 let's call this a lemma, the map from the carrier of a hyperplane to x, I'm not going to say anymore that x is non-positively curved. They really all are non-positively curved, is a localisometry. So maybe that's a lemma, maybe that's an exercise, but you'll accept it. And the corollary is that the universal cover of the carrier, which is the carrier of the universal cover, as yesterday, or as today's ago, I see there was a break yesterday. So let v be an immersed hyperplane in a non-positively curved cube complex. Then its carrier maps to x by a localisometry. You can kind of believe that from the pictures, but you have to check. And well, I suppose in particular, if x is simply connected, carriers of hyperplanes of a cat zero cube complex, x tilde, are convex subcomplexes. And in particular, the concern that you might have had that maybe you start somewhere and then you crash into yourself as you're wandering around through x tilde, that concern has gone away. Because we could have defined a hyperplane in x tilde as an immersed hyperplane. Those exist. You start off with x tilde. It has immersed hyperplanes inside. Maybe they crash into themselves. Who knows? Those immersed hyperplanes map to x tilde by a localisometry. This could have been x tilde right here. And localisometries are embeddings. So hyperplanes aren't going to crash into themselves. Their carriers don't even touch themselves. So that's how we know, or it's one of the ways of knowing, that hyperplanes exist is knowing that the carriers of immersed hyperplanes are localisometries, mapped to x by localisometries. Maybe something that you might consider if you do have the carrier of a hyperplane, then you actually have various other convex subspaces. In fact, this, which we sometimes call the frontier of the hyperplane, is convex because it's convex inside of the carrier. And the carrier is convex in the whole thing. And actually, this shaded part is also convex. That's the sort of, it's bounded by this convex object. It's going to also have to be convex. Because if you can't leave it and come back, because that would violate the convexity of the yellow thing. This is called a half space, or sometimes called a minor half space. Maybe I'll write that. A minor half space, because there's also a larger half space, which also is convex. Here's the larger half space. And here's the major half space. And here's the minor half space. They're both convex. And you can prove it by verifying that they're mapping by localisometry, or it follows from other things that we've just discussed. So in fact, just to put things into perspective, well, a subcomplex is convex if and only if it is the intersection of minor half spaces. So sort of just to put a little things into perspective over here. So that's what convexity can be interpreted in terms of minor half spaces. And let's actually say something a little bit more fundamental about convexity now. And it's the following statement. So this is now going more towards the world of groups. Let X tilde be a delta hyperbolic ket zero cube complex. And let S be a kappa quasi-convex subspace. Tell me quasi-convexity was defined last week. Thank you so much. Then there exists R such that the intersection of half spaces, of minor half spaces containing S, let me write it over here, such that the whole of S, which is the sort of combinatorial convex whole of S, which is the intersection of half spaces containing S, I'll explain that notation in a moment, is contained in the R neighborhood of S. So let me explain that notation. So this over here is N, is a carrier, right? It's short for N of a hyperplane. And this minor half space, I think I'd use the notation for all of this yellow part over here. I would call that N left, right? And maybe, I guess if you'd like, you can call this part over here N right. But who's to say which is left and which is right? So I'll just write N left now. So if you look at all of the half spaces which contain S and you take their intersection, that's the hull of S, that's actually contained in a small neighborhood of S. So the picture is the following. We have S, excuse me, yes, please? Excuse me? Do we know that hyperplanes are two-sided? Yes. So once you know that the carrier embedded in the carrier is this universal cover, from there you could see that it's two-sided. Because if it were one-sided, then there'd be a path in the carrier, the carrier is simply connected, of course, because it's living inside. There'd be a path in the carrier that goes through the hyperplane and only touches it once. And so that would violate that it's simply connected because there'd be a cycle in homology. OK, thank you. What do you say? What if the whole thing's free faces? Excuse me? What if the hyperplane carrier, all of it like your face and then it's like a free face of the KU complex. So there's like nothing. I didn't understand that. So we will talk about it afterwards. Let me draw some pictures. I have so many things I need to tell you. Every time you guys ask me a question, you're losing things. So it's painful. It's painful to be here. So you can look. Here's S. And the picture is that we're looking at all of the half-spaces that contain. Here's one of them. Here's another one. Here's another one. These are all the half-spaces that contain it. You might have seen pictures like this somewhere before. And this whole object is contained inside the r-neighborhood for some r. So here is my hull. I'll draw it in white. And that hull lies inside of the r-neighborhood of S. That's the picture that you should have in mind. The idea of the proof is somehow I'm not going to explain it. But the idea of the proof is along the lines of, well, if you traveled out more than r, then you could find a hyperplane which doesn't cross. It doesn't cross S. So because of the hyperbolicity, if you travel out far enough and you look at a cube that you're near and one of the mid-cubes of that cube is going to give you a hyperplane which doesn't intersect S. And so any point that's sufficiently far out over here is going to be cut off from S by some hyperplane. That's roughly the way the proof goes. Now the corollary of this is very, very important. Very important. It's critical. Let X be a compact, non-positively-curved cube complex with pi 1 of X hyperbolic. Let H be a quasi-convex subgroup, then there exists a local isometry from Y to X with Y compact and pi 1 of Y mapping to H. So I'll just say equals H. What this is telling you is that if you ever have a quasi-convex subgroup of a hyperbolic group that is the fundamental group of a compact cube complex, you can represent that subgroup with a non-positively-curved cube complex. This is generalizing our understanding subgroups of the fundamental group of a bouquet of two circles. Whenever you have a finitely generated subgroup of a fundamental group of two circles, that subgroup is represented by a combinatorial immersion of graphs for those of you that have studied free groups and that's critical to understand this. This is a main point in understanding their subgroups is this idea that you can always represent subgroups using an immersion of graphs. Over there it's quite easy because there are no squares around. So the existence of this is quite simple. You just look at the covering space. You look at the covering space associated to your subgroup and you choose a finite, if it's finitely generated, you could choose a little finite subgraph which the whole covering space deformation retracts to. And any immersion of graphs is a local isometry, so it's easy. This is just the generalization of that idea and it works quite nicely. And it's part of a theme where, you know, freeness gets replaced by hyperbolicity, graphs get replaced by cube complexes. Okay, so, oh, oh, it's painful. Let's cut this and maybe I'll get back to it. So it's kind of Galois correspondence for local isometry, right? Excuse me? It's kind of Galois correspondence for local isometry. I don't know what Galois correspondence means. Okay, so it's, you mean a correspondence between subgroups and covering maps? No, it's not because it's only for quasi-convex subgroups. So let's now move to special cube complexes. Okay, so what I'm going to tell you now is about a category of non-positively curved cube complexes that are, that even better echo the nature of graphs. So graphs, we feel like we can understand their subgroups or their fundamental groups. So this is gonna be a certain non-positively curved complex, non-positively curved cube complexes that are especially nice. So here goes, a non-positively curved cube complex X is special if each immersed hyperplane embeds and is two-sided doesn't self-osculate. I have to tell you what that means. And also, and no two hyperplanes inter-osculate because these have to be clarified. So I'm going to suggest what these, you know, I'm going to draw pictures of what's not allowed to happen. So no pathologies, no pathologies of the following four types. Okay, now my pictures are going to be in two dimensions, but I think they get the point to cross. Well, self-crossing is not allowed. One-sidedness is not allowed. A hyperplane cannot self-osculate and what that means is that it's not allowed to come along right next to where it was, okay, but we're very picky about what that means. Actually, we're a little careful about it. Since hyperplanes are two-sided, we can do a convenient thing. We can orient, direct all of the edges consistently. We don't have to worry about being well-defined over here. And self-osculation means that you can't have, self-osculation means two edges at a zero cube, which are dual to the same hyperplane and are directed in the same way. They're directed, they're both directed outwards, let's say. Okay, this is not allowed. 10,000 ways are not allowed? Well, the other way, if you change the orientation, it would be this way. So the two ways are exactly the same. So it's not allowed, oh, I see, the other way. This is allowed, I can draw it properly. This is okay. I don't wanna draw things that are allowed here, but I'm gonna say okay over here. Okay, that's what you're asking about, correct? Yeah, excellent. So, inter-osculation is the following. It's a pair of hyperplanes that cross each other, they cross each other in one spot and then they oscillate with each other without crossing at another vertex. Okay, this is inter-osculation. This is self-osculation. This is inter-osculation. It doesn't matter, it doesn't matter. So the key thing that matters is the idea that there's no square over there. Okay, so the key thing is that even though they are dual to one cubes, which even though they're dual to one cubes at this zero cube, they don't cross the same square at this zero cube. Okay, so inter-osculation says that there's a square over here, but there's no square over here. I'm gonna draw that square, but it's not there. So I'm gonna draw it in purple, so it's hard to see. No square, because what would happen if there were a square over there? They would cross over there, because they would both enter that square on two different one cubes right over there and then they would cross. Okay, the way when Friedrich Haglund, with whom we cooked up this definition many years ago, at the beginning we would remember it by saying, crossing pair has a square, okay? That's me, because Friedrich remembers everything after one time, and I have to make little rhymes for everything to remember stuff. And this looks similar to something we discussed earlier, doesn't it? The further more, the further more is somehow related to this. These are very artificial looking definitions. I hope you agree, I mean they certainly are. They came out of nowhere for us. Let me give some, let me mention, so at least I'll mention an exercise. I should give an example, I'll give one in a moment. So among our exercises, if X is contained in the product of two graphs, then X is special. Okay, so now at least you know that we're not talking about the empty set over here. Any graph is special, we're also going to exercise the salvetti complex of a rag is special. Now I understand I'm supposed to call them right-angled tits groups, yeah? Okay, so, and actually a cat zero cube complex is special. Let's stop for a moment, I'm not going to prove it in its entirety, but let's stop for a moment and think about this statement over here that a cat zero cube complex is special. Well, hyperplanes don't self-cross, right? We've already dealt with that. And actually, they're two-sided, the carriers look like products, right? And they don't self-osculate, those carriers embedded, okay? They don't even have the other type of osculation that you thought of, right? And they don't interosculate, that's because of the furthermore statement. I'm not going to spend more time on this, but it's because of the furthermore statement that they can't interosculate, all right? And then, well, we cooked up the notion of a special cube complex for a different purpose, but fairly quickly, we found the following, and that purpose I will tell you about in a little while, it will be our focal point, we found the following that a cube complex is special if and only if there exists a local isometry from x to r, where r is a salve di complex. Of some graph, okay? So, and actually, this turns out to be easy. It was very strange for us when, we didn't believe it at first. Well, so this direction holds because pathologies project to pathologies, okay? Namely, well, if you believe that a salve di complex is special, right, that's an exercise to do that. The hyperplanes aren't so big, things are pretty small, there's just one zero cube, so it's handleable. Well, if you have a pathology in x, like a self-crossing hyperplane, it would project to a self-crossing hyperplane in r, but r is special, so that can't happen. Likewise, the other pathologies project under a local isometry. In this direction, I will tell you that the most important thing is that the graph gamma, right, so if x is special, we're going to create a local isometry from x to a salve di complex in which salve di complex, right? Because there's one for every simple graph. So let's first notice which graph we're going to use. We use what's called the crossing graph, the hyperplanes, what's called the crossing graph. Gamma has a vertex for each hyperplane and an edge joins vertices whose hyperplanes, whose associated hyperplanes cross. So maybe I'll quickly draw a picture to get an example going over here. Okay, so this is a cube complex with six, seven, eight, nine squares over here, okay? Nine squares, don't tell me I got that wrong. And it's actually, it's special, right? You can check that it's special. This is special, okay? There's a bunch of things you need to check all the, right, but certainly you check that the hyperplanes don't self-cross, right? And you see that they're two-sided. You, nobody sees any self-osculation. And I promise there's no inter-osculation. Okay, that's the way the proofs go. What is this gamma? Let's try to suggest gamma. So maybe I'll do it like this. So there's an orange hyperplane. I'm gonna run out of colors. But everybody see that orange hyperplane? Well, I can draw edges dual to it, and I'm actually gonna mark them and direct them. Okay, so instead of drawing the orange hyperplane, I can just draw a bunch of edges labeled by, you know, single arrow that are all dual to it. And let's do that for the other hyperplanes as well, because this is gonna tell us everything over here. So there's this hyperplane here, right? And it's a sort of double triangle. And there's this single solid triangle as well. And then there's this triple solid triangle. And then I've got my hyperplane, which is dual to the triple arrows, and that's it. And then, huh, no? One more, one more, I'm having trouble seeing it. Who sees another one? The three cube has one, two, three, four. There's one three cube over there. Let me, you guys can tell me, you'll correct me. Oh, you mean one more of these? Yeah, yeah, of course, of course I didn't finish that. So the goal, I guess the goal of these sorts of lectures is to get people to the point where they're correcting you frequently. So we're almost there. So there are all the hyperplanes, and now we might as well, I recognize gamma. So the, I'm sorry, I'm not gonna draw it orange. It's drawn white over here. But one arrow crosses two, two and three cross, hyperplanes two and three cross, hyperplanes one and three, hyperplane one crosses with double triangle, and hyperplane one crosses with single triangle. Okay, so this is gamma. I also have a triple solid triangle that I left out, and thank you, and it's right over here. Thank you very much. All right, so here's x and here's r of gamma. You know, r of gammas looks something like this, et cetera. Here's it's one skeleton, and then it has a bunch of squares and it has a three cube as well, right? That's the one skeleton of our gamma. Now everybody sees a map from the one skeleton of x to the one skeleton of r, because the way that I've directed and labeled the edges determines a map, right? You just send each edge to the corresponding labeled edge and you know exactly what direction to use. And well, whenever you have a square over here, that means those hyperplanes, whenever you have a square over here, that means the corresponding hyperplanes cross, which means that you're gonna have a square to map it to. So every square has a square to map to over here. And whenever you have actually an n-cube, it means all end of those hyperplanes cross. So if you have an n-cube over here, it knows there's a place for it to map to over there. And actually this ends up being a local isometry. So we used, for instance, two-sidedness to know that we can direct things consistently, the fact that it's a local injector the fact that it's a local injection is gonna need as well that the hyperplanes don't self-cross and that they don't self-osculate. And then the idea that there are no missing squares, which is what we need to make sure that it's a local isometry, that's actually going to follow from the inter-osculation. So this is, you know, something that can be worked through, takes a little while to think it through, but it's really that simple. Okay, and I've kind of sketched out, the main point is to sort of realize what the map is. Now, what's the corollary of this? Yes, what's your question? Do you know that X is compact here? Nope, excuse me? If X is non-compact, why is gamma finite? It's not. Why is it finite? So it's already complexity in front of it, perhaps. Sure. There's a lot of people out there that want their cube complexes to be compact. We saw a wonderful one yesterday, two days ago, she's not here anymore, right? Otherwise, she would have been complaining. In fact, I actually have to warn you all that geometric group theorists love compact things because then they feel secure that they have control. Right, everybody, you're all control freaks, right? But the reality is that compactness is not natural. It isn't always the right thing. If you're studying finite, if you're studying finite presented groups and you wanna have control, the control actually comes from other elements. Usually, we're gonna see what they are tomorrow. Okay, so if you want, tomorrow we will see. Okay, but compactness is we don't necessarily want compactness, although we'll take it. If you can give me compactness, I'll take it. I love it, okay. But we don't need it, though. Okay, so, oh, am I 35 minutes over or I have another 25 minutes? Yeah, that was a good way, I did it right. So what's the fantastic corollary that we can state? If X is special, then pi one of X embeds in a rag, right? So we have our local isometry from X to this salvetti complex. The fundamental group of the salvetti complex is this right-angled art and group, which is a very trivial-looking group, right? Bunch of generators and some of them commute, okay? So if you manage to find a cube complex that is special, that cube complex, its fundamental group is somewhat reasonable. It's at least a subgroup of something that you might understand, okay? So this turns out to be so special cube complexes form a bridge from, you know, from some type of combinatorial geometry world to a kind of combinatorial group theory because right-angled art and groups, they're the sorts of groups that would make combinatorial group theorists happy. You don't even need geometric group theory to a certain extent. You don't even need geometric group theory to understand them. Okay, you could just, you know, mess around with letters. So life isn't so easy, though. Well, unfortunately, we end up getting led to worrying about virtual specialness. We end up with a question because we might find a non-positively curved cube complex. I'm going to show you where they come from tomorrow. Okay, you can make them by hand, which is great, but they actually show up for other natural reasons. And your cube complex is probably not going to be special because it'll surely have some pathologies. I mean, self-crossing happens very quickly. If you just randomly glue cubes together, forget it. You already have some self-crossing. Forget about the non-positive curvature, okay? Even if it's non-positively curved, you'll probably have some self-crossing. So one is led, though, to try to understand virtual specialness and there was a long exploration to understanding this for the last almost 20 years now, which culminated in A goal, proving that if X is compact and pi one of X is hyperbolic, then X has a finite cover that is special. It's virtually special. Okay, so X has a, this cube complex whose fundamental group is hyperbolic has a finite cover, probably a very big finite cover, but a finite cover which lives, which shows you that this finite index subgroup lives in a right-angled art group and it answers many, many questions about that hyperbolic group because right-angled art groups are linear. They live in SLNZ. They are orderable. They have all sorts of lovely properties. Okay, so this teaches us many, many things about a very, very large class of groups. And it turns out that the hyperbolic groups that are fundamental groups of non-positively curved cube complexes are quite common. And one could argue that most of them are, right? At least when there aren't that many relations, most of them are. When you have many, many relations and I have been guilty of not saying this, when you have many, many relations then you likely have property T and which I'm not going to talk about and property T is mutually exclusive with cube complexes. So a group with property T is not the fundamental group of a cube complex unless the group is trivial. All right, so where did this notion of special cube complex come from? Because I told you, even though we were very excited that if it's special then it lives in the rag because it's being special is actually, it's easier to just define special this way, not that way, that was a hard way to define it was a strange way. Where did this notion of special come from? So let me explain. So there's something that we call canonical completion and retraction and I'm just going to suggest it. So there's the following theorem about special cube complexes. Let Y to X be a local isometry with Y compact and X special. There exists a finite cover, a finite covering space of X. I will use the offensive notation such that Y lifts to this, what we call canonical completion. Moreover, there exists a retraction from this covering space to Y. Okay, so I'm remiss. Let me draw a little bit of a picture, a diagram of what's happening. So you start with a local isometry and the theorem says that there's a finite covering space of X such that the local isometry from Y to X actually lifts to an embedding. And not only that, there's a retraction back to this embedding of Y. So I'm not going to explain this for cubes but I'm going to explain this for graphs in a way that I hope will take everybody along and you'll know what exactly, what sort of thing we're talking about over here. So excuse me. You assume that the spaces are connected. Well, it is assumed but it's true. It's true, the non-connected, let's see, the non, I think that the, yeah, I mean the non-connected case follows from the connected case. But assume that they're connected, so we don't have to talk about that. Okay, so let's, by the way I appreciated that you came and asked me to check the trivial, trivial cases. I, John Stallings did that, used to do that to me whenever I told him anything and I do it to other people so we should all do it to each other all the time. So here's a lemma that a right angled art and group is residual finite, there's various ways of proving this. Well, what is residual finite? Residually finite means, everybody knows what that means? Yes, every element, every non-trivial element survives in a non, it survives in a finite quotient. What I can tell you is that 20 years ago you had to say what that meant every time you gave a talk at a group theory conference. So the world's changed a little bit, it's kind of pulled, we pulled it along over in this direction, which is nice. So why are right angled art and groups residual finite? Well, one way of proving it, you could prove it in many ways. One way of proving it is that right angled art and groups are subgroups of coccider groups. They're subgroups of right angled coccider groups, actually, and right angled coccider groups, I think are widely known to be linear. And linear groups, finally generated linear groups are residual finite. So that's one of the explanations. And then another lemma is that a retract of a residual finite group is separable. So what does separable mean? It means, well H, subgroup H of G is separable if, well, H is equal to the intersection of finite index subgroups of G, probably infinitely many finite index subgroups of G. For those of you that prefer, you can put the profinite topology on G and this just means that H is closed in the profinite topology, okay? Residual finite and this means the profinite topology on G is, that's the topology given by a basis of cosets of finite index subgroups that that topology is Hausdorff. Okay, so this is a fun lemma. It's kind of one way to, there's various ways to prove it, one way is, I guess, retracts of Hausdorff topological spaces are closed. Okay, that would be, that's how it fits into things. Okay, so I mentioned these two though in order to observe a corollary of this theorem. So we get the corollary that pi one of y is separable in pi one of the canonical of this covering space and hence in the slightly larger group, pi one of x. So if you're separable in a finite index subgroup, then you're separable in the whole group, okay? And you're separable in this finite index subgroup because you're a retract of it, okay? So this theorem about retractions and covering spaces is really aimed at understanding separability, right? But being a retract is, of course, a bit better. So let's state one more corollary before we get, before I sketch what this theorem looks like. So that corollary just to make things a little bit real and to connect up together with what I mentioned with what I mentioned before, when x is special and pi one of x is hyperbolic and h is quasi-convex, is quasi-convex, then h is a virtual retract and hence separable, right? Is a virtual retract and hence separable and all you're doing is saying, okay, I have a quasi-convex subgroup. That means that I could find a local isometry, right? If I have this special cube complex, let's say, and okay, it's compact and its fundamental group is hyperbolic, I have a quasi-convex subgroup, h, so I could find a compact cube complex y and a local isometry from y to x such that h is just pi one of y. And now, I then say, okay, well now I have a local isometry from y to x, right? I wanna study y by thinking about this local isometry, which is really a much better way to think about the subgroup. It's a much, you have real good control over everything and I passed to a, I passed off, I'm able to pass off to a finite covering space of x so that y is a retract of that finite covering space and so then I see that my quasi-convex subgroup is actually separable, okay? So there's a nice interplay over here between the geometry and certain combinatorics that we have sort of working for us and it leads to nice algebraic results. So let's now draw a picture that explains what this canonical completion is about and the best way to explain it is to explain it in the case of a graph. So let's hold blackboard over here, maybe I'll still use this one. So here's a graph, let's call this x, and I'm going to choose a local isometry from capital Y to x. So in a local isometry, it's so easy over here, it's just a combinatorial immersion, okay? So that's a nice local isometry, but I'm going to be creating this commutative diagram here. So now I'm going to draw my canonical completion, which is going to be this covering space of x. So in fact, we said that y is going to embed in this covering space, so we might as well build the covering space around y, here it is. And to create a covering space, what I need to do is I need to make this map into a local isomorphism, right? A covering space of complexes, a combinatorial map of complexes is a covering space if it's a local isomorphism, right? And that's quite simple, I just need to get the right number of incoming and outgoing edges of each type at each vertex. And there's many ways of doing it, right? You can just, whenever you see a missing outgoing A edge, you can pair it up with a missing incoming A edge, right? And you know, you're on your way, right? You just add more and more missing stuff until you've created a covering space, okay? So there are many ways of doing this, but we're going to have a canonical, we're gonna choose a kind of canonical way of doing it, so let me describe that canonical way. I mean, this idea of embedding it in a finite cover really goes back to Marshall Hall, like 1950 or so. So what you do is you, when you have that missing outgoing A edge, you're going to connect it up with a very particular missing incoming A edge. How did I choose that? I kind of, what I did is I looked at a maximal A arc and then I closed it up. So let's repeat that process. Here's a maximal, here's a maximal B arc right over here. And I'll close that up. Maybe I'll make this a little bit more interesting. Let's make this a little more interesting. That was a little more stuff on it. So here's a, so where's my next orange edge that you might want to put? So here's a maximal B arc right over here. And I'll close it up. Okay, let me finish it now on my own. So there's some trivial A arcs you might not have noticed. And here's a, here's a trivial B arc. Here's another one. Do I have a finite covering space? Not yet. I'm missing over here. And I'm missing over here. And I'm missing over here. I'm good now, right? Otherwise there'd be a riot, yeah? So what's, now there's many, a graph retracts onto any connected subgraph. So nothing to fear over here. There'll be no problem retracting it. But what's the canonical retraction that we will use? Well what you do is you map the new, each new edge to the arc that it was associated to. That's the canonical completion. That's the canonical retraction. Okay, I won't fill in the rest. And as it turns out, this procedure, which I was using around 1994, five, Frederic and I generalized it around 2002 or so, it works quite naturally when X is a salvetti complex instead of when X is a graph. It's almost the same recipe. Okay, you don't actually need to know what the recipe is, but you get this conclusion. And there are features that the canonical retraction has that allow you to have a lot of control when you're doing certain elaborate arguments that are required working with special cube complexes. There's now quite a bit of a theory of special cube complexes and groups which are special and they seem to, a group is special if it's a fundamental group of a special cube complex. And they occupy a kind of nice little spot where you can understand a lot of things about them. I'm not going to say more about them. I finish on time. That means you guys give me extra five minutes tomorrow, right? Thank you.