 Hyperplanes, a mid-cube. So now I'm going to, I apologetically told you that we're not going to care so much about the cat zero metric, even though we're going to be interested in cat zero cube complexes. And I said we're going to move towards more combinatorial way of thinking about things. The objects that it turns out we're going to care about are hyperplanes. And I'm going to start by defining them. A hyperplane in a mid-cube is a subspace that you get by restricting one coordinate to one half. So for instance, in this three-cube right over here, which we're going to conveniently identify with a product of three intervals, if we restricted one of the coordinates to a half, then we'd get that mid-cube over there. And there's two other mid-cubes in that three-cube. And well, maybe in this two-cube, there's that mid-cube right over here. And in this one-cube, there's that mid-cube right over there. You all know what mid-cubes are now. A hyperplane in a cat zero cube complex, which I will call X tilde to remind you that it's simply connected non-positively curved cube complex, is a subspace which is built from mid-cubes in the following way. It's a connected subspace whose intersection with each cube is either empty or an entire mid-cube of that cube. So maybe I'll get a picture in here of a hyperplane, or maybe a few, so bear with me for a moment. Do I really need so much clutter over here? Maybe not. All right. So here's a, I think it's a simply connected non-positively curved cube complex, right? Has anybody checked that it's a cat zero cube complex? You inspected all the links of zero cubes. And let's try to find a hyperplane, and let's do it in a suggestive way that kind of is going to help us understand what these things are. So let's start at a mid-cube right over here, this orange mid-cube right over there. And what are the rules? The rules are that if you intersect a cube in a non, if you have non, if your orange object, your hyperplane, has non-empty intersection with a cube, then actually your intersection has to be an entire mid-cube of that cube. And right now, our cubes are all closed cubes, by the way. Right now, it intersects this two-cube right over here at this point. But the rules are that you have to intersect at an entire mid-cube. So I've extended it. Likewise, that two-cube, right? And it's still not a hyperplane now, because notice that it intersects this three-cube over here, it intersects it at a point. So it has to intersect it at an entire mid-cube. And similarly for its neighboring three-cube. Well, now I've got a hyperplane. Let's draw another one. I think you probably can do it yourselves now. How about this yellow one right over here? There's a yellow hyperplane. Let's see if green is visible on this board. Yes? No? No, yes, no. Red, sometimes it works, sometimes it doesn't. But you said that the cubes are not necessarily embedded. So when you say intersection, is it just a pullback or should it be the image of it? Well, it's just a subspace. So I don't think I did anything wrong here. So I have an object. And I have a cube complex. And oh, I see your question, I understand. So what he's asking about is, well, do cubes actually embed inside of a cat zero cube complex? So it turns out that every cube in a cat zero cube complex is going to embed. And so you're going to be safe for certain concerns that you have. All right? We will get to that soon. OK, so you see there are some other interesting hyperplanes. I don't have to draw more. And in fact, I think it's not obvious. Anybody who's thought of non-positively curved cube complexes was eventually led to hyperplanes. But I think Sagiw was, I think he was first because it's tricky because there's actually a literature on things like cube complexes called median spaces that precedes the topic from other area in math by 70 years or so. So it's hard to say if people have been there already earlier since the group theorists are a little bit oblivious of that topic, although that's changed in the last 10 years. Every mid-cube lies in a hyperplane, so meaning every mid-cube of a cube within our cat zero cube complex lies in a hyperplane. Let's call that hyperplane V. Any hyperplane V separates x tilde in the sense that the cat zero cube complex minus V has two connected components, actually even stronger than that, what we call the carrier n of V, which consists of all the cubes, the union of all cubes intersecting V, all the cubes that carry its mid-cubes has the property that it is isomorphic to a product of V and a 1-cube. This carrier of our hyperplane turns out to be a convex sub-complex. I'm going to talk about convexity later, but you can understand it for what it means. Any geodesic whose endpoints lie inside of it lies inside of it. And you know about the metric on cat zero spaces that I described already. I'm going to tell you about another metric in a moment. It turns out that hyperplanes are what give cat zero cube complexes their character. The world is filled with many cat zero spaces, many wonderful spaces that satisfy this cat zero inequality. And cat zero cube complexes are among these spaces. But what really distinguishes them and makes them have extra wonderful features are these hyperplanes, these separating hyperplanes. And so let's see some examples that you're familiar with of cat zero cube complexes. Trees are the simply connected one dimensional cat zero cube complexes. And I already mentioned Euclidean space. That's Euclidean two space over there, cut up into squares. And maybe the hyperbolic plane broken up into squares. I'm sure most of you have gone ahead and played a little game of tiling the plane by something and sort of see what you get. And you learn that you have to make the squares very tiny as you're going outwards. You get a nice, if you have the tiling with five squares meeting around each zero cube is my favorite. And well, the hyperplanes in these objects or the hyperplanes in the trees, you might not have noticed them, because they're so small. They're just single points, the mid-cubes of edges. The hyperplanes inside of the tiling of the plane. Yeah, you know about that, but they're barring. And the hyperplanes over here seem just a little more interesting, et cetera. You kind of, they are going to look like some nice circle, arc of a circle that would intersect the boundary in the right way if I drew this properly. Well, idealizing this picture, I hope. And you're, you know, this statement about the carrier looking about right to you now. So you know what's going on, and you already were inspecting, testing it against that cat zero cube complex over there as well. I need to, before I go on, I need to be a little bit honest about the metric that we're going to focus on. We just suppose that in the terrain that the complex is a cat zero. Could you repeat that? In the terrain, is the cube complex always cat zero? In a cat zero cube complex, the following hold? Even for the first one. This is really true. This theorem's true. Yeah, yeah, yeah. But if you don't assume that the cube complex is cat zero. If it's not cat zero, then if it's not cat zero, but it's still non-positively curved, for instance, then you could end up with a situation like this. And you'll have a hyperplane. You might sort of know that you want to define your hyperplane. You start off and everything is looking good. You have this mid-cube, and you have another mid-cube, another mid-cube, another mid-cube, right? And then another mid-cube over here, another mid-cube. And then, whoa, you're going to crash into yourself, bang. And over here, we all know that that orangey red thing is a hyperplane anyway. We know it's true, but it didn't quite satisfy the definition. So this can fail. This can fail, if you're careful and you're ruthless about the definitions. But you do get what's called an immersed hyperplane. Well, that's really fine print. You probably can't see it. I'm going to talk about these later. They're going to be very important to us later. So it's good that you've, I think it's good that you've brought them up now. All right, so let's get to work over here now. My McGill students were waiting to watch me watch at least the blackboard. I didn't hear them giggling about it. Let's bring some stuff down, just a little hook. So I'm sorry about that. The metric that we will use on a cat zero-cube complex is the taxicab metric. I think people like to call that the L1 metric. We're not really going to use it a whole lot anyway. But let's be conscious of it. It has the distance between points equals the length of shortest path that is piecewise parallel to axes within cubes. So in contrast to the picture that we had before, where we kind of went across diagonally in order to get there quickly, now what we will do is we will force ourselves to travel in a very rigid way, always staying parallel to the axes. So what did I do? I went from here to there. So for the cat zero metric, I don't even want to draw it. I'll just make little dots with my fingers. We did it like this. Now what we're going to do is we're going to travel just parallel to the axes. And the length is whatever the sum of the lengths were. And unfortunately, we don't have uniqueness of geodesics, right? I told you the cat zero inequality actually gives you uniqueness of geodesics, lovely. But now there's even traveling around a single square, even traveling around within a single three cube, there are many ways, even if you restrict yourself to traveling in the one scale, to never mind traveling within the interior of the three cube, there's lots of ways to get as a geodesic from one zero cube to the other. But if you like the one skeleton, which I do, there's this way and there's also this way, for instance. And there are many ways, right? So here's a pink way. So it's certainly not a geodesic, it's not a unique, there aren't unique geodesics. There are geodesics, though. The truth is that I am mostly going to, well, almost entirely, focus on the metric on the one skeleton. It's going to be good enough for our purposes. And the L1 metric, or the taxicab metric, agrees with the usual path metric when we just focus on the zero cells. So let me say that in a more cleaner way. Induces the same metric on the set of zero cells as the usual path metric. And in particular, geodesics in the one skeleton between vertices are geodesics in either the taxicab metric or just the graph metric on the one skeleton. So for the most part, you can just think about the path metric on the one skeleton over here. Yes, we are not going to be cat zero. So unfortunately, we're going to use the term cat zero cube complex, but we're not going to be using the cat zero metric. What we are going to use is the enabling condition that the links of zero cubes are flag complexes. That's going to be a very powerful condition, and it's going to control everything that happens over here. So when you write x0 tilde, this is the set of zero vertices? No, when I write, yes. So this really means that. OK, but then if you have a square and the two opposite vertices, the distance is square root of 2 in the previous metric, and now it is 2. So it is not the same metric. Right, yes. So you say it induces the same metric. It induces the same metric on x to the 0 as the usual path metric on the one skeleton. The cat zero metric, forget about it. I'm only told you about it because I was pressured to do so. My roof is just going to confuse everybody. In fact, it's beautiful, and it's extraordinary that Gromov threw these out as, oh yeah, these are good examples for you guys. Do these non-positively curve cube complexes to give you many nice cat zero metric spaces? But the cat zero metric, in my experience from thinking about the cube complex as something that's just about cube complexes, if you're using the cat zero metric and you're sort of moving more towards traditional common and total group theory arguments with thin triangles and constants and whatever, it turns into a huge mess. But if you just say, I'm only going to use the flag complex that the links are flag complexes, you support that? All I care about is that cat zero spaces are contractable, and for that, I have to use the cat zero metric. No, you don't. You could just use the, and it also works when there are infinite cubes as well. OK, so it's interesting to try to, there's a sort of theme of combinatorial non-positive curvature, which is alive and kicking right now, and we'll see what wins out the geodesic metric non-positive curvature or combinatorial non-positive curvature, it's interesting, what's going to happen for art and groups, we shall see. I think I better go back to my lecture, and where was I? I did the taxicab metric. I'm going to tell you now about immersed hyperplanes and convexity. Yes, I will. I'm actually not going to use the metric a whole lot, it's going to come up for a bit now. An immersed hyperplane in a non-positively curved cube complex, you already know what it is. Let me say it in a slightly more formal way, is a component of the space obtained by starting with the disjoint union of the mid-cubes of cubes of x, and gluing them together by gluing mid-cubes of cubes of x together along sub-mid-cubes. I'll just say along sub-cubes, cubes. So what is meant over here? Let me move this guy up a little bit, bring it back down in a moment. Let's draw one, let's give myself a ton of space. They're on the floor because I throw them on the floor, right? Which floor? Are they camouflaged? OK, are they both there? Cool. Organized. I need that guy up, take his friend down. OK, so here's a picture of immersed hyperplanes. So here's a, ooh, this is too much. OK, so there's a three-dimensional, non-positively curved cube complex. And let me draw some of its immersed hyperplanes, but I'll draw them on the outside. So maybe I'll give myself a little hint to make it easier. I'll take that little square and I'll start with that. So here's this one as that little square. And then there's that one kind of mid-cube that was there. And then there's another mid-cube of this square, of this three-cube, rather, and it continues on over there. And actually this, the first two-dimensional mid-cube kind of continued like that. And that is one of my immersed hyperplanes. But there are others, right? There's also the green one with a purple one over here. Maybe I'll add it over here. And it continues like that. And I guess there's another one over here that I forgot about. I'll put it over here, see. And there's another one over there that maybe I also forgot about. OK, and perhaps there's one more that I also forgot about. There's three more that I forgot about. In the definition, do you still require that to be right? So you have to continue going until it's possible? Yes, yes. And in fact, I think you all know what they are now. In fact, if you had taken the universal cover and you looked at the hyperplanes in the universal cover and then you quotiented them by their stabilizers under the action of the fundamental group, then the components that you get when you quotient are going to be these immersed hyperplanes. The objects that you get when you quotient the hyperplanes in the universal cover by their stabilizers are going to be these immersed hyperplanes. But you can think about them downstairs in the base space just by just collecting all of the mid-cubes that are living inside, just the disjoint union of all of these mid-cubes glued together in the obvious whenever they should be glued together because they share a sub-mid-cube. And if you do that, you'll get this. So now we know what immersed hyperplanes are. And they're going to be very important for us. As for hyperplanes themselves, they have what we might call carriers. So these are immersed hyperplanes. And they come equipped with maps, their components. They come equipped with this map. The carrier Nv of one of these of an immersed hyperplane, you probably know what it is. You just kind of thicken it up. Is the cube complex obtained by gluing together ambient cubes instead of mid-cubes? So, well, I guess in this picture over here, there would actually be that purple hyperplane is actually living inside of this carrier right over here. And likewise, the others. I'm not going to draw them all. And usually they look just like products, but not always. It could be a little more complicated than that. Okay. Are you good on what the carriers of hyperplanes are? Yes, of immersed hyperplanes? The orange-red picture. What's the carrier that's going to be simply connected for us? Okay. So let's draw it because it's self-crossed. I'm going to draw it. It looks like this. Maybe it wasn't drawn in a way that makes the embedding so clear, but it doesn't matter. You'll deal with it. I think it looks like this. And I'm not going to draw it as red and orange because give me a break here. Okay? I can't control myself. All right? So an n-cube is going to kind of make appearances n times within the carriers of all of these immersed hyperplanes. Right? Once for each of its mid-cubes. All right. Now let's check some important things. The orders of the lectures is important because we are in lecture three. Yes. We are in trouble. There are my disk diagrams. Not here. Not here. Not here. Ah. Here. We will go with the flow. It's going to be okay. All right. So we mentioned immersed hyperplanes. Yes? Since you're already in lecture three, we can just take it away. No, that's not necessary. So you say that it is not always a product, the carrier? Okay. You want an example? Okay. So you've seen this example. I forgot what it's called. You've seen this? Yes? All right. Okay. We will return to it. It's important. Okay. Now we know immersed hyperplanes and it worked out for us. Let's talk about disk diagrams now. I thought the carrier lived in the universal cover. So why isn't the carrier of this infinite string of squares? Because this is the carrier of an immersed hyperplane. And that was the carrier of a hyperplane in the universal cover. This is consistent with the definition that we gave before. It turns out, right? You have to know various things. You have to believe that hyperplanes exist and embed and so forth. You pull apart things only when there are different hyperplanes in the projection. Okay. Tell me, what are you asking? Are you asking me to tell you, are you asking about the definition of the carrier of an immersed hyperplane? Maybe just of an immersed hyperplane. I guess I... You're asking what is an immersed hyperplane? So the reason why you get two distinct orange and red ones there is because they're distinct, they have distinct projections. I never use the word projection. So you're looking at the red and orange mid-cubes over there inside of that three-cube? Okay. So if you take all of... So we intuitively define what an immersed hyperplane is to start with. And then I gave you a hint about how to make a rigorous definition. The hint of the rigorous definition is that you take all of the mid-cubes. You take their disjoint union. And then you glue them together. Really, it's enough to identify sub-mid-cubes of mid-cubes. That will perform all of the gluings that you want. Okay. And, well, that three-cube is going to, in this sort of rigorous definition, that three-cube contributes three mid-cubes, a purple, an orange, and a red. And they're going to live in, well, these two happen to lie in the same immersed hyperplane, right? And this one lies in another. You're good now? I move forward. A disc diagram. We're disc diagrams defined in this conference. So a disc diagram is a compact, contractable, two-complex, let's call it D, with a planar embedding. But you know what? Let's put it in the two-sphere. Here are some examples of disc diagrams. And you know what? I'm going to stick to disc diagrams that are built from squares, because that's what we're going to be focusing on. Here's just a tree. There's a single point we sometimes call the disc diagram trivial. These are all two cells. And these are compact, contractable, two complexes. And I've embedded it in the plan in a very particular way. The boundary path of the disc diagram D, or boundary cycle, is the attaching map of the two-sphere containing infinity. There's a pointed infinity, haha. And if you look at the two-cell containing the pointed infinity, then it has an attaching map. And here it is. That's the boundary of the... There's a whole two-cell right over there, and there it is. That's its boundary cycle. Likewise here. I guess I didn't really need to do this sort of nifty way of catching it, but it's useful for this guy, because there's more than one embedding. There's more than one embedding of this two-complex in the two-sphere. This one basically has only one embedding. There's that, and then you can reverse the orientation. And this has many embeddings. And so you really, in order... Well, you could say that if you had declared what the boundary cycle was, well, I'm not going to be fussy about the orientation. Here's the trivial boundary path. Okay, it is a famous fact, it was a theorem of van Kampen that was long overlooked, which says that for any closed no-homotopic path, p to x, this is going to be, say, a two-complex or a complex, there exists a disc diagram, d, and actually a map from d to x, such that p is the boundary path of d. Okay, what do I mean? Well, if it's a very nice way of seeing the no-homotopy, so we know that no-homotopy means that it factors through a disc on the way in, but there's a more combinatorial, this map is a combinatorial map, and a combinatorial map. A map between cell complexes is called combinatorial if it sends homeomorphically to open cells. So cells, one cell goes to one cell, two cells go to two cells by homeomorphisms. A very, very nice map. So you have some complex, something really messy and complicated, and you have a path p, and you know that it's no-homotopic, well, you can actually see that it's no-homotopic by exhibiting it as the boundary path of some disc diagram. And of course, if p factors through this disc diagram on the way to x, then p is no-homotopic. And actually, it's basically a simplicial approximation theorem, if you have a no-homotopy, so you're factoring through a disc, but it's really a topological thing that's happening, you can adjust and choose and actually factor through a disc which is kind of tessellated. Oftentimes, it's still homeomorphic to a disc, but sometimes it will have singularities to it, these are called singular points. So this isn't really homeomorphic to a disc. It has cut points. All right, so we're going to use this. We're going to use this. Excuse me? Thank you. Thank you very much. Okay, so we're going to... Of course, once you start with this, you're interested in minimal diagrams, minimal disc diagrams have fewest cells. Usually people focus on the number of two cells, and they call that the area of the disc diagram, among all disc diagrams with a given boundary path. So the combinatorial group theorists view this disc diagram as the proof that P is an homeotopic. They view this disc diagram, and you can actually think of it as encoding the path P as the product of conjugates of relays if you're thinking about a group presentation, if you've seen this before. And while we like short proofs, so we prefer to find a minimal size disc diagram that shows you that it's an homotopic. Now, if you're working in a cube complex, when d to x is a disc diagram in a cube complex, d is a square diagram. It's going to just be mapping into the two skeleton. And we're going to focus on its immersed hyperplanes are called dual curves. So let's give a picture that will explain all this. There, there's a disc diagram right over there. I don't know if it's minimal or not, I didn't check. And you already know what these dual curves are. They're just what we would have called immersed hyperplanes. Here's a dual curve right over here. Here's a red one. And here's a yellow one. Maybe there's a green one to test the green on green theory. So everybody knows what dual curves are now. And these dual curves are kind of, their carriers are ladders. So for example, the carrier of the yellow dual curve looks like this. Let's see, it looks like a crooked ladder. And well, I guess in principle, they could be cylinders. It could look like this also. That could happen because it's in a disc diagram. And the disc diagram is a contractable two complex. You said it was in a cube complex. D is a disc diagram mapping to a cube complex. You think the carrier in D, not in X. That's correct. So the lingo, unfortunately, the lingo is a map from a disc diagram. So the disc diagram is living inside of the plane. And when we say a disc diagram in X, we mean a map from that disc diagram to X. And we're talking now about the dual curves, which are these immersed hyperplanes in the disc diagram. So let's state a quick theorem that if D to X is a minimal area diagram, disc diagram, then D has no, and there's four configurations that we want to draw your attention to. It has none of these. And it's statements about how, it's statements really about the carriers of dual curves. This is the most important. There are no bygones. So I'll even say no bygones. There's a little furthermore that's interesting, but I'll kind of leave it out over here. Let's talk. So I have two minutes plus three minutes because I had to erase the blackboard on my own. What else do I got? I don't have a lot, huh? It works that way. It works that way. Whatever I want, no. That's Friday. So let's talk about the, I'm going to describe the method of proof, but I have to sort of be, I have to tell you maybe how the story fits together. What we're going to find is that, here it goes, method of proof. Indira made me do this. Hexagon moves. So if you ever see, if you ever see this, either of these configurations inside of your disc diagram, you could swap it and replace it with the other. Because if you see one of these, there's actually a three cube inside of the cube complex, and you could take that disc diagram and you could push it across the three cube to get a slightly different disc diagram. And there's another important thing that you might do. If you ever see two squares in your disc diagram that are meeting along a path of length two, they're forced, since the link is a flag complex, they're forced to map to the same square inside of x. You didn't mention that it's not possible to make it. Oh, you're kidding me. That's not over there. It's over here. Thank you. If you have a pair of squares that are meeting along two edges, right, then those squares have to map to the same square inside of the cube complex x. Why is that? Well, the link, these two corners of those squares, have to map to the same corner of a square at the image of this white vertex over here. And the links are flag complexes, so there's only one edge joining two points in the link. So if you ever see two squares that are meeting along a pair of edges inside of a disc diagram, they have to map to the same square in x. But that means that, well, you could just, what it means is that these two edges are the same as these two edges. So if you ever have this, then you could have just cut it out of the disc diagram and replaced it with that. And so your disc diagram, which has this, what's called a cancelable pair, can be made smaller by just doing this simple replacement. This is a very important notion in combinatorial group theory, right over here as well. If you have a disc diagram that contains this picture, you can replace it by a disc diagram that contains exactly that picture, because these two hexagons, these two hexagons have the same boundary path. So I am going to stop now, which is an incredible display of self-control. Thanks, good self-control.