 Thank you. I'd like to thank the organizers for inviting me to give a talk, the very last talk. I'd like to thank you all for surviving and coming. That's real strength. Muy fuerte. As the last talk, I think we should also thank the organizers. I was going to parlo in italiano, but I can't. So let's thank the organizers. I think we have Alan, Bruno, I forget who else. Michelle, Dick, is that it? That's it. OK, so I'm going to try and give a talk that will be pretty low-key, won't be too technical and too difficult since it's the last one. But if that's too much and you perhaps want to understand the internet works in this room, if you perhaps would just prefer to look at some lovely pictures of cappuccinos, let me put up my Instagram account. Hopefully I can spell it correctly. Well, actually, I realize I'm not spelling it correctly. Instead of the O-S, it should be an I, but I'm American. So please admire. I'm just as happy as if you like my cappuccino photos during the talk, or if you like the talk, that's great too. My underscore cappuccinos. And if you know anyone in Italy who's looking for barista, I'm available. OK, let's get started. So I want to talk about a theorem about hyperbolic three-manifolds. And if I ever state the actual theorem, it'll be at the end. I think I want to, because it's somewhat technical and involves some three-manifold topology, which I don't think we've talked about in the last two weeks. So I mostly want to concentrate on examples. And I'm going to focus on just a simpler version of what we want to talk about. So let's start with some definitions. So I want n to be a compact three-manifold. And what I'm going to be interested in is hyperbolic structures, complete hyperbolic structures on the interior of n. And I'm going to restrict to those that are convex co-compact. So we want to look at, so convex, so I want to study. So CC of n, this is going to be convex co-compact, hyperbolic structures. Now for what I, the theorem that I ultimately proved is colds and more generality, but this is just to keep the discussion simpler and more concrete. So let me just say what this is. So what is this? So this is a complete hyperbolic structure on the interior of n that extends the conformal structure on the boundary. OK, so if you want to think about this in terms of an atlas. So here is a picture of our three-manifold. And here is hyperbolic three-space union its boundary, which we can identify with the Riemann sphere. Well, what I can think of this as being defined by is some sort of atlas of charts. So I'll draw a little neighborhood of, say, a point in the boundary. So a point in the boundary will be topologically just a half-space. And I want a chart, which maps this to a little neighborhood in H3 union its boundary, where the boundary of the three-manifold is taken to the boundary of hyperbolic three-space. So this is now isometries of the hyperbolic plane. They extend to conformal automorphisms of the boundary. So this will give us a conformal structure on the boundary of them. And those are the kind of hyperbolic structures I want to study. Now, certainly every three-manifold doesn't have such a convex co-compact structure. But it turns out that if, by a theorem of Thurston, if n has any interior of n has any complete hyperbolic structure, then it does have a convex co-compact one. And in fact, Thurston's famous hyperbolicization theorem tells you exactly, gives you an exact characterization of when these things exist. OK, and what's the other observation is that a conformal structure on a surface, oh, and I'm going to let's rule out one thing, I want to assume that, so, I don't want any tori in the boundary. So no tori boundary of n. Again, this is just to make life simpler. Oh, actually, it's ruled out by the count. There won't be any tori in the boundary of n because I'm assuming the structure is convex co-compact. That's a consequence of it. So in fact, nor will there be any sphere. So in fact, what happens is if s is in the boundary of n, then the genus of s is going to be greater than or equal to 2. And in fact, a conformal structure on the boundary of n, so if m is one of these convex co-compact structures, so in general, m will be a hyperbolic structure and n will be the manifold throughout the talk unless I get confused and switch them. Hopefully I won't. So if you have some component s in the boundary of n, then s of n is a conformal structure. But by the uniformization theorem, this is equivalent to being a hyperbolic structure. Every conformal structure supports a unique hyperbolic structure, and of course, a hyperbolic structure determines a conformal structure. And what we're really going to be interested in is if we know something about this hyperbolic structure on the boundary, what can we say about the geometry of the manifold? All right, so let's first start off with sort of a classical theorem. Oh, and let's make one other assumption now in the manifold, which will also simplify things. So I want to assume throughout the talk, I want to assume that the boundary of n is incompressible. So what this means is that if I take, so I have some component of the boundary, then the inclusion of s and n is injective on pi 1. Then we have the following theorem, which is a classical theorem. And let's, the big chunk of it is due to Bayes. There's lots of other names on convention. It says that this space of convex co-compact hyperbolic structures on n is naturally parameterized by the Teichmer space of the boundary. OK, so here is one answer to our question. How does the geometry of the boundary that determine the manifold, it determines it completely? If I give you a hyperbolic structure on the boundary, that's going to give you a unique hyperbolic structure on the interior. All right, but we'd like more precise answer to this question. And so here, so let's think of some more detailed ways we can try and understand the geometry. So let's let yama and the boundary then be an essential closed curve. OK, so if we have a hyperbolic, so if we have a convex co-compact hyperbolic structure on the interior then, or if we have a hyperbolic structure on a closed surface, then every essential curve, every curve that's not homotopic to a point, is homotopic to unique geodesics. So we can measure the length. So we can measure the length of gamma as a geodesic on. So if we have some hyperbolic structure M, we can measure it on the boundary of M, which has a hyperbolic structure as a surface. Or we can push the curve into the interior of the manifold. Or in the three manifold M, in the hyperbolic three manifold M. In both cases, we have some geodesic representative, once on the hyperbolic surface, and the other time in the hyperbolic three manifold. And so I'll denote this. So this is length in the boundary. And then here is length in the manifold. So a natural question is, how do these two lengths compare? So that's a pretty straightforward question. And hopefully we have a straightforward answer. So in one direction we do. In one direction we do. There's a pretty elementary bound due to bears that tells us that if I know what the length of a curve is on the boundary, then I can get a bound on that length in the manifold. It's called the bears inequality. And it says that the length on the boundary, two times the length on the boundary, is going to be longer than the length of the curve in the hyperbolic three manifold. OK, well, what about in the other direction? What about in the other direction? If I know, if I have some control over the length here, can I say anything? Can I get any upper bounds in the length here? And the answer is no. And there is an easy way to see this from this parameterization theorem. OK, so what I'm going to do now is, well, this isn't possible in any three manifold, but we certainly can construct three manifolds where we have two curves on the boundary that aren't homotopic as curves on the boundary. Say they may lay in different boundary components, but are homotopic in the manifold. So now let's assume that gamma and gamma prime in the boundary then aren't homotopic in the boundary, but they are homotopic in the manifold. OK, well, we could do this. It's not hard to construct a manifold of such a property. So for example, we could choose N should just be a surface across a closed interval. And then we could take gamma and gamma prime to be the same curve on the surface, but on the two different components of the boundary. Now what does this parameterization theorem say? It says we can take any hyperbolic structures we want on the boundary and realize that as a hyperbolic three manifold. So, well, so what we could do, I mean, let's assume these curves are, say, simple and disjoint. We can realize a final hyperbolic structure so that this curve is much longer than this one, for example. We can find some convex co-compact structure such that the length in the boundary of M of gamma is, say, much greater than the length of gamma prime. OK, well, then what do we get? Then we know from this bear's bound here, let's put two's here just to make it. So we can apply this. We know from this bear's bound that this is going to be greater than the length, this is going to be greater than the length of gamma prime in the manifold. But in the manifold, the two curves are equal. So what we see is that this curve is much longer on the manifold, much longer on the boundary than it is in the manifold. We're not going to get, there's no way we can possibly get a bound in the other direction. So another way of saying this is if I just tell you the length of some curve on the boundary, that you're not going to have any lower bound on, there's no way you're going to produce any general lower bound on the length of the curve in the manifold itself. So there's one very important special situation where we can say some very concrete information about how the geometry of the boundary determines the geometry of the manifold. And I've already mentioned this. That's when n is just equal to s across some closed interval. And here the surface s is a closed surface orientable. Genus greater than or equal to 2. In this case, dc of n, this is the quasi-fuchsian manifold that were discussed earlier by Jean-Marc and Sarah. And here I won't say what this theorem is, because it would take at least the whole talk. So there's the model manifold theorem of my three co-authors. So this is Brock, Marian Minsky. And what this theorem produces, it produces a combinatorial by Lipschitz model for the geometry of the manifold. OK, so I'm not going to say what this model is, but let me say a little bit about the kind of information it can give you. So the way you build this model, you need some instructions. And those instructions come from the curve graph. And what do you want to input in the curve graph? Well, so I have this quasi-fuchsian manifold, so x and y is the hyperbolic structures. So x and y are hyperbolic structures on s. So there are two points in Teichmann space. And what's the information we want to get out of them? I want to take a collection of curves. I want these x and y to determine a collection of curves. And how do they do that? Well, what I basically want to do is I want to take the shortest curves on x and the shortest curves on y. Well, I need to be a little bit careful about that, but I don't think I want to say it too precisely. But I have some curves, which I'll call gamma x. So this is, say, a collection of curves. And another collection of curves called gamma y. These are collections of bounded. They're not really bounded, but anyhow. Simple cos geodesics and y. And from this information, I can, for example, figure out what are the bounded length curves in x and y. So here's sort of a sample thing you can find. So if alpha lies on a tight geodesic in the curve graph, Cs from gamma x to gamma y, then alpha has bounded length in the three-manifold. So this is the kind of thing that you have this information about the geometry of the boundary. And it tells you about the geometry of the three-manifold. So you can get much more precise information than this. But this is just sort of a sample of the kind of thing you can prove. So what if you have a manifold that is not just a surface, an interval boundary over a surface? It's not just s cross i in the more general setting. So well, one thing we could do is we want to apply this great model manifold theorem to try and study this more complicated manifold. And the way we can do that is we can take a cover associated to the surfaces. So if s is some component of the boundary of n, let n sub s be the corresponding cover. OK, so maybe we have some longer piece of chalk here. Or at the end of the conference, it looks like we only have little bitty nubs of chalk in color. So the rest of the talk will be in color. Box where? Yeah, this box is filled with itty-bitty nubs of chalk. Or here's another box. Maybe we'll know this box. Yes, if anyone wants like little three-quarter inch nubs of chalk. Thank you. You found Francois's phone. You found the chalk. OK, well, maybe it would have been better if I had given the rest of the talk in color. All right, so we have this cover. Now, the cover is going to what does this cover going to look like? What is it going to look like topologically? What is it going to look like geometrically? Well, we can sort of answer both those questions at once. So it turns out that this cover will be one of these quasi-fuxian manifolds. Of Thurston, if m is some convex co-compact structure on n, then this is really a very special case. So ms is now the hyperbolic structure induced by taking the cover of m corresponding to this cover here, then ms is quasi-fuxian. OK, so this is a special case of what is known as Thurston's covering theorem. In this case, when it's actually a little bit easier to prove, well, it's easier to prove if we assume that our beginning manifold is convex co-compact. This is zero to zero proof. In any case, we have that this manifold is now quasi-fuxian. In particular, it's going to be a copy. It's once again going to be homeomorphic to s plus or close interval. OK, well, this is great because we have this theorem, this model manifold theorem, which tells us a lot about the geometry of this manifold. Except to take advantage of this theorem, we need to know what the two conformal boundary structures are. OK, well, we know one of them. So when I take this cover of n associated to the surface s, well, s is some conformal structure on it. And that will be the conformal boundary of one component of n of s. So if I look at this covering map, sort of at the top, it's just going to be an embedding. It's going to be an embedding of some collar of this manifold here. But the lower end, we don't know what that is. And that's what we're trying to figure out. If we want to use the information that we have here, then we need to know what that bottom surface looks like. So if x is the hyperbolic structure on s in the boundary of m, then x is one component of the boundary of this cover. OK, so what is the other component? Well, let's give it a name. And so let's let. So if I take this component s, then what can I do? I take a hyperbolic structure on m. I take the corresponding cover associated to the boundary surface. That gives me a quasi-fucine manifold by this theorem of Thurston. One hyperbolic structure is x. The other, I call sigma s of m. So this is the hyperbolic structure on the other. OK, well, I wrote this sort of suggestively. This is a map. So sigma of s, it's a map from convex, co-compact, hyperbolic structures on n to the Teichmer space of s. This is called the Skidding Map. This is Thurston's Skidding Map. OK, so we'd like to understand the behavior of this Skidding Map. So if we make some further assumptions on our manifold, we get a pretty good answer. So a manifold n is a cylindrical if every properly embedded annulus has a proper homotopy into the boundary. OK, so if you have an annulus that's properly embedded in the manifold, well, you can just homotope that to some annulus in the boundary. So for example, this is very, very much not a cylindrical, right? Because you can take a curve on s cross 0. You could take the same curve on s cross 1. And they generate an annulus that you're not going to be able to homotope into the boundary through a proper homotopy. That is a homotopy that keeps the boundary curves in the boundary. OK, so then we have, well, let's give this its name, what is called the Bounded Image Theorem. Well, this is generally attributed to Thurston, although there's a pretty significant last bit at the end that was first written down by Richard Kent. And well, what does this say? It says that the Skidding Map, so if n is a cylindrical, then sigma s, bounded image. I'm going to go into the chalk here. All right, maybe we have more. OK, so this is a very important theorem. It was a key piece of Thurston's proof of his hyperbolicization theorem. So the reason he was studying this is there was a certain gluing problem. He wanted to glue together two hyperbolic three manifolds to make another one. And this bounded image theorem played a key role. OK, so roughly speaking, and I don't know if this point could be made into an actual theorem, but roughly speaking, now what you can think of, if you have these a cylindrical manifolds, is there's some core of them? If you have any convex, co-compact, hyperbolic structure in n, what does it look like? Well, basically there's some core that's going to always be the same, or coarsely the same. And then the ends are going to look like quasi-Fuxian manifolds, where for each boundary component, the skidding image is coarsely going to be the same. So having bounded image, if you're doing, say, geometric group theory, well, you might as well think of this as being a point or something. So this gives you some starting point to describe a model for what the hyperbolic structure of one of these a cylindrical manifolds looks like. OK, so we'd like to come up with a similar theorem, but without this assumption of being a cylindrical. Let me just mention one conjecture before I move on. There's a following conjecture on Minsky's. So this theorem says that this image is bounded. So you'd like to be good to prove something stronger. And that is the following, the diameter. This is, again, in the a cylindrical case. So the conjecture is that the diameter of the image is bounded by a constant only depending on the genus. So I think one of Minsky's motivations for making this conjecture is that, well, we've learned quite a bit about the geometry of hyperbolic three-manifolds in the last 15, 20, 30, 40 years. And this is a pretty basic question that we still seem to have quite a tough time figuring out. There's various partial results, but I will not mention them now. OK, so how might we generalize this? How might we generalize this? So to state that the actual theorem that we prove involves some discussion of the characteristic submanifold, and I don't think I could. I'd be too embarrassed to discuss the characteristic submanifold in front of Dick. So I'm going to avoid that. And instead, just stick to a very concrete example, and then maybe at the end I'll say something about the more general theorem. OK, so what I want to do is talk about a very specific submanifold, the book of eye bundles. And it's going to be very non-acylindrical, but it's also not going to be a surface bundle. So it's about the books of eye bundles. So let's let sigma 1 through sigma n be compact hyperbolic surfaces with one boundary component of length l. So I just have a bunch of surfaces. They're hyperbolic. Maybe they have varying genus. But their boundary components are all the same length. So let me say that I'm going to do some geometry and topology at the same time. The book of eye bundles could be constructed purely topologically, but I want to think of it geometrically as I do the construction. OK, why do I want all the boundary components at the same length? Well, I'm going to glue them together. So glue boundary components by an isometry, form a two complex, so I'm just going to glue all these things together to form a two complex. And let's give this curve a name. The curve we're going together, let's call this gamma. That's the curve where all these things meet. OK, so in fact, you can embed this in R3 if you wanted to. I won't try and draw the picture. It's not that hard to do it, but it'll be on my personal drawing talent. But I can thicken this to form a compact manifold with boundary. So I won't really discuss this particular issue in the talk, but there's actually more than one way to do this. This is an example of a way of constructing three manifolds so I can rearrange the way I eat. So why is this called the book of eye bundles? So these surfaces are the pages, and we're putting them together to make a book. And you could rearrange the pages. And this gives you an example of three manifolds that are homotopy equivalent, but not homeomorphic. Putting the pages in different ways, I can make the homeomorphism class of this three manifold change. This gives lies to all sorts of interesting phenomena, which I will not discuss today. Well, how can I make this into a hyperbolic structure? What I want to do is I want to take this two complex, and I want to embed it. I want to take its universal cover, and I want to equivalently embed it in hyperbolic three space. So this x, it has some geometric structure to it. Because as I was saying, these surfaces that I'm using to build it, they're hyperbolic. They have a hyperbolic structure. So there's some geometry to this. And I want to see that if I make the geometry correctly, I can isometrically and equivalently embed the universal cover into H3. And then, well, when I say equivalently, I mean there's going to be some group of isometries of H3 that are going to act when restricted to the action of the universal cover of this thing will be the deck group. So I want to embed universal cover x tilde. I want to embed equivalently x tilde in H3 such that there is a group, gamma, of isometries of H3 that fixes x tilde set-wise, not point-wise, and acts as the deck group. So this universal cover will be sitting in H3. I'll have this group of isometries. And I'll have this group of isometries. And then when I restrict the isometries to the x tilde sitting in H3, it will just be the deck group. So the quotient will be this two compotes again. And if I do this carefully enough, I'll get a three manifold, H3, gamma, which is one of these convex co-compact structures on my book of iBundlesM. OK, so there's a lot of things, a lot of care I need to take when doing this. But the key thing that I want to think about, I'll draw it over here, so what does that universal cover look like? Well, it sort of looks like a tree, except it's got an extra dimension. But what do I need to do? If I want to embed, say, the regular trivalent tree in H2, it's a similar problem. I mean, if I look at this, there's going to be branching in every direction. And I want to try and fit this thing in H3. So what happens when I do the same thing with a trivalent tree in H2? What do I need? What sort of geometric information do I need to know about the tree to be able to do this? Well, the point is I need the edges to be sufficiently long. If I want to embed this tree in H2, if the edges were too short, then the thing could start hitting itself. But if they're long enough, well, what happens is any sort of path in here looks like a quasi-geoskeleton, they don't intersect. So to embed this tree, we need these edges to be long. They have to have some definite length. If we can't do this in R2, we can't do this in Euclidean plane, and if the edges are too small, then the picture will look too Euclidean, and there'll be some self-intersections that will run into trouble. So you can probably make some estimates using Gaussian A if you wanted to, but don't worry about that. So we have the same issue here. But what do I mean by edges? What is the length that I need to control? What is the length that I need to make sure is too short? So a necessary condition for embedding x tilde equivalently is that the shortest proper geodesic arc, the sigma i, must be sufficiently long. OK, you can sort of the corresponding thing in the three-dimensional picture. This is like some geodesic arc. This is like the lift to the universal cover of some geodesic arc connecting one boundary component to the other. OK, well, there's just a sort of a simple area argument you can do with surfaces. If these boundary curves are very long, then there will always be a very short arc. So this leads to the following. The length of L must be bounded. I mean, I'm sorry, the length of gamma. Length, gamma, bounded. This is the boundary. OK, well, this isn't maybe the most satisfying proof. But this is a special case of a theorem of thirst. Hopefully, this is the motivation of why it's true. So this is very different than in the case of just looking at a surface bundle. Well, what is a surface bundle? A surface bundle is a book of eye bundles with only two pages. And in that case, there's no control over the length of any curve if you're looking at an arbitrary hyperbolic structure on a surface bundle. But if you start adding pages, if your book of eye bundles is non-trivial, it has three or more pages, then the binding curve, this curve gamma, is going to have uniformly bounded lengths, only depending on the topology. OK, so now let me state our theorem. OK, so this is the relative bounded image theorem for this very special manifold. OK, so I'm not going to be able to control the entire skinning image. There's no way, because as I was saying at the start of the talk, for curves, so if I take a curve that's lying in one of the pages, I can make its length be anything I want on both sides. OK, so there's no way I'm going to be able to control the length of curves that lie on the pages. The only thing I'm going to be able to control is the length of this curve gamma. And so let me state it. So let's let S be a component of delta M. Then there exists some L greater than 0, such that the length of gamma on the skinning image, so for all, the length of gamma on the skinning image is less than L. This particular manifold is very close to being an eye bundle. It's sort of as close as to being an eye bundle as you can be without actually being an eye bundle, and this is the only curve that we have any hope of being able to control. So notice by the Bayer's inequality, this theorem applies the statement of Thurston, but we very much use the statement of Thurston to prove this. So we're certainly not giving a new proof of that. So again, this is the point. All other curves that lie on the surface, I can make them as long as I want. So there's no way I can be able to control their skinning image. They control the length of any curve in the boundary, and controlling it in the manifold. And since every other curve, I can make as long as I want in the manifold, I can make it as long as I want. So let me just say a bit about why this is true, and then maybe I'll say a bit more about what you expect more generality. So the result of Thurston is that the, so you have this binding curve, the core curve, where all the pages are glued together. And Thurston's result is you take any hyperbolic structure in the book of eye bundles, there is a universal bound on the length of that core curve. And this was supposed to be sort of why you should think that that's true, at least in this special case. I mean, his argument is different, but I mean, I don't know. I mean, probably for this special, I mean, he personally much more general. And probably for this particular situation, you could use this kind of argument to prove this statement, but that's not what he meant. OK. So let me just say a few words about why this is true. So what's our picture? So we have this book of eye bundles. So I'll draw sort of. So here's our original complex X sitting inside the manifold N. And let's say this is the surface S. And then we lift to the cover. So M sub S, the cover. And on the top, we have, OK, so here, I used X too many times, didn't I? OK, so over here, we have the original hyperbolic structure that was on the manifold M, boundary of M. And then down here, we have the skinned image. OK. And well, we have this result of Therson that says, if I look, so OK, so I have the curve gamma, which I'll draw as a dot here, and I'll draw as a curve in here. I have this result of Therson that tells us that curve gamma has bounded length. It's uniformly bounded length in the manifold. Unfortunately, we know that having uniformly bounded length in the manifold doesn't tell you anything about the length and the boundary. Doesn't give you any upper bound of length and the boundary. So what can go wrong? And so basically, there are sort of two things that can go wrong that could make this curve gamma be much longer down here than it is here. One thing is there might be some short curve, there might be some very short curve that lies below this curve, so this is some short curve gamma. And the other thing that can go wrong is there might be, this is a really different color, let's see, there might be lots of curves. Well, maybe they're not short, lots of bounded length curves. So these are the bad things. We want to rule out. We want to rule these things out. One, two, rule out both. OK, and why are these the two things that are bad? Well, this is, as I was saying before, we have this awesome model manifold there, and it tells us a lot about the geometry of three manifolds, and it comes from that, comes from that. OK, so how do you control this? Well, this is where the cores come in. So we have this sort of nice x sitting inside here. And what do we know about x? Well, we know all the things that we know about hyperbolic surfaces. So for example, we know things like the diameter of this surface is bounded away from the thin part. There's various things we know about the geometry of these surfaces. And then the other piece we know is bounded length, from this theorem of Thurston. So that's a very, very rough sketch of some of the ideas behind the proof. And let me just say a little bit about how you can do this more generally. So in general, is built three kinds of pieces. Eye bundles, air-based cylindrical manifolds, solid tori. OK, so this is the characteristic sub-manifold theory that I'm avoiding discussing. So there's a unique way of breaking your manifold into these pieces, which I am not being very precise about. But roughly speaking, what can I do? Well, I can take, say, some eye bundle, and then I can glue it to some three-manifold that's, say, some compact three-manifold with boundary. Or maybe I'll draw it slightly differently. Over here I have some, say, a cylindrical three-manifold. Oh, I'm drawing those there. So here's the boundary of it. And I can take this boundary of the eye bundle, and I could glue it to some curve in the boundary of the a cylindrical three-manifold. And then I could do the same thing over here with some other a cylindrical three-manifold. And maybe this three-manifold has some other boundary component. Maybe I'm doing some eye bundle. You can just piece these together. And what's the point? Well, the window is this part. So this is the window. The eye bundles are the window. You can see through them. You can push the curve from one side to the other. And the general version of the relative bounded image theorem says that if a curve is homotopic outside of the window, then you get this bound.