 So thank you for coming. I hope you guys have been having a productive time. It seems like you have. So in about the exercises, I wanted to say that I didn't carefully label them as to which ones are harder than which other ones. There's about nine of them. And so feel if you should skip around. If one of them seems too hard, then maybe there's some hidden problem or something. So you don't all have to solve every single problem. So what's the goal? So I want to talk about, I guess by now, people have talked about this group a lot. But let me just, so my notation is going to be this. A surface of genus G with n punctures is labeled this way. So G is the genus, n is the number of punctures. So maybe, for example, a good example is this one. This is S1, 2. So and whether they're punctures or boundary components doesn't matter that much to me. You'll see that it doesn't play a big role. And then the mapping class group, maybe I'll write it this way, is just, I guess we could write Diffio plus, or maybe just Diffio of S, mod diff not, or you could do homeo. So this is the group we want to talk about. It's a finitely generated kind of a group. And I want to, so the goal here is to talk about the course structure of this group. So in other words, in the kind of setting of geometric group theory, I want to think of this group as a course object. And in this course structure, there's kind of two different phenomena. There's sort of hyperbolicity. And there's kind of flat structure. So curvature zero structure, more or less. But none of it is exactly on the nose. All of it is sort of coarse. So it's a little tricky to study. So and I guess the work I want to talk about is not recent work. But I'll talk about this a little bit later. But some of this structure has been generalized in various settings, some of which you've heard about. And so in some sense, the mapping class group is a good example to think about. Because it's a kind of place where you see a certain kind of structure playing out in a pretty rich way. So just the motivation is both it's important to us as a group and also it's a kind of interesting example to study. And so the way I want to divide up the two talks, so that kind of hour one will be sort of, we'll talk in the surface. We'll talk about the structure in the actual surfaces. We'll talk about curves and laminations, at least briefly, a little bit of Thurston's theory of the mapping class group, just briefly. And in curve complexes, and a kind of construction called subsurface projections, which is a way of examining. So the curve complex of a surface is a pretty coarse thing to study. It loses a lot of information. But by studying lots of subsurface, you can refine what you can do using these techniques. And so I want to introduce all of these objects and give their basic properties. That's the first lecture. And then we'll have the exercises. And in the second lecture, I'll kind of maybe we'll write in the group. My goal is to give you some examples of how you actually do something with this structure. So there's a bunch of theorems about this. I want to talk about it. And what do you do with it? I want to give you concrete applications of how you can actually study the group using the structure from the first lecture. OK. All right. Please ask questions. I don't know. Like 100 people here. But feel free to ask questions. So where do we start? Maybe let's just get to know the group a little bit. So what's inside the mapping class group? So first of all, there are a lot of abelian subgroups. In other words, elements that commute with each other. So typically, if you take elements that are supported on disjoint sets, they will commute. So you take elements supported on disjoint sets. That's kind of a general fact that such things would commute. So in particular, you could take your, let's draw a genus to surface. You could take some non-trivial curves. So here's one. And here's another one. And you could take a dain twist on each of these curves, and they would commute with each other. So maybe we'll call this curve a in this b. So the dain twist on a commutes with the dain twist on b. So what is the dain twist? Let's just draw the picture in case I sort of imagine people have seen this picture, but we'll draw it once. So elements supported on disjoint sets commute, like for anything. So yeah, so what's a dain twist? Let's draw the picture very quickly. If you draw a curve in a surface and you draw a little neighborhood of it, and you define a map which takes, I don't know, take a homomorphism of this annulus, which takes these arcs that cross and twists them once around in one direction, so like this. So the map that I'm drawing is the identity on the boundary. It takes the blue to the orange. So you can extend it to the identity on the rest of the surface. That's a dain twist. If this curve is an essential curve, if it doesn't bound a disk, then this twist is a non-trivial element of the group. And in fact, it's an infinite order element. So actually, that's one of the exercises, is to convince yourself that this is an infinite order element of the mapping class group. And so we produce, so the group generated by T, A, and T, B is a copy of Z2 sitting inside the group. So that's one kind of thing. I could do, so how big a rank can I put into this group? Well, I can draw one more curve, say this one. And now I can get a Z3. But I can't put any more. If you think about where else I could put curves in this, any other curve I put in here is either going to be parallel to the previous ones, or it's going to bound a disk. So OK. And in fact, OK, so it's in fact true that the three is the largest rank it's possible in this group, no matter what you do. OK. OK. So that's one kind of thing. The other kind of thing is kind of hyperbolic elements. I'll put that in quotes. So these are what are known as pseudo-anosov. And now I don't want to get into the whole definition of this. Nobody ever wants to write down the whole definition. But roughly speaking, a pseudo-anosov acts in the following way. On your surface, you have to have a certain kind of structure. The structure is, for our talk, I think we're going to use laminations. So I want, so when is a map F pseudo-anosov? It's in the following situation. Suppose I can draw, I'll tell them what a lamination is. I want to draw two laminations on the surface. So I'm drawing a little piece of the lamination. I'm going to explain what it is in a minute. So this is some kind of object like that. It has a bunch of leaves, which is preserved by F. And then there's another one, which I have to draw in a different color. Maybe orange is good, which is transverse to the first. It's going to look like this. It's going to look like this. It sort of runs through. So everywhere in the surface, you see either empty space or transverse leaves crossing each other. And they cut up the surface into disks. Every complementary component in this picture, if I drew the full picture, will be a disk. Or in the case with boundary, it might be an annulus. So these are two transverse laminations. So there's lambda plus and the minus are transverse. Well, I was debating with myself how many definitions to really give. So I'm going to try to wave my hands about some of the definitions and just give you the basic picture. But transverse laminations, there's a measure on these laminations, which I won't discuss. And f stretches 1, say lambda plus, and shrinks lambda minus. So what happens is the way the map actually looks, how do you map the white curves to themselves and the orange curves to themselves and yet not be the identity? So what you do is, you do kind of have to preserve these regions. But then you have to stretch the white leaves, and compress the orange leaves. So for example, there's a lot of pieces here. You would take, say, a little square that lives here, maybe, and you would map it to a much longer square and a much thinner square. So maybe it would map to, let me draw it over here. So here's the white leaves and the orange leaves. And this will somehow map to, maybe, elsewhere on the surface to a much, much longer end. So there's a lot more structure in here. The stretch factors are actually the same. They're rather reciprocal factors. They are eigenvalues of a certain matrix. There's a lot of structure. But this is the basic picture. And what a hyperbolic element does, it takes every, for example, it takes any curve that you might draw on the surface other than these. And it compresses it along the orange and expands along the white. And so over time, it exponentially expands. At first, it exponentially contracts if it looks very, very parallel to the orange. And then once it stops being parallel to the orange and starts being parallel to the white, it expands. So it's like a hyperbolic matrix acting on the plane. OK, so that's a pseudo enosso. And you will see a little later the sense in which I mean that they are hyperbolic. But maybe you'll say one thing about these pseudo enosso elements, if f is pseudo enosso, then the centralizer of the group generated by f, so this is the set of all elements that commute with f. So group of all elements commute with f is a finite index extension of the group generated by f. So this is just so the group generated by f is in here with finite index. So such an element cannot belong to any z2, for example. So it's kind of the opposite. Any questions about this? OK, all right. Oh, one more thing. So there's one, well, so here's a little bit more on the list that I should say very quickly. There's also reducible elements. Let's just quickly say this. So an element is reducible, say f, is reducible if f preserves a multi-curve. So a multi-curve is, so for example, what's an example here? I could, I don't know, I could take this curve maybe and this curve, this union of two curves I could preserve. For example, I could just fix them. I could fix these two. I guess these two I can't exchange. So this pair of curves, if I fix the pair, I have to fix them individually. And then what else can I do besides the identity? Well, I could twist around this one and I could twist around that one. And I could sit inside one of the complementary subsurfaces. I could do one of these pseudo-anossobs restricted to the subsurface, for example. So a reducible one is like a step in an inductive description. There's an invariant collection of curves. And then in the corresponding subsurfaces, something else is happening that you have to then re-examine. OK, so there are reducible elements. There's also finite order, which we won't talk about because that's coarsely. Nothing's happening if you're finite order, but finite order just means f, the n, is identified with the identity. And there you can certainly come up with those. And then I guess we should state Thurston's theorem every element in the mapping class group is of these types. One of these types is of one of these types. OK, so this is called Thurston's classification theorem. OK, let's look at three. What's the first type that goes before pseudo-anossobs? You mean the abelian thing? That wasn't a type, so that's a bad type setting. So, so to speak, the abelian subgroup is an example of a subgroup, and then the elements in it are reducible, the dain twist. Yeah. But you need, I guess one thing that comes out of this, the abelian subgroups have to consist of reducible elements. OK, maybe that's enough of that. So. Sir, is there anything you can say about the other elements centralizer of that? Yeah, so first of all, there could be roots of F. Maybe F was actually the square of a smaller pseudo-anossob, so there's that. And the other thing is, so up to those, up to that cyclic subgroup of powers of F, I could just have finite symmetries of this picture. Imagine that this pair of laminations was invariant by some rotation or something. So there's some kind of finite subgroup that preserves the laminations. That's what it looks like. OK. So maybe let me state one more theorem that kind of gets us in the mood for what kind of things can happen. This is a, I guess this is Birmann-Rubotsky-McCarthy, I think, is a TITS alternative for the mapping class group. So a subgroup of a mapping class group is virtually abelian, which is to say it has a finite index abelian subgroup, or contains a free group on at least two generators. That's the TITS alternative. And it's the pseudo-anossobs that give you these. Either the pseudo-anossobs or the things that are pseudo-anossob in the subsurface give you these free groups. They really don't commute. I guess what actually happens is if you take one more statement, for example, if f and g are pseudo-anossob with different laminations, I still haven't defined lamination, have I? I'm going to try to avoid it for a little while. Maybe I'll have to later. But you sort of get the idea. So if you have two pseudo-anossobs with completely different lamination, different pairs of laminations, then there exist powers, say a power m, such that f, m, g, m together generate a free group. f and g might have some relations. But if you replace them by finite powers, then you can get a free group. And that's an ingredient in this theorem. OK. So that's the mix of kind of hyperbolicity, kind of symbolized by this, and flat structure symbolized by the Abelian groups. And then, OK. So if you think about all this kind of structure, you realize, first of all, there's Abelian subgroups. But there's a lot of Abelian subgroups. They interlock in complicated ways. If you take that Abelian subgroup up there, you could throw away a couple of the curves, leaving one, and replace them by other curves. So you could draw a picture that looks like this. So one multi-curve would be these three, which generates a Z3. And another one might be, say, this one, and I don't know what. That's another good one to draw. This one, and now I'm going to be stuck. I don't know. There's some other one that runs around like this somehow. I don't know. That wasn't a very good picture. But I keep one, and I throw away to replace them by something else. Now I have two Abelian subgroups isomorphic to Z3 that intersect along the Z. So once you start drawing pictures like that, you realize there's a pattern. There's a bunch of Z3s sitting in my group. They overlap on certain subgroups, and how do they all fit together? That becomes a question. So to study that question, you introduce an object that encodes this kind of overlapping structure, and that's the complex of curves to find it here. So complex of curves of S, C of S, so has simplices corresponding to multi-curves. That's really all you have to say. Take all multi-curves, and we're here inclusion of multi-curves corresponds to faces, the face relation. So a vertex is a single curve, and edge is two curves, and so on. So often, it turns out, for our discussion, we often just look at the one skeleton. The one skeleton is just a graph, some kind of graph, where the vertices correspond to simple closed curves. Oh, I should say these are essential simple closed curves. So that just means they don't bound a disk, and they're not parallel to the boundary. So they don't not the boundary of a disk and not parallel to the boundary of the surface, or the punctures, whichever. OK? And then we put an edge in whenever the two curves are disjoint, and a triangle when three curves are disjoint. So every simplex in this picture kind of corresponds to a z to the t for some t. This is a t plus 1. No, wait. Sorry, a t minus 1 simplex has t vertices corresponds to a z to the t inside the group. All right, so that's some kind of object. You can think of it as a metric space just by giving every edge length one, making every simplex regular. OK, so it's a metric space. And you can ask, what is the geometry of this space? Well, first of all, before you ask the geometry, you could compute the dimension. Actually, that's an exercise. It's finite, though. That's not hard to see, right? So the dimension is how many non-trivial curves can you put on the surface together? And there's a bound. So the dimension is finite. But that's one thing. But the graph is locally infinite. This is what makes it interesting, right? Because why is it locally infinite? Pick one curve. At least usually it's locally infinite. Take a curve, and as long as what's on the complement is not too trivial, there's infinitely many curves in the complement. It means that the link of the vertex is infinite. OK, so that's what makes it kind of hard to think about. The fact it's locally infinite, I mean, somewhat hard. And I think that François mentioned this in his talk. So the geometric thing that we want to use is that this curve is hyperbolic. So that's a theorem with how we measure. It says that C of s is delta hyperbolic. OK, I was going to give you a proof of this. I'm not sure if I'm trying to figure out if I'm going to run out of time or not. Oh, the answer is yes, right? If the question is, well, we ran out of time, then the answer is yes. So I'm going to, let me decide a little later if I want to give you the proof. So howie and I wrote a proof that was very long that I will not want to give you. And then there are kind of modern proofs that have successfully gotten much, much simpler and there's a proof that maybe you've all seen. I don't know, I think the shortest proof I've seen is by Sisto and Psztycki, which uses ideas of Hensel and Psztycki and Sisto, I guess, which is very, very short. And I find that our old-fashioned proof, I feel like I know why it's working. It's complicated and nasty and it has little bits and pieces and it's like an old car, but I know what makes it run. And the new proof just goes by and I don't know what happened. So anyway, I think that's an interesting feature of the way mathematics goes. But maybe I'll try to say something about the proof later, but let me mention other facts about this which are important. One of them is that the diameter of C of S is infinite. This is also a fact. This is, I think, the shortest proof. No, it was known from Kobayashi and Luoh. It was told to me by Luoh, I think. And that's kind of important if there's going to be any good for anything, right? A delta hyperbolic space with bounded diameter is, well, every space of bounded diameter is hyperbolic in a trivial sense. So it had better have infinite diameter. So let me actually explain this. And then the explanation of this will also talk about the boundary. So you all heard about the boundaries of a hyperbolic space. You should ask, what's the boundary of this hyperbolic space? And so let me say a few words about that and that will force me to talk about laminations. So how do I bring the board down? I was just at a conference where there were buttons to push. You were at this conference. It was impossible. Okay, so I want to say a few words, really just a few words about what happens at infinity in this complex. So I'll tell you what a lamination is. So S, as long as the Euler characteristic of S is negative, which is true for almost every S, right? So then S admits a hyperbolic metric, as you've seen. And then I can tell you what a geodesic lamination is. A geodesic lamination is a closed subset of S, once we've given it the hyperbolic metric, foliated by geodesics, which is to say, so, well, the picture's up there. It's a closed subset and locally, near every point, there's a neighborhood where it looks like a kind of stack of geodesics. But it doesn't fill the entire surface, so the local picture looks like a product. There's a bunch of lines, which are geodesic lines. And then there's some closed set here, which parameterizes the lines. So this really looks like a topologically 0, 1 cross k, where k is some compact set. What can k be? Well, k could be discrete. And then this would just be a finite collection of geodesic loops. Or k could actually be a canter set or something like that, or a couple of other possibilities. So some closed subset of the interval. Well, with more time, one would draw some examples of lamination and explain where they come from and so on. But let me just say a few things about these. So first of all, so facts. So there exist laminations, geodesic laminations. Well, let's see, which are minimal. That's already something. Well, which are minimal. So a closed geodesic, of course, would be minimal, but have infinitely many leaves. In fact, uncannably, if it's infinite, this is going to have to be uncannable. So I was trying to say whether to explain how to do this or not, but it's kind of a long story. Let me just give you a couple of quick pictures that give you an idea of what's going on. So one kind of lamination you might have is this. First of all, you could take a closed geodesic. That's a lamination. You could take another geodesic which accumulates onto the closed one. So it's possible to draw a picture like this, a geodesic that somehow manages to come around the surface and spiral around a closed geodesic like that. And then on the other side, it has to do something similar. So let me kind of draw that, maybe like this. That's an example. This object exists where one geodesic leaf spirals around the other one. You could actually draw this in the universal cover. You've already seen the universal cover of a hyperbolic surface. This closed loop lifts to a geodesic. If you act on this whole thing by the group, you'll get a bunch of copies of this closed loop everywhere, like this. And then, so on, infinitely many, each of them is stabilized by a cyclic group in the fundamental group associated to this loop. And then you could, the other leaf, you would see if you lifted this upstairs, it would basically be some geodesic, which is asymptotic to this on one side and also asymptotic to a different one on the other side, some sort of picture like that. Take that picture. If I choose this so that all of its translates or disjoint from each other, like that, there's going to be some kind of picture like this. Maybe there's one here. So it's not obvious where they all have to sit, but such a picture, if all the geodesics are disjoint here and never cross, would come from a picture like this here. So that's what the spiraling is when you lift the universal cover. It's just too asymptotic hyperbolic geodesics. So that's an example of elimination, which is not a simple closed curve, but it's also not minimal because, so minimal means there's no closed subset that's also elimination, but this one, of course, is not minimal because the closed curve is a smaller sub-alumination. And it doesn't have infinitely many leaves, it only has finitely many leaves. So that's not an example. So the way you, so it's not, it requires some thought where these examples come from. I'm just going to ask you to believe that they exist. But the rough idea is, one rough idea is that if you just take a sequence of closed loops that get longer and longer and longer in a sort of generic way, then they will accumulate onto a lamination of this type. Okay? That's one way that it comes about. The other one is from the pseudo-nassov. So the laminations, lambda plus and minus of a pseudo-nassov are of this type, which is to say they are minimal with uncommonly many leaves. The fact that they're minimal, another way to say that it's minimal, is that every leaf is dense. Take a leaf, any leaf in the lamination, and over there maybe, follow it around and it'll accumulate on the entire lamination. That comes from being minimal. Okay, so there exists such laminations. Anybody okay with this story? Yeah? It seems from your picture that a pseudo-nassov would not preserve the leaves of its laminations. Is that? What? Why? It preserves. So the definition was that it does preserve. It doesn't preserve individual leaves. It takes leaves to each other. Okay. Preserves a set of leaves. So yeah, the leaves move around. Some leaves are preserved actually, but most leaves, there's uncommonly many leaves and most of them are not preserved. They get moved around. Okay. Actually, let me, sorry, I sort of feel bad about not giving you any example at all. Let me give you one example, and that's the following. Take a torus, maybe, okay, take a torus, T2, not a hyperbolic surface, and draw on it a foliation, not a lamination of just lines. Draw lines at a given slope. That's easy, right? Okay. And I'm going to choose, it's a square torus, and I'm going to choose a slope, say S, which is not rational. And the fact that the slope is not rational, so I'm not going to prove that there exist numbers that are not rational, but that's also a theorem. So these, the leaves of this, of this object, right, these leaves of slope S, every one of them is dense in the torus. This is a well-known fact about irrational numbers. So if you follow one leaf around, it will be dense. Okay, so that is kind of the idea we want, except these are not hyperbolic geodesics. Okay, so one more, let's do one more thing to this picture. Let's remove a single point, this one. Now we have a punctured torus. A punctured torus, so torus minus a point, has negative Euler characteristic, and it has a hyperbolic metric, which we can draw quickly. I promised I wouldn't get stuck on this point. And here I am. So let me draw the picture of the hyperbolic version of this. In other words, you remove this point, you make it a cusp, and you get a picture that looks like this, right, with a very thin exponentially. There's the punctured torus. And in this punctured torus, oh, one more thing about it, you can also draw it in the universal cover. I'll just do that one example like this. So take just a quadrilateral, and carefully glue the opposite sides. And if you're careful about the gluings, you get exactly this picture. And these four points kind of identify to be the cusp. And then on here I have this foliation. It's missing one leaf, but there's unkindly many other leaves. So there's lots of leaves that run around in this torus. They're not geodesics. They're not geodesics. They're just these leaves from this picture. But there's a well-defined way to pull them tight to make them geodesics, so maybe this I will leave to your imagination. Pull tight. And then each of them separately will become a geodesic. They will pull apart from each other. They will not remain filling up all the space, and they will turn into one of these laminations. And in this lamination, every leaf is dense. One more feature of this, the pulling tight. I'll draw one more thing. If you take one of these leaves and draw it in the universal cover, what would it look like? It would run through this fundamental domain, and it would run through the next one somewhere, I don't know where, and it would run through the next one somehow, and so on. And it would accumulate infinity on some point. And now if you pull it tight, in other words, put your finger on these two points and draw the unique geodesic between them, that is what I mean by pulling tight. You have to go to infinity and pull tight at infinity. Okay? All right. Yeah. Yes? When you picture the terrace, you can only remove one of the... I remove one of the leaves. I do. So how does it keep the lamination closed in the... Yeah, so this tightening procedure replaces this picture by a new picture, which is still closed. But if you want a kind of... like a topological picture of what happens, this is sometimes also called the dangiois. It's related to something called the dangiois example, kind of, well, you can look up the word dangiois and you'll find this picture. But here's what you do. Take this picture right here, replace... Instead of just removing the sleeve, first before you remove it, thicken it a little bit, make it into a little... kind of bygone like this, a region like that. And now I can't... If I do that here, what's going to happen? How do I keep going? I'll go somewhere else, maybe here. So there's some kind of... I'm building an open region here. And then I follow it along using the structure of this lamination. Now I'm describing a topological operation. As I go, I have to make it thinner and thinner and thinner. So you could do that. You could just choose a way to do that and make it always follow the slope that we're making it follow and have it come around. It'll be dense, but it'll get thinner and thinner. So that's still a thing. There's an object on the surface. The orange chalk describes an open set. The complement is the lamination. So that is actually a topological picture of what happens here. Here, the orange set looks like a neighborhood of the cusp. It looks sort of like this. Everything here is kind of bounded by a pair of geodesic leaves, which accumulate. So I recommend that you draw this picture for yourself and try to think about what it might have to look like. Okay. Okay, so, yeah. You said you have to look carefully to get the one puncture torus, but what do you need to be careful about? When did I say you have to be careful? This picture of the high point, so the ideal square where you move it. Oh, yeah, that's a good question. So to get a hyperbolic structure, you have to glue this edge to this edge by an isometry, just like in Anna's talk. And this edge to this edge by an isometry. But there's not a unique isometry. So there's a one-parameter family of identifications of this geodesic with that geodesic. There's a translation. So you have one parameter here and one parameter here. You have to choose them correctly. If you choose them incorrectly, then the glued-up surface, it will still be a puncture torus, but the metric will be incomplete. It will not, in fact, be a complete cusp type of all surface. So there's a nice discussion to be had there, but that's the issue. Okay. So why did I go into this much detail? I wanted to tell you about this. Why is the diameter of this complex infinite? Okay, let me convince you why that's the case. So here's what you do. You pick, so let lambda be a minimal, I'm going to add one more adjective, filling lamination. So I told you that there were minimal laminations, but I can also make them filling. What does filling mean? Filling means that, i.e., the complement of the lamination is a union of ideal polygons. Oh, I'm doing this here. I'm talking about the surface without punctures because punctures introduce a second case to every sentence. So this way I can say true things. So S minus lambda is some open set in the surface with geodesic boundary. So it's like a surface with boundary. What kind of surface is it? In the pictures that we saw, which are gone of the pseudo enosso, we had, I at least hinted that the complementary regions of the lamination are, in the picture I drew, they were ideal triangles. You saw ideal triangles already, right, in the plane. So that's what these are. If you lift them upstairs, you'll get polygons of some finite number of sides, ideal polygons like this. So when that's the case, so I claim it's the case for the pseudo enosso ones, and it's in general generically the case for laminations, when that happens, it also means that every simple closed essential curve intersects lambda. It follows that every, once this is true, you cannot draw a simple closed curve in the complement unless it bounds a disc. So it must intersect lambda. So that's what it means to be filling. It intersects every simple closed curve. Okay. Yeah. What would be complement and non-filament? Well, for example, let me cheat. I could take this one, okay, I can cut a hole in the disc, and I can take a tube and connect this to another surface. Okay, there we go. So now here's the surface of genus three, and on this part of it there's a lamination, which is minimal, but it does not fill the surface because this complementary component is more complicated. Okay. Any other questions? Okay. Whoops, sorry. So here's a lemma. This is the lemma that I'm talking about over there. Let any, let's see, if gamma one, gamma two, gamma three, and so on is a sequence of simple closed geodesics converging to a lamination lambda hat containing lambda. Then, so then the distance in the complex of curves, let's say from the first one to the nth one, is going to infinity. So what do I mean by converging? This is just a house store of convergence. So you know that there's this, the house store of topology and closed subsets of a metric space is just the one in which things are close if they look close, right? If you take off your glasses and they look the same, then they're close. And if you, with better and better glasses, right? So, or worse than worse glasses? Anyway, so I'm not going to define the house store of topology, but that's what it means. They converge if they look like they're more and more the same. And it's not hard to see that a sequence of simple closed curves can only converge to a lamination. So it's a theorem. You have to check, but that's what the laminations are there for to encode this kind of convergence of simple closed geodesics to these sort of infinite objects. So, first of all, there exist such sequences. There always exist such sequences. Certainly they exist when we have, when lambda comes from a pseudo-anosov, we could just take one curve and apply the pseudo-anosov to it, and it's going to converge to the lamination. And this lemma is saying that if I have any such sequence, they must be going to infinity in the distance in the curve complex, okay? So, I think I'm going to not do everything I want to do in the first hour, so we'll rearrange. But let me explain why this is true, because you should see the proof of something. I mean, if you call this proofs. Okay, so here's a proof sketch. So suppose not, suppose there exist, let's give them gamma 1, gamma 2, and so on, so that the distance from gamma 1 to gamma n is not going to infinity. So if it's not going to infinity, I may as well assume that it's constant, because I can always take a subsequence where it's constant. Okay, so suppose this is just equal to some m. If this is not true, then there exists a sequence for which this is true, right? You're very clear on that. Just pigeonhole. Yeah? Yes? Okay. So, I'm going to get a contradiction. So note that, yeah? Do you eliminate the physics to be different? Oh, I do. Sorry, there should be sequence of distinct. You're right. Well, let's see. Well, here, of course, if they're converging, they have to be distinct if they're converging this. But here, you're saying here, I need to be distinct. Is that your question? Did I understand the question? Yeah, I didn't. Okay, so here, in order to converge to lambda hat that contains lambda, which is infinitely long leaves, these can't all be the same geodesic. They have to be distinct. And then we can take, if they're all bounded, we can get a subsequence which is just equal to some fixed number. And now I'm going to get a contradiction. I'm going to find out that m is, well, you'll see. The contradiction would be that m equals 0, which is a contradiction. So, here's the idea. Let's write down, so what does it mean for the distance from gamma 1 to gamma n to bm? It means we can draw the following picture. Here's gamma 1, and here's gamma n for all n. And there's a path in the curve complex of length m that terminates in gamma n. So, I'll draw it. Here it is. So, we'll call, maybe we'll call this one beta, I don't know what to call it, beta n1, beta n2, beta n3, and so on. And the last one, beta nm is gamma n. Okay? Sequence of simple closed curves that approach this one. Sorry, that land on this one. It's of the same length m. So, and I can do this for every n. So, there's kind of a picture. These guys are converging. Let me draw this sequence, right? So, these are kind of gamma n plus 1 and so on. So, there's sort of a sequence like this of length m for every n. So, let's look at the penultimate column in this diagram. So, look at beta nm minus 1. Okay? That's a sequence of geodesics. They must, after a subsequence, this is a feature of the Huster's topology, everything, it's a compact topology. So, such a thing must converge after a subsequence to some new lamination. Okay? Now, beta nm minus 1 and beta nm, which is our gamma n, are disjoint, right? So, we have a disjoint bunch of, a bunch of pairs of geodesics getting longer and longer, disjoint from each other. So, their limits cannot intersect. Their limits cannot have any transverse intersection. So, mu hat and lambda hat can only intersect on leaves, right? On entire leaves, because if they were, anything else would be a transversal intersection and that would have produced an intersection between these guys, right? So, you have these geodesics, the betas, these betas in this list and the gammas in this list are each converging to some lamination, mu hat and to some lamination, lambda hat. The laminations cannot intersect transversely. But that means, well, it means in particular that lambda must be in mu hat. Why? They can't just be disjoint because in the complement of lambda hat, there's nothing. There's just a bunch of polygons. There's no room in there for any uncountable collection of geodesics. So, it must contain something and because lambda is minimal, the only thing that it actually can contain is lambda itself. Everything else accumulates on lambda. Lambda hat was a lamination that contained lambda, right? That was the hypothesis. So, it was some, how can lambda have contained lambda? There's really only one thing that lambda hat can have. It has lambda and then in the complementary regions that look like this, maybe let me draw one with pentagons. Suppose that lambda looks like this. It has pentagon complementary regions, okay? Then, all I can do is add some diagonals. I can add a leaf that's asymptotic to the cusps. So, here I could add as many as two. So, lambda hat could have a couple of extra leaves that look like this. So, I'm neglecting to tell you the whole structure of lamination. But if you have a minimal filling lamination, the only thing you can do to add to it is add a finite number of diagonals which are accumulating onto the minimal part, okay? So, does that answer your question? Okay. Where was I? Right, so that's what lambda hat could have only been. And I'm claiming mu hat can only be the same kind of thing. Mu hat is not allowed to intersect lambda hat in any way, and so all it can be is just some other way of filling in lambda. But what does that mean? That means that we produced a new sequence, right? So, this produces a new sequence, namely these guys with distance m minus one, right, with the distance from, I guess gamma one's the initial point for everyone, and then beta n m minus one is m minus one, right? That's the inductive step. I had a sequence of length m that converged to something containing lambda, and now I have a sequence of length m minus one whose last terms converged to something containing lambda. And now I induct. So, what does it mean? Where does the induction end? It really ends with m equals zero, right? It ends with m equals zero, right? I can keep going, and what I learned at the very end is that gamma one is lambda, okay? So, that's a contradiction, because gamma one is finite length and lambda has infinitely long leaves. Okay, so that's a contradiction. Okay. How much time do I have? Like five minutes or? Five minutes. Okay. Let me round off this set of, this discussion by quoting a theorem of Clarke, which says that the Gromov boundary of the complex of curves is the set of minimal filling laminations. In other words, each of these things gives rise to a sequence of curves going to infinity, but this is a finer statement. It says that in the sense of kind of using this equivalence class to define, you know, the equivalence relation that generates the Gromov boundary is really finding quasi-geodesics that go to infinity and thinking about those as the endpoints of those in an appropriate sense. That thing is exactly identified with the minimal filling laminations, okay? And this is actually a topological space, and this is a topological space that topologies match. This is actually a homeomorphism, but I haven't told you what the topology on this space is. Anyway, the punchline is we have this infinite diameter hyperbolic space that I didn't prove hyperbolicity, I guess, in the time we have. It has infinite diameter and the boundary is really tightly connected to the notion of laminations in this way, okay? So laminations really kind of are how we get to infinity in this object. Okay, so, yeah, it's like three minutes. Let me tell you where we're going next. By the way, having proved that the diameter is infinite, one of the exercises is to find an example of two points at distance four apart, and that actually takes some work to do. In fact, I don't have it. If you ask me to do it now, I won't be able to. Okay, so we have this object. We have this action of the mapping class group on this object. Okay, all one good. This one is hyperbolic. This one is not a hyperbolic group. It has all these abelian subgroups. It stabilizes by construction, by motivation. Stabilizers of simplices are abelian, contain abelian subgroups here. So this is certainly not a proper action. So we need to, but we want, the goal is to be able to kind of talk more about this group by thinking about things like this. Okay, and so in the last two minutes, let me just tell you where we're going. So we're going to refine our discussion by considering the kind of system of all subsurfaces of the, all subsurfaces of the surface s and their complexes. So there's infinitely many, what are these? What are these? These are essential subsurfaces and they should be considered up to isotope because everything is up to isotope. So each one of them, including s itself, has a curve complex. I cheated you on the definition of curve complex and maybe I'll mention something, but, and I have, so I have an infinite thing on which the group acts. Okay, so it's some kind of huge cloud of things and I'm going to study them all together. And that's the goal, I want to explain to you how we can kind of interact, get some interaction between these objects and use that to really study this group. And maybe, actually maybe, we only have a minute left. Let me, let me just give you the, let me be clear about definition of this. So I guess if W, in almost all cases, this is just the curve complex we've been talking about. But there are a couple of special cases. So let me just draw the special cases and maybe we will discuss them in the very beginning of next lecture. So one special case is the punctured torus we had before. The problem with the punctured torus is if you draw any two distinct essential simple closed curves, they have to intersect because the complement of this one is a three-hole sphere which doesn't have any other new curves in it. So any two curves intersect. What are we going to do to define, so this is S11. What are we going to do to define this? And the answer is we do the following. We have the same vertex set, but we define edges to be curves intersecting once. Once is the least you can do and that's going to define our edges. And this is actually, if you've seen it before, this is equal to what's called a fairy graph. This is a classical object. Another one with the same problem is the four-hole sphere. And I will leave it up to you to think what to do, but you see that if you draw one curve then there's no room for another one, the same issue. There's also the three-hole sphere and actually here it's just the empty set. There's nothing in here. We're just going to stay with nothing. And then one last one is the annulus, the two-hole sphere. S02. I have to explain what to do here. It's a little trickier. Even though it's just an annulus. What could go wrong? But the idea here is not to think about curves but to think about arcs. And somehow this complex is going to be measuring twists of arcs around this. So it's going to be quasi-osometric to z, more or less. And we have to explain where that comes from. It's a special case that's always in the ways. We have to at least get it over with once. Okay, so I guess I'm out of time. So at the beginning of next time after the exercises I will talk about how to tie these together and how to make something happen.