 Okay, anybody interested can come in and sit down, we'll start the second lecture. So the first lecture was somehow a motivation overview of the field. And as I said, what we are going to do now is give you the foundations to understand some of the topics in the first lecture. We will not talk about all the applications. We will not talk again about the Oppenheim conjecture, but we'll talk about analog, more simple to comprehend things. And so in this lecture, we will talk about the geometry and dynamics of one of the simplest examples of a league group that is important in the applications, and that's SL2R. So we talked about translations, that was of course a very simple example, in particular because it's a commutative group. This is a non-commutative group, and as you've seen in the lecture, there are end dimensional versions of this. So just to fix the notation GLNR is the group of invertible N by N matrices with real coefficient SL and R are those matrices with determinant one, and of course SL2R is a special case. So today, we'll talk about the case N equal to two, and we'll deal with this group in general in such a way that you will see, not see, but that will allow you to generalize these things to those higher ranked groups, and this viewpoint is that I'm going to present is particularly useful for that. So SL2R is then more explicitly, of course, two by two matrices where the coefficients are real, and let me just write out what the determinant is here in this case. It's just this. So this group will be the main character in this lecture, and it's very intimately related to hyperbolic geometry. So what is hyperbolic geometry? Hyperbolic geometry is a geometry like the Euclidean geometry or the spherical geometry. And it is the hyperbolic plane in two dimensions. So if we look at two dimensional hyperbolic geometry is a model for the hyperbolic plane, the analog of the Euclidean plane. And it can be realized by the upper half plane, sometimes also called Poincare upper half plane, or Lubashevsky plane, depending on where you have your office. And that's the space, and we equip it with a Riemannian metric that makes this a Riemannian manifold. So I'm going to write every point in this complex upper half plane as x plus i, y, x and y are real variables, y of course is positive. And this is the element. You can also write it in terms of the quadratic form in the tangent space that gives you this line element. So the Riemannian metric for two vectors, tangent vectors, in the tangent space at z, you can write this metric as v dot w divided by imaginary part of z squared, where this here is just the standard Euclidean inner product. So you have this funny scaling factor here. And so let me make a picture. This is a picture of the upper half plane. Here's the point i, and we look at all these points. Now this metric gives you a distance, right? That's what it means. So you get the distance between two points by looking at curves that connect these two points. You can calculate the length of this curve. And then you can minimize it. And if you do that, you see that the curves that are minimal, these are called geodesics, are actually semicircles that are perpendicular to the boundary. Or if you have two points that have the same x-coordinate, the curve will just be a straight line. So this is how geodesics look like in this Riemannian surface. It's infinitely extended. And it's a beautiful geometry, of course, and you have to switch off your Euclidean mind here because points that have a small y-component can be very far away, because you see when y is small, the line element will be very large. So small Euclidean distances here might not be small hyperbolic distances. Okay, now how does SL2R come into this? When you have such a beautiful geometry, you can ask, well, what if I apply transformation to this picture, say, coordinate transformation? And you want those transformations that leave the Riemannian metric invariant. These are called isometries. And the group of isometries is precisely given by a particular action of this group SL2R. And that's why SL2R is so important for the hyperbolic geometry. So SL2R acts on H by fractional linear transformations. So this is a map from H to H, where you take a point in H and you map it to something I call JTZ, which is simply defined as AZ plus B over CZ plus D, if T is a matrix of this form in SL2R. Okay, these are called fractional linear transformations or Möbius transformations. And note here that the imaginary part of JTZ is given by Y over CZ plus D squared, exercise. So as I go along and you get bored, just do these little exercises. Think about if you have any questions for the tutorials. We'll give you in the tutorials a few problems that are related to this. Now, I talked about isometries. So why are these fractional linear transformations related to the isometries of the upper half plane? Okay, so we'll define isom plus as the group of orientation preserving isometries H. And by H, I always now refer to as H as a Riemannian manifold, so equipped with this Riemannian metric. And then there is an important classic statement that says that if T in SL2R, then JT is an isometry. And every isometry can be obtained in this way. So if F is an isometry, then there exists a T in SL2R such that F is equal to JT. Okay, now let me show you how this comes about, this theorem. I'm going to sketch this. We've given you a little bit of a reading list and there is a very nice book by Einstein and Ward that covers this. In fact, we can give you a slides of the pages and PDF of the relevant sections if this is according with UN law. The title, it's on the reading list. So have a look at the reading list. If I write the title, I will finish the lecture too late. Okay, so just as a notation, it's sometimes not so convenient to always write JT. I'll often write just T of Z or even TZ and that will always mean, or sometimes of course, because it is a matrix, I'll just write it like that. And this is understood as this action, CZ plus B over CZ plus D. And this, as I said, defines a group action which you can check. So this means that if you apply first JT2 and then you apply JT1, that's the same as applying J of the product of the two matrices. So what this means is that the map that associates with every matrix in SL2R, a fractional linear transformation JT is a group homomorphism that's onto. So it's surjective and I think these are, I always forget, epimorphisms or whatever. So it's an onto and that what you can find out is that the kernel of this, so you take a matrix A, B, C, D and you associate with it this fractional linear transformation. If you change the matrix A, B, C, D by minus A, B, C, D, so you just put a minus sign in front of all the coefficients, you get the same fractional linear transformation. And that's the only way you can get the same. So that means with kernel plus minus one. So this means that the matrix one, you can check it, right? Matrix one gives the identity transformation. Matrix minus one also gives the identity transformation. So that's the kernel and there's no other matrix that gives you the identity transformation. So here I mean identity matrix. This is identity matrix, yeah, it's a good question. Two by two identity matrix. And so this means that if you remember your group theory that now we have an isomorphism between the group of fractional linear transformations, the group isom plus, so of H with the group SL2R modulo the kernel, right? And we call this group, we give it a special name, PSL2R where P stands for projective. So what this means is that the group that we have indeed here, what this theorem really says is that the group of isometries is isomorphic to this group PSL2R. And one of the things we can talk about in the tutorials is how you can prove this. So we'll give you a long list and you can pick and choose, okay? So we don't expect you in the tutorials to do all the problems we write down on the board. We want you to pick a problem that you fancy and then you can try it and we'll go around and we'll help you if you get into difficulties. Or even before that we can help you. The next important observation is that you have a nice way of writing every group element in SL2R and therefore also in PSL2R in a very nice way and this is very useful for what we are going to do. So let me just see if I skipped something here. And let me, sorry, write first what the theorem is called. This is called eva-zava decomposition. And let us define the following subgroups of SL2R. Some of them you've seen already. So let A be the subgroup of diagonal matrices and capital N the subgroup of these upper trigonal ones. So these are called unipotents. These are not called unipotents. These were examples of unipotents in the first lecture. And then the group of rotations to, so these look like this, cos phi minus sin phi, sin phi cos phi where phi, well phi goes between 0 and 2 pi and because it's periodic let me write it like this in the way you've already seen. So I'm taking R but then modulo 2 pi z. And now what the theorem says is that every element in SL2R can be written as a unique product of these three matrices. So for any G in SL2R there exists a unique Nx, or just say N in N, A in A and K in K such that G is equal to N, A, K. Why is this useful? Well what's the dimension of SL2R? Because we have four parameters and one condition, the determinant being equal to one, so this means it's a three-dimensional space. And this decomposition here gives a nice parameterization of this three-dimensional space. Each of those elements comes from one parameter subgroup with a very nice structure. And so this is a very good way of putting a coordinate system on your league group SL2R. So for every T, X and Phi I get a point in SL2R and for every element in SL2R I can uniquely read off the T, the X and the Y. And the Ha measure on SL2R can be written as so if I have an element like this G and X, A, T, K, Phi then the Ha measure has a very nice formula which is e to the minus T dx dt d Phi. Because remember our eventual goal is to do equidistribution theory for orbits on this space and the Ha measure will be the space with a measure with respect to which these orbits can become equidistributed. Now if you remember the Riemannian element that I've just erased of our hyperbolic metric, it was this, and you can read off the Riemannian volume or area element for the hyperbolic plane. So this is d mu of the hyperbolic plane. It's just dx dy over y squared. And if you look at this equation here and you set y equal to e to the T, you see you get exactly dx dy over y squared times d Phi. So that's the nice thing here. You somehow see the Riemannian area element appear in the Ha measure and this is not a coincidence which is what I'm trying to explain now. Why does the Riemann volume element of the upper half plane appear in SL2R? Okay, and in order to do this, let's go back a little bit first to the first part of theorem 1 where I would like to explain to you how the group of isometries acts on the tangent bundle of the upper half plane. So the tangent bundle is simply the collection of all the tangent spaces. So at each point you have a tangent space and the tangent bundle is just the combined space or the bundle where the fiber at each point in the upper half plane is your tangent space. Okay, now what do I erase first? Maybe this thing here. Okay, so this gives a few hints on the proof of theorem 1A. So we will call V a tangent vector at the point Z where Z is a point in the upper half plane. Okay, so there we have a tangent space. That's a two-dimensional space. We can think of these guys living in the complex numbers. It's a two-dimensional space. They are two really, but we just identified with the complex numbers. That will make some calculations easier. So then the Riemannian metric, once I think of these as complex numbers, unless I've made a mistake and Andreas will immediately correct me if I make a mistake or anybody else. Okay, I think you can then write them like this. So this is just the standard Euclidean inner product that we had before. I said these are two vectors, and this is just the Euclidean product, but now I am interpreting V1 and V2 as complex numbers. So this should give just the square of the sum of the coefficients, not the square, the inner product. So if we look at our upper half plane, we have a point here, Z, with a tangent vector V, and now we apply a fractional linear transformation and that will map the point Z to a point TZ here, and it will also change the vector, right? And so I'm going to write that new vector as DT, differential of T times V. That's my new tangent vector. So think of a curve going through this and then you map the whole curve over there. So the new tangent vector will be the tangent vector to that curve that you've mapped. And now you can, of course, ask yourself, well, we already know what TZ is, but how do we have a formula for how the tangent vector transforms under this isometry? And yes, we have. So this is just complex analysis. The tangent vector at the transform point V is simply the differential of T at Z, and you can compute this. And the formula is so pretty and easy, and that's not absolute values. And you get this as the answer for T prime Z. So here's the formula. Now, this is the proof of theorem 1A, right? We want to show that JT is an isometry. So what does JT have to satisfy to be an isometry? The length of this guy has to be the same as the length of V with respect to the Riemannian metric that we're using, right? So let's make it a little more general. Let's take two tangent vectors at the point Z, and we look at the Riemannian metric at that point Z. And remember, this is just written here. And so now what we need to show is that this is the same as this. So this is the question. Is this the same as this? And we can just compute it. Evaluate it now, of course, at the transform point TZ. And you can show this, right? So that's another very little exercise that this is equal to that. That means it's an isometry, but even angles are preserved, right? So this is an inner product. This is actually a quadratic form on the tangent space at Z. And the angle between V1 and V2 is the same as the angle of the transform point. So fractional linear transformations are not just isometries, they also preserve angles. So this is a two-line calculation here. And again, try it in the tutorial, and we'll come and answer questions if there's a query. So I'm not going to talk about the proof of the second theorem, because what I wanted to show you here is that we understand how the isometries act on the tangent vectors. So you have a tangent vector at the space Z. You apply your isometry, you map that to the tangent space at TZ. And this is the formula on how the new tangent vector looks like. Remember, V is now thought of a complex number, right? Very good. Now we want to define dynamics on this tangent bundle. And in fact, the kind of dynamics we look at is the dynamics of the geodesic flow, which preserves the length of the tangent vectors. So you can think of the tangent vector as a direction in which you want to travel, you apply a geodesic flow and then you just follow it, right? Just like in Hamiltonian dynamics, you have momentum and position, you're sitting in some position, you define the momentum and then that specifies the initial condition completely and you can calculate the trajectory. So this is just Hamiltonian dynamics when you talk about the geodesic flow. And so the right space to study the geodesic flow is not the tangent bundle, but the unit tangent bundle. And unit just means that you're looking at tangent vectors that have length one. So let me write down. The unit tangent bundle. It's called T1 of H. In two dimensions also sometimes called unit sphere bundle. So it's the pair of points where Z is a point in the upper half plane and V is a point in the tangent space at Z. So that will be the tangent bundle and the unit tangent bundle has the additional restraint that we only look at vectors of length one with respect to the Riemannian metric. So this guy here is just as usual our Riemannian metric evaluated. All right. Now comes one of the really pretty things in this subject what normally in Riemannian geometry is a subject of geometry, the understanding of geodesic flows and so on. I will now show you how this can be translated in this particular setting to a completely algebraic formulation. And from then on we never need to worry about geodesic flows and so on anymore. We only have to worry about actions of 2 by 2 matrices on 2 by 2 matrices. Fractional linear transformations will disappear from this. Everything will become so much more easy. And the key observation is that you can identify the unit tangent bundle with this group PSL2R that I erased. The group of orientation preserving isometries and this works like this. So here is my claim PSL2R. So we map it to an element in the unit tangent bundle in the following way. So we take a matrix. So remember PSL2R was simply 2 by 2 matrices with determinant 1 where we identified plus and minus 1. So we take such a group element and we have understood the action of PSL2R on an element in the unit tangent bundle or in the tangent bundle here. We understood that these guys are isometries so we've just calculated how such an isometry acts on the vector V. So let's just take this guy and let's just act on the point I and on the unit vector that's pointing upwards. So here's the picture. So this is the upper half plane, here's the point I and this is the unit tangent vector I. That's like somehow the origin in our unit tangent bundle. The simplest point you can think of. And now we just define this map from here to there. Why not? We have an action. We know that this acts nicely on this space. So we just map this point to that point. Now what one can show in fact that this is a bijection. So you can find the inverse. Given any point in here you can find a matrix in PSL2R. And let me show you how this inverse map looks like. And what you're using there is theorem 2, the Eva-Zava decomposition. So here's another claim and that's another nice exercise that you can now do is that we know we can write every element in SL2R and therefore in PSL2R in this way. Log y, okay, phi. So this guy here is simply square root y, 0, 0, 1 over square root y. I don't like the e to the t here so much. For now you will see y. So if you choose this, so every element can be written in this way. And now just a little subtlety here. If we end PSL2R it's not unique because if we replace phi by phi plus pi you get the same element, modulo plus minus 1, right? You see this? And if you don't see it it doesn't really matter for what I'm going to do. So let's calculate this. Let's calculate it. What is this? What is this? And what is that? And if you do that calculation something very beautiful happens. So for this G as chosen here and as I said we can write every G in this way. For this choice of G you get the following answer. X plus Iy comma e to the minus 2 pi I phi, okay? That's the answer you get when you do this calculation. And that's exercise. All you have to do is look at your fractional linear transformation. Apply it to the point I. Hint, I give you the rotation matrix. K phi acts on this point here trivially. Which means it leaves it invariant. So K phi applied to I doesn't do anything. It gives you I. And then all you have to compute is what does this do to the point I? And then what does this do to the result you get from the first calculation? We can talk about it in the tutorial, yeah? Try that. That will give you a real insight into fractional linear transformations and so on. Now on the other hand on the tangent vector the rotation matrix will rotate the tangent vector. And you can just plug it into this formula here, right? C will be cosine of phi and D will be sine of phi. And just try to work it out how it looks like. Yes, yes, yes, yes. Yes, yes, yes. I know. There's probably something wrong in this formula. Divided by whatever. The right power of y. There's something here so there's a question mark. No, no, no, I don't want you to give the, I want to do it in the tutorial. Who will the modulo ask this question, right? Where are you? I recognized your voice, I didn't see you. But it's your challenge for the tutorial. Because I've written it this way, forgetting that of course a unit vector in hyperbolic geometry is not a unit vector in the usual complex plane. So that has to be divided so that that thing here actually becomes a unit vector with respect to the hyperbolic matrix. So that's a good exercise, yeah? Okay. And then on the tangent, sorry, on the tangent vector, you can then see that this guy, the other two matrices act very simply. Let me put it this way. So this is a nice calculation. Okay, so what have we seen here? Is that we now, well in fact what's going to happen is that if we call the angle of our unit tangent vector theta, then theta is related to this rotation parameter by minus 2 phi, right? Because it appears here and this is really the rotation angle. That's for the tutorial to do. It won't help you if I do the calculation for you. That's something you can do. And what have we learned now is that, well, if you give me a point in the upper half plane and a direction, a unit tangent vector, then I can give you a matrix G. Namely, I just read off here the point where we are. That gives me an X and a Y and you give me a theta. I compute the phi and then I know what G is. It's just going to be this thing. Okay, so that's why it's an isomorphism between the group PSL2R and the unit tangent bundle of H. Right. Now we have the setting on which we work. We want to define now the geodesic flow and the unit tangent bundle, but since we've now identified it with this very nice and simple group of 2 by 2 matrices, I can forget about the Riemannian geometry completely, okay? And I'm just going to define the geodesic flow in an algebraic way and then claim that it actually produces the Riemannian geodesic flow. Right. And of course, I can't go through everything because you can do it. So the geodesic flow is a solution to the Euler Lagrange equations and what the isometries help you to do is to start with a very particular solution of the Euler Lagrange equations which is very simple and get all solutions by just applying the isometries. So what do I mean by this? Let me again make a picture of the upper half plane and let's see. So I'm not going to solve these equations. I'm just going to draw pictures and I'm going to use some colors now. I can find some colors. No colors. So let's see. When we have a, let's suppose this is our starting point and we want to look at the geodesic that emerges from this point in direction i. What you can believe me if you solve the Euler Lagrange equation or if you even just solve the variational principle, you'll find that it's this curve. That's the curve that it is. And if you're moving with unit speed after time t here, you'll be at the point e to the i times e to the t. That's where you'll end up with and t is the time. And what you will see is that if you just remember your hyperbolic metric, line element, oh god, b x squared plus d y squared over y. If you calculate the distance between those points, it'll be, well, this doesn't appear and you just integrate, you know, from here to there and what you will get is that the distance, the hyperbolic distance is just t. So everything's consistent, okay? So that's a geodesic arc from going here to there. The infinite geodesic that you get by moving t to plus and minus infinity will just be this line up here to that. So that's something you can believe me properly, probably. Now let's write this and furthermore, the vector v is invariant, okay? So if I draw the resulting unit tangent vector up here, it will be the same. So here is direction, we're going in direction i. And here will be simply i again. Not quite, right? We need the right normalization factor, which I'm not just guessing. Is it e to the t or something? I don't know. Yeah, so that will be a unit vector in the tangent space with respect to the Riemannian metric, hopefully, okay? But you can check that. It has to be a unit vector. So in effect, it remains the same vector. It's just the length is distorted because of the choice of the metric. That's a geodesic flaw. Represented in the unit tangent bundle description, right? And using theta or using the vector v. Now how will this flow look like in this geodesic in the description of 2 by 2 matrices? Well, I'll just start with the identity matrix and then I will apply a t. So this will give me the matrix e to the t over 2, e to the minus t over 2. So I'm declaring that this now is the geodesic flow applied to the point i in PSL2R, the identity, and it's just defined in this way, okay? Now what you have to check to make sure that I have done this consistently with this picture is you have to go back and forth with this isomorphism. Have we got a name for this, Andreas? I don't know. I think you call it i or something. Okay. You will use it? You won't use it. Okay, let's just call it a tilde because we won't use it anymore. It's probably a bad letter. So you now just have this commutative diagram between PSL2R. We're acting with phi t. Then you have this a tilde. Well, let me call it capital psi. Oh, sorry, yeah. Okay, let's do it here. Let me first write another thing down. So this is just to illustrate that this is this geodesic under this identification. And then more generally, you can now define the geodesic flow as a flow on PSL2R to PSL2R by starting with a general group element and then simply multiplying from the right with the flow phi t. That's now the definition. And here I'm just choosing a very particular initial matrix G. That will lead to this geodesic. Oh, 80, I'm sorry, 80. And the claim is, so let me call this psi because this is like a conjugacy between dynamical systems. The claim now is that this bijection psi gives a conjugacy between this algebraically defined flow on 2 by 2 matrices and the geodesic flow. And one way to simply illustrate that that makes sense is just to look at this particular orbit here and see where the orbit of this 2 by 2 matrix curve is mapped to here. And my claim is it's going to be exactly mapped to that geodesic that I showed you. And the same is true for any other geodesic. So if you start with a point here and a tangent vector here, then you'll get this geodesic that will hit the boundary at right angle. And you again will move along this curve for time t. So if you take this point Z and vector V, you find the matrix G in the procedure that I explained. You flow, you'll get exactly this curve. And if you think about it, this curve to describe algebra in terms of coordinates here, of course it's a circle, but you still have to write quite a lot, right? Here I'm just saying you take a 2 by 2 matrix G, you just multiply 80 from the right and that's a simple calculation. That's really just linear algebra. Well, here you're doing already some geometry. You have curved curves. So that's the big advantage. So I've explained to you now that the unit tangent bundle and the geodesic flaw in the unit tangent bundle can be realized as just my matrix multiplication in SL2R. When shall I stop? Quarter past? Yeah? So now the next important... That's the calculation, yeah? That's the matrix that gives you exactly distance t. If I would have put t in, then I would have got 2t as a distance. It's a little bit like with a rotation. There was not so clever. With a rotation I put rotation by matrix with parameter phi and I found out the phi is actually half of the actual rotation angle in the half-a-half plane theta, right? Actually minus theta over 2. So it's just how you relate the geometry to the matrix is just making a clever choice and here I've just made that choice initially. Otherwise I had to half, half that too. Okay. The next important object and the dynamics of the geodesic flow are horocycles. And horocycles, for those of you who are in dynamical systems, are basically the... parametrize the stable and unstable directions for the geodesic flow. But as we will see, the horocycle flow is interesting in its own regard, in particular because for it you have the measure rigidity statements and the horocycle flow appears in many important applications and it's higher dimensional generalizations too. So in the last few minutes, let me just introduce the horocycle flow. And now the picture is the following. So, so horocycles are lines that are either parallel to the real axis or, I'm making a big picture here, or tangent to the real axis. I view this line here as a circle tangent to infinity, if you like, okay? And hyperbolic geometry, straight lines and circles are the same thing, in a sense. As you've seen here, geodesics can be straight lines or circles with a certain intersection property of the boundary being perpendicular, horocycles are the same, just that now we are not perpendicular to the boundary but tangent to the boundary. And there are isometries that can map this to this line as well as there are isometries here that can map geodesics to straight lines to circles. And fractional transformations always map circles to circles or straight lines and straight lines to circles or straight lines. And so the meaning of horocycles dynamically is that if you look at the point of tangency here, there is all geodesics that emerge from this point will intersect this horocycle at a right angle. So that's a geometric feeling. Now here, the point of tangency is infinity, so all the geodesics that run to infinity are these straight lines and you see the same thing is true here as well. And now if you think about the geodesic flow and you remember the way the Riemannian metric worked is that if you flow along this line and you compare the distance between these two, they will come close at an exponential speed in terms of the geodesic flow parameter. And this is exactly the same picture except that now this point plays the role of the point of infinity. All these will come exponentially close. And so this is in some sense the meaning that this horocycle here parameterizes the stable direction of this horocycle flow. And similarly here, these horocycles parameterize the stable direction of everything that sort of flows into this point. Now on the other hand, if you flow in the opposite way, if your geodesic flow runs down here, these points will diverge at an exponential rate. So the horocycle with a vector is pointing downwards parameterizes the unstable direction of this family of geodesics. Okay, and now I'm just going to define the horocycle, these two horocycle flows, stable and unstable in terms of actions on PSL2R. And then that will be it. So the horocycle flow. So here this is what I call contracting. There you take, that's going to be called HT. You take an element G in your, so we again, we define this flow exactly in this way. So this, yeah, so this was the definition of phi t. Now I'm just going to define HT again by right multiplication. Now by the matrix, oh sorry, HT, NT where NT is 1T01 and the expanding is just given by H minus GNT minus, where NT minus is 1T01. Now if you again do the same exercises for the geodesics, you will see that, this you see very simply here that this guy here just corresponds to a point starting here, let's say, and then it's just shifting in this direction. So this is vectors pointing upwards. This is why this is the contracting horocycle flow. If you look at this matrix and you start at the point I, you get exactly this picture. So pretend I is here, that's the point I. Then you flow along these lines and you see the equation for this circle you have to compute and so on. So in Riemannian terms it's quite complicated. Here again it's just matrix multiplication and you can convince yourself that indeed this action projected onto the upper half plane gives you a circle if you apply the fractional linear transformations. Okay, so I think I've talked enough. What I didn't cover, what I really wanted to do is to also talk about now quotients by a lattice. So if you instead of looking at H, you look at the coset space H mod gamma and this models hyperbolic surfaces that can have finite volume. And there you can define the geodesic and horocycle flows exactly in the same way because you can identify the unit tangent bundle of these surfaces provided gamma is torsion free with PSL2R modulo gamma. And now gamma acts from the left on this group. The geodesic and horocycle flows act in the same way from the right, right and left multiplication commute. So this is then a well defined flow that you can take from your upper half plane and project it onto this quotient space. And this is exactly our example of a first homogeneous space if you remember the first lecture. Okay, so now we have a setting where we can talk about geodesic flows, horocycle flows, but in an arithmetic framework. Everything becomes much easier in that framework. In particular, all the theorems about measure rigidity are now much, much simpler to formulate and don't exist by the way in sort of many variable curvature situations as far as I'm aware. Okay, and the other thing that I missed is to talk about the modular surface but I think we can come back to that and define that. I don't want to do that in a rush, okay? So we'll have tutorials this afternoon with everybody with all the other lectures together and then tomorrow Andreas will continue with talking about what? Ah, about equidistribution. So then Andreas will talk about geodesics becoming uniformly distributed, horocycles becoming uniformly distributed, and so on. Okay, thank you.