 Hello, so it's a pleasure to be here. And I unfortunately just arrived. So I didn't talk to almost any of you. So if I am too fast or tell you something you want to know more about, ask me questions. If I'm too slow and you say, we already heard about this in the past two days, then also tell me and we can go faster. So what I want to tell you about a bit in this lecture and the next one is about geometric structures on surfaces. And so for this, we first look at surfaces without any geometry, so just as topological object. So we'll look at mainly surfaces in the sense of saying that they're two-dimensional compact manifolds without boundary. And so topologically, they are classified and basically you can draw pictures of any such surface. So the simplest surface is a sphere. So the boundary of a ball. And then the next surface is a torus, which looks like a donut. And the next surface is a double torus and then so on. And if you just look at the surface without any geometric structure, so just as a topological space, it's classified by one number, which is called the genus. And that's just how many, if you start with the surface of a ball, how many handles do you have to attach to this ball so that the surface you look at is the surface of this shape? So this has genus 0, this has genus 1, and this has genus 2. And then we have genus bigger than 2. So now what we want to understand how can we endow such a thing with a geometry? And the first thing we're going to look at is the most, in some sense, simplest kind of geometry we can put up on the surface, where every point on the surface in the neighborhood of any point looks exactly the same. So we want to put a structure on the surface, which is in some sense as homogeneous as possible. And the first geometric structure we want to look at, and we mainly will look at that, are geometric structures really in the sense of Riemannian geometry or metric geometry. So we want structures where you can measure distances between points, where you can measure angles, and so on. And so if we try to do that for the surface, there's a very famous theorem, which is called the uniformization theorem, which says that any such surface admits precisely one metric, one structure of a Riemannian metric of constant curvature. And depending on what the topological invariant of the surface is, so what its genus is, the curvature takes different values. So the sphere has a metric of constant curvature plus 1. The torus has a metric of constant curvature 0. So we can think of it as being glued out of a piece of paper, which we identify two sides. We have a long cylinder, and we imagine identifying the two other sides. And as soon as we have genus 2 or bigger, the metric and the structure we can put on the surface is a Riemannian metric of constant curvature minus 1. And this will be the geometric structures we will look at in more detail. So every surface genus G bigger equal than 2 carries hyperbolic structure by which I mean a Riemannian metric of constant curvature equal to minus 1. And another way of thinking of it as I meant in the beginning is to somehow, if you don't know what the Riemannian metric is, I'm not going to give you the definition. But so you should think of this structure as modeling your surface on a model space. And every neighborhood on your surface looks exactly as a neighborhood in the model space. So equivalently, we can think of this as giving an atlas from the surface. So for every neighborhood, we map it into a nice homogeneous space in the hyperbolic plane. And we'd have two charts which overlap. So then if we look at the map in the hyperbolic plane, which sends this piece to this piece, this is given by an isometry. OK, so the first thing I want to discuss a bit, and we spend some time discussing, but how does this model space look like? And probably, yes? You didn't say oriental or geometric? Yes, actually, sorry. I wanted to say that. So two-dimensional oriental. Yeah, you still have geometric structures. But of course, something you couldn't have, things which are orientation-reversing. So for example, if you think of a Mubius band or Klein bottle, I mean, if you take a strip of paper, you can glue the Mubius band by gluing it with a half-twist when you glue together. And if you think of this long Mubius band, you can now take the ends and glue them together. Or you could take a cylinder and glue it somehow the other way around. So for example, higher genus, yeah, you can do the same thing. But I would really just talk about oriental points. OK. So the first thing I want to discuss is how does the hyperbolic plane look like? The second thing I want to discuss, which in a bit more detail, how do we actually endow a surface with a hyperbolic structure? And the third question I want to address is if we have a surface, there might not just be one way to put a hyperbolic structure, but there might be many ways. And we want to look at how does the space of all possible hyperbolic structures look like. And then there, I'm not sure if we get to that. But otherwise, I'm happy to answer questions later in private discussions about this. If we have time, I would like to discuss some examples where we go beyond hyperbolicity. So we look at geometric structures on surfaces which are not anymore hyperbolic structures. OK. So let's look at the hyperbolic planes. And this is really the space on which we want to model our surface. And the hyperbolic plane doesn't have a good representation in Euclidean 3-space. So in mathematical terms, there's no isometric embedding of the hyperbolic plane in Euclidean 3-space. So whenever we try to visualize it or discuss it, we have to compromise on something. And that's why it's very useful to have very different models on how to look at the hyperbolic plane. So the first one to discuss is what's called the hyperboloid model. And this looks at the hyperbolic plane in a similar way as we usually look at the sphere. We have the sphere. We think of it as sitting in Euclidean 3-space as the set of vectors of norm 1. So when we look at the hyperboloid model, we also look at 3-space. So we look at R3. But now we don't endow it with the Euclidean scale-up product, but we endow it with a Minkowski scale-up product. So we think of this. Sometimes people also write R12. So it means we put a scale-up product, which I will put an M for Minkowski, which what does it mean? So if we take two vectors in R3, then the scale-up product is given by taking the product of the two first components, but with a minus, and then taking the product of the second two components, and the third two components with a plus. So for the Euclidean scale-up product, we have that. And we just change the sign, this one. So if we do that, then we can look at the sphere with respect to this metric. And once the sphere with respect to metric, so now we have to make a choice where you choose plus or minus. And we will choose minus. So the hyperboloid model, we look at all vectors in R12, such that Minkowski scale-up product, or norm squared of this vector, and we put a minus. OK. So this is, I mean, you can try to draw it, or this is something that. But we actually can take the vector, or if we take minus the vector, we have the same thing. So we usually add second condition, and we just take one of these hyperboloids, namely the one where the first coordinate is positive. So this is our space. And now you can do a lot of things similar as how you would do it for the sphere. So for example, if we want to describe the tangent space at a point, so let's, and here we have zero. Here we have the first axis, and this thing goes through one. So we have any vector in here. We can look at the tangent space at this point of the hyperboloid model. And this is the orthogonal complement of this vector with respect to this Minkowski scale-up product. And then you do, in order to get now a non-symmetric by linear, a symmetric non-degenerate by linear form on each of the tangent space, which would give you a Riemannian metric is what you do is you take the Minkowski scale-up product and you just restrict it to this orthogonal complement. So this is the metric at a point B, or let me write it. So there's one thing you have to check is that this actually gives you, I mean, you start with this Minkowski thing, and this gives you a positive definite by linear form on this subspace. OK, so now if we look at geometric space, we might want to look at other things. So angles, we can always define when we have this scale-up product in the tangent space, because we just take the same formula for the sign of an angle as a nucleotide space. The sign of V and W is a scale-up product between V and W divided by the norm of V, norm of W. The other thing you can do is you can measure length of curves. So similar as you do when you measure length of a curve in Euclidean space, where you take each point, you have a tangent vector to the curve, you take its norm and you integrate. And then you can look at shortest curves, just as taking the infimum of the length of all curves between two points. And you might want to understand how do shortest curves look like in your space. And here, if you look at shortest curve between two points or also geodesics in the Romanian sense, they have a very simple geometric descriptions. They are the intersections of your hyperboloid with planes to the origin, so 0. I'm not going to draw that, but if you have a vector here and you have a tangent vector with a point V on the hyperboloid and a tangent space U at that point, you can just take the plane generated by U and V through the origin, and this will intersect. And then you can also think about how to actually parametrize these geodesics as parametrized curves, but this you can do in the exercise, if you want. So this is, in some sense, one of the most natural models of the hyperbolic plane. And we will start with this model and now derive from this one other models we have. So the first one, so the next one we want to derive from. It is what is called the Klein model. And for this, we do the following. So here we looked at vectors in R12, but now we can basically take a vector and just think of the line it generates. Every vector here generates a unique line. So we can think of the space of lines in R3, which is a protective space of dimension 2, and we can think of what the hyperboloid gives us in this model. So let's call this B. So this is a space of line, say this is line as RV in P of R12. And now what's the condition we have? So the vector was in a product of itself minus 1. And so we just have to have that if we take, let me, for this, let me write Q of V of V comma Vm, so Q, and we restrict it to L, this is negative. And for every vector in this line, we have negative. So how does this look like? So we can think of this in this picture. So in order to visualize this, we pick some affine chart. So how this looks like will depend on which affine chart we pick. And one of the simplest affine chart we can pick is the plane, which goes to 1 in the first coordinate vector. And then if we project, so we project this hyperboloid from 0, so what we get is actually a ball. But if you take a different plane, for example, 1, which is tilted, we're not get a ball, you will get an ellipsoid. So now when we look at this model of the hyperbolic plane, so how do geodesics look like? If we take what we know here, geodesics are intersections of planes through the origin with the hyperboloid. How does the intersection of a plane through the origin looks like with this thing? So what is it? Anyone? It's just a straight line. So geodesics are just straight lines. So how does the metric looks like? And so in this model, it's trickier to write down the Riemannian metric. One so it's easier to write down just the distance function. So let me write down the distance function for you. So if I have two points, p and q, I want to compute what is the distance between p and q. So what I can do, so I can take the straight line, I'll span by p and q, and it will hit the boundary into other points, p bar and q bar. So whenever I have two points, I take the line, I get four points on the line, and four points on the line have an invariant, which is called the cross ratio. And now I just write it down first, and then we will actually go and explain the cross ratio. So the distance function is I take the cross ratio of p bar, p, q, q bar. I might take the absolute value, take the log, and take the absolute value, and that's the distance, and q. And this thing, which you can define without talking about the Riemann metric, something will, if you compare it with this model of the hyperbolic plane, will precisely give you the hyperbolic distance between two points. So this gives the hyperbolic. So let me now tell you, so now we take a digression, and I'm not going to discuss the other two models I want to discuss, but we are going to say, I will give you the definition of what the cross ratio is and give you some properties of the cross ratio. And for that, we actually want to work in a slightly, it's a bit more generality. So we want to work really on C, and if you want to see the extended complex plane, so C hat, or to C union infinity, or you could also think of it Cp1, so as a complex productive space, so lines in C2. So let's define the cross ratio. So the cross ratio is an invariant of four points. So Z1, Z2, Z3, Z4. And then you basically want to write down an invariant which makes it have as big of a group of symmetries as possible. And to that, so find it as Z1 over Z2, write it as Z1 over Z3, Z4 over Z3 divided by Z4 over Z2. So this makes sense whenever these are distinct and non-zero. And you have to be a bit careful when you look at cross ratios, so there are different conventions of how to normalize. So you can do any permutation of Z1, Z2, Z3, Z4. And then there's another thing you could do in the exercises and see how does the cross ratio look looks like after permutation. But I mean, whatever you pick is basically fine. And I take no guarantee that the one I picked here and the one I picked there actually are compatible. OK, so we have the cross ratio. And this cross ratio plays an important role. And the other thing which plays an important role is the group of symmetries of the cross ratio, of transformations of C infinity or Cp1, which preserve this cross ratio. And that's the group of Mobius transformations. I also want to introduce those here. So Mobius transformation is a map, say F from C hat to C hat, given by a fractional linear transformation. So given by Z goes to A Z plus B by C Z plus D, where A, B, C, D are complex numbers. And you want that A, B minus C, D is equal to 1. OK, so you could also just have it's non-zero. It's also fine. OK, so how can you think of this group of Mobius transformations? It's very convenient to think of it as a matrix group. So we will. Question? No? Yeah? Is it really A, B minus C, half B minus C? Sorry, A the determinant. OK, so how do we think of it as a group of 2 by 2 complex matrices? So one nice thing to think about is if you think of Cp1, the point in Cp1 is a vector in C2. So if you take a vector in C2, you can take a standard basic E1 and E2 of C2. And now you can write any vector which is not E2. You can write as a vector of the form Z1. Sorry, any vector which is not E1, you can write as a vector of this form. And now if you act with a 2 by 2 matrix A, B, C, D, how do you act on it? So you send it to Ac plus B, Cz plus D. But now we want to rewrite it somehow in this form. So this is precisely A z plus B divided by C z plus D1. And so we think of the group of Möbius transformations as the group of 2 by 2 matrices over the complex numbers with determinant 1. And you actually have to model. And I mean, you see this here. If you take minus the identity of minus 1, minus 1, and act the same way as the identity. So we'll actually look at it's actually PSA2C. So it's PSA2C quotient by plus minus the identity. OK, so let me just give you two facts about the group of Möbius transformations. So that's one word about also about having this as a group. So of course, with this definition you have to check that if you take twice and write two fractional linear transformation, compote them, you can rewrite them as one fractional linear transformation. And if you do that, you realize this is just matrix multiplication in this group. So one fact, and this is again something I'm not going to prove, but you can do as an exercise. So if you look at the Möbius transformations, this Möbius transformations are conformal. These are maps which preserve angles. So you have to check this on the derivative of the map. So Möbius transformations are conformal and actually map circles to circles. And so by circles, I mean generalized circles. And one way to see the second property is, and I'm not going to write it, but it's one of the exercises on the exercise sheet, to check two things on Möbius transformations. So first, that if you have a Möbius transformation and you apply it to all the four points of which you take the cross-racial, you don't change the cross-racial. So the Möbius transformation preserves the cross-racial. So this is group of symmetry. And then the second thing you have to check is that being four points on the circle is actually a property you can express with the cross-racial. And then you get this. OK, so this was a short digression. And we might come back to Möbius transformation. We will definitely come back to this group PSI2C later. But now I want to discuss the two other very common models for the hyperbolic plane. And these are probably the models all of you have seen already, the Poincare disc and the upper half plane. So to get the Poincare disc, we do something similar as what we did to get the Klein model. So we want to use a projection of the hyperboloid to a plane. But we choose a different plane to protect them. We choose a different point from which we project. So let me draw a picture of the hyperboloid here again. So here we have 1, 0. Now we take minus 1. And now we basically project the plane 3, 0. And we project the hyperboloid by taking a ray from minus 1 to a point on the hyperboloid. And we look at where it intersects this plane. And when you do that, you will get, again, disc, actually the interior of a disc in this plane. And we think of this two plane as a copy of C, so the interior of the disc, so point of norm. It's more than one where this plane is just, I mean, if you think of the Minkowski scalar product restricted to this plane, this is just the Euclidean scalar product. OK, so if you look at the Poincare disc, you can look at how do geodesics look like. So this is something which is probably already. So geodesics are circular arcs, which are perpendicular to the boundary of the disc. So they come in, in some sense, two flavors. So we have circular arcs through 0, which are circular arcs of infinite diameter. So these are just straight lines, or there could be circular arcs of finite diameter. One reason why this model is one which is used quite often and is very nice is that you can write down the metric on this Poincare disc in terms of the Euclidean metric on C2, where you just have to rescale the metric depending on which point you have where you are in the hyperbolic plane. Let me write it down. So if you want to look at the Riemannian metric, this is the hyperbolic Riemannian metric at the point. So this is the just Euclidean scalar product at the point. So E, so at the same point, but the Euclidean scalar product is constant, so I don't write the point. But now with a factor which depends on the norm of your points, so you normalize your Euclidean scalar product by a factor which gets bigger and bigger, the closer you go to the boundary. So if you look at how things look in the Poincare disc, if you are at the point zero, everything basically looks like you're in Euclidean space. And the more you go out to the boundary, the smaller things look for your Euclidean i being the same size in the hyperbolic plane. So if you take something which hyperbolicly doesn't change, but you walk it out to the boundary for your Euclidean i, it becomes smaller and smaller and smaller. And the boundary of the disc is infinitely far away from every point in the interior. OK, so this metric is once called conformal to the Euclidean metric, so it just means that at every point it just varies by a positive constant. And this has the consequence that if you measure distances, they might be different, but if you measure angles in the Poincare disc with respect to this metric, it's precisely the same as measuring it with the Euclidean distance. And so this makes it very nice. And you could, for example, you could use that to show one of the nice properties of the hyperbolic plane, which you can try to explain to any school kid who sees Euclidean geometry, and that if you take a triangle in the hyperbolic space, the sum of the interior angles is not 180 degrees, as we learn in school, but it's actually smaller than 180 degrees. So if you take a triangle formed by three geodesics, we would have here this circular arc. And clearly, this angle is smaller than when you take the straight line, some 180 degrees. So let's, so how do I get the top board between? Oh, sorry. So now let's just take the last model, so the upper half plane. Model and that's the second. And I would say the Poincare disc and upper half plane model is probably the most common ones you see. So here, you take also, I mean, you think of it as a subset of the complex plane. Now take points where the imaginary part is positive, imaginary accessing this subset of the hyperbolic plane. And you can, again, write the metric as a conformal metric just using the Euclidean structure, where now you have to normalize by the size of the imaginary part. And think of this real line as being infinitely far away. So you write down the hyperbolic metric as 1 over imaginary squared, the Euclidean scale up. OK, so you have here, again, that geodesics are circuits perpendicular to the boundary. And one way to see that is to write down an explicit map between the Poincare disc model and the upper half plane model, which is usually called the Cayley transform. So this is a map, takes a point in the disc, send it to the upper half plane. And you can write that explicitly as, how do I do it correctly? So that minus i over that plus i. So you should check again. Make me, let me make one comment. So if you remember our fraction linear transformation or our Mubius transformation, this is a Mubius transformation. There's just one specific Mubius transformation. And this tells you that there's something wrong. But this tells you that if you, since Mubius transformation sends circles to circles, that whatever property you have on the Poincare disc actually holds on the upper half plane. OK, so let me now make one last comment about this one last thing I want to point out of this model of the hyperbolic plane. So if you want to understand what is the group of isometries of this space as a Romanian metric, actually it's very useful and convenient to do this in this models. Because you can basically think of trying and having to check certain things. I mean that, for example, geodesics are sent to geodesics. And when you, OK, distance I have to preserve, but when you try, no, you can very, not much difficulty, check that the group of isometries of both the upper half plane or the Poincare disc is precisely the group of Mubius transformations which preserve your set. So the group of isometries of H with respect to its Romanian metric is a set of Mubius transformations of C hat. I mean think of this as sitting in C hat such that F of the upper half plane is actually the upper half plane. And then you have to check basically what do you need to preserve the upper half plane. You have to preserve its boundary, which is the real line. So the real axis. Just have to look at Mubius transformations preserving the real axis. And if you look at it, which elements of these two by two complex matrices of determinant one preserve the real axis, get that this happens if and only if the coefficients are real. So you will get that this is PSI to R. And if you do the same exercise with the Poincare disc model, you get a group which, of course, I mean these are equivalent Romanian manifolds. You get a group which is the same group of isometries, but you will get it in a different way embedded into the group of Mubius transformations. You will get it embedded as what is known to be the group of PSU 11. And I'm not going to. But it's just some of you have different conditions on the coefficients of the matrix, which are not that they are real, but that you have some certain another involution which has to be preserved. OK, so this is basically all I want to say about how the hyperbolic plane looks like. I just want to make two or three comments which I'm not going to be precise about. And you might see this in other things. So there are certain combinatorial properties if you want to properties of the hyperbolic plane, which you can abstract from. And you don't have to think of Romanian manifold anymore, which capture a lot of the hyperbolic behavior of the metric. So one is the growth of balls. So if you look at balls of radius r, and you let r go to infinity, if you are in Euclidean space, they grow polynomial. If you are in hyperbolic space, the volume of balls, depending on r, grows exponentially. And there's one important property of hyperbolic space, but which you can also generalize to other spaces, even discrete spaces if you want. The other is that we saw, I mentioned here, the sum of angles, inner angles in a triangle is smaller than 180 degrees. So to talk about this, you have to have a notion of angles. But you can also characterize that some triangles in hyperbolic space are thin and thinner than in Euclidean space. And this is, again, a property you can abstract from the setting of Romanian manifolds and think of it as an arbitrary metric space or a graph. Even so, these are interesting properties in this, especially some of these properties. Currently, there's a whole group of people who are not mathematicians interested. So there are quite a few people in machine learning who start to look at hyperbolic geometry as a tool to do embeddings of graphs in a better way than doing that in this Euclidean space. And one of the key features there is that you have this exponential growth of balls. But I think it's really nice that it's a way to get other people interested in hyperbolic geometry and also to see that, actually, it's useful in aspects where you didn't necessarily think about when studying hyperbolic geometry. OK, so now we have the hyperbolic plane. Now we want to look at how can we endow a surface with a hyperbolic structure. So let me just write to a hyperbolic structure, a surface. So the first thing is, and this is in some sense the way we thought about it right in the beginning, is to create an atlas. So to take charts on your surface and put them in the hyperbolic plane and have the coordinates or transformations are isometries of the hyperbolic plane. So put this in it. So I don't want to, I mean, this is a very nice setting and really fits nice in the theory of differentiable manifolds. But in some sense, if I give you a surface, it's not so easy to give me an atlas. So I don't want to say anything more about that. So there's a second thing you could think of. And this is, if you think of the, if I give you a torus, how do you put a Euclidean structure on the torus? So in some sense, one of the easiest ways to describe a Euclidean structure on the torus is to say, well, I take a sheet of paper and I glue my torus this way. And another way to say that is to say, I look at R2 and I look at just the integer grid on R2. And then I look at the symmetries of that tessellation of R2. And then I mod out R2 by the symmetry. So I identify the quotient, I can identify the quotient with this fundamental piece. And if I act by translations of the grid, I mean this side is identified with this side and this side is identified with this side. And so in this way, in some sense, in order to give you a torus, a Euclidean structure on the torus, I can give you a tessellation or I can give you the symmetry group of a tessellation. And so the same thing we can do to get a hyperbolic structure on a surface. So we have H2. We want to find the tessellation of H2. And then we're just think of, forget about the tessellation, just think of the symmetry group of this tessellation. So we can think of tessellation or its symmetry group. And so this means if I just think of this symmetry group, so this is a subgroup of the group of isometry. So this would be subgroup gamma in the group of isometries, which we saw as PSI2R. So you can also think of giving a hyperbolic structure by giving a subgroup of PSI2R, which is discrete, which will actually be a lattice, because in the end we want a compact surface downstairs. And of course, for this group to really give you a surface, this group will have to be isomorphic to the fundamental group of that surface. And let me just write this on the surface SG. So how does the fundamental group of that surface looks like? So I can pick a base point. And then now basically I do the same as what I do for the torus. For the torus, I take two loops, which go around, somehow one as the meridian, one as the longitude. And so here I do that for every torus I see. So I have two curves for this handle and I have two curves for this handle. And so I can think of the fundamental group as being generated by pairs A1, B1 up to AG, BG. So a pair of each handle. And then I have to have one relation, which is if I go around these things, I end up at the trivial loop, which gives you a relation, saying that if you take BI and you take the commutator, so this is AI, BI, AI inverse, BI inverse, then the product of all these commutators has to be the trivial element. So this is another way I probably saw things about discrete subgroups and lattices already. I might come back to this. I will come back to this in the next hour. But now I want to describe you a third possibility. And this is to somehow try to take your surface and cut it into very simple pieces and give a hyperbolic structure on each of the simple pieces and then glue a hyperbolic structure on the surface out of the pieces. So yes? No, it has to be discrete. OK. And can you find subgroups a lot? Yes. OK, so the other is cut into simple pieces and try to put a hyperbolic structure on the simple piece and then glue it back. And for this, there's a nice tool. So basically, the way to do that is to cut a surface and cut it into what people call pairs of pans. So take Sg and cut it into pairs of pans. So what's a pair of pans? A pair of pans is this thing with three holes, right? And how do you cut a surface into a pair of pans? So you have to pick enough disjoint simple closed curves on your surface that the complement you see is just pairs of pans. So here we have one pair of pans and we go take this one and this one. And there are many different choices. Take a different choice. So what's now the key point? So we take such a pair of pans and we want to understand how can we put a hyperbolic structure on this pair of pans. And now we make it even simpler. So we cut this pair of pans. And we can cut it into two hexagons by taking a perpendicular from this curve to this curve from this curve to this curve and from this curve to this curve. So think of starting with a surface which has a hyperbolic structure. We want to decompose and understand how is the hyperbolic structure? How does it look like on the piece? And then we use this information to somehow build a hyperbolic structure on the surface where we don't have one. So if we have a hyperbolic structure, we take these geodesics and there will be, this is topologically always the same number. So there will be 3g minus 3 pairwise disjoint simple closed geodesics. If you cut along, then you get 2g minus 2 pairs of pans. And then we can take each pair of pans, take the perpendicular between two of its boundary components, and we cut it into a hexagon. OK, so if we start with a hyperbolic surface, so surface with a hyperbolic structure, each of this curve has a length. So when we cut, each of these curves has a length. And when we cut it into the hexagon, each of these curves, which correspond to the hole in your pair of pans, has a length. Namely, it's half the length of the simple closed curve you had. So A, B, C. And now there's a nice fact of hyperbolic geometry that if you look at right-angled hexagons, where you know three of the side lengths of the somewhat distributed in this pattern, there's actually a unique way, up to isometry, how you can construct such a hexagon in the hyperbolic plane. So there is a unique hexagon H2 with a unique right-angled with these side lengths. And this is, again, now makes it much easier. We start with a complicated object. Now we have really a small piece of hyperbolic plane. And this is, again, something you could prove in the exercise that you have. So this is elementary construction in the hyperbolic plane. But now, when we have that, now we can go backwards. So if I want to find a hyperbolic structure on the surface, what I do, I have my surface which has no geometry yet. But I can, I mean, it has some topology. And I can topologically decompose it into pairs of pans by just taking simple closed curves, not necessarily geodesics, because I have no geometric structure to talk about geodesics. But I cut it just topologically in these 2G minus 2 pair of pants. And I think of each pair of pans being decomposed into 2 hexagons. And so now whenever I specify for each pair of pans this length, side length, this side length, and this side length, I can construct these two right-angled hyperbolic hexagons. And I can glue them together. And I get a hyperbolic structure on this pair of pans with the prescribed side lengths, L1, L2, and 3. So given L1, L2, L3, any positive real numbers, there exists a unique hyperbolic pair of pans with these side lengths. So now I have my building blocks. I have these hyperbolic pairs of pans. And if I want to get a hyperbolic structure on this more complicated surface, I take just as many pairs of pans as I need. And I build it. I just have to make sure that if I take this pair of pans, I take one where I cut it here, these two curves have the same side length. And if I want to glue those two, I just have to make sure that these two have the same side length and so on. So if I basically describe you the 3g minus 3 curves, length parameters, I can build a hyperbolic structure, which has, for this given pans decomposition, these side lengths for the simple closed geodesic realization. OK, so this is one way to construct the hyperbolic structure on the surface. And what I want to discuss after the coffee break is in some sense how you can use this way of constructing and giving a hyperbolic structure to also understand how the space of all hyperbolic structures on a surface looks like. And let me just make one last comment here. So you might be interested in surfaces which are not closed but which have boundary or punctures or something. And there's actually another very much related way of giving a hyperbolic structure on such an object is that if you have this, say, just one mark point, for example, on the surface, you can decompose your surface not into a pair of pans, but into ideal triangles. So you can make a triangulation of your surface where all the vertices of the triangles are in your specified mark points. And if you do that, you can think of cutting your surface into some ideal triangles. And if you think of how does a hyperbolic structure on an ideal triangle looks like, so how does a hyperbolic structure look like on a triangle where all the vertices are at infinity, you realize that actually in the hyperbolic plane, there's up to isometry, again, a unique ideal triangle. So you don't specify anything because all the sides have infinite length. So on each of these triangles, you have a unique hyperbolic structure. And now you can think of the hyperbolic structure on your surface, again, as being glued out of these triangles. And so you have to understand how can you glue two triangles and what is the parameter to glue two triangles. And there is a nice, if you look at this picture of the hyperbolic plane. So if you have two triangles, you want to identify one of the edges and glue another triangle to have an ideal quadrilateral, you see four points on the circle. And the cross-racial of these four points on the circle, where again, you think of the circle as a line or extended line r and infinity, is precisely invariant of how you can glue two ideal triangles. And there are two things which are, I mean, one thing which is very difficult for pairs of pants is if you have one pair of pants decomposition and you change to another pair of pants decomposition, how do lengths and things change? And this is something which is much easier for gluing things out of triangles. So there's very nice formulas and very nice structure. If you think of how does the cross-racial change of these four tuple of points, when you somehow think of the four tuple of points thought in a different way, so one of the symmetries of the cross-racial, which gives a lot of nice structure for surfaces with puncture. OK, so let's have a coffee break.