 Thank you very much. Of course, I thank the organizers for this kind of invitation. Here's the title of my talk, Modular Groups over the Quaternions. Let me see if it works. I mean, since the time of the great Kal-Fiddler Gauss, one of the most fascinating and important publishing mathematics is the Modular Group, and it's action by isometries on the upper half plane. This group, which consists of web use transformations, I don't know where's the pointer, consists of web use transformations, AZ plus B over CZ plus D, which therefore is the group of matrices with integer coefficients, a determinant one, quotient, the advantage of this group is fantastic. We all know it has implications in things like number theory, Fermat's theorem, coding, infinitely many things. And therefore, it is natural to, well, this is the beautiful picture. It is the picture of the Modular Group. So this is the desolation. Where's the pointer? This one? Yeah. OK, I see. OK, so you see this region here. This region is the fundamental domain of this group. Namely, if you take this region, which is actually a triangle, ideal, one pointer, infinity, and two angles of 60 degrees of area pi over 3, if you take this fundamental triangle and propagate it on the reaction of PSL2Z, we all know, I'm telling you things that you know, but in order to motivate my talk, well, you tessellate with mosaics, beautiful isometric mosaics, the upper half plane. So this is a picture of it. This group actually generated by two transformations, translation by one. And this transformation, which is, you see, this point is i, the fundamental imaginary unit. This rotation 180 degrees around that one. So if you compose a translation followed by inversion, this point is fixed. And therefore, the composition of this one with that one is a transformation of order 3, which has 6 points. So one can see immediately that, just from these generators, generate a group, which is isomorphic to the free product of a group, a cyclic group with two elements by the cyclic group of three elements. And you want the fundamental group of the collective sum of projective space with the length of space. I like that viewpoint because it's very geometric. And we have that as a group. And therefore, it is natural since C is not the only field. The quaternion is another field. Not commutative, but it's a division algebra. We, a natural question, we ask, what can we do in the setting of the quaternions? And so for this, we define the quaternions. We have a problem. The problem is that too many h's, h's. So please keep track, h for quaternions, h for Hamilton, h for hyperbolic, h for Hurwitz, h for the what? Hermitian, h for the form that appears in your notes for the quadratic groups. So it's a universal h. So be careful of that. So you have this h, Blackboard h, is the division algebra of the quaternions, which naturally identify with R4 associated to this quaternion, the point we call it x0, things that you know, which is R4. And we define the hyperbolic half space like that one. Notice that here, since there are three imaginary units, it's better to choose a real line as the vertical axis. Otherwise, there's no democracy. You have i, j, and k. So you use that one, and to be more precise, this is the, so here's another h. So this would be the half space, consists of quaternions whose real part is positive. This would play the role of the upper half plane. Now you have to twist your head and look the other way around like that, to the right or to the left, and that is, and of course we provide this space with hyperbolic metric, which is this is a Poicaré metric. And as such, you know, it's a complete metric, simply connected, it's unique, complete, simply connected space of culture, of calcium culture minus one, the bottle we all talk about, so I don't need to repeat it. And I will be saying things that we have been, that you have seen during the school, it's not nothing wrong to repeat them, even if you know them. And so you have that. And now consider, obviously the analog is the group, consists of two by two quaternionic matrices. So A, B, C, D, and quaternions, it's called GL2Z. And well, we mimic the Muevius transformations using the quaternions and they're very careful which side you multiply. If you are dyslexic, you are in trouble. So you have this, this is our favorite group, homography, a group of homographies, Muevius transformations. And so we've been very careful with the usual thing, we take the one point compactification of the quaternions which then R4 becomes the sphere, S4, and we act this way. Be careful, this is to the right. You cannot do QA, you have to do AQ, et cetera. And this is quaternionic linear fractional transformations associated to each matrix, we associate that. And of course, we standard conventions for infinity. Exactly like you do in the case of, okay. Well, if you take the set of four matrices here, you can see, you can identify that with GL, you can think of these as eight by eight matrices if you write the, no? And therefore you have that, the following definition. So there's a canonical inclusion of F, this group of matrices into GL are A, inverted matrices eight by eight, because H times H is a canonical isomorphic to, as a real vector space, therefore you have this definition, all these in the book by John Parker, you want to own many other places. And then you have this theorem, the set of quaternionic matrices, a group with respect to the composition operation, and you have a canonical map whose kernel is plus or minus identity, okay. So you have this completely algebraic thing. And well, now you may wonder what's the analog of the set of Moebius transformation that thick the disk. Well, we know by our complex variables that Moebius transformation that thick the disk is exactly this form, is a Moebius transformation where we have to take the absolute value of A less than one, otherwise inverse the orientation, take the inside to the outside. And so here you can see immediately that this group is parametrized by the disk times S one, okay, things that we know very well. And we want to sequence the analog. Here's another H that has nothing to do with any of the other Hs. So you take this matrix, this Hermitian matrix, well you take as in the lesson in the book by, you take the Moebius transformation that preserves that, and there's this theorem. The set of Moebius transformations with preserved the disk is exactly that one. And I mentioned that just to show you how this is exactly that, you see? C minus A, Q minus Q zero is exactly the same formula, except now this e to the i theta, which is a complex number of modulus one, you need two quadrionic numbers, one on the right and one on the left. And this is a beautiful formula to express all the asymmetries of the disk in a concrete way. Well, we have proofs of that in several papers. We give that simply observing that this group depends on 10 parameters. You count the parameters, this V is unitary, so it's three, U is unitary, E three, three plus three makes six, plus a point in the interior makes four C's plus four, and then you just count the parameter, and you see that this is the group of, on the other hand, the ball, the unit ball, let's keep going here. Okay, so this way of transformation preserves the Poincare metric of the disk. The Poincare metric of the disk is that one, given by that one. Exactly mimics the case of the complex. And you see that these Moebius transformations is isomorphic to the group of isometries of the ball, and therefore, during the school, you saw the analog of that in the complex ball, and this was the subject, was the protagonist of all the talks, practically all the talks, but this is that, okay? Well, now you can quantify that. You see that obviously the set of transformations that preserve the ball is a subgroup of, okay? One of the fundamental facts is the way of transformation that I put at the beginning, AQ plus B multiplied by CQ plus D inverse, in that order. We see that with a conventional facility, this is precisely concise, with a set of all conformal transformation of the sphere of dimension four, and that means that precepts orientation and precepts angle. Now there's a famous theorem by one of our common hero, because we are here, there's one man that is called Henri Poincaré, so it should be a common hero. I know that for this one, it's obviously, we know that, but therefore Poincaré proof that you can, any conformal map of the sphere, you can extend to the theory as an isometry of hyperbolic space of five dimension. And therefore you see there are many relations of groups, subgroups with other, and this interconnection of groups is what makes this theory incredibly rich. So you have this, you have the set of conformal mappings of S four, since, can be extended to isometrics of the disk, hyperbolic disk of dimension five, it acts simply and transitively on the space of frames. The space of frames is equal to five, has dimension five for the base, and then the frame is a SO four, so this group has dimension 15, and on the other hand in the lectures by Pepe, and you have another interpretation of that, you take the action on the quaternionic plane, it's action on the left lines. There are right lines, left lines, like as I mentioned, the joke of politics, like politics, I choose the left, what was the left, the left lines, and these groups acts, these things was given already in the lecture by Pepe, and therefore we see that it can be interpreted as a projective action in the quaternionic plane, and therefore the quaternionic plane compactified becomes S four, and well, now there's a fantastic injection, so you have that, the group of things that preserve the ball is a super therefore or super conformal, you have all these inclusions, and these things were given, I think are in the lecture notes, I don't know if I'm in the lecture notes by you, and well, he had repeated what I'm saying, the Poircale extension, this is just, I go very fast on that, just equality of rules, read it, it's very things that are natural, different actions of that, this is important, any orientation preserving conformal map or the force sphere has the property that it sends the set of complex structures at one point, and the set of complex structures in another point, because preserves angles, and therefore there's a natural extension into the set of PSL3C, and that's what makes it so rich, the interplay between twister spaces, conformal geometry, hyperbolic geometry, in five dimensions, complex climate groups, they are all in one same, and that's what makes the theory so incredibly rich, and many of those things have already been exposed during the workshop. Okay, but now we are going back to the real source of my talk, which is this, so we want to consider the isometries of the half space, remember they have a space consisting of quaternions whose real part is positive, the upper half plane, upper, take it with a grain of salt, the right hand side plane, call it the upper half plane, and well, via the Cayley transform, look at the Cayley transform, usually you have Q plus I, but since we are declaring the right hand side plane, is the actual, the actual Cayley transform is a little different, and well, we have the, we want to store two isomorphic groups, the ones that preserve the upper half plane, or the unit ball, is the same thing because the Cayley transform changes one to the other, and so we can do any of the two. Well, we have this theorem, the set of conformal maps of the upper half plane, the same thing, those question marks because it's cut and paste, it's probably referred to my paper, to my paper with Pepe. Okay, so, well, we want to see so which ones are preserved, the upper half plane, and now, there are a little bit of history here because some things will overlap, very little of what I do here, will actually overlap because it has been done. Okay, so, in the first place, as was explained already by John Parker, is this person, Donald Valin, I only talk about his mathematics, who introduces in 1901 the idea of using Clifford numbers to define Muebius groups, and using Clifford matrices, but this was taken by Alphors in 84, and he defines, he gives conditions of matrices, allow the Clifford numbers, the quaternions are a special case of Clifford numbers, and so, he gives conditions of the Clifford numbers, and as these conditions are complicated, as John Parker said, they have to preserve vector space, or dimension four, and the conditions are rather complicated, there are two conjugations, et cetera, and very recent, relatively recent, Graziano Gentili, Shintia Bizzi, two Italian mathematicians from Florence, found a simpler condition, which is equivalent to Alphors condition, so it's the following condition, what? This one is in my uvascular, from the 40,000 and lower. I think they refer to that. I don't, no, they say bad, they do some other things, but I, well, maybe John Parker, but historically, I learned it from them, but they refer to you. Yeah, you're right. So this is the condition, so, I should maybe call it the GBP conditions, or the BGP conditions. Okay, so, the condition is extremely simple, is this commut, this relation, so you take the matrix, this matrix, this is a, and in the bottom, make the square of that matrix itself, and you take the set of it, matrix A, B, C, D, that satisfies this condition. And, this is equivalent, if you put it to this condition, maybe you are the same you gave, these conditions, so, all these three conditions are equivalent, having this commutation relation, or having these conditions, you know, these three conditions. This is sort of like, you see, this is sort of like a determinant. Okay, so this is the condition, so, the group of invertible yarn transformations that define the, BGP conditions, consist of orientation, precision hyperbolic isometries, and, we just call it, Weber's transformations, and so we will use these conditions all over, to see, because what I want to do, what the title of my talk, I want to define a modular, group over the quaternions, so for that, I need the analog of the integers, and of course, we don't have candidates for the integers, but we have to be careful to, and we have to, things don't commute, okay, so, I repeat the same thing, all over, you know, the upper half plane, these groups act, act on the upper half plane, and therefore the infinity is the, will be, the subspace of dimension three, of unit of imaginary quaternions, plus the point of infinity, that is three, that will play the role of the real axis, in the case of the modular group, okay, I have that, okay, so, of course, people think about that, so groups, we charge generalizations, they pick out a modular group, well-considered by, McLeishland, Waterman, Wielenberg, so they have a, something similar to the Lipschitz integers, and also Ruth Keller has, who will speak now, later, has applied these quaternions to study, hyperbolic five manifolds, okay, and, well, this is a little bit of history, there are more than that, it probably completes, but it has, so, the first thing that makes a difference, is, well, you have the, the affine group, of, web, of web use transformation that preserve the upper half plane, so it consists of that, so it's a group of dimension seven, is the one, this group is the one that fixes, the whole plane, real part of Q equals one, you know, so it's the affine group, perdone? That would be another issue, exactly, there are too many edges though, unfortunately, okay, so the affine group is the maximal subgroup which fixes the point infinity, the same thing happens in dimension three, it's a real group or, lead group of dimension seven, is a conformal group, and is, this group is obviously equivalent to the conformal group of infinity, namely the conformal group of dimension three, you have that, so I'm telling general properties, and then we have the analog of Iwasawa, the composition for this group, and it is the theorem, the matrices A, B, C, D, which preserve the upper half plane or the real half plane, can be decomposed in this, you remember how every, for SL2R, you have this composition, diagonal matrices, even potent matrices, and cos i theta, cos i theta, but it is the analog, and this is beautiful, because in particular, these conditions tell you that, sorry, these conditions tell you that this group, you see the group of six alpha beta, is another representation that I haven't seen anywhere of SO4, so SO4 consists of quaternions like that, so this is Iwasawa, I haven't seen it anywhere, but I suppose it's known, like many others, so this is Iwasawa, you have to give a proof, we gave a proof, and okay, and now we come to the matter, now my conference is starting now, the previous was just background, so what are the integers? So we need rings within the quaternion, which are discrete, and which mimic the integers, so there are two magnificent candidates, are the Lipschitz integers and the Hurwitz integers, both of which are very rich, so a Lipschitz quaternion, or Lipschitz integer, is a quaternion whose components are integers, so this is A, B, C, A plus B, G plus C, J plus D, K, where A, B, C, D are integers, this is obviously close to the quaternionic multiplication and addition, and therefore force a subring of the division algebra of quaternions, and well, you say well now, I take A, B, C, D matrices, quaternionic matrices with entries here, and that's it, well, it's okay, that preserves S4, isometry of H5, but they don't preserve the upper half plane, because they must simultaneously, apart from they may not satisfy the conditions, the BGP conditions, okay, and therefore, this is actually a subgroup of a much richer group, which is the so-called Hurwitz quaternions, two great mathematicians, Lipschitz and Hurwitz, not only because of Lipschitz functions, they are great mathematicians, and therefore the Lipschitz integers consist of a quaternion written in this way with A, B, C, D are simultaneously all integers or all have integers, no mixed, no mixing, and that's easy to see, just by computation, there is actually also a ring, it's close on the multiplication and addition, and it's another candidate to construct, okay, so you can take the group of web use transformations with entries here, these are discrete subgroup of isometries of why is discrete, as I learned from your park it because it's the final of a discrete group, I remember your anecdote, the set of integers is discrete, and so, but I wanted to be, preserve the upper half plate, so, what are the good candidates for that? Well, you see here is the clue, the modular group, the classical and beautiful modular group generated by two simple things, one translation and one rotation around one point, so what we're gonna mean is that construction, and so we take, first we have to analyze the quaternion translations, so we consider translations of this form, one W, C, or W, where W is an imaginary quaternion, the real part is zero, this clearly translates and leaves invariant the right half side plate, so it's a beautiful candidate, now it's generated not by one, but by three linearly-dependent translations, so that's a beautiful group of translations into the by itself, so you take the group of translations consist of these imaginary quaternions, consist of translations in which omega is imaginary, and then, of course, this acts freely on H3, has a fundamental domain, what I call a chimney, well, if you take H4 divided by this, what you get is a flat torus cross R, one ends a cusp, and the other is a trumpet, going down is a trumpet, you see, and that's a very, very beautiful, but non-compact, and then, now comes the analog of that guy here, is this translation, which becomes, in this case, really beautiful, really just inversion of quaternions, so I'm going to mimic what we do for the border group, and I'm going to take the group generated by these translations, and this inversion, and our theorem that I will prove, I will sketch a proof, is that the group of quaternionic fraction transformations, which preserve the upper half plane, is exactly generated by that, so that the conditions, the simultaneous conditions, BGP conditions, plus, are equivalent to that generated, it's exactly the candidate for the border group, but now it becomes very rich, the theory, because we have three translations, and we have so, well, now T here is an analysis of T, T is an inversion, or an inversion around, so this is the imaginary quaternions, we have something I call the igloo, is a half sphere, this is one, and Q goes to Q minus one, is an antipodal map, respect to that point, is one of beautiful isometry, makes the analog of that one, is further to, and that's what we play a very... Yeah, exactly, I'm taking, so not a fundamental object, you know the igloos in the north pole, before the climate change, they had a, before all the polar bears died, had this thing to open to have light, this cube, what's the definition of C? C is a spherical cube, central one, of unit size, but let me, instead of doing that, let me, I hope I have a picture, here, this thing here is the, wait a minute, I'm going too far, okay, you see here, this will be a very important thing, you know, we'll play the role of one of the most beautiful objects that we have in nature, which is this, you know, you'll agree that we love this thing, we love the modern orifices, now we replace it by this cube, this segment, we'll replace, and that's play an important role, so C is this, but I gave the question, but just think of that, it's a spherical cube, and it's an intersection of half the spaces by, okay, you have that, and now comes, composition, recompose translations with inversions and see what happens. For the modular group, we know, we'll think of order three, but what happens in this case, what happens is, this fantastic thing, you see, because very complicated, I want to put your attention into that, that, if you compose that, followed by that, you get something of order three, and if you see, you look carefully, this is the analog of that, but you know what, you take the square, the cube is not identity, it's almost identity, but it's not identity, and wow, you arrive to the last matrix, the right matrix is identity, because remember, the matrix and the negatives are the same, and so that means that it's order six, I was surprised because I thought it would be order three, and what happens is something really amazing, so if you take this plane here, the plane generated by one and I, you see variant of that translation, so one and J, one and K, depending if W is I, J of K, and therefore actual display exactly like the modular, but you know what, it reverses, that's what the six becomes, and then, now you can see that we're going to play like the model group, we're going to start rotating along the axis, the dimension axis, propagating, but the group begins extremely rich, so look at the relations, so you have elements of order two, elements of order four, because now in the cube, the cube has eight vertices, so you have rotations there, you have 12 middle points of edges, and have six middle points of faces, and so you start playing around, you have different orders, and you have fantastic group theoretical things, and geometric things, and so this is, now I gave the definition, Lipschitz modular group, is the group generated by the three imaginary translations and the inversion of the sphere, and the beautiful thing, this comes with the matrix, with the entries of the Lipschitz integers that preserve the upper half plane, it's a theorem, okay, don't treat it, but you have that, not difficult, but you have to wheel, okay, you have that, and so I mean, depends what you take for omega, remember omega is an integer, an imaginary integer, and depending what you take, you have different orders, and you have all these, you have fixed points, you know this is a fixed point here, for the modular group it's a fixed point of order three, here you have axis of order six, you have rotation around two dimensional, you have a very rich geometric, so this is the definition, the Lipschitz quadratic modular group, finally, I come to my topic, because she is generated by the inversion T and the translation, and now you realize if you translate by one, that's not in our group because that takes minus one to, no, doesn't preserve the upper half plane, so this is a real theorem that you need to prove, and now something beautiful happens, you see there are really not what are the isometries that fix that, nah, I mean the quaternionic units, the Lipschitz units, are these ones, and you see now you have this group, the group of diagonal matrices where U is a Lipschitz unit, acts, belongs to our group, and therefore acts in our group, and is the group that fixes the horizontal, well, obviously I don't know horizontally say, the hyperspace real part equal to one, and it acts by isometry there, clearly you can prove easily that this group is isomorphic to the Klein group of order four, so it's C2 plus C2, and you can see the geometry of that, a rotation along the axis, and it's a finite group, and now we have something we didn't have in the standard modular group, is the affine Lipschitz, you see, this is the Lipschitz parabolic group, is a set of all elements of that form where B is a Lipschitz integer and U is a unit, so this is the group that preserves the hyperplane that, and this is actually the maximal parabolic subgroup, so the option of these will be that if you take the upper half plane, quaternary half plane, and divide by the modular group just defined, this will be a orbital, or finite hyperbolic volume, and one end, and the one end is a flat orbital, is the quotient of the of the horizontal cycle at height one, divided by this group that I gave you, the quotient becomes actually a cube, topologically, and now you have many I mean, the model orbital has only two conic points, has one cosp, and two conic points, they are well known, but now you take the quotient, it's extremely rich, has many edges and ridges and points and beautiful geometry, we can compute the volume, we give a complete, I won't have the time, we give a complete description, detailed description of the orbital structure, okay, this is the first group that we key, and well, here you can see that, that's the same thing, you know, okay, but now we enter into the Hurby's group, remember the Hurby's group is a richer, contains as a as a sub ring, they leaps it, so you have the, well, you see now the units, the inverted elements in the Hurby's integers consist of these elements here, all 16 combinations are possible, so these are numbers that are such that they are invertible within the integers and it's already a very symmetric, this group is afforded 24 and it's known as the binary integral group, these elements can be seen as the vertices of the 24 cell, the 24 cell is a complex, regular for polytopics, boundary is composed by 24 octahedral cells, you can read six meters in each vertex or Togo Nali, and the faces give an ortho system, as they want to study by Louis Kaderheff, and if you use the Kader transform you take, and you have this beautiful football, one of these that you learn from Phyllis Klein, you know he's a regular solid whose faces are octahedral, beautiful object, no invariant and this is going to be part of our game here and now you have the analog all methods of that type where you is a Hurby's unit preserves the upper half plane and it's for order 12 and now how we define the Hurby's border group, the same game as the border group we take key, which is the analog of that and we translate but we translate with imaginary part of Hurby's Syntegers, it's a group what is that group, that contains a super of what previous so this is my, what is the structure, what is the orbital, you know, what can I say about that, what is for the mental domain, what is presentation, what is Kaderheff any question you have for, you have asked for the modular group, you can ask now, and of course it hasn't escaped to us, that we can mimic many of those theorem, and that's what we are doing exactly, so you have that and well, you have now the analog of the affine group you see, for the modular group there's no, the only affine group is translations there's not very interesting, but in case of that one you have to reach the group, you have that and now you see these matrices here you take you Hurby's unit you know of the 24, and you construct these matrices, I don't know how we came down with that, they are 24 matrices and this exactly by using the Kaderheff the vertices of the 24 cell interplay between the 24 cell hyperbolic geometry and the constructions of many hyperbolic manifolds, what I mentioned for manifolds and orifolds using will be arithmetic of course using these groups, so you have that well I call we call them the Pauli matrices because they are finite there's no reason and I say why we call them the Pauli and they are 24 elements in the 24 cell so I consider the group that preserves this famous cube three-dimensional cube in the half sphere you know and I want the group, because I want to find the fundamental domain, I want that okay, so now this is teleographic projection of the 24 cell the picture is due to my collaborators Fabio Blacci from Fienzi, Florence Juan Pablo Diaz from my institution, Cuernavaca so this is the 24 cell which in Schlafly terms is 343 and this is this is the theorem this is the remark is the theorem that these two groups and define the Hurwitz model of orifolds and the okay, but what about taking this cube everything is above this cube namely this figure the analog of that and take its its inversion you think that's white, that black and you consider well, this object here is a it's a coaxial polyhedron it's a convex polyhedron with one vertex at infinity and all angles are rational multiples of 2 pi our hero and reflect and you have desolation of H4 but so this is the this is I didn't find the last time, this is what I have so this part, this here on the bottom is R3 but actually it's the space of imaginary quaternions and then you take the half sphere three dimensional sphere that and then you you see, you see this is why I tried to put there and you take by reflexive side and you get a non-orientable kaleidoscope of hyperbolic and what we do, we divide this into little pieces to see the fundamental domains of the Hurwitz group and see if you can manage to find the Cayley graph to give presentation this is like a too easy presentation beautiful but and we will see that all there will be three probes so very complicated and that's the reason I describe to you the Unitary groups because they, to see elements of finite order know how they go and then you have that so this is just this is just a description of the polygon well we have vertices one dimensional faces, two dimensional faces three dimensional faces and one solids, you know these are the scripts you can compute the order characteristic is 12 well that's what I say, I say it by words if you take p and tp this satisfies the precarate conditions so you propagate and if you take the inverse that one, the bottom and now you have a subgroup of index two, now you have a finally we have hyperbolic order for the dimension four and then of course what we will do is use the famous theorem to unfold it and to get infinitely many examples of hyperbolic manifold that's easy to say than done and it will be the analog of certain things that are under construction like congress, subgroups, etc and so now so this is the oriental order fold you take the double pyramid I go fast about that I just told you that take this pyramid with vertex at infinity over the cube in the e-glue inverted you have a double pyramid and then take this group of orientation, words of order two and take the plane, take the question and that's our first fundamental hyperbolic order and now comes the actual computation so this formula we know very well in the case of of C when we know that to prove that PSL to our isometries of the upper half plane the first thing you do is to prove this formula not the imaginary part which of course now is the real part satisfies this condition, this remains true but now you have to be a juggler of commutation because you have no commutativity it's very much not trivial to do it the main point of this theorem is to have guess that it was still true in the quaternions and then you have to juggle so the proof is this you make computation, computation continue the computation use the BGP conditions continue and here we finish so just pure computation but I tell you the merit is this is still true to have phase, this is still true even in no commutative case so this has important consequences because it will tell it's the key for this corollary if you take a point and apply the or ellipsis the height of the orbit the imaginary part namely the real part remains bounded and that's the key thing to prove what is rigorously what is the fundamental domain and this is the fundamental domain you prove that you can bring under the modular group any point to this fundamental cube the pyramid of this once you are there you are not finished because you still have things that move points namely the unitary group and that means that the fundamental domain will actually be the union of two cubes because you still have the unitary group it's not a fundamental domain that thing you still have things that take something inside to something inside and then the final option is this is the fundamental domain and here you will be able to cut and piece fundamental domains are beautiful if you can cut and glue maybe disconnected and then you cut with scissors, hyperbolic scissors cut and paste and what you get actually is beautiful as fundamental domain will be the core of the rhombic rhundotic ahidra this is the work of art of Mojuan Pablo and this is obtained from the cube by crowning each face with a pyramid so here you put a pyramid in such a way that these two are like that and so what you have is this rhombic rhundic ahidra has 12 faces and which are rhombus and this is one of the few things that desolate you can see because you desolate it by cubes you take the middle points and refine it and that's beautiful because it tells you how the geometry of the Lipschitz modular group looks like but priorities that is have difficulty so here you have this is the fundamental domain so the union of two cubes here and the other is the union of two pyramids for the hulwits those are the fundamental domain then you can and then this is the way of color-paste to obtain that it's a beautiful design by Juan Pablo and then we understand the geometry of the orbital fundamental domain instead of playing with the modular group now we play with this one and we have that and these are the theorems I told you the characterization is given by the groups we define just by translation and inversion concise exactly of the Moebius transformations which present the upper half plate and with entries in the Lipschitz or hulwits conditions once you approve that we use the existence of the fundamental domain and all the things that we develop well this is the analog for the hulwits and now we still can refine our fundamental domain to a finer thing that has both symmetries the so-called coccitor decompositions and so this cube we have pyramids, simplex which you can think of some color that will fill each of the fundamental domains and 48 of them will fill and then many of them will fill the 24 cell and that therefore will make the we link these groups with the 24 cell and therefore with hyperbolic manifolds which are canonically well constructed by Radcliffe and Rood by Chank et cetera connected with things that are already done but are described in an arithmetic in an arithmetic way algebraically so this is the delta L I will tell you quality this is this piece the cube, the fundamental cube has 48 of those triangles and therefore you propagate those ones, that will be the fundamental domino piece ok this will allow us to compute volumes to to do many things related so because the compositions has 48 atahedra so what we do is the following so this is the cube ok how I want to show you well these are the notation et cetera et cetera it's just a notation but what I want to show you is ok this is this little piece of cocciter is a polytope with these angles and that's how you propagate and this will be a piece filled initially with the fundamental domains and from this everything will be for this we will get the cell cover covers because what we want we are finite in the cell groups that are actually many forms of orifices but we will get extremely detailed in our papers you know say 60 pages get a tail full of the orifices et cetera ok I say that even worse you take this little piece well maybe I'm going to go a little faster because I'm getting my time is I spare about time trying to show you I'm doing something non-formal so with this little piece of domino we fill the 24 cell we know how to compute volumes we compute the volume and the volume is equal to 4 pi squared over 3 and with this little piece also we will describe the cocciter I mean K-lay graphs just by looking at the fundamental domains I will read the relations and describe give an explicit presentation like that one of the different groups ok well this is a ok you take the tiny little little fundamental domain this is the tragedy that I show you and the group generated by reflection of that certain group now you have this group each of them is a super group of index respectively 4, 12 and 3 and from that you can compute therefore you see the orifice the ellipsis orifice has volume p squared over 72 and the volume of the is 2 pi over 216 so you see 3 times that is that of course because it fits 3 times so that's how the volumes of our we can explicitly compute the volumes we can characterize all the different properties these numbers appear here how many subdivisions you have number theoretically is very pretty I think you have to I don't know if this works ah this is a beautiful picture I mean just to tell you I mean I learned from Thorsten that you get imagine you in mercy hyperbolic space you see the desolation you see something beautiful this is schematic of course Royce Nelson is very kind so I asked him if I could use and he's the courtesy of him he has a very beautiful page so this is more or less what you see if you get our group and this is of course only schematically and now I don't know okay so the group is generated by T and T so what are the relations these are you see elements of order 2 elements of order 6 of course by series of theorem it's not surprising that but these are the relations and these have to do with the different things you can do with a cube you translate rotate twist, shake and roll and then you get all these relations and this is a very complicated very much more complicated than that one of course here and there something we haven't done and we have to do is get free products you know this thing have been done also by the English school like a McLeod McLeod's line probably Bearden by the way some of the formulas we got coincide with the book of Bearden which is happy you know it's correct therefore and also this is a generator and the whole this is a little bit more complicated look at what a beauty I'm sure well these relations probably are superfluous so it would be nice to get a minimal set of relations we haven't been able to do that but we got the group that's already something and to prove that you have to do you see many interesting things symmetries of the cube arrive you know and the binary of the cahedral group arrives as a sort of group of the has an enormous amount of geometry this just show you how this would be used excuse me theorem I should have written theorem because it was a lot of work and here I have to thank Juan Pablo I mean Juan Pablo came to my house with a toy with tinker toy and all my grandchildren want to play don't touch that because we use it for model so we're all the time playing with also Fabio knows ok so here so what about silver covers so this will be subroutes will find an index free will give you hyperbolic ok let me just perhaps just so what are the minimal orders of like you know the order group has a super for the 6th classical no gamma 0 that becomes the fundamental group I mean I think somebody spoke about the 6th manifold no the puncture total of the puncture well here the minimal has these orders you know this all comes from the geometry of the fundamental domain then here's the connection with the work of Rasky a chance excuse me I don't know they are normal subroutes they are just finite index so I don't know probably not but we haven't thought about it so here is so using this silver covers I just finished with that using the silver covers you have hyperbolic manifolds of finite volume and the cosps are flat manifolds torus and therefore you can can you say the tetapa you get what you can attach T2 cross T2 whose boundary is T3 to have a compact manifold and many times this manifold happens to be S4 and this way you get solid tori embedded in S4 whose complement is hyperbolic at a Thorston and these they did by computer and we now have these we think this algebraic could be very useful and Juan Pablo Diaz co-author and the one who wrote the beautiful pictures there is given as example of Thoright a benniness for whose complements are hyperbolic manifolds and part of this work is also the one I'm presenting now and I think I'm finishing many things happening here well only to tell you that if you take the 24 cell think of it in the bowl so at infinity you have 24 vertices and 24 faces that are octahedral and there's such a notion as opposite of the octahedral and then what you do you reflect and translate and you glue the faces to present to you by gluing and pasting a concrete visualizable hyperbolic manifold not orifold but manifold a finite volume how many costs has Pablo five exactly I already computed once but I have five and it's beautiful but now you can use the challenge of in their paper they use a congress group for their two we can use our group we haven't done it we'll do it and of course CITLU hasn't escaped to us the capabilities that we have like Bianchi groups modular forms anything that you think can you do that we have thought about it it's under construction so with that I finish thank you very much for me