 Thanks. So before I get started, I just want to mention two things. The first is I was cruising around the web the other day, and there are two papers I noticed that are based on mini courses much like what I'm giving here. And if you look at these, they will expand a little on the sort of things I'm talking about here. And also, they give lots of references to where you can find the actual proofs of things. And so if you want a little bit more of the same nature, these two papers are both good places to look. As I said, they're both expository. They're not full of proofs, but they're nice overviews of these groups. OK, the second thing I want to say, and I wrote this up there here because I knew I'd forget if I didn't write it up here, is I call this mini course Art and Groups. And now I've written Art and Tits Groups, where there's this movement going on to change the name from Art and Groups to Art and Tits Groups. And the reason is Art and Groups are, well, first of all, as I will explain in a few minutes, they grow out of Coxer Groups. They're very closely related to Coxer Groups. And really, the theory, the general theory of Coxer Groups and Art and Groups is really due to Jacques Tietz, not to Coxer or Art and yet his name doesn't appear anywhere on here. I mean, it's never used. And so there's this sort of movement of let's start calling these Art and Tits Groups. And I'm all for it. It's just that I never remember. I'm always forgetting. I've been working on Art and Groups for too long, and I always forget to say Art and Tits Groups. So we're talking about Art and Tits Groups, but we'll call them Art and Groups for short, OK? All right, OK, so good. That's your intro. So my goal in the first lecture is to tell you what Art and Groups are. I know many people have maybe never encountered anything, except possibly a right-angled Art and Group. So what are they? Where do they come from? Why are they interesting? And then I'm going to talk about a sort of a long list of open questions regarding Art and Groups, all right? That's the main goal of the first lecture. The second lecture, although I may fit a little bit of this into the first lecture, so you can do your problems. But the second lecture, I'm going to talk about geometric objects, geometric techniques that are being used, have been used and are currently being used to try to answer some of these questions. So what kind of progress have we made and how have we done it using, by the way, cat zero cube complexes? So that's why I'm very pleased that Danny Wise gave his first lectures before mine. And I'm hoping you are all there, because these objects we're going to be playing with are cat zero cube complexes. All right, so let's start with what are Art and Groups? Well, the story really starts with coxeter groups. So let's actually begin with what a coxeter group is. I find that many more people have run into coxeter groups than art and groups, because they appear in so many areas of mathematics and combinatorics, representation theory and geometric group theory all over the place. Coxeter groups sort of show up naturally in many, many, many contexts. So a coxeter group, there's two ways we can define it. One is just in terms of a presentation. So a coxeter group is any group of the following form. We have some finite collection of generators. Every generator has order 2. So si squared equals 1 for all i. And we get to specify the order of si times sj. So si times sj, we get to specify some mij equals 1. We're here, mij can be 2, 3, et cetera. And it's also allowed to be infinity. But when it's infinity, this relation's really just not there. But we'd like to have an mij for every i and j. So either it can be finite, in which case we have a relation, or it can be infinite, in which case there's no relation. So that's it. That's a definition. But really, a more interesting way of looking at them is geometrically. So geometrically, yes, i different from j, sorry. This is for all i, and this is any i not equal to j. Absolutely. Geometrically. So it turns out that any coxeter group can be realized as a reflection group. So we can realize w as a group of a discrete group generated by reflections. Let's just go on, generated. OK, so let's think about that for a moment. Draw a picture here. So if I have a, let's just draw it in R2 for a moment. Let's say I have two reflections. So perhaps I'm going to reflect si over this and sj over this. All right, so obviously si and sj have order 2, right? Reflect, reflect, you're back where you started. So that's good. And what does this say? Well, if I do a product of two reflections like this, what I get is a rotation through twice this angle. Product of two reflections gives you a rotation through twice the angle. And I want this to be a discrete group. So I want that reflection, sorry, rotation to be some integer, 2 pi over some integer. So in particular, I want this to be pi over an integer, and that integer is exactly mij. So what that mij is reading is the angle between the reflection hyperplanes, okay? So that's the way to picture what's going on here. How do you get mij equal to infinity? Well, I don't want to go into formal detail of exactly what I mean by reflection, but let me just say we allow for things like affine reflection. So our reflection might fix some affine subspace. And if it fixes some affine subspace, then inside that affine subspace, the reflection hyperplanes could be parallel. This is actually combed off to in another dimension. But it's fixing this affine subspace, and we have si sj. And in this case, if we take si times sj, it's a translation. And a translation has infinite order. So in this case, we would get mij equal infinity, all right, when you have a picture like that. All right, ready to got it? Okay, so, okay, so let's, we want to have a simple to specify a coccidic group. So we encode this information in a graph, gamma, where gamma consists of the following. The vertices of gamma are exactly in one-to-one correspondence with the generators, all right? One vertex for each generator, labeled by the generators, okay, and the edges, well, it's actually a labeled graph. I should say a labeled graph, excuse me. Edges in gamma, well, there's an edge from si to sj precisely when mij is finite, is not infinite, all right? And we label it, mij. So I labeled graph, meaning I'm going to keep track of those mijs by labeling edges. So this is whenever mij is not infinite, is less than infinity, okay? If they're infinite, there's no edge, all right? Okay, so usually when we want to specify a particular coxer group, what we do is we draw the graph, we just draw the graph, and then say that the associated coxer group, we denote associated, all right? So usually we work this way, we start with the graph, and then we associate to it a coxer group. So this is any finite graph with label integer labels, okay? Yeah, yeah, question? I guess in definition of W, it implies that the mij are symmetric, right? Oh yeah, I didn't say that. Absolutely, let's see, does that follow from this definition? They're definitely symmetric, mij, sorry, equals mji. Maybe that's, is that follow from the fact that it's, anyway, definitely symmetric, definitely symmetric, mij equals mji, all right? Okay, okay, so, all right. Okay, so that's a coxer group. So, oh no, let me give you an example because we need to have a couple of pictures in our head as we go along, so let's look at a couple of examples of gamma. Well the easiest one of course is just this with say two generators, let's call them S and T, and we're looking at the coxer group with two generators and this one relation, and I guess it's, I think I'll leave it for you to see that this is just what's called the dihedral group of order two n, so two n. It's exactly, it's exactly, actually maybe I will draw you this picture because we'll use it later. So again, I have these two reflections where this is pi over m, right? Okay, and I start reflecting around and I get some more reflection hyperplanes, all right? And if I just look at a point in one of these regions, these are called chambers and look at its orbit, I get a two n-gon, two m-gon, okay? And what I've got, what the coxer group is is the symmetry group of this two m-gon, all right? And that's the dihedral group of order two m, all right? So it's just the symmetry group. And by the way, this picture I drew here is something called the coxer cell associated to the coxer group, sorry. Sorry, I didn't put in the coxer group here. W gamma is equal to the dihedral group, sorry. The coxer group is the dihedral group of order two m, all right? Okay, so we'll come back to coxer cells later but that's your first image of a coxer cell. Okay, so all right, let's do, so that one is a finite group, all right? On the other hand, clearly if I just take this with no edge, where I have Mij equal to infinity, so again, S and T, but this time Mij equal to infinity, in that case, I get W gamma is what's often written as d infinity and that's just the, is there a name for this? So now you're taking reflections like this if you want, right, it's the infinite dihedral group, I think is what it's called, right? It's just the infinite dihedral group, so it's an infinite group. This is a finite group, this is an infinite group. Okay, there's a really important point here and let me give you one more example so you can see it, which is that that's not the only way a coxer group can be infinite. So let me, let's look at this one. Three, three, three, all right? Let's say R, S and T is my vertices, all right? What is that coxer group? Well I want three reflections, R, S and T, and any two of them have pi over three is their angle between them. Well it turns out that can be realized as what's called an affine reflection group, namely in the plane, let me draw it up here. So in a plane if we take, just in Euclidean space, if we take an equilateral triangle and look at the reflections across each one of those faces, all right, then each one of these is pi over three, all right, and we take a reflection in each one of the faces, then that turns out to give you exactly the coxer group you want. But let's start doing that, we start reflecting on what happens when we get another triangle and then we get another triangle, we get another triangle, we reflect, reflect, reflect, reflect, reflect, we tile the entire plane with triangles. So the coxer group in this case is in one-to-one correspondence with the entire triangulation of the plane by this, by this. So it's infinite, so W gamma equals, W gamma, I'll just say tiles, moving this thing around by W gamma tiles the entire plane, hence is an infinite group. Because we're gonna care in a little while a lot whether our coxer group is finite or infinite. And the point I wanna make here is just because there are no infinities, M-I-J infinities doesn't mean the coxer group is an infinite, can still be infinite, okay? All right, any questions? I shouldn't ask that, people just, we'll be here all day. The coxer group on C was not accurate. The symmetry group is bigger than the B2M. The entire symmetry group is bigger. I'm just looking at the group generated by those three reflections. The first example of the polygon, I see you said something, well, anyway. Oh, you may be probably right. So it's not the full symmetry group, but it is the dihedral group of order 2M. You're right, there are some missing symmetries. Okay. It's not in the middle, yeah. Yeah, right, I understand, you're right. It's not the full symmetry group. It's contained in the symmetry group. It's generated by those reflections on these sum. Okay, so I'm hoping a lot of you have seen coxer groups because that's not the subject of this talk. The subject of this talk is art and group. So now what is an art and group? All right, so let's move on. All right, as in the case of coxer groups, there's two ways of defining an art and group. The first is simply by a presentation. And let me just say, actually before I start, that for every coxer group there will be an art and group. So let's say we already have specified the graph. So we've given a graph and now we're gonna associate to it. We already associated a coxer group. Now we wanna associate an art and group to it. So assuming we've already specified our graph, then a gamma is again the same generating set. Now we do not want generators to be ordered to anymore. So we drop that relation entirely and we rewrite the second relation. So the second relation I'm now gonna write like this. S-I dot dot dot alternating I-J I-J with M-I-J terms not pairs. So if M-I-J is three, I would stop right there. M-I-J letters equals the same thing in the other order. Now it turns out that in the coxer group case, that relation and that relation are identical because I could take the inverse of this and put it on the other side. When the coxer group, the inverse of S-I is S-I and I just get that relation up there. So if we S-I equals S-I inverse, this guy is the same as this guy. I don't have S-I inverse anymore and I have to write it this way. There are two completely different relations in this situation, all right? Okay, so there it is and that seems kind of arbitrary. So where do these come from and why are they interesting? Well, it comes from the geometric description. So let's look at what they are geometrically. I should probably push everything up here. On the ground. Oh, that. Yes, thank you. This is the hardest part of giving these talks. Let's figure out how to use the boards. All right, so let's go, I should... Okay, so, all right. So now we wanna give a geometric description of this group. Okay, so let's go back to our coxer group. We have W gamma acting as a reflection group on Rn and the first thing we're gonna do is complexify. So just tensor Rn with the complex numbers. So we get an action on Cn. This is just the obvious thing, just take the tensor product with the complex numbers. But now notice something interesting happens. So when we were working in Rn, the reflection hyperplanes were co-dimension one, real hyperplanes. Now they're co-dimension one, complex hyperplanes, which is co-dimension two in real terms, yes? So what happens now, if we remove these hyperplanes, we get some fundamental group. Things can go around the hyperplane. We're removing something of co-dimension two. It's like removing a line in R3. And now we get stuff, you know, fundamental group. So let's define the following. So each reflection are in W gamma. And by the way, these are not just the generators but also conjugates to the generators act as reflections. So all the reflections I'm looking at. Each reflection R fixes a complex hyperplane, which I'll call HR in Cn. And what we want to look at, so let me say the associated what's called hyperplane complement. Well, first I'm going to write down something that's not quite right, but almost right, is defined as follows that we get. So I'm going to do script H gamma. It's going to be my hyperplane complement. And it's going to be what I get from taking Cn and removing all of these reflection hyperplanes. So the union over all the reflections take out all the reflection hyperplanes, all right? I'm cheating, but I don't have time to explain this in detail. In the finite Coxer group case, it's exactly this. In the infinite Coxer group case, in order for the theorems, I'm going to state to be true, you actually have to restrict to a big open cone in here. It's not the whole Cn. It's some big open cone in here and do the same thing. It's a technicality that's not going to play a real role in what I say, but I'm just saying this is exactly the picture if you have a finite Coxer group, all right? In general, you might have to restrict to an open cone. So what do I want to say about this? Oh, yeah, OK. So then the point is this is just true in general. Then W gamma acts freely on H gamma because it turns out we've removed everything which has a non-trivial stabilizer when we do this, all right? Everything which has a non-trivial stabilizer lives in one of these. So it now acts freely on this. And the main point is that A gamma is the fundamental group of what we get when we mod out by W gamma. All right, so these spaces arise naturally in algebraic geometry, and particularly in singularity theory. These spaces simply come up when you're trying to resolve singularities and people were interested in understanding their topology. And that's where the original interest in art and groups came from, from people who were trying to understand these spaces that arose, OK? So sorry? The out-in-brain group is a perfect group. Yeah, well, that's the example I'm going to do. Just give me a second. OK, so let's do the classical example. That's exactly what I want to do. So let's take a look at this to understand it better. So here's the classical example. I guess I'll do it here. OK, so let's start with a coxer group that everybody understands. So W gamma is the symmetric group on N letters. OK, you can now leave it to you to convince yourself that is a coxer group, OK? So this is the sort of classic example of a coxer group. And there's an obvious action of this on CN, namely, simply permute the coordinates, all right? So by permuting coordinates. All right, and what are the reflections here? Well, the reflections are just interchange two coordinates. That is a reflection, all right? That is a reflection. And the reflection hyperplane, the thing that's fixed when you interchange two coordinates, is the subspace where those two coordinates are equal to each other, OK? So let's write that down. So I need to save that board. I can use this one. Let's use this board. All right, I won't erase this. OK, so let's see. Yeah, OK, so the reflections, actually, let's put this here. The reflections are, let's call it Rij, sorry, tau ij. Equal interchange ith and jth coordinate, OK? And therefore the hij, which h tau ij, that is to say the hyperplane fixed by, yeah, where did I define, yeah, this, HR. So the hyperplane fixed by this is exactly the set of z1 to zn in cn such that zi equals zj, yeah? That's the hyperplane, all right? OK, so what is the hyperplane complement? The hyperplane complement is the set of z1 to zn in cn such that ci is not equal to zj for all i not equal to j, yeah? OK, well, that is a well-known space. It's called the configuration space of n points in the complex plane. So another way of looking at this instead of in the complex plane. So instead of looking at this as a point in cn, I can look at it as n points in c. And what we need is that those are distinct, n distinct points in c, all right? Yeah, so we want to think of these as being a collection of n points. Here's the complex plane, and I have more space here. Well, that's OK. I have n points in here. And the only rule about where they are is that they can't run into each other. They have to be distinct, yes? All right, so what's its fundamental group, all right? So we claim that a gamma is, well, I'm telling you that a gamma is the fundamental group of this thing. All right, so let's figure out what the fundamental group is. Oh, sorry, mod w gamma. By the way, what does mod w gamma do? Well, it's only a question of whether we remember what order these points are in, or if we mind out by w gamma, we just don't care what order the points come in. Same set of points, we're just not going to worry about what order, yes? So either order, you can have configuration space of n ordered points or n unordered points. Either one makes sense. I'm going to look at unordered points, where I don't care what order. All right, so now I want to take the fundamental group. So I fix some base point. In other words, I fix, I'm going to start here. That's, I declare I'm starting here, OK? And now I let time go from 0 to 1. And as time goes from 0 to 1, these guys are allowed to move around, all right? But they're not allowed to run into each other. So let's watch them move around. So let's have a little movie where this is time going down here, this is, you know, time goes from 0 to 1. And these guys move around, and they have to end up back where they started, at least up to permutation. We don't care if they get permuted. But they have to end up in the same set of points, yeah? OK, so let's watch them. Maybe this guy goes like this. And maybe this guy, oh dear, you can't run into this. So let's be careful. You better not run into this. And maybe this guy goes like this. And this guy, oh dear, goes like this. That's something, right? I mean, I'm watching these move around at any time. What I'm seeing is n distinct points in the plane, all right? What is this group? The braid group, yes. No, we all know this group. This is the braid group on n strands. So this is the classic example, and it is the one that Arton first did. This is the Arton braid group, OK? All right, so hopefully you see where these came from and why people were interested in them, all right? OK, so that's sort of your intro to the basic idea of an Arton group, right? Yeah. You have two different definitions for the Arton group. Is it supposed to be obvious, or is it an interesting result that you have to give them? It's not supposed to be obvious. This is a theorem. Yeah, this is definitely a theorem. Yeah, there are a lot of theorems in here. I'm going to quote, not prove. That's right. You definitely have something you need to prove here. I mean, in this particular case, it's not hard to check that this thing has the expected presentation that looks like this, all right? But yes, in general, there's definitely something to prove. All right, so let's move on. All right, so Arton groups, I like to say, come in two flavors. We classify them into two groups. So Arton groups, I'll just say, come in two flavors. You're allowed to use slang for her. Anyway, they're two, classified into two different groups. The first one I like to call finite type Arton groups. Some people call them spherical type, either finite or spherical type. I tend to call them finite type Arton groups. And what are they? Those are exactly the ones where the coxotr group is finite. So any Arton group, any A gamma associated to a finite coxotr group is called a finite type Arton group. On the other hand, we have infinite type. Type A gamma are those associated to infinite. Warning, the Arton group itself is never, never, never finite. The Arton group is always infinite. In fact, the generators have infinite order. So I'm not saying the Arton groups are finite. I'm saying it is an Arton group associated to a finite coxotr group. OK, all right. So if you've ever studied coxotr groups, I think I first learned about the details of coxotr groups out of Ken Brown's book on coxotr groups. I don't know if any of you know it. Anyway, there are all kinds of books out there. He starts out with all about finite coxotr groups, develops the whole theory of finite coxotr groups, and then later in the book, he expands that to give the general theory of infinite coxotr groups. But the bottom line is, wow, everything that works for finite coxotr groups works for infinite coxotr groups, too. I mean, that's not, I'm exaggerating slightly. But I mean, essentially, all the basic theory and the tools you need go through equally well for the finite and the infinite. It's exactly the opposite for Arton groups. We have these amazing tools that we can use to study these. And we understand an enormous amount about them. And we're totally in the dark when it comes to these, with the exception of some very special ones like right-angled ones. There are a few special cases where we have other tools to work with. But essentially, these are, so let me say a little bit about why. So we know a great deal about finite type, a gamma, and very little about infinite type. OK, so why? Well, the reason is I'm saving this board for something that I don't want to erase later. That's OK. I might need them both. It's possible. So I don't dance when I. OK. So, yeah. OK, so I claim we know a lot about finite type and very little about infinite type. And the question is why. And the reason is that finite type, a gamma, have something called a garcide structure. And that's another entire course to tell you about what garcide structures are. Let me just say it's an extremely nice normal form for words, for elements of the group. Extremely nice way of writing an element in terms of the generators that allows you to, for example, it gives you a lot of combinatorial information. For example, it gives you a bi-automatic structure if you know what those are. And gives you just really a handle on certainly anything algebraic. And it turns out a fair amount of geometric stuff as well. So which gives, well, not so much geometric, but certainly algebraic, which gives nice normal forms and lots of combinatorial stuff like bi-automatic structure, et cetera, et cetera, et cetera. And this has really been behind almost everything we know about those groups. The infinite type has nothing of that sort. And that's the problem. We don't have a place to start. We don't have anything. So infinite type, infinite type, gamma. Well, there have been. John McCammon has sort of generalized garcide structures a little bit to some affine ones. But basically, no, we don't have anything like that. So no such tool as a garcide structure. However, the most effective techniques we've had so far for studying these are geometric techniques. So actions, building interesting, useful complexes that these things act on and using those to tell us something about the group. And that's what my second lecture is going to be about. Well, a little bit. Hopefully, we'll get to some of it in this first lecture. Most of the second lecture is going to be about some of those constructions and how we use them to learn about these groups. Because that's kind of what I'm interested in. So before we get to that, though, I want to tell you about what do we know and not know about these groups. And I'm going to break this up into, so I'm going to put it over there. So here, so next, I'm going to put here are some conjectures slash questions. And most of them are about infinite type, but a few of them will be about finite type. So I don't want to specify in advance yet about what I'm going to talk about. So I'm going to give you a long list of open questions. And I'm going to put them here because I want to be able to refer back to them from time to time. So we're going to save them on this board over here. So the first is a set of what I'm going to call old conjectures. These go back to, so these groups were first really studied extensively by Deline and Bricecorn and Cytoback in the 70s. So these conjectures go at least that far back. I think some of them go even farther back than that. But they go back 50 years. So these have been conjectures that have been around for a long time. And I'm going to put a star by them so I can refer back to these conjectures, these old conjectures. Yeah, 50 years is old for you guys, right? OK, so here's some conjectures. OK, if you're a geometric group theorist, you're not going to believe this. We do not know if these groups have solvable word problem. We think they do. So I'll give you a preview of what's coming. It turns out these conjectures are all known for finite type. But they're not known for infinite type. So I will say afterwards which ones are known. So solvable word problem. I don't just mean that we don't have a nice solution, like some nice bi-automatic structure. We don't even know if it's solvable. We don't even know if there is an algorithm to solve the word problem. That's crazy. Two, a gamma is torsion-free. We can't find any torsion, but we neither can we prove that they're all torsion-free, all right? Three, here I have to assume a gamma is irreducible. So if a gamma is irreducible, and what I mean by that is that I can't write it as a product of two smaller arty groups. So assume I can't split it up as a direct product of two smaller arty groups, then we know something. We conjecture that the center of a gamma, well, we know in the finite type case that it's cyclic, that there is something in the center if a gamma is finite. For example, is finite type, excuse me. In the braid group case, if you take the braid that does a whole 360-degree twist of the whole thing, you can check that that's in the center, that that commutes with everything. You can exercise, draw a picture. So they do have a cyclic center if they're finite type. We know that. And we think the center is trivial if a gamma is infinite type. All right, four. And these should really go in the opposite order, but I'm going to do them in this order. We think that a gamma has a finite k pi 1 space. So everybody knows what a k pi 1 space is. It's a space whose fundamental group is your group, a gamma, and the universal cover is contractable. There's no other homotopy. The universal cover is contractable. So these are extremely useful k pi 1 spaces in homotopy theory and computing homology groups and all kinds of, they're basically the topological way to work with a group. So in fact, we not only think that there is one, but we think we know what it is, and I will explain this later. Let me just say, namely, the Salvetti complex. So by the way, finite, I mean finite Cw complex. Sorry. By finite, I mean there's a finite Cw complex, which is a k pi 1 space complex for a gamma. Now you heard, if you were in Danny's talks this morning, what the Salvetti complex was for a right-angled Arton group. Turns out there's a Salvetti complex for every Arton group, which I will explain to you later what it is. And we think that is, in fact, a k pi 1 space, always for everything. And the last one, 5, is all right. The last one goes back to this description of this hyperplane complement. So I already said that the interest in a gamma came originally from the fact that it shows up as the fundamental group of this hyperplane complement. I said that. Well, the conjecture is that this guy is a k pi 1 space. Now it's not a Cw complex, but in the sense that it has the right fundamental group and its universal covering space is contractable. So it already carries all the topological information one would want to study this group. So this last one is the h gamma, w gamma is a k pi 1 space. Maybe I should say a k a gamma 1 space. Would that be better? It's a k pi 1 space for a gamma. How's that? k a gamma 1 space may be a better way of saying it. It's a k a gamma 1 space. And by the way, since we already know it has the right fundamental group, all we need to prove is that its universal covering space, this is the universal covering space, is contractable. That's what's left to prove, because we know it has the right fundamental group. This one, you may not find the most interesting, but it's the most famous. And it has a name. It's called the k pi 1 conjecture. So for example, in that I put up those papers, there was one that Louis Paris paper that was about the k pi 1 conjecture. That's the conjecture he's talking about. So this is the most famous conjecture regarding these things. So as geometric group theorists would probably find the first few more interesting, but the people who are interested in these for other reasons were particularly interested in these later ones. OK, so that's the list of, yeah. So let me say what we know. So as I said, in the early 70s, Deline and a paper by Breeskorn and Sito. So these are two separate papers. They appear in the same journal, though, I think. Or it's around the same time. Early 17s, Deline and Breeskorn and Sito proved that all of these conjectures for finite type. OK, and since then, and this is part of what I'll talk about in my next talk, we've proved that they hold for certain classes of infinite type, but they're all still open for an arbitrary infinite type. None of them is known in complete, completely. We've almost solved the trivial center one, almost. Anyway, I'll talk about this in the second talk. All right, so these conjectures have been around for ages. And we're sort of just now beginning to figure out ways of dealing with them. OK, there's another set of conjectures, which I'm also going to put up here. And then good, then I'll still have 10 minutes left to tell you about Deline complexes. That is newer, and I'm not going to say. I don't need to save, so they can go on this board. And these have to do with, I'm a geometric group theorist, so I like to know, is a group hyperbolic? Is it cat zero? Those kind of questions that we ask, right? So what that means is that does it. So we're asking, does the group act properly co-compactly on some really nice space? A hyperbolic metric space, or a cat zero metric space, or a cubicle cat zero, even better metric space? So we want to know geometric properties, all right? So let me list some new questions. And these are all questions, because we don't have answers to any of them, really. All right, so one, is a gamma hyperbolic? Well, we know the answer to that one, turns out. And the answer is yes, if and only if. This is one of the exercises on your exercise sheet, is to prove that the answer is yes, if and only if a gamma is the free group, meaning if and only if gamma is a discrete, has no edges. Has no edges. If it has no edges, then the art group is a free group. And so they are clearly that hyperbolic, OK? That's the only case, all right? So that's an exercise, OK? So we know the answer to that one, actually, all right? So we could then ask, is a gamma cat zero, meaning does it act properly compactly on a cat zero space, or better still, is a cubically cat zero? Can we make it act on a cat zero cubical complex? Cubically cat zero? What do you call these? Something that acts on a cubical cat zero? Co-compactly-cubulated? Cubulant? Co-compactly-cubulated? I mean, just write it like that. Whatever. So acts nicely on a cat zero space, or nicely on a cubical cat zero space, one that comes from cube complex, all right? We don't know, all right? So even for braid groups, we only know. So it's conjecture that braid groups are cat zero, but it's only known for braid groups on at most six strands. It's known to be true up to six strands. Past that, we don't know. And in fact, there are some really interesting recent papers showing that, nope, most of the braid groups even. Anything braid group of more than four strands is not cubically cat zero. So the question here is, who knows? Don't know. We don't know even for braid groups. And the question here is often not. Many, many cases, the answer is no. Co-compactly-cubulated, but it's not known whether or not they're cubulated. When I, OK, all right. So I don't know that, when I say cat zero, I mean acts properly co-compactly, all right? And I want the same thing here. Acts properly co-compactly. Usually when you say something's cat zero. So fine, I'll put in co-compactly. I want proper co-compact actions, OK? For these two questions. All right, there's a bunch of names here. Should I try and write them all up, or just say them? Let me see, I can't even pronounce them. Hey-tell is the H-A-E-T-T-E-L. Hey-tell, did I pronounce that one right? He's got a bunch of results. He's got results on both of them. And then you can help me with this one, Danny. There's a paper by Wang, Jan Kiewitz, and Piotor. But tell me how to pronounce Piotor's last name. You know, I don't know how many. OK. Prtitsky, I'm sorry. Prtitsky, Prtitsky. Prtitsky. OK, whatever. I'm very bad at it. I could write it up here for you. Anyway, there are a bunch of recent papers showing that, in fact, even the braid groups, most of the braid groups and a lot of other art and groups cannot be cumulated. That's really bad news. And let me tell you why it's bad news. We don't know how to construct cat zero spaces that aren't cubicle. Or we don't have much, you know, we really constructed cat zero spaces out of something other than cubes is really hard. So if they're not cubicle, we're going to have a hard time answering this one. Because we're going to have to find weird cat zero spaces that don't come from cube complexes. And, you know, so that's a hard problem. All right, so OK. So let's keep going. So let's see, what else did I have? Two more, three. Oh, questions about, OK, these are, if you're not into this stuff, don't worry about it. There are weak versions of hyperbolic, like a cylindrically hyperbolic. I think somebody mentioned that in an earlier talk. So is a gamma, I'm going to leave space here for a reason, a cylindrically. So that means there's an action on a hyperbolic metric space that isn't as nice as you'd like. It's not proper, but it has some nice conditions, all right? So it's a weaker version of this. So it turns out that as a center, it can't possibly be. So we first have to bode out by the center to have any hope here of getting anything. So if it's finite type, it has a center. We have to kill that to even hope that it might work, all right? We more or less have an answer to this one, all right? This one appears to be yes, all right? So it was first done for the finite type by Calvez and Wiest. And then Indira, Chatterty, and Marta, is that how you pronounce it, did it for something called FC type, which we'll talk about later, and then Rose, Morris, Wright, and I have almost finished the picture, but there's still some technicalities, a few cases where we can't deal with yet. But the answer looks like it's going to be yes, and we've almost answered this one. So hopefully yes on all of those, all right? And the last one is, oh, I don't even want to say what these are. Nobody's mentioned hierarchically hyperbolic here. We won't worry about it. So there's something called hierarchically hyperbolic group, which I absolutely won't define, but what it's really about is trying to construct an analog of the curve complex for these things. So the braid group can be viewed as a mapping class group instead of as an art group, and it has a curve complex, and curve complexes are really useful. So the question is, is there something like a curve complex for other art groups, at least for the finite type? And that's an interesting question. Is there an analog of the curve complex for a gamma? And so there's some work on this by a paper with tons of authors, Komplito, Gebhardt, Wiesten. Thank you. Gonzalez-Menezes, yes. And then another one recently by Rose Morris Wright, who's anyway. There's some work being done on this, but we're far from a real answer to it. So I'm not going to go into these. These are beyond the scope of this talk. I simply want to say that there's a lot of interesting things to think about. Some of them are very basic and combinatorial, and are old, have been around forever. And then there's all these new questions involving what the geometry of these things look like. So I think they're interesting groups to study. I have only five minutes. So that actually brings us to the second part of the talk. But I wanted to give you a little bit of it because the exercises are between the two talks. So if you look at the exercises, the front page are some mostly fairly easy questions about basic questions about playing with some art and groups. And the second part is playing around with one of the cat zero cube complexes we're going to use in the second half of the talk. So what I'm going to do now is define it. And then in the second half of the talk, we will use it and see what it's good for. So as I said, so this is really part two. It's the geometric techniques that we've used to study art and groups. But before we do that, I need a little bit of terminology. So back to basic theory of art and groups. So first of all, some terminology, which is, or notation if you want. So suppose a gamma is an art group. I just want to give a name to the generating set. I'm going to refer to the generating set as s. So of course, it's equal to the vertex set of gamma. But I just want to be able to call it s. It's s1 to sn, whatever. So that's the generating set. Then for any, t contained in or equal to s, and that includes the empty set is allowed here. Any subset of s, we define, this is just notation, we let at be equal to the subgroup of a gamma generated by t. All right. Well, t is a bunch of vertices inside this graph gamma. And we could also look at the subgraph spanned by t. In other words, connect them with edges if and only if they were connected in gamma with an edge. So the question is, how are these related? And the answer is, it's what you think it is. So fact, and this is due to van der Lek. I forget when 80s or 90s, somewhere like that. He showed that if del t is the subgraph of gamma spanned by t, then in fact, in the obvious way, a t is naturally isomorphic in the obvious way to a del t. It is, in fact, the art group given by that graph. That's not surprising. Well-known for coxer groups, that's true. So if I restrict to any subgraph, I have some complicated graph gamma. And if I restrict to some subgraph in it, then the subgroup generated by those generators is the art group for that subgraph. That's all I'm saying. OK, so oh, these are called, these are known as, I'll usually just write like this. They're known as special subgroups. And more generally, they're conjugates. So if I conjugate one of them, say g, at, g inverse, are called parabolic subgroups. OK, so I just want to be able to use this notation. OK, so let me define for you, I'm going to go about two or three minutes over. We're just about there. We just need to give you the definition of this. OK, so back in the 1990s, Mike Davis and I, we were tried to prove a k pi 1 conjecture. All right, so what did we want to prove? We wanted to prove that this space, this universal covering space, was contractable. So what we did is we constructed a cube complex with the same homotopy type as this. We showed we could retract this thing onto a certain cube complex. So let's see. So we defined a cube complex, which I'll denote d gamma because we referred to it as the Deline complex. And the reason is that Deline had used something kind of like this in his work back then. We defined the Deline complex, a cube complex such that, and this part I won't explain how we did it. This is a whole other. It's homotopy equivalent to this. We basically retracted this space onto some cube complex. So we found this cube complex we could put inside it. So now proving this is contractable is a matter of proving a cube complex is contractable. Well, by the way, we already know it's simply connected because it's homotopy equivalent to this guy, which is the universal cover. It's simply connected. So I've got a simply connected cube complex. And I want to prove it's contractable. And the answer is just prove it's cat zero because every cat zero space is contractable. So there was an obvious sort of next step. So I claim, so suffices to prove D gamma is cat zero. And if you remember, I'll go back over this at the beginning of the next talk. But if you remember from Danny's talk to prove it's cat zero, we need that it's non-positively curved and simply connected. We already have simply connected. So all we need is non-positively curved. And we have great techniques for checking whether a cube complex is non-positively curved. So that is part of that is on your exercises. And I will simply define the complex and then let you go. I would say the first, as I said, the first part of the exercise sheet is just simple things on art and groups. The second thought is all about this complex. So what is this complex? So let me define it for you. D gamma is the following. So I'll start with the vertices. The vertices are in one-to-one correspondence with cosets, where A is any element in A gamma. And the important thing is that A t is finite type. So t is contained in S and A t is finite type. OK, notice, by the way, this is only interesting when the full A gamma is infinite type. Otherwise, it's not really interesting. So let's assume that the full A gamma is infinite type. So we've got an infinite type thing. But inside, there can be plenty of subgraphs that generate finite type. For example, any single edge generates a finite type. So it can have lots of subgraphs that generate finite type things, just because the whole thing is infinite type. Doesn't matter. So those are the vertices. Edges is just when one coset is contained in another and differ by a single. So if I can add a single generator to t and still get something finite type, then I connect those two. So they have to have a common representative. And they have to differ by a single generator. Differ by a single generator. And finally, what are cubes? Well, cubes are a generalization of this. They're what I call intervals. Namely, let's write AAT. So now let's say I differ by more than one generator. So where AAT is contained in AAR. And what is this cube? So what do I mean by this? Well, I'm going to look at everything that's between here and here. And I claim I'm going to see a cube. Because if I start out with little AAT, I can add one guy that's in here and was not in. I can add one guy, t union s1. And then maybe I add another guy. And then maybe I add both. And then maybe I add a third guy. So what happens here? Eventually, I'm going to end up at AAR. Here I've added one, an s, i. And maybe then I add another one, union s, i, s, j. And eventually, I've added enough to get me all the way to here. And if you look at all the different ways of adding this one first versus that one versus that one, what you see is the one skeleton of a cube. So if I look at all the things that live between here and here, I see a cube. I fill in that cube. Fill it in. Put a cube in there. So that's where my cubes are. I'm out of time, so I'm going to leave it at that. There is another way to describe this as a simplicial complex where you just think of this as a partially ordered set and take geometric realization. The problem is that description isn't going to help us. We really need the cubicle description to do anything with this. So the question's going to be, is this or when is this cat zero? That's part of that. The beginning of that question is on your exercise sheet. So this is going to play a central role in the second half. I'm going to quit here and have fun.