 subgroup, a finitely generated subgroup of F2 is co-dimension 1, if and only if, this is a free group, if and only if it's infinite index, OK? So you can do that exercise while I do my exercise, and sorry, I forgot about this. I got punished. Is there one more? I will leave that. We'll leave that for me to take a short break during the talk. I want to talk a little bit about finiteness, properties, the dual. So well, we had a group. We found some co-dimension 1 subgroups. We made a wall space. We created a dual cube complex. And now we have a group acting on this cat zero cube complex. But often the dual arising from the construction is locally infinite, is infinite, is dimensional, has cubes of arbitrarily high dimension in it. And among the exercises, if you have three pairwise crossing walls, there's a three cube. If you have four pairwise crossing walls, you'd expect to see a four cube and so forth. So you'll be forced to look at that a little bit and consider that a little bit. And actually, we heard about Thompson's group. So let's see. Locally infinite, maybe the simplest example would be like a bass ser tree. That's really one of the simplest versions of this dual cube complex is really the bass ser tree. And it's actually the case. It's also an exercise of lemma. What's the difference? Theorem. Everything is just an exercise, I think. Right? It's an example. All right, I see. Yeah, the EX means example. If you have an amalgamated product, then the amalgamated subgroup has co-dimension one. And likewise, provided that we're going to assume that C is not equal to A or B. So we're assuming that it's a genuine splitting. Likewise, if you look at an H and N extension, then the edge group is co-dimension one. And really, the bass ser tree ends up being, it really is just a dual to a wall system that you're creating with this co-dimension one subgroup. That's really all that's happening. That's why you're getting a tree. And it's a very nice type of co-dimension one subgroup. It's a co-dimension one subgroup, which it's a very nice type of co-dimension one subgroup. It's a co-dimension one subgroup. It's a little bit heuristic, but this is fairly close to the truth. You had your group, and you found the co-dimension one subgroup, giving you walls that don't cross their translates. And so the dual is going to be a tree. That's really the connection with the bass ser tree. But unfortunately, typically what's going to happen is that the group is not going to act freely on this dual cube complex, this bass ser tree. And we're really, really very concerned with the stabilizers of the vertices. And we know that in this case, they're conjugates of A and B, conjugates of the vertex groups. They're called vertex groups. They're conjugates of A and B, or the conjugates of A in the H&N extension case. Does this notation mean A star CT equals D? This is sometimes notation for an H&N extension. So have you heard of the notion of H&N extension? OK, so then you'll learn about that, and you'll find out. It's a generalization of an amalgamated product. But if you like co-dimension one subgroups, then you can first do co-dimension one subgroups, and then work your way down to the case where you get a one-dimensional cube complex. And then you'll know about bass ser trees. You could skip it more or less. Do the both C&D are proper subgroups or one of them? It's not necessary. In this case, it's not necessary for them to be proper subgroups. You can just have any both, because they do. Sure, sure. OK, so I'm not going to use this notation very much as we proceed. So I'll just continue. This works backwards, too. If you get the dual complex as a tree, then you get it splitting. If the dual complex is a tree, that's the most important way that it works. If the dual complex is a tree, then you get an action on a tree. And Sarah's book is, if you've got an action on a tree, then it's a splitting. You can write a book about it. So we saw another, so I just wanted to mention, the connection to the bass ser tree, just not to be remiss. And we saw briefly another, so this is sort of the classical case where you have locally infinite behavior, because usually the action on the bass ser tree, well, the bass ser tree usually has a locally infinite vertices, unless the vertex groups were finite, or the edge groups more generally were finite index in the vertex groups. But that doesn't happen very often. Usually the edge groups are not finite index in the vertex groups. And so the vertices in the bass ser tree are locally infinite, which is one sort of pathology, to a certain extent. A more interesting case, it could be infinite dimensional. So there are some tubular groups that exhibit that behavior. But what's more exciting, locally finite, locally finite, yet, but infinite, dimensional, I'm correct, for Thompson's group. And I think the cube complex that you described, which was described not using this dual cube complex construction, it actually is a different way of getting to cube complexes, which is more closely related to what people call state complexes. So there are lots of pathological examples and pathological behaviors around. I am interested in, as you know from the special cube complexes, I'm interested in the group, not acting on a cube complex in an arbitrary way, but actually being the fundamental group of a cube complex. So I want it to be acting freely. And even though I said that I'm open-minded and that not everything has to be compact, truth is that I prefer compactness, OK? So well, here are some, so we're looking for theorems that are going to give us control over the action of the group on the dual. So I want to describe a few to give you some sense of what's out there. So Mika, the first thing that he did and to a certain extent the last that he did in this topic until he came back to the topic some years later, interested in the Crop-Holler conjecture, which is something a little bit esoteric, he proved the following. Let G be hyperbolic and H be a co-dimension one subgroup that is also quasi-convex than the dual, right? So there's more things to do over here. You have to choose a calligraph, pick your walls, choose what your partitions are going to be. But it doesn't matter what choices you make. The dual is G-co-compact, right? In particular, it's finite dimensional, right? If it's G-co-compact, the group is acting, permuting the cubes around, the maximal dimension of a cube, there's a bound to the dimension of a cube over here. Now, here's another theorem that gives us some sense. Suppose that for each, it'll give us some sense of towards freeness of the action, right? This is towards co-compactness of the action. Suppose that for each non-trivial element, little G, there exists a wall that cuts G in a sense that I'm not going to make totally precise. I'll draw your picture. Then G acts freely on the dual. So this is now a setting. Let me maybe clarify. So what's the setting? I better specify that I'm sorry. So the setting over here let G act on a wall space. So I have a wall space. There's a group which is acting on it. So the G is acting on the set. And it's permuting the partitions around. It's permuting the walls around as it acts. And then, as you can imagine, there's an induced action of G on the dual cube complex. And we're interested in, OK, what sort of thing could guarantee that the group will act freely on the dual to this wall space? And this is one of the criteria that comes in handy quite often. So what do I mean by cuts? In a sense, there's a few ways of thinking about it. How about this? How about this? Let's choose a point in the wall space. Let's do it like that. So we have the identity times a point. So let S be some fixed point. And what we'll do is we'll look at all translates of that point by powers of G. And we want there to be some wall that separates them into the translates, which are negative, and the translates, which are positive, OK? We're a little more general than that. We'd be OK with an orange wall that did that. And so eventually, the sufficiently positive translates are going to be on one side of the wall and sufficiently negative translates are going to be on the other side of the wall. So maybe I'll say that. Your task is choice one or fix the least? Yeah, just choose some point. Just so that we can identify group elements with just we're looking at an orbit. OK, we'll just choose to some orbit. So this definition won't depend upon the choice, OK? So in practice, our wall space, often S has extra structure, e.g., S might be is a geodesic metric space, as is the case when S is up epsilon. It's a nice geodesic metric space. And a nice way of thinking about this is, so cutting G, W cutting G, it's W cutting G. So let me say it like this. Then we would G maybe has an axis, so maybe a nice geodesic, and that the element G is stabilizing. And the picture that we really like is something like this, where our wall cuts a nice geodesic axis. And the axis kind of goes far away from the wall on each side. And that's really the motivating case. And it's not very difficult to prove that when every element is cut by a wall, then it is impossible for a zero cube to get stabilized by a group element. Because you try to imagine a zero cube which is stabilized, and then you reach some type of contradiction in the definitions of the way a zero cube behaves. I'm not going to get lost in it right now. Instead, I want to convey sort of one more thing. And I think together, they will give you a pretty good intuition about when a group will act freely. So here's another condition. Suppose that we're still in this setting that G is acting on a wall space, but we're going to assume that S is a metric space. S is a nice metric space, like the caligraph is. Let's suppose that the number of walls separating two points, so this is measured inside of the wall space, of course, is greater than or equal to, I don't know, a distance between p and q minus b for some a and b. Suppose for some constants a and b, we had this property, which I like to call linear separation. Linear separation. Then actually, you get not only does G act freely, then the action of G on the dual cube complex has a quasi-symmetrically embedded qi embedded orbit. So let me maybe say it like this, so there's a map from a caligraph of G to the dual cube complex, there exists a qi embedding from the caligraph to the dual cube complex. It's not a quasi-symmetry between them. The cube complex is probably much, much bigger than the group because of all of the surprise zero cubes that showed up. I mean, in my mind, I'll draw a picture of that in a moment. Talking about freeness now, what you should absorb from this is somehow that the intuition. So in detail, what is the assumption on G? G lets G be a finally generated group so that we can give it some nice metric. The culture over here, by the way, we usually just care about finally generated groups. Maybe countable, but usually just finally generated. So G is a finally generated group, and it's acting on a wall space so that we can compare the number of walls with a metric. There's a metric on the wall space. So the picture for that criterion, maybe I'll reuse this picture over here. The picture for that, I'll draw a new picture. So the picture for that criterion is that we've got p and q. And there's a certain distance between p and q in our wall space. And we want the number of walls that are separating p and q to be roughly proportional to the distance between them. Now, it's a tricky, I mean, this is saying that it's lower bounded by the distance between them. But the way the finiteness properties go, you can never expect more than a linear number of walls as candidates. So this is the best. This is the best you can hope for. Is it actually? So does what act by isometry? G act on the cube complex. It's acting by isometry. On the wall space with metric structure. Yes, thank you. But you need some faithfulness to embed the calligraph in C. You need something about. So all of this is the sort of generalizations of the group acting on its calligraph. And you're saying that if it didn't act faithfully over here, then you wouldn't be able, you wouldn't expect that the walls are going to lead to faithful actions on the cube complex. Well, might. I don't know. But I'm going to assume that it's acting. Usually what we do is we act properly. So we will often add a little bit of finite stabilizers as the situation that we normally are considering. But you don't have to make, when you're first learning about this, don't make everything so difficult by thinking about grand generalities. Think of two cases. One is that you have a group acting on a calligraph. And the other case that you should be thinking of is a group acting on hyperbolic space. And you're going to have walls in hyperbolic space coming from co-dimensional one-subgroups in hyperbolic space. That's the other type of situation that you should be thinking about over here. Now, if you just require that the number of walls separating p and q is going to infinity as the distance between p and q goes to infinity, you'll still get something. So maybe I'll just say that over here. If the number of walls separating p and q goes to infinity as the distance between p and q goes to infinity, so as two points are getting farther and farther away from each other, the number of walls separating them is increasing, then that will be enough to know that the map from epsilon to c, or if you prefer the orbit instead of being quasi-symmetrically embedded, it will be metrically properly embedded. So the orbit of the cube complex is going to be sufficiently spread out inside of the orbit of g. And the cube complex is going to be sufficiently spread out. This is a minimal amount that we want. But that's probably the one that you should be looking at. The intuition is really, I'm getting a little bit out from where I wanted to focus. The intuition is that if the walls cut the wall space very well into small pieces, then the group will act nicely, meaning freely, properly, with finite stabilizers on the dual cube complex. And that's what both of these statements are really telling you. And cutting it up means what you think it does. So that's the intuition. But you have to be very, very careful. But unfortunately, this is a soft statement, so nobody can criticize me about this because it doesn't mean anything. But unfortunately, it's hard to tell. So you can be fooled, easily fooled, why? Well, you have these points, p and q. And it's looking good. Looks like there are lots of walls separated in them, right? But maybe it was all the same wall. Hey, they're not even separated. So this is a pitfall that often occurs when we're studying this, when we're trying to recognize that we found enough walls to actually have an interesting action of g on the cube complex. We're looking to see that we've cut it up many, many ways, but you can get into some trouble. And these statements are all just trying to pinpoint what's happening so that the intuition, which should be guiding you, isn't going to give you too much guidance and send you in the wrong direction. So well, we would like to see that there really are actual walls separating them. It shouldn't just look like you're chopping it up into little pieces. Or this is enough as well to ensure it. Let's look at an exciting thing that happened a few years ago that increased the interest in the topic for people who were not yet believers. So Kahn and Markovich proved that the fundamental group of a hyperbolic manifold contains many, many quasi-convex co-dimension 1 that has to be the case closed surface subgroups. So the setting over here is let m be a closed hyperbolic manifold. Now, it's a general fact that actually another example is that if I think I stated this already, that an orientable n dimensional manifold subgroup, a closed orientable n dimensional manifold subgroup inside of an orientable n plus 1 dimensional manifold, did I state it last hour? No, I didn't. So if you have an inclusion of fundamental groups, of closed orientable manifolds, an n manifold fundamental group in an n plus 1 manifold fundamental group, then you get a co-dimension 1 subgroup, which is the motivation for the name over here, for the terminology. Although during the break, someone was complaining that the trivial subgroup is a co-dimension 1 subgroup in a cyclic group, which is a co-dimension 1 subgroup in a free group of rank 2. So you don't want to get too attached to notions of dimension here. But this is the motivation for the language. So these are all co-dimension 1. So the dimension didn't drop here, but then it dropped over there. So Kahn and Markovic gave us answering a long-standing problem about hyperbolic 3-manifolds. They gave us a quasi-convex co-dimension 1, quasi-convex surface subgroups, which of course are co-dimension 1. It's a general all-purpose fact. And so in fact, there's various ways to think about why this is the case. But we then can deduce, so using sufficiently many, maybe 2, 3, maybe 10 of them, of these surface subgroups and applying the dual-cube complex construction, we obtain a free co-compact action of the fundamental group of this 3-manifold on a Katsuo-cube complex. So this is corollary to their discovery. The co-compactness of the action is just coming from Suggiv's observation. The freeness of the action is coming from the fact that there are many. You could choose so many surface subgroups that when you look at the hyperbolic 3-space, in fact, what they showed was that for any two points in the boundary of hyperbolic 3-space, they found a closed-surface subgroup. So this is the universal cover of our manifold. Kahn and Markovic found that for any two points, you could find a surface, probably very hygienist, whose universal cover maps pi 1 injectively into the manifold. But when you look at the induced map between the universal covers, so this is the universal cover of my surface over here, and it has a nice yellow boundary at infinity, I guess. You could think of it as hyperbolic plane. And it very nearly, so if you choose a circle that separates the two red points on the boundary, Kahn and Markovic can choose this surface S so that its universal cover comes so close to this orange hyperbolic plane over here that actually its boundary kind of looks like that. Just kind of travels right along nearby it. And this yellow boundary circle of the universal cover of our surface, which you're looking at and you're saying, that's going to be a very good wall, this surface manages to separate these two points at infinity. So with that, you can actually choose sufficiently many of these. And by some compactness argument, you can actually, and of course, you can use all of their translates, as we discussed. Maybe you've chosen seven of these. Plus all of their translates, you actually get into a situation where you're going to cut this up sufficiently well. And so you know that the action is also going to be free. It's not just co-compact, which is a generality from Sagiv's theorem, but it's actually going to be free because you've cut hyperbolic space up sufficiently well. And you could do it like that as well. So we know that our three-manifold is actually the fundamental group of a compact cube complex. And from there, of course, we hope that we could show that that cube complex has a special finite cover, which is what we were talking about yesterday. And that's what happened in the end. That was the plan. So one would like to hope that every closed hyperbolic manifold is pi 1 of a compact, non-positively curved cube complex. But the main thing that we are missing, and we're probably very far away from that, is to understand where the walls are going to come from. People knew to be interested in finding these walls even before they knew what to do with them. They wanted to find these surface subgroups, very useful to cut a three-manifold along a surface subgroup. And there obviously hasn't been as much interest yet in finding ways of cutting four-manifolds. Along it doesn't have to be, it won't be along three-manifolds. It'll be along something, along some quasi-convex co-dimensional subgroups. And to find enough of them so that your four-manifold, its universal cover is going to give you a nice wall space, and you could do the same thing. And so your hyperbolic four-manifold is also going to be the fundamental group of a non-possibly curved cube complex. This is all conjectural. This is what I believe mostly because I want it to be true, because it means that cube complexes are the glorious answer to many things. But that's what I've felt for about 20 years, just because I didn't know how to understand anything else. So now for three-manifolds, by the way, maybe we could spend a little time talking about that because we heard about three-manifolds yesterday. For three-manifolds, so theorem almost all compact three-manifolds have the property that pi 1 of m is the fundamental group of a non-possibly curved cube complex, but often not compact. So some exceptions, well, nil and sol and the PSL2 tilde, I think I haven't thought about that in a long time. The more serious exceptions are, which are a little bit interesting, certain closed graph-manifolds. But if you, for instance, took a few hyperbolic three-manifolds and you glued them together along their tori, so I'll just write H for hyperbolic. These are little hyperbolic pieces. If the JSJ decomposition were non-trivial, and for instance, all of the pieces were they called pieces yesterday in the JSJ decomposition or blocks, pieces, I think? Well, if all the pieces in the JSJ decomposition are hyperbolic, then this is actually a fundamental group of a non-possibly curved cube complex, not necessarily compact, but it's still virtually special. So the specialness is really working very nicely together with hypervelicity over here. I should probably spend a few minutes mentioning, listing some other examples. We heard about coxsweter groups. So every coxsweter group, finally generated coxsweter group, is equal to the fundamental group of x, where x is a non-possibly curved cube complex. This is Nibbleau and Reeves. And also, if I'm not mistaken, Pierre Pansou didn't publish it. And they were using the wall space construction. I'm not sure if they had the words wall. They knew of the terminology wall space already, but it was pretty clear what the essence of Saqib's construction was, and they used the reflection walls that I mentioned just briefly before. However, x is not compact. Oh, and I said something wrong. This isn't correct. A coxsweter group has torsion, of course, and a non-possibly curved cube complex, as Ruth mentioned, its universal cover is contractible. But if it's compact, you're certainly not going to have. These are finite dimensional. Finite dimensional. So let me state this correctly. So I'll say is virtually, so it has a finite index subgroup, which is the fundamental group of such a thing. And let me state it a little bit more accurately. Let's call the group G. G acts properly on a finite dimensional cat zero cube complex. So that's a little bit more correct. And actually, here, too, there exists G prime contained in G such that G prime acts freely. Let's call this cat zero cube complex x tilde. There exists G prime contained in G, so that G prime acts freely. And in fact, x tilde mod G prime is special. So all coxsweter groups using Suggiv's construction on the walls, you get an action on a cat zero cube complex, which the dimension goes up quite a bit. But it's something organized that you can understand. And then you can pass to a finite index subgroup and see that your coxsweter group is virtually inside of a rag, like this. This is the sort of continuing theme if you've heard of hyperbolic arithmetic lattices of simple type. They also fall to this approach. Also, on cat zero cube complex and are virtually special. So there were discussions of arithmetic lattices at the beginning of last week. Is that correct? But you were probably talking about bad lattices that had property T. Is that right? Yeah, those are dead already. So I have about three or four minutes to wrap things up. So maybe I'll describe one more suggestive example, which is really my interests are to convey the intuition of when this procedure of using Sageev's construction is going to work. So maybe it's a good idea also to mention an example in regards to the coxsweter groups. So cat zero two complexes were discussed here last week. The week before last week. Maybe. Not really? OK. Let me then change the, I won't try to do that. I'll change the topic entirely and close with one with another thing of a different flavor. So let me talk about a combination here. This is with Tim Shue. If G is equal to an amalgamated product where G is hyperbolic and C is quasi-convex. And A and B are virtually fundamental groups of non-positively curved, of compact, let's just say compact, special cube complexes. Then G is a fundamental group where X is a non-positively curved cube complex that is compact. So sometimes you can take two non-positively curved cube complexes and you can glue them together. That's maybe the simplest case. You might just have cube complex number one and cube complex number two. And then create an amalgamated product of them. And well, if you glue them together, we know that if you're gluing them together along local isometries, then maybe you might be lucky and you can actually, the object that you get might already be a non-positively curved cube complex. But usually it's not. Usually it's a huge mess. OK. But in a very good situation where X1 and X2 are special and the amalgamated subgroup, C, this is A and this is B, when C is quasi-convex, you can actually get this whole group, the fundamental group of this new space to be acting on a cat's ear cube complex. So how would you do that? You find walls because there isn't another way of doing it. You somehow find walls. And so you'd look inside of this object and you'd say, it's almost like you're looking if it were a three-manifold. If this were a three-manifold, what you'd be doing is you'd say, OK, there must be a surface in here somewhere and you hope to find one. And the same thing is happening over here. You're looking for kind of immersed walls inside. Maybe you found one. You have to be very careful about it. Hopefully it's going to correspond to something which in the universal cover of this object is going to give you an honest wall. And hopefully it'll be quasi-convex. And then maybe you will have to find another one. And perhaps you'll observe that, oh, there's already one waiting for you right over here. Because that edge group right over here is going to be a nice wall. And then you have the universal cover. You have the picture of all of these walls cutting it up. And that's how you'll get an action of the amalgam on a cube complex. And this is how the subject has been progressing. And there's lots of things that we don't know. There's lots of things as we drop hyperbolicity, then it's trying to generalize this statement over here. It has a generalization to a relatively hyperbolic setting. But there are, as you increase the sort of group that you want to apply this to, because hyperbolic groups are amazing. But those are the easy groups that are easy to understand. What about the difficult groups? So then as you stretch this, things get messier and more complicated. And you try to extend what we've done in the hyperbolic case. And it's going to get more and more complicated, as that's the way math goes. So thank you very much for your attention.