 So, thank you, thanks to the organizers for inviting me here. In summer 2010, I was at this conference, or similar conference. Some people here remember it, and I was a baby mathematician. I had just finished my master's, was going to start a PhD, so it's a great pleasure to be back here five years later and feeling all grown up. I'm a real mathematician. So I'm here to talk to you about the volume of complete ante-de-sitter three manifolds. And yeah, I will actually give a result that also deals with some other locally homogeneous space, I will be precise later. But I would like to start by motivating the result by some more general questions. About quotients of homogeneous spaces will be my motivation. So I think last week that Dick Canary will complain about people that start their talks with a let G be a semi-simply group. But he's not here anymore, so let G be a semi-simply group. Sorry? Okay, sorry. So H will appear later. So okay, now you don't have to be scared. You can think of SLNR, SNC, and actually very soon G will become SON1 times SON1. But I will come back to that. For NO1, I'm just looking at some fairly generally group. And X will be a homogeneous space. So the quotients of the group G by some subgroup H. So it's equivalent to saying that G is a manifold with a transitive action of the group, X is a manifold with a transitive action of the group G. And I will assume that G, the action of G on X preserves a volume form that I will denote vol X. So there are plenty of examples of things like that. There are particular lots of examples where G preserves a pseudo-Riemannian metric on X. And when it preserves a pseudo-Riemannian metric, it also preserves a volume form. Okay, so I have to introduce some terminology. I will also denote by gamma a discrete subgroup of G, and sorry, here are some definitions or some terminology. So if gamma acts, you can look at the action of gamma on X. If this action is properly discontinuous, this assumption is what you need for the quotient to be a nice topological space, namely an orbifold. Then the quotient of X by gamma is what you call a Clifford Klein form. So in the title I said locally homogeneous space, what I mean is actually Clifford Klein form, a space which is the quotient of a homogeneous space by a discrete subgroup of transformations. And if moreover, so when I have a Clifford Klein form, so I can look at the volume form on X, but since it's invariant by gamma, it goes down to a volume form on the Clifford Klein form. I can integrate it and I get what I will call the volume of X over gamma. If this volume is finite, then I will say that gamma is a GX lattice. So it's, I think it's completely non-standard terminology. The point is that if the isotropic group H is compact, then G preserves a Riemannian metric on X and being a GX lattice is the same as being a lattice in G. So it's, in some sense, a generalization of the definition of a lattice in G. Okay. So it's, I don't know, it dates back from 100 years to trying to describe the spaces. It's one of the most natural ways to geometrize the manifold, identify it to the quotient of a homogeneous space. And so I will, I want to ask several questions concerning the study of the Clifford Klein forms. So the first question is, well, those lattices, do they exist? So I will give some answers later, but the second question, well, if they exist, if you have a gamma, a GX lattice, maybe you can deform it in the group G, and maybe when you, you can deform it so that it remains a lattice. Or maybe when you do such a deformation, the only way to do it is to conjugate gamma. So it's essentially a trivial deformation. So if every deformation of a lattice in the space of lattices is trivial, you will say that the lattice is rigid, and the question is, are they rigid? And the third question, which is the one I will address, is, okay, suppose I have a lattice, a GX lattice, and suppose I can deform it non-trivial, so it is not rigid, then is the volume of X over gamma going to vary with the group, or is it going to stay the same? And so if it stays the same, I will say that the volume is rigid, and so the third question is, is the volume rigid? So those are fairly general questions that make sense in my setting, and maybe I will, the first thing I will do is remind you that the answer is well known in the, what I call the Riemannian case. So the Riemannian case is when H is compact, so when H is compact, as I told you, G preserves a Riemannian metric on X, and being a lattice in G is the same as being a GX lattice, so this column, then later I will go to the non-Riemannian case. The point is that, yes, the point is here it's just recalling the classical theory of lattices in, in E groups, and so the first, the answer to the first question is always yes, so the main theorem is due to Borel, who proves that every semi-simple E group admits some lattices, those lattices are arithmetic, but then there is a whole work to prove that there are also sometimes non-arithmetic lattices by Gromov, Piotrowski, Shapiro, Thurston, Deline-Mostow for complex hyperbolic lattices, or more recently, De Raux-Parker-Pauperte, as we saw yesterday, but yeah, essentially every semi-simple E group admits a lattice, what about the rigidity of this lattice? Well, it's also now a classical result that the answer is essentially yes, by a theorem of Mosto for Lie groups of rank one, and Margulis for higher rank Lie groups, so the answer is yes, unless there is one case where lattices are not rigid, and it's when G is essentially PSL2R, you know that there's this theory of Taichuner space, or Fricker space, where you can deform the hyperbolic structure on a closed surface, so you get all deformation space of lattices in PSL2R. And finally, the third question, well, the answer is always yes, but why is it always yes? Well, for every Lie group, it comes from the rigidity of the lattice, so if the lattice is rigid, its volume is rigid, and the only case which is not solved is the case of PSL2R, but here you know that, so you're essentially looking at the volume of a hyperbolic manifold, and you have the Gauss-Bonne formula that tells you that this volume is essentially a topological invariant, so it does not vary when the lattice is varied. So PSL2R case comes from the Gauss-Bonne formula. So, okay, what I will be interested in is the non-Riemannian case. So in non-Riemannian geometry, in particular in pseudo-Riemannian geometry, so here I'm assuming that H is non-compact, well the picture is much more complicated, and there is no categorical answers like this, so yeah, in some sense, in the non-compact case, you can, yeah, it's also the Gauss-Bonne formula, to be a bit, yeah, essentially, yes. Well, there are several ways of saying it, the Gauss-Bonne formula, saying that every hyperbolic pair of pants has the same volume, or something like that. Well, so in the non-Riemannian case, what is the answer to the first question? Well, sometimes it's yes, sometimes it's no, and sometimes we don't know. So for instance, I can look at this homogeneous space SO2N2 quotient by SO2N1, so experts might have recognized the ante-de-sitter space of dimension 2N plus 1, so later what ante-de-sitter space is, and here the answer is yes, you can find ante-de-sitter lattices. There are some discrete groups acting on, properly discontinuously on ante-de-sitter, with a finite co-volume, and even co-compact subgroups. But if I look at another space, which is the de-sitter space, so this time it's SON plus 1, 1 quotient by SON1, so it's de-sitter space of dimension N, then here the answer is no, it was noticed first by Calabi-Marcus in 62. So okay, and the answer for the first question, you have very seemingly, very similar homogeneous spaces, one admits some lattices and the other one does not. So what about the rigidity in the case, in the case where you know that there are some lattices? So I don't think there is a general theorem, I don't think there is a homogeneous space for which we are able to prove that every lattice is rigid. But in some examples we know that the lattices we know are rigid, so for instance, how do you build a lattice, ante-de-sitter lattice, so something acting properly discontinuously co-compactly on this? Well in SO2N2 you have the group UN1, which acts properly and transitively on this homogeneous space. So if you take now a lattice gamma into UN1, then it will act properly discontinuously co-compactly, well or with finite co-volume on this homogeneous space, and this lattice is rigid. So this is the only way for N, so N bigger than 2, this is the only way we know to construct lattices and they are rigid. This is a, this comes from a result of Raghunathan and, sorry, yeah locally rigid. So you cannot deform it, yeah maybe it's a bit confusing about rigidity or local rigidity. You cannot deform this lattice. And, but I can look, here there is some kind of symmetry, UN1 acts properly, transitively on this homogeneous space, well you also have, you can also look at SO2N2 quotient by UN1. Well, but yeah by the way I say rigid, it's not completely true because you have a, UN1 has a center, so you have a lattice in UN1, you can deform it by adding some representation in the center, but this is essentially the only way you can deform, so it's almost rigid. And also you can look at this homogeneous space. On this space you have a proper action of SO2N1, so it's just the, just switching the picture here and you can take a lattice in this space and this one is not rigid. You have deformations of this lattice that becomes the risky dense in SO2N2. And so this is due to Johnson-Millson, so they prove that you have deformations that become the risky dense. And by Nick Hassell proved that those deformations give you, still give you lattices in this homogeneous space. So here again the answer to this quite to rigidity for non-Riemannian homogeneous spaces is fairly mainly open, but the picture is that sometimes it's rigid, sometimes it's not. And the answer to the first, the third question, so you have some situations where this is relevant to ask, because these deformations do the, does the volume vary? Well the answer is, I don't think the question has really been addressed before, but that's what I want to do here in a particular case. So here they don't preserve a subspace here, but they, they commute, they commute with the action of U1, because there is a center here, and this action of U1 is, gives you a killing field. So a way to see, if you have a anti-deceit geometry whose group, holodomic group is actually in UN1, it means that it has a time like killing vector field. There is a, you can tell what the geometry of such a space is. I think it acts transitively so, but not, yeah, not isotropically, if you want. So now I am going to, the second part of my talk, so I'm going to restrict which geometry I'm interested in working on. So I'm going to talk about quotients of SON1. I will say SON1, but I will put this little zero, because SON1 has two connected components, and I'm just looking at the connected component of the identity. So now, so X is not any homogeneous space. So now the space I'm looking at is SON1. So SON1, you might know him by many other names, it's essentially the isometric group of the hyperbolic n-space, and it's also the group of conformal transformations of the boundary of the hyperbolic n-space, n minus 1. So it's a group we saw. I'm just calling him by his name, but we saw it several times last week. But here I'm considering this group as a space, as a homogeneous space. And what is the group of transformations acting on X? Well, G will be X times X acting by left and right multiplication, meaning that if I take two elements in SON1, they will act on an element of SON1 by multiplying on the left by G and on the right by H minus 1, because I want a left action. So I need this minus 1. And so what, so this section by left and right translation preserves an X volume form, just because X is a unimodular group. But more precisely, it preserves the killing metric. So I won't give you a precise definition, but it's a natural geometry to consider, because on this space you have a pseudo-Riemannian metric, a natural pseudo-Riemannian metric, which is called a killing metric, which is preserved by both left and right translations. So it's preserved by G. And in particular, the volume form associated to this killing metric is preserved. So the signature is, the positive part is the dimension of the symmetric space, the Riemannian symmetric space. So it's n, because the symmetric space is hn, and the anti-symmetric part is a complement. So it's nn minus 1 over 2, and the signature is n comma nn minus 1 over 2. Well, it's not a Riemannian metric, otherwise it would be in the Riemannian setting as before, but it would mean that the group is compact. Anyway, so this is a homogeneous space I will consider, and this is, for this homogeneous space I want to compute the volume of, the co-volume of lattices. But first, let me tell you why this would be interesting to study this homogeneous space. So there are two reasons. The first reason is that a particularly interesting example is a case where n equals 2. So when n equals 2, SO2 1, let me give a name to the killing metric, I will call it Kappa. So SO2 1 is also PSL2 R, so it's an isometric group of H2. And it's also, with its killing metric, the killing metric is a Lorentian metric of a constant negative sectional curvature. So this is a model for what is called anti-disseter geometry. So this is the anti-disseter space, which is just the Lorentzian analog of hyperbolic space, the three-dimensional space, Lorentzian space with constant negative sectional curvature. And the group, the product, so G, well G is a product of two copies of SON1, is exactly the group of transformations, connected component of the identity in the group of transformations of ADS3 preserving this Lorentz metric. So every locally homogeneous, every manifold with the Lorentz metric of constant sectional curvature minus 1 is locally modeled on ADS3. And you can even say more, you can say that it is a quotient of ADS3. So it's a result by a clingler and you also need a bit, a result of Kulkarni and Raymond to state what I'm going to say. So that every closed ADS3 manifold is up to finite cover, a quotient of ADS3 of a gamma. So the point is maybe it does not surprise you. It's like if I was saying every closed hyperbolic three manifold is a quotient of H3. But since it's Lorentz geometry, it's more complicated than that. But yeah, essentially it boils down to proving that every closed anti-dacetyl three manifold is geodesically complete. And so that's why in the title I put complete anti-dacetyl three manifolds because I want to look at quotients of ADS3. But the point is every closed anti-dacetyl manifold is a quotient of ADS3. So it's a good motivation to study those quotients and understand their volume. And the second motivation, so yes, you could look at, no, I'm looking at the action of both PFL2R on the left and on the right. The action, if you only act with SO2 on the right, you get an action preserving the Riemannian matrix. You get Thurston's sixth geometry. But here I'm looking at the whole group of Lorentz and transformations. Yeah, the quotients are safe at spiral spaces. You can get non-trivial actions on the right. Yeah, yeah, it's, well, you, okay, so I'm not sure I will have time to discuss precisely the geometry of the quotients. Well, I am going to, actually I'm going to explain later that these quotients are, well, up to finite covers are all circle bundles of a hyperbolic surface. But they don't necessarily have an isometric action of SO2. That's what I mean, the quotient. Yes, yes, the base could be an orbit fold. That's why I say, yeah, those are, okay. These manifold will be cyphured fiber spaces with hyperbolic base, which a way of saying it is that up to finite covers there are circle bundles of a hyperbolic surface. But yeah, with non-zero class and this, you have a converse. Conversely, every such cyphured manifold has a geometry like that. Okay, so, so the second reason why this example is interesting because it is quite well understood. And actually it's one of the only, one of the best understood examples of non-Riemannian homogeneous spaces. So in general, it's well understood and I think I have to hurry up a little bit. So I will give you two big theorems that essentially tell you what the topology of such a manifold is. What are the groups that act properly, discontinuously on the ADS3 and more generally on SON1 and how they act. So the first theorem is started with the work of Kulkarni and Raymond in 85. So it was for the case N equals 2 but it was generalized by Kobayashi and Kassel. So the theorem is that GX clefort line forms at the form. So when I say GX clefort line form but now J and X are really these specific homogeneous spaces. So they have the following form. So it's quotients of SON1 by J times rho of gamma where J and rho are two representations of gamma into SON1. So far I didn't say anything. I just say okay I have a group. You can look at the left projection, right projection. I get two representations in SON1. The product gives me a representation in SON1 times SON1 but the theorem is that J is discrete and faithful. So it's up to something. It's up to finite cover. You have to assume that the group gamma is torsion free but you can do this up to finite cover. And it's up to switching the factors because of course if you have a J rho and you look at the action of rho J you also get a properly discontinuous action. But the point is that in particular the group gamma is a Kleinian group in the sense that it's a group that acts properly discontinuously on the hyperbolic n-space. No, yeah I'm just looking at properly discontinuous actions. So Coulcarnier and Raymond proved it when n equals 2 so far until the C2 geometry and then Kobayashi in a more general setting proved faithfulness and Kassel proved faithfulness I think and proved discreteness. So this tells you which group acts properly discontinuously. And now it doesn't tell you every which representations give you a properly discontinuous action because if you take any rho maybe the action won't be properly discontinuous. For instance if you take rho equals J then you will fix a point in SO n1 so the action won't be properly discontinuous. But Gerito and Kassel found a very nice criterion for properness. So I have to introduce a definition that we already saw last week in Bertrand's course. We will say that J strictly dominates. So now J gamma is a discrete group, J is a discrete and spaceful representation. J strictly dominates rho and I will denote it like that. If there exists a map F from hn to hn so I remind you that SO n1 is still the isometric group of hn. So if there is such a map which is J-rho-equivariant and contracting that is lambda-lipschitz, some lambda smaller than 1. And the Gerito-Kassel criterion is the following. So I will also add the name of Salon because actually Salon was approved in one direction. So you have to assume that gamma is finitely generated. So J and rho are two representations of gamma into SO n1 with J discrete and spaceful. And the theorem is that J times rho of gamma acts properly discontinuously on SO n1 if and only if J dominates rho strictly. Sorry, isn't the same as finitely generated? Okay, sorry, yeah. Yeah, Alberto gave us examples. Okay, so you need gamma to be, well you need J of gamma to be geometrically finite. Yeah, n equals 2, n equals 3 also? No, not even n equals 3. Okay, sorry. I thought Alberto gave, you gave a counter example for n equals 4, no, of a group which is finitely generated but whose action on H4 is not or S3 is minus its limit set is not geometrically finite? Yeah, I can put it, okay. Well, okay, well, yeah, okay, so you, okay, sorry, sorry about that. Anyway, now I am going to assume that gamma is a lattice because for the volume of this thing to be finite you will need gamma to be a lattice. So, okay, so now we have a precise criterion. So now gamma always denotes a discrete group, J discrete spaceful representation, rho another representation which is strictly dominated by J and now I am going to assume that J of gamma is a hyperbolic lattice and I am going to tell you what the volume of the quotient of SO n1 is. Okay, maybe I forgot to say something was that in this theorem, so what Salon did is he was the first to notice that domination here is a sufficient condition and he used that to give new exotic examples of quotients of ADF 3 and François Guerrito and Fanny Cassell proved that this sufficient condition is actually necessary. So in particular, this condition is an open condition. So if you fix J and you have a rho which is strictly dominated by J, you can deform a little bit rho and this gives you a non-trivial deformation, but sometimes it gives you non-trivial deformations of your lattices, lattice, but seen as a lattice in the, as a discrete group of SO n1 times SO n1. So this gives you non-trivial deformations. So here I'm in a situation where there will be some, there are some lattices and they are, there are some lattices that are non-rigid. And so to state the theorem, I have to give a definition, I have to define the volume of a representation of such a group gamma. So I assume that now that J of gamma is a lattice, so the volume of J is just, well, just the volume of the hyperbolic three-manifolder, hn quotient by J of gamma. And to define the volume of rho, I will use the definition, the generalization of the definition Bertrand gave last week. So the volume of rho will be the integral on hn of a J of gamma of the pullback of the volume form of hn by some map f, which is J rho equivalent. So to define the pullback of a volume form, let's say it's piecewise and smooth. And this works well if J of gamma is co-compact. If J has some cusps, you have to be more careful, but in our setting, you can add that f is lip sheets. So if you look at any lip sheets equivalent map like that, then you can define this number. This does not depend on the choice of the map f. And this defines you the volume of the representation rho. And the theorem is that the volume of the locally homogeneous space SON1 quotient by J times rho of gamma is equal to, you have a factor here which is the volume of SON times the volume of J plus minus 1 to the n times the volume of rho. Okay, so this tells you, yeah, this is the volume of this homogeneous space. And maybe, so there are some choices in how you normalize the volume of SON, how you normalize the volume form of hn, how you normalize the volume form of SON1. Well, there is a natural way to make this compatible. You use this natural ways and this formula is correct. But the point is the interesting part is how it varies when you change the group acting and the representation. And remember that the question I was interested in in the first place is whether the volume is rigid. Well, no, it's not an absolute value. It's important. Sorry. This can be zero. This must be nonzero. So this is nonzero because since you have a J rho equivalent contracting map, then the absolute value, that's what you say. Sorry. The absolute value of the volume of rho is always strictly smaller than the volume of J. So it's okay. This is always positive. And as a corollary, you get that in this situation, the volume is rigid. It's rigid except for one case unless n equals 2 and J of gamma, non-compact. So yeah, I will be a bit brief about that. So this is, well, this boils down to proving that the volume of a representation of a hyperbolic lattice is something rigid. And this is due to Besson-Portois-Galot. So you have to prove that when rho varies, its volume stays the same. So Besson-Portois-Galot did it for the compact case for when J of gamma is co-compact. And Kim generalized it to a non-compact case. But the non-compact case, you have some more subtleties. In particular, when n equals 3, this is not true that the volume of a representation is rigid, but it is rigid when you restrict to representations that are strictly dominated by your lattice. And for n equals, sorry. Well, both. Yeah. So I think it's, so Incan Kim and Shuyun Kim, I don't know how to pronounce their name. Okay. Sorry? There, sorry? Are there minimum volumes? Well, it's, so in all, anyway, in all the examples that are known here, well, for n big, this, sorry, this, the volume of rho is zero. So you just get essentially the volume of a hyperbolic three manifold, n manifold. So the question is whether there is a minimum volume of hyperbolic three manifold? I guess, yes. But I don't know what it is. And yeah. And for n equals 2, there is no minimal volume. But because all what I described are quotients up to finite cover, actually, because here I assume, yeah, maybe I didn't, yeah. Didn't very explicitly say it, but for this to describe all the quotients, you, well, this only describes all the quotients when gamma is a torsion free. But you can actually, in some examples, you can act by a lattice only on the left. And then you can add a compact action of a finite group on the right. And if you do that on SL, PSL2R, you see that you can divide the volume by, on the right you have SO2. You can quotient by every, any finite subgroup of SO2 and divide the volume by anything you want. The point is that, yeah, okay, the volume of representation of a lattice, oh, yeah, I think so. Yeah, but you will increase your class when you quotient. Oh, yeah, yeah. You can do PSL2R. And on the left, you quotient by a lattice, co-compact lattice. And on the right, you quotient by the subgroup e to the i 2 pi over p. And this will divide the volume by p and multiply the Euler class by p. This is the group, the U1 action acts properly on this space. So if you take a district subgroup, it acts properly discontinuously. It's, there is something quite subtle in the way the volume behaves with the, the volume is not directly related to the Euler class of the circle bundle. It's related to the Euler class of the circle bundle and the length of the fiber. So, yeah, it's, I wrote it in my thesis explicitly which circle bundles admit which structures with which volume. Okay, if you normalize it so that the, if you normalize it so that the length of the, the length of time like geodesics are pi or 2 pi, then yes, there's probably a minimum. I think this is the only way to get something arbitrarily small. You're right. Okay. But so yes, and interestingly, but I, I don't have time to state something precise. If you take n equals 2 and j of gamma not co-compact, then there are some representations row. You can vary the volume of the representation row even into, inside the space of representations that are strictly dominated by j. And this way you get a family of a non-uniform lattice, anti-decitaire lattices whose volume vary between a zero and a two times the volume of j. And, but this is the only case in our setting. But it's interesting because I don't think it has been, there are no examples elsewhere of such a phenomenon. Particularly, it's completely non-remanient phenomenon. Okay. So how much time do I have left? So I wanted to give a, an idea of the, the strategy of the proof. I probably won't have much time. But I will, at least I will tell you why, why it is possible to do it and why Gerito Cassell's theorem that I'm just erasing helps you. So what I'm going to explain is that these quotients of S O N 1 are actually bundles or vibrations over hyperbolic three manifold. So this is a, this was the third part idea of the proof. And so I will, maybe I will just state this lemma which is due to Gerito Cassell actually. So it's a corollary of their work. This manifold J times rho of gamma quotient by S O N 1 is a vibration over a hyperbolic, the hyperbolic manifold H N over J of gamma with fibers of the form. Well, they essentially look like the group S O N inside S O N 1 up to the action of the isometric groups, up to left and right multiplications. So in particular, if you, if you look at anti-deceit geometry, so here you have S O 2 1 or PSL 2 R, inside you have S O 2, which is a circle. And with this construction, you get, you prove that these, this manifold is a circle bundle. So here I have to assume that gamma is portion free, but otherwise you get a a vibration or you get a vibration of an orb before, which is essentially what a safer bundle is. But essentially you get a circle bundle over a hyperbolic surface. With fibers, and the fibers are images of a compactly group of S O 2, and S O 2 is a time-like geodesic in PSL 2 R. So you get a vibration with fibers, time-like geodesics. And the fact that the manifold is a cypher bundle was noticed by Kulkarni and Raymond as a corollary of their proof. But the work of Gerito Kassel gives you explicitly this circle bundle structure. Okay, so let me tell you how you construct this vibration, because I think it's interesting and it gives you good insight on how you can study this geometry. Well, you have this map F, which is J-row equivalent and contracting, and you can define a map Pi from S O N 1 to H N. In the following way, you look at Pi of J. J is a S element of S O N 1, so it's an isometry of H N. You can look at J composed with F. This is still a contracting map from H N to H N. A contracting map has a unique fixed point, and you can map G, sorry, not J, G to the unique fixed point of G composed with S. And if you do that, exercise, it's an exercise to check that the reverse image of a point X has the form K X G, sorry, step X G. So step where step X is a group of transformations preserving X, so it's conjugate to S O N, and up to some elements G in the fiber. The fiber is just step X. And that Pi is also gamma equivalent, which means that if you look at Pi of J of gamma, so J-row acts on S O N by multiplying on the left by J of gamma, on the right by row of gamma minus 1. If you look at the image of this, you get exactly J of gamma Pi of G. So this map Pi is a vibration with fibers of the form S O N, and it goes to the quotient. It goes down to a vibration from this to this. And yeah, anyway, the proof is a bit, it's technically called a differential geometry, but to compute the volume, you have to compute the differential of Pi, and then you get, you understand the volume form on the top with respect to a volume form which is made of, well, with respect to the volume form on the bottom. Then you first, you integrate along the fibers. You get a volume form on the bottom, and this volume form is, you have to prove that it's precisely the volume form of H N plus minus 1 to the N times the pullback of the volume form of H N by F. So essentially, once you have this picture, you integrate along the fibers and then on the quotients. In the few minutes I have left, I just wanted to state some open questions that are, essentially one open question, right by this work, and talk about the values. So the question is, what values exactly the volume can take? And despite the fact that we have a very nice formula, well, the answer to this question is still unclear because we lack examples. So the answer in, if N equals 2, N equals 2, then you can take gamma to be a surface group, the Pi 1 of a closed surface of genus G, let's say a closed surface. J is Fuxian, or maybe I should have stated the case N equal 2 because it's anti-docitor geometry, and then you get, oh I forgot to say something else important, then you get that the volume of J times rho of gamma quotient by SO21 is up to something, the Euler class of J plus the Euler class of rho. And it's a theorem of Salin that rho can take any non-extremal Euler class here. So here you know the values that this volume can take. Okay, so something I forgot, important I forgot to tell is that Laburi recently found an alternative proof, and so maybe I should have, there is this corollary that the volume is rigid and you may ask can it just, it could just be a consequence of a Gauss-Bonne formula. But the point is that there is no generalization of the Gauss-Bonne formula in our setting because these manyfold quotients of SO and 1, they have a flat connection on the tangent bundle which means that all their primary characteristic classes vanish. And so the volume cannot be a primary characteristic class. So there cannot be a simple proof like that, that it is an integer. But what Laburi noticed is that the volume is a secondary characteristic class. So a characteristic class coming from the Chan-Simons theory. And so in particular the Chan-Simons theory gives you the rigidity of the volume. But it's much more subtle and I'm not an expert on secondary characteristic classes. But essentially it uses the fact that this tangent bundle, when you have a quotient of SO and 1, the tangent bundle comes with two connections. One which comes from parallelism on the left, one from parallelism on the right, and one has a holonomy J, the other one has a holonomy rho. And when you have a bundle with two connections then you have a Chan-Simons invariant. And so that's how he gets this. So this gives you a proof in the case n equal to, this gives you another proof of the rigidity. With a bit of work you can actually get the formula. And it may be able to, it may be possible to generalize this to any n. But there is some work to do. Okay, so I think I have to stop here. Yeah, I didn't finish what I wanted to say on the values of the volume. But I'll just say that essentially for n higher than 3, the only examples that are known, in the only examples that are known, rho is the deformation of the trivial representation, so it has volume 0. So if anyone finds a lattice, a hyperbolic lattice in h3, and a representation of this lattice which is uniformly dominated by the lattice and which has non-zero volume, I would be very happy.