1/2 Mapping Class Group and Curve Complex
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Transcript: English(auto-generated)
00:17
So thank you for coming. I hope you guys have been having a productive time.
00:22
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. So you should skip around. If one of them seems too hard, then maybe there's some hidden problem or something.
00:42
So you don't all have to solve every single problem. OK, so what's the goal? So I want to talk about, I guess by now, people have talked about this group a lot.
01:00
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
01:21
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 it's diffio of S mod diff not. Or you could do homeo.
01:41
So this is the group we want to talk about. It's a finitely generated countable 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 setting of geometric group theory, I want to think of this group as a course object.
02:02
And in this course structure, there's two different phenomena. There's hyperbolicity, and there's flat structure. So curvature zero structure, more or less. But none of it is exactly on the nose. All of it is sort of course. So it's a little tricky to study.
02:28
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.
02:40
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. OK, and so the way I want to divide up the two talks,
03:04
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. So we'll talk about curves and laminations, at least briefly.
03:23
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
03:43
is a pretty coarse thing to study. It loses a lot of information. But by studying lots of subsurfaces, you can refine what you can do using these techniques. And so I'm going 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,
04:00
maybe we'll write in the group, I will, 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 one to talk about, and then 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.
04:21
OK, all right, please ask questions. I don't know, there's like 100 people here, but feel free to ask questions. OK, so where do we start? Maybe, let's just get to know the group a little bit.
04:41
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 this joint sets, they will commute. So you take elements supported on disjoint sets.
05:07
That's kind of a general fact that such things will commute. So in particular, you could take your, let's draw a genus-2 surface, you could take some non-trivial curves. So here's one, and here's another one.
05:22
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 and this B. So the dain twist on A commutes with the dain twists 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,
05:40
but we'll draw it once. So elements supported on disjoint sets commute, like for anything, right? Yeah, so what's a dain twist? We'll 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,
06:05
take these, take a homeomorphism 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.
06:23
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.
06:40
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 TA and TB is a copy of Z2 sitting inside the group. So that's one kind of thing. So how big a rank can I put into this group?
07:00
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 it's in fact true that 3 is the largest rank that's
07:24
possible in this group, no matter what you do. So that's one kind of thing. The other kind of thing is kind of hyperbolic elements. I'll put that in quotes.
07:46
So these are what are known as pseudo-anasov. 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-anasov
08:01
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 when is a map F pseudo-anasov? In the following situation. Suppose I can draw, I'll tell them what a lamination is.
08:22
I want to draw two laminations on the surface. So I'm drawing a little piece of the lamination. I'll 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.
08:43
And then there's another one, which I have to draw in a different color. Orange is good, which is transverse to the first. It's going to look like this. It sort of runs through. So everywhere on the surface, you see either empty space
09:02
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, would be a disk. Or in the case with boundary, it might be an annulus. So these are two transverse laminations.
09:22
So there's lambda plus, lambda minus are transverse. Well, I'm going to write, I was debating with myself how many definitions to really give. But so I'm going to try to wave my hands about some
09:40
of the definitions and just give you the basic picture. Transverse laminations, there is a measure on these laminations, which I won't discuss. And F stretches 1, say lambda plus, and shrinks lambda minus.
10:00
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, say, and compress the orange leaves.
10:23
So for example, you would take, 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, well, let me draw it over here.
10:41
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. Okay, so there's a lot more structure in here. The stretch factors are actually the same. They're rather reciprocal factors.
11:01
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. It first exponentially contracts
11:21
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. Okay, so that's a pseudo-onosov. And you will see a little later the sense in which I mean that they are hyperbolic.
11:42
But maybe you'll say one thing about these pseudo-onosov elements. If F is pseudo-onosov, then the centralizer of the group generated by F, so this is the set of all elements that commute with F,
12:02
it's a group of all elements that 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.
12:22
Okay, so it's kind of the opposite. Any questions about this? Okay, all right. Oh, one more thing. So there's one, well, so here's,
12:41
there'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's reducible, say F, is reducible if F preserves a multi-curve. So a multi-curve is, so for example,
13:05
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.
13:21
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 sub-surfaces. I could do one of these pseudo-anosols restricted to the sub-surface, for example.
13:42
Okay, so a reducible one is like a step in an inductive description. There's an invariant collection of curves, and then in the corresponding sub-surfaces, something else is happening that you have to then re-examine. Okay, so there are reducible elements. There's also finite order, which we won't talk about because that's coarsely,
14:02
nothing's happening if you're a finite order, but finite order just means f to the n is identified with the identity. And you can certainly come up with those. And then I guess we should state Thurston's theorem.
14:23
Every element in the mapping class group is of these types, one of these types, of one of these types. Okay, so this is called Thurston's classification theorem.
14:42
Okay, oh, there's like three. What's the first type that goes before pseudo-anosyl? You mean the abelian thing? That wasn't a type, so that's a bad type setting,
15:01
so, so to speak. The abelian sub-group is an example of a sub-group, and then the elements in it are reducible, the Dane twists, yeah. But you need, I guess one thing that comes out of this, the abelian sub-groups have to consist of reducible elements.
15:21
Okay, maybe that's enough of that, so. Yeah. Is there anything you can say about the other elements, such as the order of f? Yeah, so first of all, there could be roots of f. Maybe f was actually the square of a smaller pseudo-anosyl, so there's that.
15:40
And the other thing is everything, so up to those, up to that cyclic sub-group of powers of f, I could just have symmetries, 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 sub-group that preserves the lamination. That's what it looks like.
16:02
Okay. So, oh, 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 Biermann-Bubatsky-McCarthy, I think, is a Tietz alternative
16:25
for the mapping class group. So a sub-group of a mapping class group is virtually abelian, which is to say it has a finite index abelian sub-group
16:40
or contains a free group of powers of f. Free group on at least two generators. That's the Tietz alternative. And it's the pseudo-nosovs that give you these, well, either the pseudo-nosovs or the things that are pseudo-nosov on a subsurface give you these free groups. They really don't commute.
17:03
I guess the state, what actually happens is if you take, one more statement, if f and g, for example, if f and g are pseudo-nosov with different laminations, I still haven't defined lamination, have I?
17:21
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-nosovs with completely different laminations, different pairs of laminations, then there exist powers, say a power m, such that f, m, g, m together generate a free group.
17:45
Okay, 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 actually, that's an ingredient in this theorem. Okay, so that's the mix of kind of hyperbolicity,
18:02
kind of symbolized by this, and flat structure symbolized by the abelian groups. And then, okay, 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.
18:20
If you take, so if you could 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,
18:43
say, this one, and I don't know what. That's another good one to draw. This one, and now I'm gonna be stuck. I don't know, there's some other one that runs around like this somehow. That wasn't a very good picture, but I keep one and I throw away two,
19:02
replace them by something else. Now I have two abelian subgroups, isomorphic to z3, that intersect along a z. So once you start drawing pictures like that, you realize there's a pattern, right? There's a bunch of z3s sitting in my group. They overlap on certain subgroups, and there's a, how do they all fit together, right? That becomes a question.
19:21
So to study that question, you introduce an object that encodes this kind of overlapping structure, and that's the complex of curves. Now define it here. So complex of curves of s, c of s,
19:51
so has simplices corresponding to multi-curves.
20:02
That's really all you have to say. Take all multi-curves, and the, where here inclusion of multi-curves corresponds to faces, the face relation. Okay, so a vertex is a single curve, an edge is two curves, and so on.
20:20
So, okay, so often we look, 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
20:41
simple closed curves. So that just means they don't bound a disk, and they don't bound, 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,
21:04
or the punctures, whichever, okay? 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, right?
21:21
This is a t plus one, no wait. Sorry, a t minus one simplex has t vertices, corresponds to a z to the t, okay? Inside the group, all right.
21:41
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, okay? 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,
22:01
you could compute the dimension. Actually, that's an exercise. It's finite though, that's not hard to see, right? That you should, so the dimension is how many parallel 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.
22:23
This is what makes it interesting. Okay, 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
22:42
is not too trivial, there's infinitely many curves in the complement. It means that the link of the vertex is infinite. Okay, so that's what makes it kind of hard to think about, the fact that it's locally infinite, I mean, somewhat hard. And I think that Francois mentioned this in his talk, that this curve, so the geometric thing
23:02
that we want to use is that this curve is hyperbolic. So that's a theorem with Howie Mazur that says that C of S is delta hyperbolic.
23:26
Okay, I was gonna give you a proof of this. I'm not sure if, I'm trying to figure out if I'm gonna 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.
23:43
So I'm gonna, let me decide a little later if I want to give you the proof. The thing about this, 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
24:01
that have successfully gotten much, much simpler, and there's a proof that maybe you've all seen, I don't know, by, I think the shortest proof I've seen is by Sisto and Piotitski, which uses ideas of Hensel and Piotitski and Sisto, I guess, which is very, very short. And I find that our old-fashioned proof,
24:23
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, so anyway, I think that's an interesting feature
24:40
of the way mathematics goes, but let me be, maybe I'll try to say something I have to prove 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,
25:01
Bruno was known, was from Kobayashi and Luo, was told to me by Luo, I think. And that's kind of important if there's gonna be good for anything, right? A delta hyperbolic space with bounded diameter is, well, every space with bounded diameter is hyperbolic in a trivial sense, so it had better have infinite diameter.
25:22
So let me actually explain this, and then in the explanation of this, we'll 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.
25:44
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, and it was impossible.
26:10
Okay, so I wanna say a few words,
26:20
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,
26:47
and then I can tell you what a geodesic lamination is. The geodesic lamination is a closed subset of S,
27:03
once we've given it the hyperbolic metric, foliated by geodesics, which is to say, well, the picture's up there. It's a closed subset, and locally, near every point, there's a neighborhood where it looks like
27:20
a kind of stack of geodesics, but it doesn't fill the entire surface, so the local picture looks like, well, there's, 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, you know,
27:41
topologically zero one cross K, where K is some compact set. What can K be? Well, K could be discrete, and then this would just be a kind of finite collection of geodesic loops, or K could actually be a Cantor set, or something like that, or a couple of other possibilities. So some closed subset of the interval, okay?
28:05
Well, in a kind of more, with more time, one would draw some examples of laminations 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.
28:20
So there exists laminations, geodesic laminations, well, let's see, which are minimal. That's already something. Well, which are minimal.
28:41
So a closed geodesic, of course, would be minimal, but have infinitely many leaves. In fact, uncountable. If it's infinite, it's gonna have, for this, it's gonna have to be uncountable. Maybe actually, so I was trying to decide
29:01
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, okay? You could, that's a lamination. You could take another geodesic, which accumulates onto the closed one.
29:21
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. Okay, that's an example. This object exists where one geodesic leaf
29:43
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, right? Like this.
30:05
And then so on, infinitely many. Each of them is stabilized by a cyclic group in the fundamental group associated to this loop, okay? And then you could, the other leaf you would see if you lifted this upstairs,
30:20
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 are disjoint from each other, like that, there's gonna be some kind of picture like this.
30:41
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, okay? So that's what the spiraling is when you lifted universal cover. It's just two asymptotic hyperbolic geodesics. So that's an example of a lamination
31:01
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 a lamination, but this one, of course, is not minimal because the closed curve is a smaller sublamination. 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,
31:21
it requires some thought where these examples come from. I'm just gonna 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,
31:42
okay? That's one way that it comes about. The other one is from the Suda and Asav. So the laminations lambda plus and minus of a Suda and Asav are of this type,
32:04
which is to say they are minimal with uncountably 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, it'll accumulate on the entire lamination. That comes from being minimal.
32:21
Okay, so there exists such laminations. Everybody okay with this story? Yeah. It seems from your picture that a Suda and Asav would not preserve the leaves of its laminations. Is that?
32:42
Why? It preserves, so the definition was that it does preserve, well it doesn't preserve individual leaves. It takes leaves to each other. Preserves a set of leaves. So yeah, the leaves move around. Some leaves are preserved actually, but most leaves, there's uncountably many leaves
33:01
and most of them are not preserved. They get moved around, yeah. Okay, actually let me, sorry, I sort of feel bad about not giving you any example at all. Let me give you one example, example, and it's the following. Take a torus, maybe, okay, take a torus, T2, not a hyperbolic surface, and draw on it
33:21
a foliation, not a lamination, of just lines. Just draw lines at a given slope. That's easy, right? Okay, and I'm going to choose a square torus. Again, 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
33:40
that are not rational, but that's also a theorem. So these, the leaves 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'll be dense. Okay, so that is kind of the idea we want,
34:02
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 a torus minus a point, has negative Euler characteristic, and it has a hyperbolic metric, which we can draw quickly.
34:23
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.
34:40
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.
35:02
And then on here, I have this foliation. It's missing one leaf, but there's uncountably many other leaves. So there's lots of leaves that run around in this torus. They're not geodesics, okay? 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.
35:20
So I can, maybe this I will leave to you, to your imagination, pull tight. And then they will, 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, okay?
35:40
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 kind of, 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.
36:01
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.
36:21
Yes. When you picture the terrace, you remove one of the... I remove one of the leaves. I do. So how does it keep the lamination closed in there? Yeah. So this tightening procedure replaces this picture by a new picture, which is still closed.
36:40
But if you want a kind of, like a topological picture of what happens, this is sometimes also called the dangeois. It's related to something called the dangeois example, a kind of, well, you can look up the word dangeois and you'll find this picture. But here's what you do. Take this picture right here, replace, instead of just removing this leaf,
37:02
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 gonna happen? How do I keep going? This will go somewhere else, maybe here.
37:22
So there's some kind of, there's like a, I'm building an open region here, okay? 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, okay?
37:40
So you could do that, right? You could just choose a way to do that and make it always follow, you know, 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. Okay, there's an object on the surface. The orange chalk describes an open set.
38:03
The complement is the lamination, okay? 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.
38:21
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?
38:41
This picture of the ideal square where you glue it. Oh, 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, so that, just like in Anna's talk, and this edge to this edge by an isometry,
39:01
but there's not a unique isometry. So there's a one-parameter family of identifications of this geodesic with that geodesic. Right, 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'll still be a punctured torus, but the metric will be incomplete.
39:20
It will not, in fact, be a complete cusp hyperbolic surface. So there's a nice discussion to be had there, but that's the issue. Okay, so, oh, so why did I go into this much detail?
39:47
I wanted to, I want 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.
40:01
You pick, so let lambda be a minimal, I'm gonna 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.,
40:25
the complement, the complement of the lamination is a union of ideal polygons. Oh, I'm doing this, here I'm talking about a surface without punctures,
40:41
because punctures introduce a second case to every sentence. So this way we'll, 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,
41:04
of the Sudar and Asav, 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, ideal. If you lift them upstairs, you'll get polygons
41:22
of some finite number of sides, ideal polygons like this. Okay, so when that's the case, so I claim it's the case for the Sudar and Asav ones, and it's in general, generically the case for laminations. When that happens, it also means that every simple closed essential curve
41:51
intersects lambda, right? It follows that every, once this is true, you cannot draw a simple closed curve in the complement unless it bounds a disk, so it must intersect lambda.
42:01
So that's what it means to be filling it. It intersects every simple closed curve. Okay, yeah? What would the complement mean for the lamination? Well, for example, let me cheat. I could take this one, okay,
42:22
and I can cut a hole in the disk, 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.
42:42
Any other questions? Okay, so, 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
43:04
is a sequence of simple closed geodesics converging to a lamination lambda hat
43:26
containing lambda, then, so, then the distance in the complex of curves,
43:40
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 Hausdorff convergence. So you know that there's this, the Hausdorff topology on closed subsets
44:02
of a metric space is just the one in which things are close if they look close, right? You look, if you take off your glasses and they look the same, then they're close, and if you, with better and better glasses, so, or worse and worse glasses, anyway. So, I'm not going to define the Hausdorff topology,
44:20
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 exists such sequences,
44:43
there always exists such sequences, certainly they exist when we have, when lambda comes from a pseudo-inosive, we could just take one curve and apply the pseudo-inosive 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
45:01
in the distance in the curve complex, okay? So, I am running, 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 proof.
45:21
Okay, so here's a proof, proof sketch. So suppose not, suppose there exists, say, let's give them, say, gamma one, gamma two, and so on, so that the distance
45:41
from gamma one 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, right?
46:00
If this is not true, then there exists a sequence for which this is true, right? Everybody clear on that? Just pigeonhole, yeah, yes? Okay, so, I want to get a contradiction. So, note that, yeah?
46:23
Do you, I mean, should zigz to be different? Oh, I do. Sorry, they should be sequence of distinct, you're right. Well, well, let's see.
46:41
Well, here, of course, if they're converging, they have to be distinct if they're converging this, but here, I need to, you're saying here, I need to be distinct. Is that, was that your question? Did I understand the question? Yeah, I didn't, okay, I, yes. Okay, all right, yeah, so. So here, in order to converge to lambda hat,
47:00
that contains lambda, which is infinitely long leaves, these can't all be the same geodesic. They have to be distinct. And then, we get it, 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 will be that m equals zero, which is a contradiction, so.
47:23
So here's the idea. Let's write down, so what does it mean for the distance from gamma one to gamma n to be m? It means we can draw the following picture. Here's gamma one, 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,
47:42
so I'll draw it, here it is. So, maybe we'll call this one beta, I don't know what to call it, beta n one, beta n two, beta n three, and so on. And the last one, beta n m, is gamma n, okay?
48:03
Sequence of simple closed curves that approach this one, that, 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 one, and so on.
48:23
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 n m minus one, okay? That's a sequence of geodesics.
48:42
They must, after a subsequence, this is a feature of the Hausdorff topology, everything, it's a compact topology. So, such a thing must converge after a subsequence to some new lamination, okay? Now, beta n m minus one, and beta n m,
49:03
which is our gamma n, are disjoint, right? So we have a disjoint 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.
49:22
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?
49:40
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, that means in particular that lambda must be in mu hat.
50:05
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
50:23
is lambda itself. Everything else accumulates on lambda. Lambda hat was a lamination that contained lambda. Right, that was the hypothesis.
50:42
So it was some, how can lambda hat contain 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,
51:02
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 laminations. 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,
51:21
which are accumulating onto the minimal part, okay? So, does that answer your question? Okay, so, 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. Right, mu hat is not allowed to intersect
51:41
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?
52:02
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
52:21
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? I can keep going, and what I learn at the very end is that gamma one is lambda, okay? So that's a contradiction, because gamma one is finite length
52:40
and lambda has an infinite length, 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,
53:03
this discussion by quoting a theorem of Klareich, which says that the Gromov boundary of the complex of curves is the set
53:20
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.
53:41
It says that in the sense of kind of using this equivalence class to define, 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 the appropriate sense. That thing is exactly identified
54:02
with the minimal filling laminations, okay? And this is actually a topological space, and this is a topological space, the topologies match. This is actually a homeomorphism, but I haven't told you what the topology on this space is. Okay, but the, anyway, the punchline is we have this infinite diameter hyperbolic space that I didn't prove hyperbolicity, I guess,
54:21
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, I have three minutes. Let me, yeah, let me tell you where we're going next.
54:48
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.
55:04
In fact, I don't have it, and I don't, if you ask me to do it now, I won't be able to. Think about it. Okay, so, okay, so we have this object. We have this action of the mapping class group
55:23
on this object, okay, oh, one good. This one is hyperbolic. This one is not a hyperbolic group. It has all these abelian subgroups. It has, it stabilizes, right, and by construction, by motivation, stabilizers of simplices are abelian, contain a big abelian subgroups here.
55:41
So this is certainly not a proper action. So we need to, but we want, the goal is to be able to kind of learn 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 gonna refine our discussion by considering the kind of system
56:02
of all sub-surfaces of the, all sub-surfaces of the surface S and their complexes. So there's infinitely many, what are these? What are these? Let me show you. These are essential sub-surfaces,
56:24
and they should be considered up to isotopy, because everything is up to isotopy. Okay, so each one of them, including S itself, has a curve complex. I cheated you on the definition of curve complex,
56:41
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 wanna study them all together, and that's the goal, I wanna explain to you how we can kind of get some interaction between these objects and use that to really study this group, and maybe,
57:02
actually maybe, we only have a minute left, 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.
57:22
So let me just draw the special cases, and maybe we will discuss them at 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,
57:41
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. So what are we gonna do to define, so this is S one, one. What are we gonna do to define this? And the answer is, we do the following. We have the same vertex set,
58:01
but we define edges to be curves intersecting once. Once is the least you can do, and that's gonna define our edges, okay? And this actually, if you've seen it before, this is equal to what's called a fairy graph. It's a very classical object.
58:23
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,
58:42
and actually here it's just the empty set. Like there's nothing in here. We're just gonna stay with nothing. And then one last one is the annulus, the two-hole sphere, okay, S zero two.
59:00
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 gonna be measuring twists of arcs around this, so it's gonna be quasi-isometric to z, more or less, and we have to explain where that comes from.
59:22
It's a special case that's always in the way, so 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.