2/6 CAT(o) Cube Complexes
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Transcript: English(auto-generated)
00:17
Hyperplanes.
00:24
A mid-cube. So now I'm going to, I apologetically told you that we're not going to care so much about the cat zero metric, even though we're going to be interested in cat zero cube complexes.
00:41
And I said we're going to move towards more combinatorial way of thinking about things. The objects that, it turns out, we're going to care about are hyperplanes, and I'm going to start by defining them.
01:01
A hyperplane in a mid-cube is a subspace that you get by restricting one coordinate to 1 half.
01:33
So for instance, in this 3-cube right over here,
01:41
which we're going to conveniently identify with the product of three intervals, if we restricted one of the coordinates to a half, then we'd get that mid-cube over there.
02:05
And there's two other mid-cubes in that 3-cube. And well, maybe in this 2-cube, there's that mid-cube right over here. And in this 1-cube, there's that mid-cube right over there. You all know what mid-cubes are now.
02:21
A hyperplane in a cat zero cube complex, which I will call x tilde to remind you that it's simply connected, non-positively curved
02:43
cube complex, is a subspace which is built from mid-cubes in the following way. It's a connected subspace whose intersection
03:06
with each cube is either empty or an entire mid-cube
03:25
of that cube. So maybe I'll get a picture in here of a hyperplane, or maybe a few.
03:40
So bear with me for a moment.
04:23
Do I really need so much clutter over here? Maybe not. All right. So here's a, I think it's a simply connected, non-positively curved cube complex, right? Anybody check that it's a cat zero cube complex? You inspected all the links of zero cubes.
04:43
And let's try to find a hyperplane. And let's do it in a suggestive way that is going to help us understand what these things are. So let's start at a mid-cube right over here, this orange mid-cube right over there. And what are the rules? The rules are that if you intersect a cube,
05:05
if your orange object, your hyperplane, has non-empty intersection with a cube, then actually your intersection has to be an entire mid-cube of that cube. And right now, our cubes are all closed cubes, by the way. Right now, it intersects this 2-cube right over here
05:23
at this point. But the rules are that you have to intersect at an entire mid-cube. So I've extended it. Likewise, that 2-cube, it's still not a hyperplane now,
05:41
because notice that it intersects this 3-cube over here. It intersects it at a point. So it has to intersect it at an entire mid-cube, and similarly for its neighboring 3-cube.
06:05
Well, now I've got a hyperplane. So let's draw another one. I think you probably can do it yourselves now. How about this yellow one right over here?
06:22
There's a yellow hyperplane. Let's see if green is visible on this board. Yes, no?
06:41
No, yes, no. Red, sometimes it works, sometimes it doesn't. But you said that the cubes are not necessarily embedded. So when you say intersection, is it just a pullback, or it should be the image of it? Well, it's just a subspace.
07:02
So I don't know. I don't think I did anything wrong here. So I have an object. And I have a cube complex. And oh, I see your question. I understand.
07:21
So what he's asking about is, well, do cubes actually embed inside of a cat0 cube complex? So it turns out that every cube in a cat0 cube complex is going to embed. And so you're going to be safe for certain concerns
07:42
that you have. We will get to that soon. OK, so you see there are some other interesting hyperplanes, right? I don't have to draw more. And in fact, I think it's not obvious.
08:12
Anybody who's thought of non-positively curved cube complexes was eventually led to hyperplanes. But I think Sagiv was, I think, he was first.
08:25
Because it's tricky, because there's actually a literature on things like cube complexes called median spaces that precedes the topic from other area in math by 70 years or so. So it's hard to say if people have been there already
08:42
earlier, since the group theorists are a little bit oblivious of that topic, although that's changed in the last 10 years. Every mid-cube lies in a hyperplane,
09:07
so meaning every mid-cube of a cube within our cat0 cube complex lies in a hyperplane. Let's call that hyperplane v. This hyperplane v,
09:27
any hyperplane v, separates x tilde in the sense that the cat0 cube complex minus v
09:41
has two connected components. Actually, even stronger than that, what we call the carrier N of v,
10:07
which consists of all the cubes, the union of all cubes intersecting v,
10:25
all the cubes that contain, that carry its mid-cubes, has the property that it is isomorphic to a product of v and a 1-cube.
10:53
This carrier of our hyperplane turns out to be a convex sub-complex.
11:01
I'm going to talk about convexity later, but you can understand it for what it means. Any geodesic whose endpoints lie inside of it, lies inside of it. And you know about the metric on cat0 spaces
11:23
that I described already. I'm going to tell you about another metric in a moment. It turns out that hyperplanes are what give cat0 cube
11:45
complexes their character. And the world is filled with many cat0 spaces,
12:00
many wonderful spaces that satisfy this cat0 inequality. And cat0 cube complexes are among these spaces. But what really distinguishes them and makes them have extra wonderful features are these hyperplanes, these separating hyperplanes. And so, let's see, some examples
12:25
that you're familiar with of cat0 cube complexes. Trees are the simply connected one-dimensional cat0 cube complexes. And I already mentioned Euclidean space.
12:42
That's Euclidean tooth space over there, cut up into squares. And maybe the hyperbolic plane broken up into squares. I'm sure most of you have gone ahead and played a little game of tiling the plane by something
13:11
and sort of see what you get. You learn that you have to make the squares very tiny as you're going outwards.
13:23
You get a nice, if you have the tiling with five squares meeting around each 0 cube is my favorite. And well, the hyperplanes in these objects, well, the hyperplanes in the trees, you might not have noticed them, because they're so small.
13:41
They're just single points amid cubes of edges. The hyperplanes inside of the tiling of the plane, yeah, you know about them, but they're boring. And the hyperplanes over here seem just a little more
14:06
interesting, et cetera. They're going to look like some nice arc of a circle
14:25
that would intersect the boundary in the right way, if I drew this properly. You're all idealizing this picture, I hope. And this statement about the carrier looking about right to you now, so you know what's going on, and you already
14:40
were inspecting, testing it against that cat 0 cube complex over there as well. I need to, before I go on, I need to be a little bit honest about the metric that we're going to focus on. We just reported that in the theorem,
15:01
that the complex is a cat 0. Could you repeat that? In the theorem, is the cube complex always cat 0, or? In a cat 0 cube complex, the following hold.
15:23
Even for the first one? It's really true. This theorem's true. Yeah, yeah, yeah, the theorem's true. But if you don't assume that the cube complex is cat 0? If it's not cat 0, then if it's not cat 0,
15:44
but it's still non-positively curved, for instance, then you could end up with a situation like this. And you'll have a hyperplane.
16:01
You might sort of know that you want to define your hyperplane. You start off, and everything is looking good. You have this mid-cube, and another mid-cube, another mid-cube, another mid-cube, and then another mid-cube over here, another mid-cube, and then, whoa, you're going to crash into yourself, bang.
16:25
And over here, we all know that that orangey red thing is a hyperplane anyway. We know it's true, but it didn't quite satisfy the definition.
16:45
This can fail. This can fail, if you're careful and you're ruthless about the definitions. But you do get what's called an immersed hyperplane.
17:01
Wow, that's a really fine print. You probably can't see it. I'm going to talk about these later. They're going to be very important to us later. So it's good that you've, I think it's good that you've brought them up now.
17:22
All right, so let's get to work over here now.
17:45
My McGill students were waiting to watch me erase the blackboard. I didn't hear them giggling about it.
18:01
Let's bring some stuff down, just a little hook.
18:38
So the metric that we will use on a Cat0 Cube complex
19:16
is the taxicab metric.
19:22
I think people like to call that the L1 metric. We're not really going to use it a whole lot anyway, but let's be conscious of it.
19:45
It has the distance between points equals the length of shortest path that
20:12
is piecewise parallel to axes within cubes.
20:27
So in contrast to the picture that we had before, where we kind of went across diagonally in order
20:41
to get there quickly, now what we will do is we will force ourselves to travel in a very rigid way,
21:04
always staying parallel to the axes. So what did I do? I went from here to there. So for the Cat0 metric, I don't even want to draw it. I'll just make little dots with my fingers. We did it like this.
21:23
Now what we're going to do is we're going to travel just parallel to the axes. And the length is whatever the sum of the lengths were.
21:42
And unfortunately, we don't have uniqueness of geodesics. I told you the Cat0 inequality actually gives you uniqueness of geodesics. Lovely. But now there's even traveling around a single square, even traveling around within a single 3-cube.
22:04
There are many ways, even if you restrict yourself to traveling in the 1-skeleton, never mind traveling within the interior of the 3-cube. There's lots of ways to get as a geodesic from one 0-cube to the other. But if you like the 1-skeleton, which I do,
22:22
there's this way and there's also this way, for instance. And there are many ways.
22:42
So here's a pink way. So it's certainly not a geodesic. There aren't unique geodesics.
23:00
There are geodesics, though. The truth is that I am mostly going to almost entirely focus on the metric on the 1-skeleton.
23:23
It's going to be good enough for our purposes. And the L1 metric, or the taxicab metric, agrees with the usual path metric
23:45
when we just focus on the 0 cells. So let me say it in a more cleaner way.
24:01
Induces the same metric on the set of 0 cells as the usual path metric. And in particular, geodesics in the 1-skeleton
24:24
between vertices are geodesics in either the taxicab metric or just the graph metric on the 1-skeleton. So for the most part, you can just think about the path metric on the 1-skeleton over here.
24:41
Yes, we are not going to be Cat0. So unfortunately, we're going to use the term Cat0-cube complex, but we're not going to be using the Cat0 metric. What we are going to use is the enabling condition that the
25:02
links of 0-cubes are flag complexes. That's going to be a very powerful condition, and it's going to control everything that happens over here. Okay? So when you write X0-tilde, this is the set of 0 vertices? No, when I write? Yes. So this really means that.
25:23
Okay, but then if you have a square and two opposite vertices, the distance is square root of 2 in the previous metric, and now it is 2. So it is not the same metric? Right. Yes. So you say it induces the same metric?
25:40
It induces the same metric on X0-tilde as the usual path metric on the 1-skeleton. The Cat0 metric, forget about it. I only told you about it because I was pressured to do so. My roof is just going to confuse everybody.
26:07
In fact, it's beautiful and it's extraordinary that Gromov sort of threw these out as, oh yeah, these are good examples for you guys. These are non-positively curved cube complexes. They give you many nice Cat0 metric spaces.
26:22
But the Cat0 metric, in my experience, if I'm thinking about the cube complexes, something that's just about cube complexes, if you're using the Cat0 metric and you're sort of moving more towards traditional combinatorial group theory arguments with thin triangles and
26:41
constants and whatever, it turns into a huge mess. But if you just say I'm only going to use the flag complex, that the links are flag complexes, you support that? All I care about is that Cat0 spaces are contractible and for that I have to use the Cat0 metric. No you don't. You could just use the flag, you could just use the,
27:01
and you also don't, it also works when there are infinite cubes as well. Okay, so it's interesting to try to, there's a sort of theme of combinatorial non-positive curvature, which is alive and kicking right now. And we'll see what wins out
27:22
the geodesic metric non-positive curvature or combinatorial non-positive curvature. It's interesting, what's going to happen for art and groups? We shall see. I think I better go back to my lecture. And where was I?
27:43
I did the taxi cab metric. I'm going to tell you now about immersed hyperplanes and convexity. Yes, I will. I'm actually not going to use the metric a
28:01
whole lot, but it's going to come up for a bit now. An immersed hyperplane in a non-positive curved cube complex, you already know what it is.
28:21
It's, let me say it in a slightly more formal way, is a component of the space obtained by starting with all of the,
28:43
with the disjoint union of the mid-cubes of cubes of x and gluing them together by gluing mid-cubes of cubes of x
29:00
together along sub-mid-cubes. I'll just say along sub-cubes. So what is meant over here?
29:21
Let me move this guy up a little bit. I'll bring it back down in a moment. Let's draw one, let's give myself a ton of space, in fact.
29:43
They're on the floor because I throw them on the floor, right? Which floor? Are they camouflaged? Okay. Are they both there? Cool. Organized.
30:19
I need that guy up.
30:22
I'll take his friend down. Okay, so here's a picture of
30:40
immersed hyperplanes. So here's a, ooh, this is too much.
31:15
Okay, so there's a three-dimensional, non-positively curved cube complex.
31:21
And let me draw some of its immersed hyperplanes, but I'll draw them on the outside. So maybe I'll give myself a little hint to make it easier. I'll take that little square and I'll start with that.
31:42
So here's this one. There's that little square. And then there's that one kind of mid-cube that was there. And then there's another mid-cube of this square. Of this three-cube, rather.
32:09
And it continues on over there. And actually this, the first two-dimensional mid-cube kind of continued like that.
32:23
And that is one of my immersed hyperplanes. But there are others. There's also the green one with the purple one over here.
32:40
Maybe I'll add it over here. And it continues like that. And I guess there's another one over here that I forgot about.
33:03
I'll put it over here. And there's another one over there that maybe I also forgot about. Okay, and perhaps there's one more that I also forgot about. There's three more that I forgot about.
33:23
Yes, yes. And in fact, I think you all know what they are now. In fact, if you had taken the universal cover and you looked at the hyperplanes in the universal cover, and then you quotiented
33:41
them by their stabilizers under the action of the fundamental group, then the components that you get when you quotient are going to be these immersed hyperplanes. The objects that you get when you quotient the hyperplanes in the universal cover by their stabilizers are going to be these immersed hyperplanes.
34:01
But you can think about them downstairs in the base space just by collecting all of the mid-cubes that are living inside, just the disjoint union of all of these mid-cubes glued together in the obvious whenever they should be glued together because they share a submid-cube. And if you do that, you'll get this.
34:23
Okay, so now we know what immersed hyperplanes are. And they're going to be very important for us. As for hyperplanes themselves, they have what we
34:43
might call carriers. So these are immersed hyperplanes.
35:04
And they come equipped with maps. They're components. They come equipped with this map. The carrier envy of one of these, of an immersed
35:25
hyperplane, you probably know what it is. You just kind of thicken it up. It's the cube complex obtained by gluing together
35:48
ambient cubes instead of mid-cubes.
36:03
Okay, so, well, I guess in this picture over here, there'd actually be that purple hyperplane is actually
36:25
living inside of this carrier right over here. And likewise, the others. Okay, I'm not going to draw them all.
36:41
And usually they look just like products, but not always. It could be a little more complicated than that. Okay, are you good on what the carriers of hyperplanes are? Yes, of immersed hyperplanes? The orange red picture, what's the carrier? Is it going to be simply connected? Okay, so let's draw it, because it's self-crossed.
37:00
I'm going to draw it. It looks like this.
37:22
Okay, maybe it wasn't drawn in a way that makes the embedding so clear, but it doesn't matter. You'll deal with it. I think it looks like this.
37:43
And I'm not going to draw it as red and orange because, give me a break here. Oh, I can't control myself.
38:01
Alright? So every, an n-cube is going to kind of make appearances n times among these, within the carriers of all of these immersed hyperplanes. Once for each of its mid-cubes. So, let's check some important things.
38:36
The orders of the lectures is important because
38:41
we are in lecture three! Yes, we are in trouble.
39:23
Where are my disk diagrams? Not here. Not here. Not here.
39:41
We will go with the flow. It's going to be okay. Alright, so we mentioned immersed hyperplanes. Yes?
40:00
No, that's not necessary. So, you... It's not always a product, the carrier. Okay, you want an example? Okay! So, you've seen this example?
40:29
I forgot what it's called. You've seen this? Yes? Alright? Okay. We will return to it. It's important.
40:41
Now we know immersed hyperplanes and it worked out for us. Let's talk about disk diagrams now. I thought the carrier lived in the universal cover, so why isn't the carrier of this an infinite string of squares?
41:01
Because this is the carrier of an immersed hyperplane and that was the carrier of a hyperplane in the universal cover. This is consistent with the definition that we gave before, it turns out, right? You have to know various things, you have to believe that hyperplanes exist and embed and so forth.
41:20
So, you pull apart things only when there are different hyperplanes in the projection. Tell me, what are you asking? Are you asking me to tell you... Are you asking about the definition of the carrier of an immersed hyperplane? Maybe just of an immersed hyperplane. You're asking what is an immersed hyperplane?
41:41
So, the reason why you get two distinct orange and red ones there is because they have distinct projections. Are they projected to the exact... I never use the word projection. So, you're looking at the red and orange mid-cubes over there, inside of that three-cube.
42:03
We intuitively define what an immersed hyperplane is to start with and then I gave you a hint about how to make a rigorous definition. The hint of the rigorous definition is that you take all of the mid-cubes, you take their disjoint union and then you glue them together. Really, it's enough to identify sub-mid-cubes of mid-cubes
42:25
that will perform all of the gluings that you want. Well, that three-cube is going to, in this sort of rigorous definition, that three-cube contributes three mid-cubes, a purple, an orange, and a red. And they're going to live in... Well, these two happen to lie in the same immersed hyperplane
42:42
and this one lies in another. You're good now? I move forward. A disk diagram. Were disk diagrams defined in this conference?
43:04
So, a disk diagram is a compact, contractable 2-complex, let's call it D,
43:20
with a planar embedding. But you know what? Let's put it in the 2-sphere. Here are some examples of disk diagrams. And you know what? I'm going to stick to disk diagrams
43:45
that are built from squares, because that's what we're going to be focusing on. Here's just a tree. There's a single point we sometimes call the disk diagram trivial.
44:02
These are all 2-cells. And these are compact, contractable 2-complexes and I've embedded it in the plane in a very particular way. The boundary path of the disk diagram D, or boundary cycle,
44:21
is the attaching map of the 2-cell of the 2-sphere containing infinity. There's a point at infinity.
44:40
And if you look at the 2-cell containing the point at infinity, then it has an attaching map. And here it is. That's the boundary. There's a whole 2-cell right over there, and there it is. That's its boundary cycle. Likewise here.
45:05
I guess I didn't really need to use this nifty way of catching it, but it's useful for this guy, because there's more than one embedding. There's more than one embedding of this 2-complex in the 2-sphere. This one basically has only one embedding.
45:23
There's that, and then you can reverse the orientation. And this has many embeddings. And so you really, in order, well, you could say that if you had declared what the boundary cycle was, well, I'm not going to be fussy about the orientation.
45:40
Here, the trivial boundary path. OK. It is a famous fact, a famous theorem of van Kempen that was long overlooked, which says that for any closed null homotopic path,
46:07
p to x, this is going to be, say, a 2-complex or a complex, there exists a disk diagram, D,
46:24
and actually a map from D to x such that p is the boundary path of D.
46:52
OK, what do I mean? I mean, well, it's a very nice way of
47:01
seeing the null homotopy. So we know that null homotopy means that it factors through a disk on the way in, but there's a more combinatorial. This map is a combinatorial map, and a combinatorial map.
47:22
So a map between cell complexes is called combinatorial if it sends open cells homeomorphically to open cells. So one cell goes to one cell, two cells go to two cells by homeomorphisms. A very, very nice map. So you have some complex, something really messy
47:42
and complicated, and you have a path, p, and you know that it's null homotopic. Well, you can actually see that it's null homotopic by exhibiting it as the boundary path of some disk diagram.
48:10
And of course, if p factors through this disk diagram on the way to x, then p is null homotopic. And actually, it's basically a simplicial approximation
48:22
theorem. If you have a null homotopy, so you're factoring through a disk, but it's really a topological thing that's happening, you can adjust and choose and actually factor through a disk which is kind of tessellated. Oftentimes, it's still
48:41
homeomorphic to a disk, but sometimes it will have singularities to it, where these are called singular points. So this isn't really homeomorphic to a disk. It has cut points. All right. So, we're going to use this. We're going to use this.
49:01
Excuse me? Thank you. Thank you very much. Okay, so we're going to... Of course, once you start with this,
49:21
you're interested in minimal diagrams. Minimal disk diagrams have fewest cells.
49:43
Usually people focus on the number of two cells, and they call that the area of the disk diagram, among all disk diagrams with a given boundary path.
50:18
So the combinatorial group theorists view this disk
50:23
diagram as the proof that P is null homotopic. They view this disk diagram, and you can actually think of it as encoding the path P as the product of conjugates of relators, if you're thinking about a group presentation, if you've seen this before. And while we like short proofs, so we prefer to find
50:45
a minimal size disk diagram that shows you that it's null homotopic. Now, if you're working in a cube complex,
51:24
when D to X is a disk diagram in a cube complex,
51:40
D is a square diagram. It's going to just be mapping into the two-skeleton. And we're going to focus on its immersed hyperplanes are
52:10
called dual curves. So let's give a picture that will explain all this.
52:53
There, there's a disk diagram right over there. I don't know if it's minimal or not. I didn't check. And you already know what these dual curves are.
53:03
They're just what we would have called immersed hyperplanes. Here's a dual curve right over here. Here's a red one, and here's a yellow one.
53:24
Maybe there's a green one to test the green-on-green theory. So everybody knows what dual curves are now.
53:40
And these dual curves are kind of, their carriers are ladders. So for example, the carrier of the yellow dual curve
54:04
looks like this. See, it looks like a ladder, a crooked ladder. And well, I guess in principle, they could be cylinders.
54:26
It could look like this also. That could happen because it's in a disk diagram.
54:42
And the disk diagram is a contractible. You said it was in a cube complex. D is a disk diagram mapping to a cube complex. You think of the carrier in D, not in X.
55:02
Right, that's correct. So the lingo, unfortunately, the lingo is a map from a disk diagram. So the disk diagram is living inside of the plane. And when we say a disk diagram in X, we mean a map from that disk diagram to X.
55:21
And we're talking now about these dual curves, which are these immersed hyperplanes, in the disk diagram. So it's just two-dimensional cube complex over here. So let's state a quick theorem
55:52
that if D to X is a minimal area diagram, disk diagram,
56:14
then D has no. And there's four configurations that we want to draw your attention to.
56:25
It has none of these. It's statements really about the carriers of dual curves.
56:54
This is the most important. There are no bigons. So I'll even say no bigons.
57:03
OK. There's a little further more that's interesting, but I'll kind of leave it out over here. Let's talk. So I have two minutes plus the three minutes
57:25
because I had to erase the blackboard on my own. Right? What else do I got? I don't have a lot, huh? So it works that way. It works that way. Whatever I want, no. That's Friday.
57:43
Let's talk about the, I'm going to describe the method of proof, but I have to sort of be, I have to tell you maybe how the story fits together.
58:02
What we're going to find is that, here it goes. Method of proof. Indira made me do this.
58:21
Hexagon moves. So if you ever see this, either of these configurations inside of your disk diagram,
58:41
you could swap it and replace it with the other. Because if you see one of these, there's actually a 3-cube inside of the cube complex, and you could take that disk diagram and you could push it across the 3-cube to get a slightly different disk diagram. And there's another important thing that you might do.
59:05
If you ever see two squares in your disk diagram that are meeting along a path of length 2, they're forced, since the link is a flag complex,
59:22
they're forced to map to the same square inside of X. You didn't mention that it's not positive in here. Oh, you're kidding me. That's not over there. It's over here.
59:45
Thank you. If you have a pair of squares that are meeting along two edges, then those squares have to map to the same square
01:00:00
inside of the cube complex X. Why is that? Well, the link, these two corners of those squares, have to map to the same corner of a square at the image of this white vertex over here.
01:00:20
And the links are flat complexes, so there's only one edge joining two points in the link. So if you ever see two squares that are meeting along a pair of edges inside of a disk diagram, they have to map to the same square in X. But what it means is that these two edges are
01:00:45
the same as these two edges. So if you ever have this, then you could have just cut it out of the disk diagram and replaced it with that. And so your disk diagram, which has
01:01:00
this what's called a cancelable pair, can be made smaller by just doing this simple replacement. This is a very important notion in combinatorial group theory. Right over here as well, if you have a disk diagram that contains this picture, you can replace it by a disk diagram that contains exactly that picture.
01:01:20
Because these two hexagons have the same boundary path. So I am going to stop now, which is an incredible display of self-control. Thanks, good self-control.