffordd yn peribolic, sy'n gweithio'r cyfnod o'r algebrae, mae'n gweithio'r algebrae yma. Yn ymddangos, mae'n gweithio'r ffordd y ffordd o'r rhai cyfnod o'r rhai cyfnod, sy'n gweithio'r rhai cyfnod, sy'n gweithio'r ffordd o'r algebrae, mae'n gweithio'r ffordd o'r algebrae, oedd yw'r cwrdd ymlaen? Can y gallu gweithio'r ffordd o'r arbennig? Mae yw'r cyfnod ac iddyn nhw, yn cael gweithio'r cyfnod, Ac mae'n gilydd yn gweithio ar hyn o'r ffordd. Mae'r ddechrau'n iawn i'ch gael o'r gweithio. Mae'r ddechrau ar y casgliad yma, yr hyn o'r casgliad, yr hyn o'r casgliad yma, ond yna'n gweithio yma o'r Loxodromic, a'r ymlaen o'r ysgrifennu. Felly, ydych chi'r pwysig o'r gweithio, yn gweld i'n ffaithio, mae'n gweithio ar y ffaith, fel mae'n gweithio ar ei threif, ond mae'n gweithio ar y ffordd. I do the absolutely stupid thing of looking at this big space, which I can't possibly imagine even on alternate Thursdays, and I cut out all this locus where all of these people are parabolic, and look at what's left. Is that my discreet and faithful representations? Another fact is that in even dimensional hyperbolic spaces, to be nondiscreet, you must have an infinite order elliptic, if you're a whisky dense. Absolutely, absolutely. So it should be a mapping class group invariant polyhedral. I say polyhedral, I mean a very, very loose sense. I don't think you can represent a hyperbolic three-manifold group into SU21. But I think it's not possible to do it into SU21. Anyway, I'm running between the two of you. I hope the rest of the audience is at board. Sorry. You have these raw key examples, right? Not faithfully, yes. Then that would work. But I don't think that happens. Yeah, does he? He's busy. Maybe another slightly more vague question. So far, there's very little in the way of, when I say analysis, I'm thinking of how alfos and bears develop the classical theory of quasi-Fuxian groups. There is a notion of quasi-conformal mappings on the S3, or quasi-conformal mappings with respect to this CR structure. They're not good enough to get all of your deformations, because a quotient of a vertical Heisenberg translation and the quotient of a non-vertical Heisenberg translation are not quasi-conformally conjugate. So if you have such a path of representations, then you couldn't realise this by quasi-conformal mappings. So there are good things. I want to say here you should see Loftin and Macintosh, where they have been introduced cubic differentials and certain differential equations, which is a good starting point. And so they, well, I won't go into that. But that's a good starting point. But it only tells you a small subset of these deformations. Can you push this all the way to the boundary in some way? How about quasi-regularity? I haven't really thought about these things. The topology of that space, not off the top of my head. The question was, do I know anything about the topology of the space of convex co-compact representations? So what I do know is that your manifold is going to be a disc bundle over your surface. And so you have an Euler number associated to that and different Euler numbers can arise. So you would have a different component for every Euler number. Because that's a topological invariant. In particular, there are examples of Anonidon-Gazewski, I think, where it's actually a product. So you have Euler number zero. There are also examples where you have a wild knot. So a little bit like some of the things we were hearing from Gabriela last week. And so those cannot be in the same component as something with a topological circle boundary. So I think there are many, many islands of representations. And I don't know what the shape of each island would be. What are the examples? So I think we were just talking about surface groups there. So you want other groups other than surface groups? Well, I was thinking in surface groups there. So anyway, so there's lots of questions there. Finally, one of my favourite questions, and I've been asking this question for at least 10 years now, probably more like 15 years. So Gazewski and I and Alicia and people and various other things have constructed examples of continuous paths of representations of punctured surface groups that interpolate between, for these, this is from people who were around last week, Arffwxian and Seafwxian. So I've got a path here. At this end, my group is Arffwxian. At this end, it's Seafwxian. The limit set of all of these is going to be a topological circle. And its house-dwarf dimension should vary probably real analytically with respect to my parameter if I've done it set it up correctly. So then I want to think about the house-dwarf dimension of the limit set of gamma t. At this end, the house-dwarf dimension is 1. It's a round circle. The way you measure house-dwarf dimension in using this CR geometry that we've heard about means this is a strange thing when you first see it that here the house-dwarf dimension is 2 because you need to have quadratically many little balls to cover your line. So I have a real analytic function interpolating between 1 and 2. I know that as soon as I leave here it's strictly bigger than 1 because there's a version of Bowen's theorem due to Chengbo Yue and various other versions I've seen. What does this function look like? Is it monotone? In particular, do I approach 2 from below? Or if I approach 2 from above, what's the maximum value and why? I think so, yes, yes. So yes, my hand has just pointed out there's some papers I think or let said it bounded away from 3 if that chimes with my memory as well. And this is certainly an application of can you get good analysis techniques to work in this case. My guess is it's probably going to approach 2 from below but just because I can't imagine why one group in the middle should have a higher house-dwarf dimension than the others. That seems to be a monstrous thing but my guess is it could well approach 2 quite shallowly. The wild limit sets, I don't know what the house-dwarf dimension of those would be. So that's another question. See, there's questions and questions and questions and they're keeping going on. But this is a general area where I think it's very fertile there's lots and lots of questions and if you know the real hyperbolic techniques well for any of these questions then there's a good chance that something will move across into the complex hyperbolic world. And of course you can always combine this with Pepe's question and say well okay let's just embed this in CP3 and see what happens there. So that's the end of my questions. Just a quick answer to your question. If you take this quotient, if you have an open set with a group which acts, say without torsion free so that this is a manifold, which automatically is going to have a projective structure and the compact complex manifold with a projective structure are classified. They are either complex hyperbolic or complex affine. Then you only have to see that in this setting you cannot have something which is complex affine. So are there more problems? Todd is nominating Sorin from afar. So do you have some questions? So this is exactly the question I ask at the end of my talk about completeness of some GX structure, but let me insist then. Sorry about this, I will be brief. So this is somehow the complex version of results which are known in the Loren setting about completeness manifolds locally modeled. Let's take the simplest example. Here would be the complex Euclidean group acting on C3 or here G would be SL2 of C times SL2 of C acting on X is SL2 of C by left and right translations and preserving the Keating quadratic form. So they are complexified version of the ADS3 for which Klingler proved completeness. One interesting feature here is that you have some Pseudor imanian metric which is preserved. The real part of this holomorphic imanian metric is a Pseudor imanian metric of signature here C3 or in dimension N it would be NN for which for example the flat case here is a special case of Marcus conjecture since you put on the blackboard Oslander conjecture maybe I will recall people Marcus conjecture which say that complete let's say compact flat a fine manifolds a parallel volume are complete. So here in fact X is RN and G is SLNR semi direct product with RN so this is an old conjecture maybe 50 years old now. A fine manifolds are complete as soon as you have a volume which is preserved by the olonomy. It's known to be true if you replace here SLN by the linear group preserving a Lorentz quadratic form so flat Lorentz manifolds on compact manifolds are known to be complete by career theorem but for Pseudor imanian orthogonal group for Pseudor imanian metric this is still an open question and the point here is that if you look to those complex manifolds locally modeled on this complex affine basis you have a feature here which is specifically complex geometry comes from specifically from complex geometry you have complex curves which are geodesics so one could try to understand the geometry of these Riemann surfaces of course they have a translation structure as all geodesics have and if you want to have compact manifolds you can projectivise the tangent bundle so you get this holomorphic foliation by Riemann surfaces in the projectivised tangent bundle and these Riemann surfaces have some complex affine structure because it comes from the translation structure of the geodesic which was and you take quotions by homothesis so you have complex affine structure along these geodesics namely these geodesics cannot be copy of P1 of C so in the classification of Riemann surfaces they are either covered by the unitary disc or by C so the first step would be to show that these geodesics are parabolic they are covered by C and here there are tools coming from the theory of holomorphic foliations tools which were part of them were developed by Brunella or Candel if you assume you have too many leaves covered by the disc you can uniformised by the disc in families these geodesics so you have something which is negatively curved which would be somehow in contradiction with the flatness of your holomorphic Riemannian metric so this would be an extra tool to be used here in the complex setting and solve a baby part of the Marquess conjecture and also I recall that Tolo Zann have examples showing that if you have these kinds of geostructures which are uniformised if you have quotions of open sets in the model which are compact the open set will be all of the model so it goes into the direction of Marquess conjecture and shows that in the second case those which are complete those of geometric structures which are complete they will form a closed set and results of Fanny Castle, Gerritol, Gishard and Anna Wienhard show that these geometric structures which are complete form also an open set so there is those which are complete in the deformation space is a union of connected components so maybe there are exotic ones but far away from the examples we know so this would be a question and sorry I already asked it in my talk but I wanted to be more precise Thank you Have we got some more questions? I think common serrators had come up in Ruth's talk and limit sets have been sort of functioning all over the place so this is a certain question which so let's look at the general context so G is now a semi-simple league group P minimal parabolic or such that so G mod P is the first and the boundary so what you should be thinking of is a very concrete example is G equal to hyperbolic plane cross hyperbolic plane and then G mod P is the circle at infinity cross the circle at infinity so even in this context all the stuff about limit sets does go through so if you have gamma contained in G being Zariski dense then lambda gamma contained in G mod P is the smallest non-empty closed invariant set all right so I mean if you feel like thinking of rank one that's fine if you like thinking higher rank that's also fine all right so there's this theorem which Ruth had mentioned during her talk this is a characterization of arithmeticity gamma contained in G is arithmetic lattice gamma contained in G is arithmetic if and only if the commensurator of gamma is dense in G so this is by Margolis now how about gamma contained in G Zariski dense but not a lattice so what do you mean by how about this commensurator of gamma can commensurator of gamma be dense so this has to do with another topic that has been coming up in relatively recent times it's sort of sold under the terminology thin groups and that's sort of a little offensive to Kleinian groupies because we've been dealing with thin groups all our lives okay so okay so it just says that you take some gamma which is a risky dense which is of infinite co volume that's called a thin group so this has been sort of a topic that has been discussed and sort of promoted largely by and started off by Sarnac and a whole bunch of people in the league groups world are talking about it so question what are commensurators of these things so here's a proposition that if the limit set of gamma contained in G mod P the first and bug boundary is not equal to G mod P then so this is question star let's say so then answer to star is no okay so if the limit set is not everything then one can conclude that the commensurator is indeed discrete yeah so what we were dealing with in the morning today and lots of people have dealt with are the case where only in the three-dimensional PSL2C case thin groups with limits at the whole sphere are there examples in other contexts so the finitely generated so what's an example so let's give a very concrete version of this example in dimension two PSL2R where it's not known dimension two PSL2R and you sort of look at the simplest possible group where this proposition does not apply so gamma so gamma naught is equal to say SL2Z or PSL2Z and gamma is normal in gamma naught of infinite index so this implies that the limit set of gamma is the entire circle at infinity any normal subgroup has the same limits as the whole group is commensurator of gamma equal to gamma naught you don't know so even a very specific example in no example do we know this because these are infinitely generated groups so for example you can take SL2Z pass to a congruent subgroup so that you have you have homology so take the second congruent subgroup pass to its commutator subgroup so that's sort of very functorial nice algebraic constructions as arithmetic as you can get but so EG gamma is equal to gamma2 gamma2 so gammaP where this is the pith congruence subgroup of PSL2Z so but this is infinite index it's sort of as arithmetic as you can get in the infinite co-volume world we don't know whether its commensurator is SL2Z or not okay good we are due to be having our final talk in two or three minutes time so I would suggest that we have a two minute break but it's clear that there are lots and lots of problems if you have a problem that you want to ask and you didn't get the opportunity then please can you write it up and send it to us and we will include it in the list of problems and hopefully we can if we had somebody young and dedicated perhaps we could actually maintain a problems page for the group around us I am not young and dedicated to that sort of thing but perhaps as one of you who is interested in that that would be a fantastic thing if we could have a problems page for people who are interested in the sorts of things at this conference okay good