 Hello, everyone. Thanks for coming. This is our basic notion seminars. For those who might not know, we try to have the seminar every month, once a month, and it's meant as a sort of general audience and on a particular topic. My pleasure to introduce Ruth Keller-Hass, who has been visiting ICDP, is leaving tomorrow, fortunately, but if you like to talk to her after the talk, she's still going to be around. She is from the University of Freiburg in Switzerland and will talk to us on the many aspects of hyperbolic volumes. Well, thank you very much, Fernando. Do you understand me with this microphone? Oh, yes. First, thank you, Fernando, that I can spend the last part of my short sabbatical here at Priest and on the very last day of my stay here, it's an honour for me to speak a little bit about hyperbolic volume, a topic which bothers me since many, many years, and again, I went back to some hard computations, which I even here tried to complete. So, I took, I take this serious basic notions seminar, so if it is too elementary, please give me a sign and I can accelerate. If not, then give me another sign and I will explain a little bit more. Okay, so let's start with a basic notion of hyperbolic space in order to speak about volume of a domain, a measurable domain, or a polyhedron in hyperbolic space. Before you read this, let me motivate why we do not consider euclidean space. In euclidean space, as you know from your studies, when you have a polyhedron, a tetrahedron, a simplex, a triangle, a quadrilateral, then you can associate to the vertices of this object, if it is polyhedral, vectors, and to compute the volume of a tetrahedron, for example, it is basically an algebraic task. You look at the gram matrix of your vectors giving you your polyhedral object, and then the determinant of the gram matrix gives you the volume. So the message is in euclidean space polyhedral volume, and therefore other tasks for computing volumes of euclidean space forms is not so difficult. There are still many difficult issues there, but that is the reason why I want to consider non-euclidean spaces of constant sectional curvature, which means either the standard unit sphere, so we take triangles, tetrahedra, simplices on the standard n-dimensional sphere, or we look at the negatively curved space, its dual hyperbolic space of constant negative curvature minus one. So in order to to consider volume computations, we have to choose a suitable model to deal with volumes, and there are, maybe you know that, several models how to visualize as a euclidean being hyperbolic geometry in n dimensions, which makes our life easy when we want to encode a polyhedral object, the tetrahedron, through suitable parameters. So that's why I start my first slide with the linear vector space model of Lorentz and Minkowski, which deals with representing hyperbolic n-dimensional space as some, I knew that I'm going to mix it up, as it's a connected space, simply connected space, hyperbolic space, as vectors in Rn plus one, but equipped with a symmetric bilinear form of signature n1. So we have n positive signs, n1 negative signs, we have a norm which is associated to this bilinear form, and we look now at vectors in Rn plus one equipped with this bilinear form, so that's the Lorentz-Minkowski space of signature n1, such that the norm squared is minus one, or if you take the norm, it's imaginary, and we take the upper sheet of this hyperboloid model and ask that the last coordinate is positive. So we take here vectors, ending in this sheet, and all these vectors give us points of hyperbolic space. This has a lot of advantages, this model for the standard hyperbolic space, because when we look now at geodesics, paths where we walk on, with minimal distances, or geodesic hyperplanes, and we can encode them with vectors, or normal vectors, very easily, and we are very close to the other constant curvature case, the sphere, which we can embed in Rn plus one. So there we have also a vector space model, and the geodesic k-dimensional planes in this Minkowski linear model for hn is you take a linear space through zero of dimension k plus one, you intersect it with this upper sheet of the hyperboloid, and the intersection gives you a k-dimensional geodesic plane. So for k equal to two, you take a two-dimensional plane through zero, you intersect, and what you get here, you can see it, the intersection with the plane gives you a geodesic line. So that's very convenient, and when you take a hyperplane, so we take an n-dimensional oh, here should be n plus one, we intersect a hyperplane with hyperbolic space, we can associate to this hyperplane a normal vector, say, it is orthogonal to your hyperplane, it is not a hyperbolic vector, it lies outside our cone, and we can assume that this vector has Lorentz norm plus one. And in this way, for each geodesic hyperplane, we have a normal vector, which is here or there of norm one, and we get two oriented half spaces. And let's say in the future, we take the negatively oriented closed half space defined by this hyperplane or the normal vector. This is the entry to look at a polyhedral object, which is the intersection of finitely many closed half spaces, meaning we have finitely many normal vectors, unit normal vectors, giving you a polyhedral object, and that also gives us a gram matrix, the matrix formed by the products, Lorentz products of these, of all pairs of normal vectors appearing when you look at these hyperplanes. This is very convenient when you want to do polyhedral geometry, and also when you want to treat isometries. Having in mind or being motivated by studying spherical geometry, where we know the orthogonal group gives us, when restricted to the sphere, the full group of isometries. Here, the full group of isometries is the connected component of the unit element of this positive or projectivized Lorentz orthogonal matrices, Po0n1, which are defined by regular matrices with real coefficients, satisfying this, here also I made a mistake, that should be a j, with j, this matrix where you have the nth identity matrix, and here minus one. So the fact, which we have also for the orthogonal group On, and the sphere is every isometry being represented by such a matrix can be written as a finite composition of reflections with respect to hyperplanes. And in this model, this fact can be easily proven in a course, in a third year on the graduate geometry course. So message, every hyperbolic isometry is a finite composition of reflections with respect to hyperplanes. So when we look at groups of isometries, or groups acting properly discontinuously on hyperbolic space, meaning groups, which when you take a compact subset of your hyperbolic space, properly discontinuously acting means this compact set is basically pushed away from itself, and that gives us discreteness in any ordinary sense. So when you look at discreet groups of hyperbolic isometries, whose orbit spaces gives us orbit faults, or when they are even without fixed points, manifolds, then the easiest case to look at is, let's look at discreet groups generated by finitely many reflections. These are called hyperbolic coxset groups. So a hyperbolic coxset group, I hope you can read it. Yama in isome hn is a discreet of isometries, I don't repeat it, generated by finitely many reflections. Many reflections with respect to hyperplanes. And it is not difficult to see that when you have such a discreet reflection group, that its fundamental domain, which contains for every orbit one point added in the interior, or if you close by taking the boundary parts of the boundary of your polyhedral object, such fundamental polyhedra are characterized by dihedral angles. So if you take two intersecting hyperplanes, the intersection is of co-dimension two, you take a normal plane anywhere in the intersection, and you see the dihedral angle as two-dimensional spherical angle. All these dihedral angles are of the form pi over k. So they lead to polyhedra or coxset or polyhedra in hn with dihedral angles. Angles pi over k k, an integer bigger than one. Here comes a very natural explanation that such polyhedra, when they are very nice, formed by few bounding hyperplanes, all dihedral angles of the form pi over two, pi over three, pi over four, etc. They belong to very nice discreet group actions and their quotient spaces are modeled according to these polyhedra. And it turns out, I hope I can explain this to just a little bit today, that coxset or polyhedra with very few bounding hyperplanes, so in n dimensions, that means at least n plus one hyperplanes has to be a simplex, a triangle, a tetrahedron. In three dimensions, it has four bounding hyperplanes. An n-dimensional simplex is bounded by n plus one bounding hyperplanes. So when you take a coxset group with a minimal number of generating reflections, then this gives you very nice polyhedra and it turns out, nowadays we know that, up to a certain high dimension, that such coxset or polyhedra with a minimal number of bounding hyperplanes and large dihedral angles give us minimal volume quotient spaces. So this is a natural object, very basic to define and let's now pass to volume, because I said they give us small volume objects, that the metric is preserved. You have a constant curvature space, so your distances are preserved. The metric? It comes here on this slide. Good question. So I did not speak about the metric in coordinates in the upper half space, in the Lorenz-Minkowski space. There, my main point was linearity. We have vectors, we have gram matrices and hyperplanes, polyhedral objects. Now, I go a little bit more into the quantitative aspects, so I need the metric and the volume element. And there is an isometry preserving the geometric structure of hyperbolic space, which passes to other models where we see angles. And with one isometry, I will not give you the details. One can pass to the famous upper half space model of Poincaré. I just give you here a two-dimensional picture. So we have here R2 and we look at all points with a second coordinate strictly positive. What is nice about this model also, like in the Lorenz-Minkowski space, where we have this this sheet of the hyperboloid, here we see the boundary at infinity. When you add infinity, you have here the ball model of Poincaré. And the advantage in this modified model, where we also see the geometry and the curvature of the space somehow, with a little bit of fantasy, is that the line element for hyperbolic geometry becomes very easy to read off. Here you see the line element of Euclidean space. And you divide by this Nth coordinate, which is always norm zero, and you divide by the square. So you disturb the Euclidean metric by dividing and you get the line element of hyperbolic geometry. Knowing this, you know nearly everything, but actual computations of course are something else. But from that, you can also deduce by looking at the Euclidean volume element, that the hyperbolic volume element is the distortion of the Euclidean one in that sense. By the way, I see that some take notes. If you want to get the slides, I put my email address at the end and I can send you the slides. Okay, so you see now looking at this volume element, if you take a domain which is integrable, which is measurable, which has a finite measure and you want to know its volume, you have to parametrize your domain. It can be very complicated. And then you have to perform this n-fold integration. That can be very unpleasant. Usually it is highly unpleasant. And if you do the same thing on the sphere, it's even, I think, even more unpleasant. But we have advantages here in a hyperbolic space. In contrast to the spherical space, we have a boundary at infinity. Why do I say at infinity? Well, if you take with this line element, you can compute the distance between two points. And one does it like that. You take a vertical line starting, say, from zero to infinity, vertical line, the xn axis. And you take two points here, p and q. So these are two nice hyperbolic points. And we want to know the distance, the hyperbolic distance. And with this line element, one can compute in a really not difficult way that this hyperbolic distance is given by the natural logarithm function at the absolute value of the n-coordinate of p divided by the n-coordinate of q. So if you exchange the two points, you get the same distance, of course. And that means, in our aim to understand hyperbolic volume computation, one-dimensional volume is given by the characteristic function, the logarithm, the natural logarithm, is responsible for hyperbolic lengths, one-dimensional volume. That's completely different to the Euclidean space, of course, but also different to the spherical space. When you think of the sphere, you have an angle, which is defined by two points on the circle. The distance is given by the arc length. And if you use the cosine function or the tangent function, the arc tangent or the arc cosine is much more complicated than just that expression. So hyperbolic geometry has some advantages from the quantitative point of view. But now, look at a point, say, Pn, which goes to the boundary. Pn goes to zero. Then this goes to infinity. And that means that these points on the boundary, if you go to infinity or to the boundary below with n-coordinate zero, they correspond to points at infinity with infinite distance. You have now the impression this is very bad, not at all. This is very nice. Because when we study this line element again, when we restrict to a plane at constant height, h or a, positive height, meaning you divide here by a squared, and this part, it disappears, if there is no changing. Then the restriction of this hyperbolic line element to a horizontal plane moved up by a distance a is Euclidean. We call these hyperplanes horospheres, and they carry in a natural way up to a multiple, a Euclidean structure. That is very nice to have in hyperbolic geometry, Euclidean geometry and also spherical geometry, by taking balls of constant hyperbolic distance from a given point. Okay, so this is my first message. Hyperbolic distance is given by the natural logarithm. What can we say about characteristic volume functions in higher dimensions? And there is, there we have to distinguish between even dimensions and odd dimensions. Maybe you know the theorem of Gauss Bonnet for a space of constant curvature, non-vanishing constant curvature, which says that even dimensional volume is proportional to the Euler characteristic. I'm not sure whether you know all this. Okay, so since time is running, I will not speak about the even dimensional case because the philosophy there, due to Ludwig Schlafly and Knaeser, is we can reduce polyhedral volume in even dimensional space constant curvature non-zero to the computation of lower dimensional spherical volumes. Okay, so let's pass to three dimensions and how to compute volumes for polyhedral objects in higher dimensions and I would like to say odd dimensions. As I said, you can try to integrate over your polyhedral object or whatever you want to know. You can perform that, but in most cases you will not be successful. In most cases, of course, there are cases you can't be successful. There is a wonderful expression when you look at the polyhedron, so don't forget this is a convex object, finite intersection of closed half-spaces where two bounding hyperplanes intersect, when they intersect, then they intersect at the co-dimension two hyperplane and there you have a nice dihedral angle in the normal space. So a polyhedron, let me draw a picture, say a tetrahedron bounded by four planes, a hyperplane one opposite to the vertex one, etc. And we have, so here a hyperplane, here a hyperplane and they intersect in this edge. Normal to this edge, we have planes, tangent plane, a normal plane and here we see an angle. Let's call this intersection of these two hyperplanes opposite to one respectively to with the tetrahedron t f and here we have this angle alpha f inside our object, our tetrahedron and the Schlafly in a spherical case proved in the middle of the 19th century by giving three different proofs, very interesting. He proved that when you have a polyhedral object like that it's very simple, it could be much more complicated and you move a little bit the vertices such that the combinatorial structure is fixed, so it stays a tetrahedron, then of course the tihedral angles change a little bit, we have an infinitesimal change of the angles and of course the volume changes and Schlafly proved in a spherical case that the volume change, of course you can use that in Cartesian coordinates, but in parameters given by the tihedral angles, the volume change is as follows with respect to these tihedral angle changes, you take this change of alpha f and you multiply it with the length of this edge or in higher dimensions with a hyperbolic volume of this co-dimension two-phase, here should be a P, I'm sorry, and you add up all these expressions, so for this edge, for this edge, this edge, this edge and you multiply by this constant depending only on the dimension, in the spherical case you have to multiply it by one over n minus one, so here comes the curvature, non-zero curvature in and what this tells us is when you want to compute non-nuclidean polyhedral volume these are one-dimensional integrations, one-dimensional, not n-dimensional, this is much simpler and this volume differential formula has a lot of implications and it has also been generalized to other spaces in the same Eremannian case and it's really fantastic, so let's now look at the volume computation of this three-dimensional hyperbolic tetrahedron, here we have one-dimensional edge lengths, these are logarithmic expressions, logarithmic expressions in terms of these vertices which you can represent in coordinates of your space and to integrate the logarithm well, Lopacevsky in the 19th century could do it in a case of a very special tetrahedron which is a so-called also scheme, so we have only three dihedral angles which are non-right, so here we have right dihedral angles, I write them sometimes like this or since we are also speaking about polyhedra which appear in terms of discrete reflection groups or coxet theory or Lie algebras, one also could write this orthogonal tetrahedron in this way, here we have hyperplane H1, H2, H3, H4 and this graph means hyperplane H1 and H2 intersect under the angle alpha, H1 and H3, so this hyperplane and H3 is this one, they are orthogonal so they are not connected, not connected nodes in the graph means orthogonal hyperplanes and so this is a very nice graphical picture to write down polyhedra when they are not too complicated, so a polyhedron in three dimensions bounded by 200 faces, I would not use this description but it's anyway, it's combinatorial, very difficult to encode a polyhedron in a suitable way, in an efficient way when it's not of a simple combinatorial type. Okay, Zolobacevsky, we lived at the same time like Schlafly but somehow mathematically they didn't know so much about one another which is a shame, they didn't have internet and email like today and Zolobacevsky looked at these so-called also schemes, this term was invented by Schlafly, Zolobacevsky looked at the hyperbolic case, Schlafly only at the spherical case and in contrast to the spherical case, Zolobacevsky could find a closed volume formula in terms of these three parameters defining the object up to isometry, alpha, beta and gamma, alpha, beta and gamma by introducing an additional auxiliary parameter which is very complicated, a square root of trigonometric functions and then you take the arc tangent and we don't know about the geometrical interpretation of theta but it was very useful to find this formula and what is this Kirillik L, Lobacevsky function, slightly modified, it reads like that, it's an integral over a log, we are not surprised anymore that a simple integral over a natural logarithm appears because line segments, edge lengths are logarithms, are logarithms and so one could write this integral in a complex way also very easily and one can also write it as imaginary part of the so-called dialogarism function, so logarithm, dialogarism function and it can be written if you take the imaginary part as something which is related to some L series of a certain character sometimes or the Riemann's data function if x is one half, if k is well sometimes one can, so what does it mean to have a dialogarism, dialogarisms appear in many many contexts in mathematics but also in physics and they are defined and like a polyhedral volume in a inductive way there we have the co-dimension two phases and then volume etc, here we take the first logarithm, this is the log of one minus z up to a sign, that's okay, that's a length and then we take the nth logarithm as an integral over some suitable path in the complex plane over the n minus one dimensional polylogarism d log t or d t over t and in this inductive way the classical polylogarism is defined and it can be written as a series of that kind which is for number theory very important and the Lopacevsky functions appear as imaginary or real part if you take arguments on the unit circle and they have these expressions, now there are modified polylogarisms nowadays which appear in algebra, k-series, in Mahler measure theory, in number theory and they are they can be they were modified in such a way that they satisfied in a clean way certain functional equations so the modified dialogarism of Bloch-Wigner satisfies a five term relation where here this R2 is the classical cross ratio of four points yeah and they are in this context they are famous conjectures of Don Tzagie who is sometimes often here and they are very difficult and proven only for very small degrees for the trilogarism they are still valid okay no no I did not speak about where no no they have the unibolia otherwise you have to you have to make some manipulation with the functional equation yeah we should not be surprised that these functions appear because of Schlaff's differential formula where we also have this inductive process in going up with dimension to compute polyhedral volume oh yeah let's go back to Lopacevsky's formula as I told you we have this nice property of hyperbolic space that we have points at infinity where we have a horospherical neighborhood and let's assume that this point here 1 tends to the to a boundary point at infinity and this this point also tends to a point at infinity what is very nice is even if this is very brutal if you look at the neighborhood at the neighborhood of these vertices tending to infinity I write it like that these neighborhoods still have finite volume so we can look at polyhedra in hyperbolic space with as many vertices on the boundary at infinity as possible we still have finite volume and now comes the important point if we do this if we construct these boundary neighborhoods by looking at a boundary component given by a horosphere which is Euclidean basically we can use Euclidean geometry here to compute the volume of these boundary neighborhoods this is a very nice and Lopacevsky's formula in that case when these two vertices go to infinity means gamma has to tend to alpha because beta plus alpha plus where he is here I made a small mist I don't know it's okay yeah beta gamma and the right angle it will become Euclidean this means the Euclidean triangle exists if and only if beta plus gamma plus pi over 2 is pi yeah so in that case we get here alpha pi over 2 minus alpha and alpha and the volume formula of Lopacevsky reduces to a very simple expression it's one half of the Lopacevsky function at alpha in the case of a doubly asymptotic three-dimensional also scheme and John Milner derived in another way but it's very simple now by this Lopacevsky result an ideal tetrahedron an ideal tetrahedron what is this this is a tetrahedron all of its vertices are on the boundary at infinity so so we are here on the boundary at infinity it is characterized by three pairs of dihedral angles at opposite edges and the fact that we can see them in a horoscope which is Euclidean makes that alpha plus beta plus gamma is pi and this formula is very simple by means of Lopacevsky's formula and it is very important and used in snappier in a modern software package to compute the volume of compliments of knots or links on S3 these provide very nice smooth hyperbolic free manifolds they are non-compact and we can decompose them very often in ideal tetrahedron and this formula is at the basis when you have such a triangulation in ideal tetrahedron to compute volumes of compliments of knots and links so that's why hyperbolic volume also in small dimensions is very interesting then of course there are other polyhedral objects here is one pyramid which is non-compact and Wienberg derived this formula and the graph of such a pyramid is this one you can exercise at home what i mean by that infinity means the dihedral angle is pi over infinity so zero and here the message is the volume is not so much more complicated structurally but this is pyramids these pyramids they are non-compact you have an apex at infinity and then you have a face opposite to this apex and in this later i will consider pyramids where the neighborhood around this apex at infinity is topologically a product of two simplices okay still have some time to to go further is this somewhat clear okay let's go to dimension five i said even dimensions this i do not want to say evident but up to dimension four and six quite clear now in a long long time ago i looked at such also schemes so right angled simplices all vertices which can be on the boundary at infinity are on the boundary at infinity and they have a maximum of right dihedral angles pi over two so they have a six bounding hyperplanes they have a very simple graph and the fact that the two vertices are at infinity means algebraically this condition cosine squared the sum of the cosine squared equals one this is a two parameter family an infinite family it is important but not so important that i can say any five-dimensional polytope has a volume which is a sum or a difference of these that would be beautiful but this is very difficult to show i'm not sure whether this is even true but its volume is given by this expression only in terms of the dihedral angles no theta no arc tangent no cosine nothing just alpha beta gamma the sum and here comes a wonderful constant the Riemann zeta function at three which provided the constant of integration when looking at Schlafly's differential formula this function again i repeat this is the real part of the trial logarithm on the unit circle and it has this expression here as an infinite sum there are not many other closed formulas in dimension five and i think i'm one of the few ones who has this interest still today even more today than before uh also i proved that many years ago uh now let's go to these coxetopolyhedra when we have an angle pi over k then in the cox in the graph of the polyhedron i just write here not pi over k for the dihedral angle i write here k if k is equal to three which appears in the theory of coxet groups very very often then one just writes an edge without the three because yeah we do not want to write much we want also not to speak too much so we omit it the five over two means we have two pi over five and here we write five over two yeah i hope this is clear very simple graphical notation so this volume formula which i had here when you look at this condition it is satisfied cosine squared pi over three plus one half plus one fourth gives one and the volume is a rational multiple of zeta of three so a near-rational number we do not know whether this number is transcendental for this graph for this group which does not give a discrete reflection group we have here five over two but we have somewhat a quasi discrete group it satisfies the condition and we get this expression so the lobachevsky function of degree three appears by using cutting and pasting methods i could even manage to compute other volumes for example of this beast which is a very nice one believe me it is a simplex and also scheme which is not compact but a finite volume and the volume is quite small this one i do not want to speak about it it's too late but this one keep this in mind it is non-compact it has one vertex at infinity it is a simplex and it has this value actually let me now go back to these coxet of polyhedra and hyperbolic reflection groups it turns out when they are generated by a very small number of reflections or when their polyhedra fundamental polyhedra have a very simple combinatorial structure a simplex or a pyramid then if we go up with these weights here we go up that means the dihedral angle becomes small so when we have the country many angles which are big close to pi over two by Schlafly's volume differential the volume goes down has to go down so here we have many right dihedral angles pi over two we have four pi over three and we have just one pi over four one might think hey this tetrahedron or this simplex gives us is the polyhedron model of an orbit space of very small volume and Thierry Hild was my student for his thesis in Fribourg ten years ago my god the time is running he proved that among all non-compact hyperbolic quotient spaces so of the form you have a properly discontinuous group action you look at the orbit space and it has a fundamental domain what is its volume among all non-compact hyperbolic space forms this group provides us the space form of absolutely minimal volume and this volume is this irrational number this orbit fold so it is hn h5 modulo gamma by a famous lemma of Selberg it has a cover it has it is covered by smooth manifolds and actually a Radcliffe enchants could construct a cover of this of some gluings of this one which has the smallest known volume today and this volume is seven zeta of three over four I have no idea whether this is very small the smallest at all this manifold because we do not have a compulsion value in even dimensions we have the Euler characteristic the absolute value is minimal when it is one in all dimensions we the Euler characteristic is zero but so far this provides us minimality in some sense in a compact case with a Emry also a student at Fribourg I did some similar work in a compact case by restricting to arithmetically defined groups which I do not want to mention but in the message is when we look at singular space forms arithmetic or not arbitrary but non-compact we know the minimum in the arithmetic case we know the minimum and that's basically what we know in the dimension five now I showed you also this Wienberg formula for pyramids I should now say naively if one goes up in dimensions one would think maybe coxet polyhedra so groups generated by reflections in these coxet polyhedra they have a very simple presentation if the polyhedron has a very small number of faces of boundary hyperplanes the group is generated by this minimal number and the and the relations are very simple so hyperbolic coxet groups are characterized by very simple presentations and naively by looking for minimal volume space forms in higher dimensions even or odd a good thing is to look at coxet polyhedra with with small weights two three four but not ten two three four dihedral angles pi over two pi over three maybe pi over four unfortunately hyperbolic coxet polytops polyhedra do not exist anymore in dimensions beyond 995 we know that nothing to do we have concrete examples up to dimension 21 Richard Borcher's with Lee algebraic methods he constructed a coxet polyhedron hyperbolic of finite volume in dimension 21 with more than 200 facets so I cannot draw the coxet graph volume is not known so 21 is the maximum of dimension we have then we go down to 19 there's one example 18 one example they are all non-compact and if you want to know about compact coxet polyhedra the story ends in dimension eight when you ask the question what about simplices coxet simplices they have the simplest presentation on the level of the reflection groups coxet simplices exist only up to dimension nine and in the compact case compact coxet simplices only up to dimension four that's the reason why regular polyhedra with a nice symmetry they do not exist in high dimensions but objects with a high degree of symmetry usually are responsible for small volume so we are looking a little bit in a dark space when we want to search for small volume and one thing to deal with this problem is to pass to arithmetic considerations so in simplices do not exist up to higher dimensions let's look at pyramids with a apex at infinity and some basis say a simplex or say a product of simplices so an apex over a product of simplices these are pyramids which were classified by Pavel to Markine who is in Durham was a student of Wienberg he classified them and he showed they exist up to dimension 17 and all of them starting from dimension 11 they are all somewhat arithmetic they belong to some number field to some quadratic form associated to a number coefficients in a number field I really do not want to go into that together with two other students I had we classified them up to commensurability so we looked their orbit spaces when we go up to their covers can we describe them and I gave a talk two years ago here so every somehow at home among these pyramids we also have infinite sequences subject to this condition and just recently I could produce some formula because this condition just helped and for two coxsets of pyramids with diagram four four three three four four believe me it's a pyramid just just take these just take these graphs in that we are in dimension five a pyramid has seven one two three four five six seven and we take four four three three four four uh when you look at this sub diagram it is a Euclidean triangle pi over four plus pi over four plus pi over two games pi so we have a product of two Euclidean triangles uh this is a perfect pyramid in hyperbolic five space and the volume where this volume formula is this multiple of eight of three yeah another example satisfying this condition is that one now I want to come to the end because in connection with hyperbolic volume I mentioned mathematical physics with the polylogism I mentioned not theory complements of links I mentioned the Euler characteristic topology I mentioned group theory presentations now I want to go a little bit to a little bit rationality or number theory but there is much more in there than I can tell today in dimension five I have these two the six three three three six so there we have six three three three three six and this one uh this one satisfies the condition cosine squared of alpha plus cosine square beta plus cosine square here is one this one doesn't satisfy that but uh in our commensurability classification we also could see that they are not commensurable so it is not at all clear that the volume ratio is a rational number not at all but with Schlafly's volume differential I could numerically perform this simple integration this is not so difficult and when I plug this in in Paris which is not so fantastic for numerical integration I got really instantly eight over 35 and I was puzzled I rarely had such a result and uh I'm pretty sure that this is true but uh still I cannot prove it and these rationality questions are very very tough if it would be true this means that the volume of this one has to be that value because this volume I got here as a result of this precise formula but if this is true what Paris shows me I have also this one now there are strong hints that this eight over 35 is probably correct because um maybe I stopped with this slide when you take a regular simplex a regular object it has a high big symmetry group vertices are equivalent under this symmetry group of the object the symmetry group operates acts transitively on the vertices and the simplex a tetrahedron or a equilateral or a nice regular triangle can be decomposed in a certain number of congruent smaller pieces and these are also schemes or truncated also schemes and in dimension five I just discovered by chance maybe in January or maybe it was in December I don't recall it was called no maybe December that the so-called birectified simplex in dimension five so this is a five simplex which has been its vertices had been all truncated off in a similar way and in a very severe way in such a way that the truncation the truncating hyperplanes even intersect themselves and uh this is not a rectified simplex so this means we would just truncate off the vertices in a nice way we birectify the truncation uh is severe and this gives still a very nice symmetric object it can be decomposed into these pyramids and into six factorial pieces and this is perfectly hyperbolic and the characteristic truncated simplex is this one so uh then one could one can deal with volume of that object in a different way this is an arithmetical object and there are there is the hope that one can prove that this eight over thirty five might be correct but this is not a theorem this is just a question and I think with this I stop with the last remark so far we do not have a single volume formula in seven dimensional non-nuclidean space not a single one that's why I stop with dimension five and I thank you for your attention thank you