 I wanted to finish the thing that I promised I would prove at the end of last lecture. It was this theorem that was on the blackboard by the time the hour went out. So you remember what Titanium Space is from Dick's lecture. It's this space of hyperbolic metrics on the surface that remember how they want to sit on the surface, like close, by the way, Dick. This is a fantastic metaphor, the thing with the child that you have to put in each other. Anybody who has had to do this to an unwilling child cannot fail to understand modular spaces afterwards. So this is the surfaces that remember how they want to be mapped to each other. The quotient is the surfaces that don't, that are the same just when there exists an isometry between them. And the mapping class group is this group acting on the surfaces. So I was claiming that this quotient is actually a well-defined thing, that there is such a thing as a modular space. So the proof goes something like this. Let's look at the surface. The genus four will do. And let's draw enough curves on it that the curves, in a sense, record all of the topology that's going on. So I can start by drawing the curves of my favorite pants, the composition. Every complementary component is a pair of pants. And if you think about it, once you know where all these curves are mapped to in the surface, to maybe other curves, you know a great deal about your map already. In fact, the only thing you don't know is the amount of twisting that is going on along each of those curves. So to record that amount of twisting, let's throw in some more curves. Here's one. Here is a collection of curves. And I think I'm not done. So each of the blue curves has to be intersected once by the new curves I'm drawing in. And this new collection of curves now subdivides the surface into disks. And that's essentially what I need in order for the collection of curves to determine by their images a mapping class. Whenever they're getting the images. So point one. Do I need one more? For the lengths of those to determine a point in modular space, I may need some extra stuff yet. But right now it's enough to subdivide into disks. So point one is there exists a collection of curves, Cn, curves in S, such that phi of C1, phi of Cn determine phi for any phi in the mapping class group. So that's nice. And we want to prove proper discontinuity. It's something about compact sets. It says whenever I go into the large space, take a compact set in there, the discrete group is going to move the compact set away from itself, completely disjoint from itself, except maybe for a finite set of exceptional elements acting. So let K in time in a space be compact. And by compactness I can notice that there is an upper bound on the lengths of all those Ck for matrix in capital K. There exists L, a positive number, such that for any metric G in the compact set K of the modular space, the length of the metric G of the curved CI is at most L for any i, and I can make curves. Good. However, if I take one surface, then there's only a bounded number of curves that have length up to L. There's these curves, if my surface is in K, there's some other curves, short curves, fairly short curves, but there's a bounded number of curves that have length up to L in a fixed surface. In fact, if I let my surface vary a little bit, let's say over a compact subset of that space, that's still true. There's only a finite number of curves, let's say gamma, curve in S such that there exists a metric in K such that the length under the metric G of C is at most this previous bound L. This set of curves is finite. Now, if you think what it means for a mapping class to not move K off itself, so if phi in the mapping class group satisfies K intersected with K non-empty, then necessarily phi of C1, phi of C2 up to phi of CN all belong to this finite set capital E, this finite set of curves. So there's only finitely many curves that I can send each of these CI to, and so by combining these finitely many possibilities, there's a finite, possibly large, but finite number of ways that I can send the whole system to curves of bounded length. So by the first point, let me put a star here, there exists only finitely many such mapping classes as phi. I will write one more thing about mapping class groups on the blackboard before I go on, and then leave mapping class groups for the rest of the lectures. So the thing I wanted to mention is, remember yesterday I said Dane twists generate the mapping class group. In fact, I don't even need all the Dane twists, the Dane twists along these curves and in fact I think even certain sub-collections of these curves will suffice. So that's a big theorem. Dane is the first name attached to it and there's licorice on others. And one big ingredient that goes into the proof is the following exact sequence. If I have a, the mapping class group of a surface minus a point, that's kind of more complicated than the surface itself. For example, I could have the surface sitting here put my finger into the point that we are removing and then drag this along some part in the surface and back that will be a deformation of the surface. It will be a mapping class of the complement of the point. That's not trivial. However, if I pull my finger out, then the surface won't remember anything about the point. It can slide back to where it was. So there's a quotient map here. This surjects onto the mapping class group of S. So a simpler mapping class group. And the kernel is exactly the set of paths, well, the classes of paths that I could have followed with my finger. So this is one fundamental group of S with respect to the point P. This sequence is exact. If you really think about it, the hard part of proving this is exact is proving that the second arrow here is well-defined. Because when I have a, when I have a homotopy between two paths in Pi 1, then maybe this homotopy pulls the path past the point P itself, the base point, and you have to think carefully about what happens then. But if you can make sure that the arrows are well-defined, this is pretty much obvious. And I'll just say this. This relationship lets us relate bigger mapping class groups to simpler ones, and that lets you prove theorems like this theorem by induction, which is its induction on the complexity of the surface, genus and boundary components, and you have to use this kind of induction pretty much any time you want to prove something serious about surfaces. So let's just use induction, complexity of S. And this, I should say what this is called, it's called the Berman exact sequence. Somehow Berman is somebody who lived several generations and lived in, but that's still how it works. Still is. As far as I know. Bad, bad, bad things. I mean, the mapping class group itself is you have competing definitions, right? You could have things that are, maps that are supported on a compact subset of the surface and that are the identity outside, general homomorphisms. I think the theorem is still true for infinite genus surfaces but don't quote me. So restricting to mapping classes with bounded support of course. Earthquakes. They do occur in Italy, right? So I'm completely switching here now. You can forget about mapping class groups. So an earthquake of a hyperbolic surface is in pictures the following thing. You have, let's say, a closed geodesic and I'm taking a simple one here and you do this thing that we have been doing yesterday in fact already. Of sliding, cutting along, opening and sliding by a continuous amount that you get to choose an amount of length. And that's really how earthquakes work. If you've ever been in an earthquake, they don't shake like this. They shake horizontally and at the fork line you really see the ground shift actually horizontally at the surface. So if you think in terms of the objects we've been defining, an earthquake, so there's a length in R by which we shift, defines, if we have the surface S, it's metric G and the choice of line along which to do earthquake, you can map from R to the tightener space taking L to the metric, GL. And tightener space, if you remember, is a disk of some dimension so I can draw it like some sort of line, some sort of curve inside the tightener space. And it's on purpose that I do not draw it as a straight line. The reason, in a sense, is as follows. If you have a very long curve and a long means complicated, it wraps along the surface in complicated ways and try to do earthquake along that, the new surface will be immensely different in tightener space. And in fact, so you do many turns of earthquakes along this curve that is already itself very long. So it's an extreme deformation of the surface. And I claim that it would have been more convenient in every reasonable sense to first expend the necessary time to travel in tightener space to a different place where the curve looks short instead of long and then do a billion earthquakes along this very short curve because it deforms the surface less in any reasonable sense to do a gain twist along a short curve and along a long curve. So that's the sense in which if I take very distant points along this earthquake curve in tightener space, then there are short cuts. There are much shorter ways of getting from here to there. And these ways go through places where the curve looks much, much shorter. So in a sense, those earthquakes are not geodesics in tightener space. Okay. If we go to the quotient, if we go to the quotient, which is modular space, let me draw a modular space as this little cartoon. I'm drawing it like this because it has these, those are called casts or thin parts. Those are the places where the surface has some short curve and there's a big chunk here and the chunk has corners or singularities. These are surfaces that have symmetries that are sent to themselves by some non-trivial mapping classes. So it's the same point of modular space, but a non-trivial asymmetry of tightener space above should think of this as a singular locus. Well, this is going to, if I do it along a closed curve like this, this is going to produce some sort of, some sort of loop inside modular space. And yesterday, if you remember, this is an interesting example where we did earthquake without saying it. We did earthquakes along very, very long curves. I draw the same pictures yesterday again. In this situation, let me try to be a little more realistic than yesterday. The realistic way of drawing these things is you only have one line after a while. It really compresses exponentially fast together. The naked eye doesn't see any width here. I mentioned yesterday that as we glue the top curve to the bottom curve, top boundary component to boundary component, we get many, many different surfaces for different amounts of twist. And in the glued up surface, what this looks like as I vary the amount of twist is of course earthquakes. And this statement that I gave that the c-stall varies a lot over time is kind of a puppet show version of deep results nowadays. So for long curves, let's call c the curve and o sub c the orbit closed loop in modular space. For long curves, o sub c, o sub c visits a lot of modular space and this, as I was saying, is a puppet show version of theorems that say the earthquake flow, Mordes, is ergodic, is in fact mixing, and here I should say, for this to actually make sense, I need to put a cotangent bundle here of Mordes. So that's a theorem that says you pretty much visit everything. And, okay, here I should give some names. So, as Mirza Hani, Eskin, and I'm bound to be unfair to a number of people, so I'll just stop here unless somebody helps me. There are other names attached to these results, but the way this is proved is by proving its semi-conjugate, the horocycle flow, which is another well-known flow. I guess Eskin proved something about the horocycle flow and Mirza Hani proved the relationship. So these very deep modern results are the mathematically positive things that you can say in terms of how crazy those curves are. They equidist tribute, and you can think of these results as saying if you take a random collection of long pairs of pants and a random gluing of them, then the surface you get is random in a pretty fairly equidistributive way. It's a fair way of picking a surface. All right, let me not go into that. Okay, so I've been talking about long curves for a while, and what I want to do now is introduce laminations. They've shown up already in Bertrand's talks, at least, maybe in others. So what is a geodesic lamination? Intuitively, what they are are just limits, very long curves, very long, simple closed geodesics on S. So by limit, let's say we mean Hausdorff limits. So the set of accumulation points, as I take a sequence of longer and longer curves, let me draw one example in a semi-realistic way. Here's an ideal, I'm sorry, a right-angled octagon. I can glue top to bottom and left to right so as to obtain a one-spunctured, a one-hole torus. The edges that carry no symbol go to the boundary. If those were genuine vertices, I would have identified top and bottom and left and right and got no torus in the usual way. Here it opened up. And here's an example of a lamination, or what a lamination might look like. It will have a lot of arcs going, let's say, from top to bottom, and also another, a lot of arcs connecting adjacent sides. Here I've drawn four, right? So there's a unique way of connecting these four to those four without creating any crossings, and a unique way of connecting the ones at the bottom to the ones at the top, preserving the order. And that's already a very long curve. And I could have added more arcs, but in fact, if I added more and more, things would accumulate in a way that I would never see much more than these arcs. The arcs would distribute in a way that looks like r cross a cantor set. So locally, cantor set. And in the r direction, I have genuine geodesics. They are disjoint from each other. They have these little gaps. And these laminations can also come with some extra data that we'll come back to later. I could have a transverse measure that essentially counts the number of strands that I cross, except it's kind of a real number of strands. If I have a very long curve, maybe it has 17,000 strands here and 19,032 strands there, I can normalize this data to a measure that takes value 1.932 here and 1.7 there. And these are on arcs across the lamination. So transverse lamination, sorry, transverse measure is a function, segments transverse to lamination, to a given lamination to the positive reels. That's additive in the sense that if I take a segment and then another segment afterwards, the total segment has some measure and that's the same as I slide the segment up and down. And again, if I do realize the lamination as a limit of simple closed curve, this transverse measure is just the normalized intersection number with the long curves and there's a limit that we can take. So that's some extra gizmo that I can put on the lamination. And importantly, okay, let me be honest, there's a glorious tradition in this subject to not give proofs and it has been started by Thurston and we will honor this tradition today, but I'll give you a hint at why the details do not really matter. They do matter, of course, they are in the papers, but they are distracting and you don't want to talk about them in a talk. So importantly, the terminations mostly one-dimensional. So what does this mean? A cantor set, there's the famous cantor set where you remove, you start with a segment, remove the central third, then remove the central third of each remaining bit, then continue removing central thirds and so on. So you have these gaps that become smaller and smaller as you go and the dust that remains is the cantor set. The cantor sets that we have here are also cantor sets, so they are homeomorphic to the ones I just described, but the gaps fill up enormously more of the space. In fact, the nth gap has length e to the minus n or something. For the cantor set where you remove the central thirds, the nth gap has length roughly one over n. So the gaps between leaves of the lamination decrease exponentially. The reason for this is the following. If you look at the complement of this crazy lamination, each complementary component is a perfectly nice ideal polygon. It does not remember in which crazy way it's glued to the other polygons. It's just one ideal polygon with this ideal spike that goes in between two leaves and wraps around the surface in this crazy way. And there's only a finite number of those spikes because the number of spikes is essentially a count of the area of the surface. So the complement of the lamination lambda is finitely many copies of this exponentially decreasing spike. And if I look at any particular place on the surface, let's say here, and maybe the spike runs through my little magnifying glass here. Maybe it runs through again. It comes back, wraps around the surface, and crashes through again and again and again. But each time it returns, it has to have traveled a definite amount of length. So the spike has grown exponentially shorter each time. And so these gaps that I look at essentially vanish to nothing in linear time. So for that reason, the convergence, when I talk about lamination as limit of simple closed curves, is extremely strong, extremely fast, and that beats every other estimates that you might worry about when you try to do limits. OK? So from now on, in the Glorious tradition, I will just pretend that laminations are simple closed curves. Here come some examples to show you that they are not just a product of our imagination, but that nature really throws them at us. For example, let's take a representation, rho from pi1 of S into ESL2C, quasi-Fuxian. So what does quasi-Fuxian mean? It means you start with a Fuxian one. So this is really doing much of what PPSR has been doing and others too. I start with a Fuxian representation, which is one acting on this copy to PSL2C. That's the isometries of H3, right? Hyperbolic three space. So here's the Bohr model of hyperbolic three space. Inside it sits hyperbolic two space. And I start with a Fuxian representation that preserves this copy of hyperbolic two space, so it's really a real value representation. And it has this limit set, which is the whole circle. As I take an orbit of a point, it accumulates, becomes denser and denser near the boundary, and the accumulation, the limit set, is just this circle. Then as I deform a little bit, so that's called Fuxian, I deform a little bit in the space of representations. And what I will get is, as Dick was saying, it's still a quasi-isometric embedding of the fundamental group. And it gives us a limit set, but the limit set will immediately start to look very crazy. In fact, it's Hausdorff dimension will immediately start to increase up from one. So the limit set is something I must attempt to draw it. It's fractal. It has these spiraling places. Everywhere there's a little spiraling point. There's a spiraling point whenever you have an element in the, in pi one of s. That is not a pure translation in H3, but in fact a translation with a little amount of turning. Because you apply this element, it should preserve the limit set. You apply it again, it should preserve the limit set. You apply it again, and so on. So there's this kind of spiraling phenomenon that you see in a sense everywhere at the same time. Of course, most of these spirals look very small to our eye. But here's the limit set. It goes around, it closes up. It's a quasi-circle. And then what I can do is take the convex hull of this object. Let's pretend I'm in the projective model, the client model where straight lines look straight. I take the genuine convex hull in the ambient three space, convex hull of that set. Then what I'm going to see is some pieces of planes, some ideal polygons on the top. This convex set that I'm building as a convex hull has no, no extremal points except the points on the limit set. It cannot have any extremal points, other than the ones I'm taking the convex hull of. So everything in there is either on a genuine flat face or maybe on an edge, a bending edge, or maybe on some kind of accumulation set of edges and those are the laminations. So there's a top side that's bent in this crazy way. And if I unbend it, I just recover a copy of H2. So for the intrinsic metric, this is H2 but bent. And on the bottom, there's the same thing going on. There's a different surface, a different copy of H2, also bent in another completely crazy way, also bent. OK, so what I hoped, right, so these are bent copies of H2 and the fundamental group of the surface acts on this convex set preserving it because it preserves the set of points to take the convex hull off. So if I look at the intrinsic metric of H2 here on the top boundary surface, it's in the quotient, it's a surface, it's a hyperbolic surface, homeomorphic to S with some bending data. So what I hope I convinced you of is that there is no reason to hope in general that these bending locus will be nice. It will usually be one of those crazy laminations. And so every quasi-Fuchsen representation gives us not one, but in fact two, usually quite crazy geodesic laminations. So they are a very natural object to worry about. In the quotient, this is sometimes drawn as a little cartoon like this. In the quotient we have the surface and the boundary of the convex core that's bent. I should respect the colors here. I'll use green for the top, so copies of S. Sorry, bent, bent surface. So now that I've, I hope convinced you of the existence of those laminations, there's a bending angle which plays the role of the measure, right? If the bending occurred along a single line, I could use the angle here as a measure, transverse measure, associated to this particular line. And here I can take limits in an appropriate sense and say that this collection of lines has a total bending of this or that many irradiance. So these come with a measure. I did not start with a bending. That's the point. Nature gave me a bending. When I threw a roll at it, you could also start with a bending. You could definitely do that. And then nature would return another bending on the bottom. If you start with a surface and bend it, nature would... Or you could start with two bendings and ask if they can be realized. That's a hard question. Right. And in the same way that we could do earthquakes along determining whether two bendings, two bending measures, can be realized by a quasi-fuxion group. I think the status of this question is that it's conjectured that yes, uniquely, the yes part is known thanks to Bono and Otal, and the uniquely is a conjecture. Is there just anything wrong with this? Well, it's yes in the sense that you'd like to control the space of quasi-fuxion groups, but whatever data of the correct dimensionality you can find on the boundaries. Maybe a conformal class at infinity, maybe the intrinsic metric, the intrinsic hyperbolic metric here on the top, maybe the bending information on the top, and a similar information on the bottom, but surely any combination of those information, one at the top and one at the bottom, should give you a unique quasi-fuxion group and some of those conjectures are known, some are not. All right. So I wanted to emphasize that in the same way we could earthquake along a simple closed curve. We can also earthquake along an elimination. It's just that we should be a little bit, when you have an elimination, there is not an immediate neighbor to any given face. There's an infinite collection of intermediate neighbors and they all shift a little, little, little, small, tiny bit compared to each other. And this limiting thing is considered an earthquake along elimination. And it's well-defined because of those estimates. There's not much going on, actually. All right. So theorem. I'm supposed to prove theorems about hyperbolic surfaces. So here's a nice theorem. We're going to spend the remaining time proving it. Is that realistic? So for any G and G prime metrics on the surface, points of tight middle space, there exists a unique lamination, lambda, such that left earthquake on lambda takes G to G prime. And I should also, there's also a mu, of course, such that the right earthquake, I can always twist this way or that, right? And it's going to be different laminations that lead to the take G to G prime. So that's a theorem originally by Thurston. But there's an extremely interesting proof by MESS, which is the one I want to talk about. And the proof, it's really a one-picture proof. I'm going to draw more than one picture in the remainder of the talk. But the pictures are really a build-up to the final picture. And it's really a proof that you can hold in your mind as one picture. I promise. So proof. And here I have to rely on a lot of what Todd did this morning. So we have to study the geometry of the group G. So that's PSL2R. And another name for it in this context is antideciderspace. So it's a space. It's PSL2R. So let's see. It's have A, B, C, D matrices. And they require A, D minus B, C to be positive. And then I'm mod out by project by scalars. This is a form of signature 2, 2. So I can look at this as a quadric, the main in P3 bounded by one quadric. P3 projective 3 space is bounded by a quadric. And it's the quadric everybody draws all the time. So I think we antideciderspace people are annoying because there's only one picture we really know regardless of what we talk about. Eventually this appears on the blackboard. But what I'm saying is that the inside of this hyperboloid is a group. And I want to study the group action a little bit. So here's the identity. Let's place it at the center. So that's the A plus D equals one slice, by the way. A plus D equals one. So out at infinity I have the traceless matrices. This disk, if you wonder, is the symmetric matrices. And you can study many, many more subsets of matrices. But what I want to emphasize is the structure of the boundary. So what is the boundary? It's the space of degenerate of rank one matrices. And a matrix of rank one has a kernel and has an image. If I look at all the matrices with the same kernel, as the given one, I'll get a line. So this is rank one matrices. Those are matrices with a given kernel. So they foliate like this. And there's another foliation by matrices according to their image. So each of these lines is supposed to be tangent to the boundary hyperboloid. And that's what makes it kind of challenging to draw. It's a ruling. It's a ruling by lines according to the matrices of given dimension. And if you remember the definition of how the group acts on itself, so there's this action of G cross G. Namely, we, two elements A, the way they act on a matrix is just by multiplying by A on one side and by the inverse of B on the other side. I have to take the inverse. So this stays a left action for the product group G cross G. OK, there's something special about this, about this A and B, namely if M, let's see, let's look at the left action. A sends M for the left action. If I have a matrix with a certain kernel, the new matrix operated on by A has the same kernel. However, its image has shifted by A. M and AM have same kernel and image shifted by, if I look at the action on the other side, sorry, this goes to M, the inverse. It's the other way around. The image hasn't changed. If I decompose with something, then apply it. It's the generate matrix M. Same image shifted by B. OK, so I can look at this boundary torus. The top is glued to the bottom. And this boundary torus is decomposed, carries a grid, which I can turn 45 degrees to make it look like an orthogonal grid. I have horizontally a copy of P1 of the circle, P1R, and vertically another one. And the left and right actions act on the two coordinates sort of independently is a big point. All right, now let's say a little bit more about the action of one pair AB. So notice if A is a diagonal matrix, I let it act on x, y, z, t, and I act on the other side by B inverse. Then I get x times alpha over beta, t times beta over alpha, y times alpha beta, and z over alpha beta. So the AB action has eigendirections, the four matrices with only one non-zero entry. If x, y, z, t had only zeros except one entry, then its image would be proportional. So those are the eigendirections. I can place them. I know the image and kernel of alpha, the image and kernel of A, the image and kernel of B. So I can place all these things. And what I should see is my convention, we're going to see some sort of tetrahedron. So all four points are rank one matrices. The four white points are the four matrices that I drew up here. I arranged in a square in exactly the same way. The blue and red lines are lines from the ruling, and they are preserved, like any line between any of the four points. And then there are two lines across ADS that are axes, and there is a direction in which those things act. There's a strong axis and a weak axis, so basically the points are ordered according to the eigenvalue, and the strong axis goes from the smallest eigenvalue to the biggest, and everything else is accordingly. So what I want to get is that this intersection point, if I see it as a point of the pair in blue and red coordinates, points of the strong arcs, so n points, strong axis coordinates, image of B, sorry, yeah, attracting point of A, attracting point of B. This is being seen as elements of P1R and elements of P1R. So this being said, I can now... OK, what's so special about attracting points of A and attracting points of B in view of the way the surface group is going to act on ADS? So we now let, like not did, let pi1 of S on empty the city space by rho1, rho2, so gamma.x, so yeah, gamma is just rho1 of gamma, m rho2 of gamma inverse, so those are the holonomia representations of G1, G2. Where are we standing? We have two hyperbolic metrics in tight-minute space, so two representations of pi1 in PSL2R. We let one act on the left and one act on the right. There's this important fact that, let's say here is a fundamental domain for rho1. Here's a fundamental domain for rho2. It looks different. It looks maybe stretched in one direction. There's this important fact that since we have a homeomorphism between the surface G1 and the surface G2, we can lift it to H2 and it extends to the boundary, so important. There is a boundary map. There exists a boundary map continuously extending the lift, the natural identity from S, G1 to S, G2. S is one surface. It has, I can look at it in doubt with two different metrics, G1 and G2. There's this identity map. It lifts to a quasi-asymetric map from H2 to itself, and this extends to the boundary. So I have this homeomorphism between the two boundaries, and where this leads when A and B are replaced by rho1 of gamma and rho2 of gamma is that these endpoints of the strong axis are just given by the extension map. So if sigma is this map, I can view, this is the same P1 projective line, cross-projective line as before, the red-blue tiling, the red-blue grid. The graph of sigma, there's a graph of a homeomorphism, self-homomorphism of the circle, sigma that I can draw in this grid. So this is sigma. And if I tilt this back 45 degrees and overlay it again on the boundary of ADS, the white curve, the graph of sigma will go exactly to the closure of the collection of endpoints of strong axes of the elements. Now, I will do the same thing that we did before. Maybe it's time to make a pause here. In case you've lost track, what I'm saying is that there is a natural map, given the two hyperbolic metrics that I want to connect by an earthquake, there's a natural map from this circle at infinity of the universal curve of the surface to the torus at infinity of ADS. It's a slightly crazy map. It's much like the PSE artist maps on the boundary of H3. It's something that we want to take the convex hull of and hope to have laminations thrown at us. Here is the image of sigma in the red and blue ruling, and I'm drawing it. It's a fact that if sigma is not the identity or a projected map, it will always have stretches where it looks almost vertical and stretches where it looks almost horizontal. This boundary map that extends the natural map between the disks is, by essence, nasty. It has its holder property. If you don't know what that is, that's like lip sheets, but with a power. If you move the argument by epsilon, then maybe the map moves by square root of epsilon, which is much bigger. Its holder would certainly not better. Certainly not better anywhere, in fact. And now we take the convex hull. So it looks like before. There will be one copy of H2 green and brown, I think. I'm sorry? What do they look like? No, no. The convex hull with respect to the projective ambient structure. This sits in R3. There's a P3 of R. There's a natural notion of straight line. I take the convex hull in the simplest possible sense. So take convex hull, the graph of sigma. It is, like before, a convex set with a boundary that falls apart into two bent copies of H2. Its boundaries, its boundary components, bent copies H2. Why H2? Well, if you take one plane that slices through this, it's going to give you an ellipse. So there's a natural hyperbolic metric on that. You can reduce it to, I mean restrict it to the ideal triangle or whatever you have and glue those together. It's locally hyperbolic, but it's bent. OK. So this is the final picture, then, that the plane contains the whole proof in itself. What do I have now? These copies of H2 go to the quotient. They yield two new hyperbolic metrics. So there's a top one, H1. So it's going to H top and H bottom on S. So we now have four hyperbolic metrics. There's H top. There's H bottom. And there's G1. And there's G2. OK. And they all do different things in the picture. G1, remember, acts, it's something we know. We've built it into the picture. But it acts in this kind of ghost fashion. We don't see a copy of H2 that it acts on. We only see a circle of lines that it acts on. So it acts on this circle of lines. It's kind of a ghost for us. And G2 acts on the other circle of lines, twisting in the other direction. H top does not really act on a copy of H2. It acts on a bent copy of H2. There's the green one and the brown one. So I claim that there is locally an H2 structure on the inside of each face. Because each face is a flat plane in projective space, which, if extended, would slice through the boundary of ADS along an ellipse. So I take the projective and take the hyperbolic metric that has these ellipses boundary. Or in other words, I can define a distance function inside ADS by the following formula. Whenever I have two points, I define the line through them. It intersects the boundary in two more points. I get four points. Let's take the log of the cross-ratio times a half over and that's the hyperbolic. It's not a metric on ADS because for vertical lines I won't have two intersection points. The cross-ratio will be imaginary and so on. But in restriction to a plane that intersects along the ellipse, it does give a metric and in fact it's a hyperbolic metric. And those hyperbolic metric, metrics living on each of the green tiles, tie together to give a copy of H2 that's bent. OK. So the question is how do these four hyperbolic metrics relate to each other? There are two known ones that act like ghosts and there are two very explicit ones sitting in the picture but which are given to us by nature. Well, the following happens. What you want to do, what do you do if you want to realize a hyperbolic plane with the G1 action? We do not currently have one. We have this circle with the G1 action but we do not have a hyperbolic plane with the G1 action. What you're going to do is take the top surface and unbend it. And as you unbend it, here's what you see in the boundary. You could unbend naively. Here's what you see in the boundary. Here's maybe a point of the graph of sigma. And as you unbend, this point would move up. If you unbend the hyperbolic, it would move up which is bad because that means it goes away from the red leaf that it was sitting on. And what we want is to realize an action which at infinity looks exactly like the action on the red leaves. So what we do is we unbend but at the same time, earthquake along the bending locus so that instead of moving up, the point will move along its leaf. So this point P here is the point P there if I unbend the edge on top here. If I unbend, I fix. They are infinite. It is actually a lamination. So I would have to do a limited argument like I said in the great tradition of the person we want. So you choose one tile and you unbend the neighboring tiles while shearing along the bending locus so that the end point here stays on its good old red leaf. And you do this and then you do this for the neighbors of the neighbors and for the neighbors of the neighbors of the neighbors. And as you go, you have to do this shearing, this earthquakeing along all the top bending locus. And what you found is that there is an earthquake along the bending locus that leads you from H top to G1. So shear, I should say earthquake, along top bending. OK, how do I get G2? Well, I do exactly the same. I unbend but I shear in the opposite direction. So the point will stay on its blue leaf. So earthquake negatively along the same lamination and you get to blue. And if you put these two things together, you get a long earthquake that goes from G1 to top to G2 and that's one earthquake. And if you take everything I said and go through it, it also proves uniqueness because the fact that your earthquake always, your earthquake always moves things to the left says that the things bend positively. So you would have to have a convex set anyway and there are not so many convex sets that are invariant. So combining these two things, these two arrows together, one in the positive direction and the other in the negative, I get one big long earthquake along one lamination that takes me from G1 to G2. And if I take it for the bottom surface, I get an earthquake of the other handedness taking me from G1 to G2. Am I over time?