4/6 CAT(o) Cube Complexes
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GeometryCubeComplex (psychology)Musical ensembleTheoryHardy spaceCartesian closed categoryMusical ensembleSpacetimeMereologyConvex hullRight angleCovering spaceMoment (mathematics)Field of setsPositional notationConvex setCombinatoricsCharge carrierHyperplaneObject (grammar)Isometrie <Mathematik>Many-sorted logicCurveLinear subspaceHyperbolische GruppeNeighbourhood (graph theory)Term (mathematics)Lemma (mathematics)Plane (geometry)Perspective (visual)Local ringCubeEinbettung <Mathematik>Cohen's kappaLecture/Conference
09:00
Real numberImmersion (album)Object (grammar)HyperplaneSubgroupSpacetimeCategory of beingCubeHyperbolische GruppeNeighbourhood (graph theory)Graph (mathematics)Compact spaceCombinatoricsSquare numberPoint (geometry)Right angleField of setsCircleLine (geometry)Correspondence (mathematics)Dimensional analysisMultiplication signNear-ringIsometrie <Mathematik>MereologyFilm editingMusical ensembleGraph (mathematics)Proof theoryLocal ringFundamentalgruppeFree groupFinitismusCharge carrierConvex hullNatural numberHomologieExistenceCycle (graph theory)Sign (mathematics)Arithmetic meanCurveRandom matrixLecture/Conference
17:55
Iterated function systemCubeMultiplication signProduct (business)Field of setsOrientation (vector space)Square numberMoment (mathematics)Vertex (graph theory)Projektive GeometrieRight angleDirection (geometry)HyperplaneCharge carrierGraph (mathematics)Isometrie <Mathematik>Musical ensembleFocus (optics)Local ringOscillationAxiomSinc functionSet theoryCollisionGraph (mathematics)CoalitionArithmetic meanLecture/Conference
26:50
TriangleHyperplaneLocal ringField of setsMany-sorted logicCubeGamma functionPoint (geometry)Doubling the cubeIsometrie <Mathematik>Vertex (graph theory)Proof theoryArrow of timeGraph (mathematics)Graph coloringPlane (geometry)Right angleSquare numberCoalitionLecture/Conference
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HyperplaneMany-sorted logicIsometrie <Mathematik>CombinatoricsGamma functionConsistencyMusical ensembleGroup theoryTheoryGeometryCubeFundamentalgruppeField of setsCompact spaceSquare numberRight angleDirection (geometry)SubgroupGeometryInjektivitätExtension (kinesiology)Point (geometry)Generating set of a groupCommutatorPlane (geometry)State of matterElement (mathematics)ModulformMetric systemLocal ringLecture/Conference
39:07
CurveRight angleCovering spaceCategory of beingField of setsCompact spaceMusical ensembleCubeFundamentalgruppeSubgroupHyperbolische GruppeSign (mathematics)Price indexCurvatureTheoremCanonical ensembleComplete metric spaceMany-sorted logicFinitismusPresentation of a group3 (number)Social classLecture/Conference
43:27
SpacetimePositional notationComplete metric spaceLecture/Conference
44:16
SpacetimeDiagramIsometrie <Mathematik>TheoremLocal ringEinbettung <Mathematik>Lecture/Conference
45:08
Einbettung <Mathematik>Musical ensemblePrice indexFinitismusNetwork topologySet theoryLemma (mathematics)CubeBasis <Mathematik>SubgroupRight angleTheoremGroup theoryEndliche GruppeProfil (magazine)Direction (geometry)Order (biology)3 (number)Hausdorff spaceMultiplication signElement (mathematics)WeightResidual (numerical analysis)Graph (mathematics)QuotientMany-sorted logicArithmetic meanProendliche GruppeTopologischer RaumSpacetimeLecture/Conference
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Isometrie <Mathematik>SpacetimeRight angleArc (geometry)SubgroupGeometryMusical ensembleCanonical ensembleFinitismusCompact spaceMaxima and minimaImmersion (album)Vertex (graph theory)CubeGoodness of fitNumber theoryTheoryReal numberField of setsComplete metric spaceGraph (mathematics)Parameter (computer programming)TheoremLocal ringIsomorphieklasseCombinatoricsSubgraphResultantFundamentalgruppeCovering spacePrice indexProcess (computing)TrailConnected spaceFlow separationNetwork topologyDiagramFiber bundleBeat (acoustics)Many-sorted logicClosed set3 (number)Lecture/Conference
Transcript: English(auto-generated)
00:19
Let's call this a lemma.
00:23
The map from the carrier of a hyperplane to x. I'm not going to say anymore that x is non-positely curved. They really all are non-positely curved. Is a local isometry.
00:45
So maybe that's a lemma, maybe that's an exercise, but you'll accept it. And the corollary is that the universal cover of the carrier, which is the carrier of the universal cover, as yesterday,
01:13
or as two days ago, I see, there was a break yesterday. So let V be an immersed hyperplane
01:27
in a non-positely curved cube complex. Then its carrier maps to x by a local isometry. You can kind of believe that from the pictures, but you have to check.
01:41
Well, I suppose, in particular, if x is simply connected, the carriers of hyperplanes of a cat0 cube complex, x tilde,
02:11
are convex sub-complexes.
02:21
And in particular, the concern that you might have had that maybe you start somewhere and then you crash into yourself as you're wandering around through x tilde, that concern has gone away. Because we could have defined a hyperplane in x tilde as an immersed hyperplane.
02:41
Those exist. You start off with x tilde. It has immersed hyperplanes inside. Maybe they crash into themselves, who knows? Those immersed hyperplanes map to x tilde by a local isometry. This could have been x tilde right here. And local isometries are embeddings.
03:00
So hyperplanes aren't going to crash into themselves. Their carriers don't even touch themselves. OK, so that's how we know, or it's one of the ways of knowing, that hyperplanes exist, knowing that the carriers of immersed hyperplanes are local isometries, mapped to x by local isometries.
03:26
Maybe something that you might consider, if you do have the carrier of a hyperplane, then you actually have various other convex subspaces.
03:44
In fact, this, which we sometimes call the frontier of the hyperplane, is convex, because it's convex inside of the carrier. And the carrier is convex in the whole thing.
04:00
And actually, this shaded part is also convex. It's bounded by this convex object. It's going to also have to be convex. Because you can't leave it and come back, because that would violate the convexity of the yellow thing.
04:22
OK, this is called a half-space, or sometimes called a minor half-space. Maybe I'll write that. A minor half-space, because there's also a larger half-space, which also is convex. Here's the larger half-space, and here's the major half-space, and here's the minor half-space.
04:43
They're both convex. And you can prove it by verifying that they're mapping by local isometry, or it follows from other things that we've just discussed. So in fact, just to put things into perspective,
05:02
well, a sub-complex is convex if and only if it is the intersection of minor half-spaces.
05:27
So just to put a little things into perspective over here. So that's what convexity can be interpreted in terms of minor half-spaces. And let's actually say something
05:41
a little bit more fundamental about convexity now. And it's the following statement. So this is now going more towards the world of groups. Let X tilde be a delta hyperbolic ket 0 cube complex.
06:12
And let S be a kappa quasiconvex subspace.
06:27
Tell me quasiconvexity was defined last week. Thank you so much. Then there exists R such that the intersection
06:46
of minor half-spaces containing S, let me write it over here, such that the whole of S,
07:00
which is the combinatorial convex hull of S, which is the intersection of half-spaces containing S, I'll explain that notation in a moment, is contained in the R neighborhood of S.
07:22
So let me explain that notation. So this over here is n, is a carrier, right? It's short for n of a hyperplane. And this minor half-space, I think I'd use the notation for all of this yellow part over here,
07:43
I would call that n left. And maybe, I guess if you'd like, you can call this part over here n right. But who's to say which is left and which is right? So I'll just write n left now.
08:01
So if you look at all of the half-spaces, which contain S and you take their intersection, that's the hull of S. That's actually contained in a small neighborhood of S. So the picture is the following.
08:25
We have S, excuse me, yes, please? Excuse me? Do we know that hyperplanes are two-sided? Yes. So once you know that the carrier embedded,
08:44
and the carrier is this universal cover, from there you could see that it's two-sided because if it were one-sided, then there'd be a path in the carrier. The carrier is simply connected, of course, because it's living inside. There'd be a path in the carrier
09:00
that goes through the hyperplane and only touches it once. And so that would violate that it's simply connected, because there'd be a cycle in homology. Okay, thank you. What are your states? What if the whole thing's free spaces? Excuse me? What if the hyperplane carrier,
09:20
all of it, there's your space, and then it's a free face of the key complex, so there's nothing? I didn't understand that. So we will talk about it afterwards. Let me draw some pictures. I have so many things I need to tell you. Every time you guys ask me a question, you're losing things. So it's painful.
09:42
It's painful to be here. Okay, so you can look. Here's S, and the picture is that we're looking at all of the half spaces that contain, here's one of them.
10:03
Here's another one. Here's another one. So these are all the half spaces that contain it. You might have seen pictures like this somewhere before. And this whole object is contained
10:24
inside the R neighborhood for some R. So here is my hull. I'll draw it in white. And that hull lies inside of the R neighborhood of S.
10:43
Okay, so that's the picture that you should have in mind. Okay, the idea of the proof is somehow, I'm not going to explain it, but the idea of the proof is along the lines of, well, if you traveled out more than R,
11:01
then you could find a hyperplane which doesn't cross. It doesn't cross S. So because of hyperbolicity, if you travel out far enough and you look at a cube that you're near, and one of the mid-cubes of that cube
11:21
is going to give you a hyperplane which doesn't intersect S. And so any point that's sufficiently far out over here is going to be cut off from S by some hyperplane. That's roughly the way the proof goes. Now the corollary of this is very, very important. Very important. It's critical.
11:40
Let X be a compact, non-positively curved, cube complex with pi 1 of X hyperbolic. Let H be a quasi-convex subgroup.
12:08
Then there exists a local isometry from Y to X with Y compact and pi 1 of Y mapping to H.
12:29
So I'll just say equals H. So what this is telling you is that if you ever have
12:43
a quasi-convex subgroup of a hyperbolic group that is the fundamental group of a compact cube complex, you can represent that subgroup with a non-positively curved cube complex. This is generalizing the idea.
13:00
So this generalizes our understanding subgroups
13:20
of the fundamental group of a bouquet of two circles. Whenever you have a finitely generated subgroup of a fundamental group of two circles,
13:45
that subgroup is represented by a combinatorial immersion of graphs, for those of you that have studied free groups. And it's critical to understand this. This is the main point in understanding their subgroups
14:01
is this idea that you can always represent subgroups using an immersion of graphs. Over there, it's quite easy because there are no squares around. So the existence of this is quite simple. You look at the covering space associated to your subgroup, and you choose a finite, if it's finitely generated,
14:21
you could choose a little finite subgraph, which the whole covering space deformation retracts to. And any immersion of graphs is a local isometry, so it's easy. This is just the generalization of that idea, and it works quite nicely. And it's part of a theme where freeness gets replaced by hyperbolicity, graphs get replaced by cube complexes.
14:53
It's painful. Let's cut this, and maybe I'll get back to it.
15:01
So it's kind of Galois correspondence for local isometries, right? Excuse me? It's kind of Galois correspondence for local isometry. I don't know what Galois correspondence means. OK. So you mean a correspondence between subgroups and covering maps?
15:21
No, it's not, because it's only for quasi-convex subgroups. So let's now move to special cube complexes. So what I'm going to tell you now
15:42
is about a category of non-positively curved cube complexes that even better echo the nature of graphs. So graphs, we feel like we can understand the subgroups of their fundamental groups. So this is going to be a certain non-positively curved
16:02
cube complexes that are especially nice. So here goes. A non-positively curved cube complex X is special if each immersed hyperplane embeds
16:29
and is two-sided, doesn't self-osculate.
16:43
I have to tell you what that means. And also, and no two hyperplanes inter-osculate. These have to be clarified.
17:04
So I'm going to draw pictures of what's not allowed to happen. So no pathologies of the following four types.
17:27
And my pictures are going to be in two dimensions. But I think they get the point to cross. Well, self-crossing is not allowed. One-sidedness is not allowed.
17:52
A hyperplane cannot self-osculate. And what that means is that it's not
18:02
allowed to come along right next to where it was. But we're very picky about what that means. Actually, we're a little careful about it. Since hyperplanes are two-sided, we can do a convenient thing.
18:21
We can orient, direct all of the edges consistently. We don't have to worry about being well-defined over here. And self-osculation means two edges at a zero cube,
18:48
which are dual to the same hyperplane and are directed in the same way. They're both directed outwards, let's say. OK, this is not allowed.
19:00
And does it wait enough? Well, the other way, if you change the orientation, it would be this way. So the two ways are exactly the same. So it's not allowed. Oh, I see the other way. This is allowed. If I could draw it properly.
19:33
This is OK.
19:44
I don't want to draw things that are allowed here. But I'm going to say OK over here. OK, that's what you're asking about, correct? Yeah, excellent. So interosculation is the following.
20:04
It's a pair of hyperplanes that cross each other. They cross each other in one spot,
20:25
osculate with each other without crossing at another vertex. This is interosculation. This is self-osculation.
20:40
This is interosculation. It doesn't matter. It doesn't matter. So the key thing that matters is the idea that there's no square over there. OK, so the key thing is that even though they
21:01
are dual to one cubes, which even though they're dual to one cubes at this zero cube, they don't cross the same square at this zero cube. OK, so interosculation says that there's a square over here, but there's no square over here.
21:26
But it's not there, so I'm going to draw it in purple, so it's hard to see. No square.
21:42
Because what would happen if there were a square over there? They would cross over there. Because they would both enter that square on two different one cubes right over there, and then they would cross. OK, the way when Friedrich Haglund, with whom we cooked up this definition many years ago,
22:05
at the beginning, we would remember it by saying, crossing pair has a square. OK? That's me, because Friedrich remembers everything after one time, and I have to make little rhymes for everything to remember stuff.
22:23
And this looks similar to something we discussed earlier, doesn't it? The further more is somehow related to this. These are very artificial-looking definitions,
22:43
I hope you agree. I mean, they certainly are. They came out of nowhere for us. Let me give some, let me mention, so at least I'll mention an exercise. I should give an example. I'll give one in a moment. So among our exercises, if X is contained
23:03
in the product of two graphs, then X is special. OK, so now at least you know that we're not talking about the empty set over here, right? Any graph is special.
23:23
Also an exercise, the Salvetti complex of a rag is special. Now I understand I'm supposed to call them right-angled Titz groups, yeah?
23:44
And actually, a cat zero cube complex is special. Let's stop for a moment. I'm not going to prove it in its entirety, but let's stop for a moment and think about this statement
24:01
over here, that a cat zero cube complex is special. Well, hyperplanes don't self-cross, right? We've already dealt with that. And actually, you know, they're two-sided. The carriers look like products, right? And they don't self-osculate. Those carriers embedded, OK?
24:22
They don't even have the other type of osculation that you thought of, right? And they don't inter-osculate. That's because of the, it's because of the furthermore statement. I'm not going to spend more time on this, but it's because of the furthermore statement that they can't inter-osculate, all right?
24:46
And then, well, we cooked up the notion of a special cube complex for a different purpose. But fairly quickly, we found the following. And that purpose I will tell you about in a little while. It will be our focal point.
25:01
We found the following, that a cube complex is special if and only if there exists a local isometry from X to R
25:23
where R is a Salvetti complex of some graph. And actually, this turns out to be easy.
25:47
It was very strange for us when, you know, we didn't believe it at first. Well, so this direction holds because pathologies
26:02
project to pathologies. Namely, well, if you believe that a Salvetti complex is special, right? And that's an exercise to do that.
26:22
The hyperplanes aren't so big. Things are pretty small. There's just one zero cube. So it's handleable. Well, if you have a pathology in X, like a self-crossing hyperplane, it would project to a self-crossing hyperplane in R. But R is special. So that can't happen. Likewise, the other pathologies
26:42
project under a local isometry. In this direction, I will tell you that the most important thing is that the graph gamma, right?
27:04
So if X is special, we're going to create a local isometry from X to a Salvetti complex. And which Salvetti complex? Right? Because there's one for every simplicial graph. So let's first notice which graph we're going to use. We use what's called the crossing graph.
27:27
The hyperplanes, let's call it the crossing graph, gamma has a vertex for each hyperplane.
27:53
And an edge joins vertices whose associated hyperplanes
28:13
cross. So maybe I'll quickly draw a picture
28:23
to get an example going over here.
28:55
So this is a cube complex with six, seven, eight, nine squares over here.
29:02
Nine squares. Don't tell me I got that wrong. And it's actually special. You can check that it's special. This is special. OK, there's a bunch of things you need to check all the, right? But certainly you check that the hyperplanes don't
29:21
self-cross, right? And you see that they're two-sided. Nobody sees any self-oscillation. And I promise there's no interoscillation. OK, that's the way the proofs go. What is this gamma? Let's try to suggest gamma. So maybe I'll do it like this.
29:49
So there's an orange hyperplane. I'm going to run out of colors. But everybody see that orange hyperplane? Well, I can draw edges, a dual to it.
30:05
And I'm actually going to mark them and direct them. OK, so instead of drawing the orange hyperplane, I can just draw a bunch of edges labeled by a single arrow that are all dual to it.
30:22
And let's do that for the other hyperplanes as well, because this is going to tell us everything over here. So there's this hyperplane here. And it's a sort of double triangle.
30:40
And there's this single solid triangle as well. And then there's this triple solid triangle.
31:01
And then I've got my hyperplane, which is dual to the triple arrows.
31:21
And that's it. And then one more. I'm having trouble seeing it. Who sees another one? The three cube has one, two, three, four. There's one three cube over there. You guys can tell me.
31:40
You'll correct me. Oh, you mean one more of these? Yeah, yeah, of course. Of course, I didn't finish that. So I guess the goal of these sorts of lectures is to get people to the point where they're correcting you frequently. So we're almost there. So there are all the hyperplanes.
32:02
And now we might as well recognize gamma.
32:24
So I'm sorry, I'm not going to draw it orange. It's drawn white over here. But one arrow crosses two. Hyperplane's two and three cross. Hyperplane's one and three. Hyperplane one crosses with double triangle. And hyperplane one crosses with single triangle.
32:43
OK, so this is gamma. I also have a triple solid triangle that I left out. And thank you. And it's right over here. Thank you very much.
33:01
All right, so here's x. And here's R of gamma. R of gammas look something like this, et cetera.
33:20
Here's its one skeleton. And then it has a bunch of squares. And it has a three cube as well. Right, that's the one skeleton of our gamma. Now everybody sees a map from the one skeleton of x to the one skeleton of R.
33:41
Because the way that I've directed and labeled the edges determines a map, right? You just send each edge to the corresponding labeled edge. And you know exactly what direction to use. And well, whenever you have a square over here,
34:03
that means those hyperplanes, whenever you have a square over here, that means the corresponding hyperplanes cross, which means that you're going to have a square to map it to. So every square has a square to map to over here. And whenever you have actually an n cube, it means all n of those hyperplanes cross.
34:21
So if you have an n cube over here, it knows there's a place for it to map to over there. And actually, this ends up being a local isometry. So we used, for instance, two-sidedness to know that we can direct things consistently.
34:40
The fact that it's a local injection is going to need, as well, that the hyperplanes don't self-cross and that they don't self-osculate. And then the idea that there are no missing squares,
35:07
which is what we need to make sure that it's a local isometry, that's actually going to follow from the interosculation. So this is something that can be worked through. It takes a little while to think it through, but it's really that simple. I've sketched out the main point
35:20
to realize what the map is. Now, what's the corollary of this? I have a question. Yes, what's your question? Do you mean that x is compact here? Nope. Excuse me? If x is non-compact, why is gamma finite? It's not. Why is it finite?
35:41
So it's already complexly infrared in the cross. Sure. There's a lot of people out there that want their cube complexes to be compact. We saw a wonderful one yesterday, two days ago. She's not here anymore, right? Otherwise, she would have been complaining. In fact, I actually have to warn you all that geometric group theorists love compact things
36:04
because then they feel secure that they have control. You're all control freaks, right? But the reality is that compactness is not natural. It isn't always the right thing. If you're studying finally presented groups
36:22
and you want to have control, the control actually comes from other elements. Usually, we're going to see what they are tomorrow. So if you want, tomorrow we will see. But compactness, we don't necessarily want compactness. Although, we'll take it.
36:40
If you can give me compactness, I'll take it. I love it. OK. But we don't need it, though. So am I 35 minutes over or I have another 25 minutes? Yeah, that was a good way I did it, right?
37:02
So what's the fantastic corollary that we can state? If x is special, then pi1 of x embeds in a rag.
37:28
So we have our local isometry from x to this Salvetti complex. The fundamental group of the Salvetti complex is this right-angled Artin group, which is a very trivial-looking group, right?
37:40
A bunch of generators and some of them commute. So if you manage to find a cube complex that is special, that cube complex, its fundamental group is somewhat reasonable. It's at least a subgroup of something that you might understand. OK. So this turns out to be special cube complexes form
38:15
a bridge from some type of combinatorial geometry world
38:32
to a kind of combinatorial group theory,
38:41
because right-angled Artin groups, they're the sorts of groups that would make combinatorial group theorists happy. You don't even need geometric group theory to a certain extent. You don't even need geometric group theory to understand them. You could just mess around with letters.
39:02
Life isn't so easy, though. Well, unfortunately, we end up getting led to worrying about virtual specialness.
39:24
We end up with a question, because we might find a non-positively curved cube complex. I'm going to show you where they come from tomorrow. You could make them by hand, which is great, but they actually show up for other natural reasons.
39:41
And your cube complex is probably not going to be special, because it'll surely have some pathologies. I mean, self-crossing happens very quickly. If you're just randomly gluing cubes together, forget it. You already have some self-crossing. Forget about the non-positive curvature. Even if it's non-positively curved, you'll probably have some self-crossing.
40:01
So one is led, though, to try to understand virtual specialness. And there was a long exploration to understanding this for the last almost 20 years now, which culminated in a goal proving that if x is compact and pi 1 of x is hyperbolic,
40:35
then x has a finite cover that is special,
40:44
is virtually special. So this cube complex, whose fundamental group is hyperbolic, has a finite cover, probably a very big finite cover, but a finite cover which shows you
41:03
that this finite index subgroup lives in a right-angled Artin group. And it answers many, many questions about that hyperbolic group, because right-angled Artin groups are linear. They live in SLNZ. They are orderable. They have all sorts of lovely properties. So this teaches us many, many things
41:20
about a very, very large class of groups. It turns out that the hyperbolic groups that are fundamental groups of non-positively curved cube complexes are quite common. And one could argue that most of them are. At least when there aren't that many relations, most of them are.
41:41
When you have many, many relations, and I've been guilty of not saying this, when you have many, many relations, then you likely have property T, which I'm not going to talk about. And property T is mutually exclusive with cube complexes.
42:05
So a group with property T is not the fundamental group of a cube complex, unless the group is trivial. All right. So where did this notion of special cube complex come from?
42:21
Because I told you, even though we were very excited that if it's special, then it lives in the rag, because being special is actually easier to just define special this way. Not that way. That was a hard way to define it. It was a strange way. Where did this notion of special come from?
42:41
Let me explain. So there's something that we call canonical completion and retraction. And I'm just going to suggest it.
43:01
So there's the following theorem about special cube complexes. Let y to x be a local isometry with y compact and x special.
43:31
There exists a finite covering space of x.
43:42
I will use the offensive notation such that y lifts to this what we call canonical completion.
44:06
Moreover, there exists a retraction
44:23
from this covering space to y. So I'm remiss. Let me draw a little bit of a picture, a diagram, of what's happening. So you start with a local isometry,
44:43
and the theorem says that there's a finite covering space of x such that the local isometry from y to x
45:01
actually lifts to an embedding. And not only that, there's a retraction back to this embedding of y. So I'm not going to explain this for cubes,
45:23
but I'm going to explain this for graphs in a way that I hope will take everybody along and you'll know exactly what sort of thing we're talking about over here. So, excuse me? Can you assume that the spaces are connected? Well, it is assumed, but it's true.
45:47
It's true. The non-connected, let's see. I think that the non-connected case follows from the connected case. But assume that they're connected,
46:00
so we don't have to talk about that. By the way, I appreciated that you came and asked me to check the trivial, trivial cases. John Stallings used to do that to me
46:21
whenever I told him anything, and I do it to other people. So we should all do it to each other all the time. So here's a lemma that a right-angled artin group
46:42
is residually finite. There's various ways of proving this. Well, what is residually finite? Residually finite means, everybody knows what that means? Yes, every element, every non-trivial element
47:01
survives in a finite quotient. What I can tell you is that 20 years ago, you had to say what that meant every time you gave a talk at a group theory conference. So the world's changed a little bit. We pulled it along over in this direction, which is nice.
47:23
So why are right-angled artin groups residually finite? Well, one way of proving it, you could prove it in many ways. One way of proving it is that right-angled artin groups are subgroups of coxeter groups. They're subgroups of right-angled coxeter groups, actually. And right-angled coxeter groups, I think,
47:41
are widely known to be linear. And finally generated linear groups are residually finite. So that's one of the explanations. And then another lemma is that a retract
48:02
of a residually finite group is separable. So what does separable mean? Well, a subgroup H of G is separable
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if, well, H is equal to the intersection of finite index subgroups of G, probably infinitely
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many finite index subgroups of G. For those of you that prefer, you can put the profinite topology on G.
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And this just means that H is closed in the profinite topology. Residual finiteness means the profinite topology on G is the topology given by a basis of cosets of finite index subgroups, that that topology is Hausdorff. OK, so this is a fun lemma.
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There's various ways to prove it. One way is, I guess, retracts of Hausdorff topological spaces are closed. OK, that's how it fits into things. So I mentioned these two, though,
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in order to observe a corollary of this theorem. So we get the corollary that pi1 of y
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is separable in pi1 of the canonical of this covering space, and hence in the slightly larger group, pi1 of x.
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So if you're separable in a finite index subgroup, then you're separable in the whole group. And you're separable in this finite index subgroup because you're a retract of it. So this theorem about retractions and covering
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spaces is really aimed at understanding separability. But being a retract is, of course, a bit better. So let's state one more corollary before I sketch what this theorem looks like. So that corollary, just to make things a little bit real
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and to connect up together with what I mentioned before. When x is special, and pi1 of x is hyperbolic,
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and h is quasi-convex, then h is a virtual retract,
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and hence separable. And all you're doing is saying, OK, I have a quasi-convex subgroup. That means that I could find a local isometry.
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I have this special cube complex, let's say. And OK, it's compact, and its fundamental group is hyperbolic. I have a quasi-convex subgroup, h. So I could find a compact cube complex, y, and a local isometry from y to x such that h is just pi1 of y.
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And now I then say, OK, well now I have a local isometry from y to x. I want to study y by thinking about this local isometry, which is really a much better way to think about the subgroup. You have real good control over everything. And I'm able to pass it off to a finite covering space of x
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so that y is a retract of that finite covering space. And so then I see that my quasi-convex subgroup is actually separable. So there's a nice interplay over here between the geometry and certain combinatorics
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that we have working for us. And it leads to nice algebraic results. So let's now draw a picture that explains what this canonical completion is about.
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And the best way to explain it is to explain it in the case of a graph. So this whole blackboard over here, maybe I'll still use this one.
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So here's a graph. Let's call this x. And I'm going to choose a local isometry from capital Y to x.
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And the local isometry is so easy over here. It's just a combinatorial immersion.
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So that's a nice local isometry. I'm going to be creating this commutative diagram here. So now I'm going to draw my canonical completion,
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which is going to be this covering space of x. So in fact, we said that y is going to embed in this covering space. So we might as well build the covering space around y.
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Here it is.
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And to create a covering space, what I need to do is I need to make this map into a local isomorphism. A covering space of complexes,
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a combinatorial map of complexes is a covering space if it's a local isomorphism. And that's quite simple. I just need to get the right number of incoming and outgoing edges of each type at each vertex. And there's many ways of doing it. You can just, whenever you see a missing outgoing A edge, you can pair it up with a missing incoming A edge.
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And you're on your way. You just add more and more missing stuff until you've created a covering space. So there are many ways of doing this. But we're going to have a canonical.
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We're going to choose a kind of canonical way of doing it. So let me describe that canonical way. This idea of embedding it in a finite cover really goes back to Marshall Hall, like 1950 or so. So what you do is you,
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when you have that missing outgoing A edge, you're going to connect it up with a very particular missing incoming A edge. And how did I choose that? What I did is I looked at a maximal A arc, and then I closed it up. So let's repeat that process.
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Here's a maximal B arc right over here. And I'll close that up. Maybe I'll make this a little bit more interesting.
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Let's make this a little more interesting. It has a little more stuff on it. So where's my next orange edge that you might want to put? So here's a maximal B arc right over here. And I'll close it up.
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Okay, let me finish it now on my own. So there's some trivial A arcs you might not have noticed. And here's a trivial B arc. Here's another one. Do I have a finite covering space?
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Not yet. I'm missing over here. And I'm missing over here. And I'm missing over here. I'm good now. Otherwise there'd be a riot. Yeah? So what's, now there's many, a graph retracts onto any connected subgraph.
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So nothing to fear over here. There'll be no problem retracting it. But what's the canonical retraction that we will use? Well, what you do is you map each new edge to the arc that it was associated to.
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That's the canonical completion. That's the canonical retraction. Okay, I won't fill in the rest. And as it turns out, this procedure, which I was using around 1994-5,
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Frederic and I generalized it around 2002 or so, it works quite naturally when x is a salvetti complex instead of when x is a graph. It's almost the same recipe.
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Okay, you don't actually need to know what the recipe is. But you get this conclusion. And there are features that the canonical retraction has that allow you to have a lot of control when you're doing certain elaborate arguments that are required working with special cube complexes.
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There's now quite a bit of a theory of special cube complexes and groups which are special. And they seem to, a group is special if it's a fundamental group of a special cube complex. And they occupy a nice little spot where you can understand a lot of things about them.
01:00:04
I'm not going to say more about them. I finished on time. That means you guys give me an extra five minutes tomorrow, right?