 Drugam pri svoju organizatorja, da je to reputno, da je smo po svoju. Tako da j夙ako u vsej vub, da je prav dobro. Če jaz postavim, da smo razovnega v bouncedi v progras bo hlištite z Daniela de Sandrini in Hanna Wiener najno z Heidelberga. Tako, da sem tukaj pričačen, da imam vzelo na to, da je vse občas. Iskaj sem tukaj početnout? Zvukam. Zato, da smo zelo na zelo na zelo na genus G, Bigger or equal to, connected, orientable, blah, blah, blah. I guess we come from different point of view, but I think we all like and associated to sigma, we have our tekmore space, which I bet we know the importance. And being the last day, I will use a lot of introduction that all the friends before me did, so I'm using all many things that people already introduced, so I don't have to justify so much why you should be interested in those objects, but let's do it quickly. So we can consider tekmore space, so you can define in many different way, but you can define as this marked hyperbolic structure on your surface. So hyperbolic structure, I will write it using the XG structure notation, so H2 PSL2R structure on sigma up to homotopy. And as people before me, in particular, for example, Tengren, already noticed, we can also consider this space as the space of representation is somehow nice, so in this case, discreet and faithful. And up to conjugation, of course. Those representations are called fuchsian, so this space is often called fuchsian space. The point is that we have PSL2R, we have a natural embedding into PSL2C. And so we can try to see what happens when we deform away from fuchsian representation. So this is the picture that you have in mind, so you have your H2 inside your H3, and then you can try to see what happens when you deform. And when you deform, now we explain better, so you get another space that I like it a lot, which is the quasi-fuchsian space. So you can consider, as in this case, representation into PSL2C. I can't just put discreet and faithful, that would be too many, but I restrict to, let's say convex compact up to conjugation. And you can, so, suppose that we take a point here, then you can try to see what is the manifold associated, and Jean-Marc, yeah, he considered this space, and he gave a full introduction to this, and so in this case we can see what's happened to the quotient, and we have that this is our surface cross R, but then we also have a domain of discontinuity in the boundary. So this is H3, and this boundary is Cp1. So the action is not nice on all Cp1, but there is a domain of discontinuity, so we have our limit set in Cp1, and then elsewhere, so this is the set of accumulation point of your representation, and then elsewhere the action is nice. So if you consider Cp1 minus the limit set, this is your domain of discontinuity, and you can then try to understand what is this quotient, and in this case what you get is that you get two surfaces, because you can define quasi-fuxian space, so you can see as the space of representation where the limit set is a topological curve, and so it split your Cp1 into omega plus and omega minus, and the quotient of each one is your surface. So what we are going to do today is, and again so here you can see as the space of marked hyperbolic structure, in this case on sigma cross R, which is a nice structure. So our point of view today is to consider, so we have this SL2R, you can also think as SP2R inside SL2C, which you can think as SP2C, and then our Cp1 is not only the projective space of C2, but you can also think as the space of Lagrangian of C2. So what we are going to do now, what I'm going to do, is to try to understand what happened in the next step. So what about in general more general SP2 and C. So we will define, or we will recall the notion of, somehow the analogous of tec molar space, which is the notion of each in, each in representation in SP2 and R, and then you can put SP2 and R into SP2 and C, and the shard and vinar define a domain of discontinuity in the Lagrangian space, and we will try to understand what is the homomorphism type of this quotient. And I will try to also justify why this is nice, because it can also help to understand the quotient of some other symmetric spaces. So you can put symmetric spaces into the space of Lagrangian. Any question? Ok, feel free to stop me at any time. Ok, so what I'm going to do is that I'm going to give a quick introduction to symplatic groups, just fixing the notation, and then recalling the notion of anosoph representation, anosoph representation into SP2 and R, and then I'm going to try to study or tell you something about the space of Lagrangian. In particular we will try to see the SP2 and R orbit into the complex Lagrangian, and the SL2C orbit into the space of Lagrangian, and we will give a, I don't know, I like this, we will give one of these, but in orbit we will have a nice hyperbolic interpretation in terms of theta eta. Ok, but let's start, symplatic representation. Ok, so let's start with this notation. So a symplatic space is a pair where we have a 2n dimensional k vector space. For us k will be always R or C, and omega will be our non-degenerate skeucimetric bilinear form on the vector space. And we will consider the symplatic group as the set of transformations that preserve the form. So in fact the example that we will always consider is when your vector space will be k to the n and so in that case SP. And you can express the omega k of x, xy, you can express as x transpose d identity minus d identity. Y, and so your symplatic group we will then denote SP to nk. In fact we will often identify this case. So one example that we will use a lot is that we will consider vk as C4 and we will think C4 as the space of homogeneous polynomial into variable which is what Tangren was considering when he was thinking about the symmetric product when he was looking at this uniquely reducible representation to define anosev. And in this case, so this will be, the symplatic form is defined, you can give a general formula but just to give you an example because we will use it. So the symplatic form is defined by the following and I think here I have a minus. It can be wrong. Ok, so we will need to consider some particular subspace. So a subspace of your symplatic space is called isotropic or omega k isotropic if l is contained in the orthogonal and it's called a grunge if it is exactly the same. We will remark all one-dimension isotropic, all one-dimension subspaces are isotropic. Ok, we will use it soon. So, in order to define anosev we will define anosev with respect to some particular case so I need to give you some definitions so we will denote qi or qi of u k, the stabilizer of a one i-dimensional isotropic subspace and we will define i's i as the set of i-dimensional isotropic subspace which you can identify as sp mod ds qi. Ok, if you were here last week with the mini course of funny you have seen not this particular case but many other similar other homogeneous spaces and other homogeneous spaces that will come into play will be the remanian one so we will have we will denote as xr and the space sp2nr mod un so this is the maximal compact here and in the complex case you have sp2nc and this is spn but then we one other space that funny also consider is that we don't have to just stick to the remanian case so we also have some other affine affine symmetric space which in this case will be our xpq which will be sp2nr mod u pq and we are going to see lots of examples so just bear with me for a while now while I give you the definition or I recall the definition that probably you already know and now we have all the ingredients to define so whenever you have a representation gamma you take a word hyperbolic group just think pi1 of a surface and a representation is qi anosov if there exist a row equivalent ci from the boundary of your group into this i's i so into this space of isotropic subspace which satisfies some properties so the map need to be continuous it need to be dynamic preserving and what that means it means that whenever you have an element of infinite order you want to map attracting fixed point to attracting fixed point so you want to map row of the attracting and repelling fixed point gamma will go into the attracting repelling fixed point of the image and it need to be transverse and the transverse what that means is that whenever you have two different point then you can consider this sum ci and ci transverse that will need to give you the full space and the last and a little more tricky property I'm not going to define precisely so you need to have this exponential contraction expansion and again so if you were here last week funny explain that somehow in the original definition of anosov that is due to Françoise is this fourth property is given in terms of in terms of a flow so you use a flow on a certain bundle over some flow space and then so the original definition is due to Françoise and then Gishard and Wienard generalize to general word hyperbolic and then recently there is work of Gerito, Gishard, Kassel and Wienard and Kapovič-Lib party where they find a different characterization in terms of the carton projection that I gave and we don't need it today and so I'm going to not go into much into the details here let's see what and I'm going to for a particular case we will see precisely what that will be but we will do later when we will be more concrete just a few remark I can just probably say loud if you have a parabolic subgroup then a representation is anosov with respect to a parabolic subgroup if and only if is qi anosov for all the qi contained in your parabolic subgroup and a representation is anosov with respect to the minimal parabolic if and only if it is qi anosov for all the qi from 1 to n and these these representation are very nice as you have seen from many other talk so in particular you can see that the kernel is finite the image is discrete and they form an open subspace where out of gamma act properly discontinuously so they are very nice and again so since I'm going slower than I imagine I'm not going to write all of that and ok so you can ask ok so let's see some example and again so you have seen many but so the first example is the one that probably by now you know very well is the case of each representation and how you define that so you consider you consider a representation so you have a fuxian one so François also recall it yesterday again so you have a discreet and faithful representation and then you can I want to consider in SL and I want to consider the unique reducible representation into SP2nr and so those representation are called fuxian you consider the component in the character variety that contains fuxian representation or if you want a representation is called each representation if you can deform if there exist a continuous deformation into a fuxian representation and François prove that each representation qi for all i ok, so then another set of nice representation are maximal one and those you have seen in bias talk so again I am just going quickly we don't need those but yeah so here you have for every those representation you have a theoretical number and you look at representation where this toledo number is maximal and what is nice is that in this case burge jotsi labori vinard so maximal representation in this case are qn there are lots of still open questions the other fact is that each representation are maximal so each representation are maximal so in fact in particular they are q1 and nozov and so one can ask are there other maximal representation which are non each in that are q1 and nozov for example so now there are still things that are not well understood and another example is you can you can embed so deformation so you can deform SL2R embedding just to give you a sense of this qi so you can consider as a symplactic space plus R2 with small your standard symplactic form on R2 plus epsilon time where epsilon is plus or minus 1 and then you can do different things so you can embed so let's consider i to be a fuxian then you can consider the embedding where you put a fuxian and the trivial representation in the other and so this this representation will be q1 and nozov and not maximal but then you can do you can embed diagonally where epsilon is minus 1 or when epsilon is plus 1 so in this case your representation is maximal but in this case is q2 so q1 in this case is q2 so it's q2 and nozov but not maximal and you can do many other things by putting SL2C inside SP4R but yeah so the definition I need to give is the set of representation that we will consider today which is what you can call it quasi ičin or quasi fuxian however you want and those will be the formation of representation that you can deform to an ičin one or representation that you can deform to a fuxian one I need to tell you how so you need to fix one parabolic subgroup or one of these qi and so you define qi quasi ičin representation as the set of representation that you can deform inside qi and nozov to a fuxian one is qi qi quasi ičin if you can deform inside the set of qi and nozov representation to so you have your the complexification of a fuxian one to a fuxian or ičin representation ok, so now important part domain of discontinuity I will just focus on the case of q1 or qn nozov for representation because those will be the one that I will consider and so what you do is that supposing that you have a q1 a nozov representation then we saw that this give us a boundary map from the boundary at infinity of your group into so as I said it now but ičin of vk is the projectivization of vk so all one dimensional subset and ičin will correspond to the set of Lagrangian so you have this boundary map so you can define the set of bad points and now I am going to define and that will be a subset of the set of Lagrangian and similarly if you have a qn a nozov then your boundary map will take value into the Lagrangian so your set of bad points will be a subset of the projectiv space and how we define this bad set as follows so you define k of x1 as the union of k of x1 t a point in the boundary of your group and where k of l where l is a point in the projectiv space you can define as the set of Lagrangian which contain this point so you can define k of a Lagrangian subspace as the point in the projectiv space which are contained in your Lagrangian and so that's how you define that set and then what Gischar Wiener proved is that omega c1 which is Lagrangian of pk minus k c1 and omega cn as the set of points that are not into this bad set so you can see that these set are gamma invariant and gamma or rho properly discontinuously and co-compactly fact that the action is co-compact is very nice because it will allow us to just study the quotient for a Fuxian representation what happen at the Fuxian locus and then we will know that the homomorphism type will not change the homomorphism type of principle and yeah if you have a question please ask what I'm going to do now I'm switching to the second part where we will study complex Lagrangian so we will study the sp to and our orbits and so if I forgot anything oh yeah so you can ask when those discontinuity are empty so there are cases that they can be empty but if if the man is big enough so in particular strictly bigger than the dimension of the boundary of your group these discontinuity are known empty and what we are going to do we are going to study the quotient of the homomorphism type of the quotient and so this theorem which I'm not sure I'm giving you the right reference but yeah the one I have is due to somehow he's in satake book but I think he's older but so you you have you have the composition of your set of Lagrangian as these spaces r0, rn where i is the set of Lagrangian such that the dimension of w intersected w bar is i and then rn so is the unique closed orbit and correspond to the real Lagrangian or it can be identified with real Lagrangian r0 will be the union of certain how did I call I just say that r0 is the union of rn plus 1 open orbit and each of these open orbit can be identified as you remember the space x pq that I introduced before so it can be identified as one of these subspace where p plus q is equal n and p goes from 0 to 1 and then for the I is an intermediate case and it will be a bundle over is a bundle over is the space of isotropic subspaces of our real one where the fiber you can identify as u x p prime q prime where p prime goes from 0 to n minus i and p prime plus q prime is n minus i so I wanted to give you the full proof but maybe I will give you a sketch of these because I think it's very nice so what you do, let's start it's clear the statement of the theorem let's start by studying which one I want, yeah r0 so what you do is that you given your complex given your symplactic form you associate an Hermitian form and then you define these spaces as the spaces where this Hermitian form takes appropriate values so given so you define h as follows, that's maybe not as i wc of v bar v prime and so this is an Hermitian form and then this Hermitian in r0 then you can see what happened to h restricted to your Lagrangian and so the you have this identification with so you have this h pq as the set of Lagrangian, such that the Hermitian form as signature as signature pq and then you see that these you can identify as your the affine symmetric space that I introduced at the beginning and when you have instead something in r i so a remark I didn't make is that so you have let's say you can consider you have this space which we know has dimension i and remark is that it's real so it comes from the complexification of a real isotropic subspace and these give you the projection into is i of r2k to n and then what happen is that you can consider whenever you have a point a point here so you can consider the orthogonal mod z and that will be 2n-1-i symplactic space so your symplactic form restrict to a symplactic form on this space and then you can see that you can see that then a point here is uniquely determined by a point you can see that the fiber basically you are trying to study the fiber and then you can see that this space is uniquely determined by something so you know what happened because of the case studied before so p plus q equal n minus i and then the last one is that so you know that this is easier so you know that this space is compact now sl2c orbit and so the idea to study that is so in this case if you want to study instead the sl2c orbit in this case I will look at Lagrangian in C4 and in general will be way different because in C4 this space is dimension 6 and that's the same as the dimension of sl2c so what happened is that you can see this space as I don't know L0 union L1 union L2 where L0 is the unique open orbit and so here I will think C4 as homogeneous polynomial into variable and then what is L0? L0 is the set of Lagrangian such that there is no point in W with a triple root and what is nice is that we will identify this open orbit as the set of regular ideal hyperbolic tetrahedra that will make studying the quotient at least for me easier and then L2 is the unique closed orbit and correspond to the set of Lagrangian where somehow all point in the Lagrangian have a common double root and the other space is the space in between so it will be the set of Lagrangian where all the polynomial these correspond also to the set of Lagrangian where there is no a common root in W and so here so L1 will be the set of polynomial set of Lagrangian where all point in the Lagrangian have a common single root but how we ended up thinking or studying those spaces it is because spaces like that comes from these set of bad points here so I'm going to consider a fuxian representation and study this boundary map and see how these bad points comes as these any question before ok so now I stick to a fuxian representation so discrete and faithful and I will consider inside sp21c let's say sp40c ok so then the boundary map in this case can be defined quite explicitly so you have c1 and c2 and so I will consider small I will just consider small the boundary as rp1 and so the image here will be p of c4 and c4 remember that is my set of polynomial so it is the class of the polynomial given by c2 I need to give you a point in the Lagrangian of c2 and that will be the class of bx minus ay cubed bx minus ay squared x so in fact if you see this will be the image of the c2 in this particular case and so in this case in fact I can extend this map to cp1 just by keeping this definition you can small the set of bad point will be the set of c1 so the set of Lagrangian that contains a polynomial with a triple root which is exactly this so the complement will be so and you can also define this set and so so what happen is that here you have in identification with cp1 cross cp1 and in fact you should think this cp1 you should think to that as Lagrangian of c4 so if you do the same in sp6 then you can do something similar and here you have cp1 cross Lagrangian of c4 sorry Lagrangian of c2 in this case and how you do that so you have two points and you define so in this way you define in this way so bx bx minus a y square x or whatever if b is equal cd and otherwise what you do is that you do that and here you have bx a y and bx minus ok so and in this way so you understand exactly the set of bad point the one that we will need to throw away and these will be our rp1 cross cp1 and now we want to study everything else so we want to study these L0 as these the set of so the set define there so the point is that for every Lagrangian there is always a point with a double root here you have always a double root and using the SL2 action we can consider this double root as being zero and infinity so let's say basically now I want to prove the following I want to see that as the set of regular ideal tetrahedra and so I'm going to use the 3 degree of freedom and so any point here has a point with a double root that zero and a single root that one and so I use I use the SL2C action to suppose Z0 is equal to zero and Z1 is equal to infinity then I want to study this space, which is the space of all Lagrangian which such that x to y belongs to that and this space is a momorphic to cp1 and so I use the last degree of freedom so basically you look at all the polynomials such that the symplectic form with this one is zero and is zero with itself and so by using this last degree of freedom for SL2 we just end up to study so this L0 is the orbit of the Lagrangian x squared y and x cubed plus y cubed and then I just say quickly so you use the discriminant to understand all the polynomial with double roots and you notice that you have exactly four points and if you calculate the cross ratio the cross ratio of these four special points for this case give you the best cross ratio so 1 minus square root of 3i over 2 so you have these regular tetrahedra and also the single roots and this is actually the same regular tetrahedra the dual one and using that we were able then to study this quotient because so you have then this set of Lagrangian as this set of tetrahedra plus this cp1 cross cp1 which you can think as the possible generation of your tetrahedra because your tetrahedra having this nice symmetry this maximal volume you cannot degenerate to a flat one so you can only degenerate to a line or to just a point and the set of this small domain of discontinuity here it will be this set of tetrahedra plus your rp1 and so it's clear now how to define a projection into h3 union so from here you can define a projection into h3 cross cp1 and in this case you have a projection into h3 union cp1 minus rp1 and so these give you a projection in h2 and so then you end up having to study the fiber of this map and actually the fiber you basically have to study all the tetrahedra with the body center so here you send it into the body center or into the first factor and then you want to study all the tetrahedra with body center on a fixed line so this is for manifold that it look like this in fact you have supposing that this is the line where your body center end and so at the extreme you have the cp1 and here so tetrahedra with the fixed body center with the fixed body center and in this space which is the unit tangent bundle of these orbifold and in fact these three singularity correspond to tetrahedra with a very particular symmetry and so the idea is that then these quotient will correspond to a fiber bundle over a surface with fiber there is still one point that we are not sure if it's an orbifold but I think it's just cp2 connected some cp2 bar because you have this construction in the same construction that you can get from cp1 and now I'm over time and I'm going very fast so that's a bit what I wanted to say but thank you very much