6/6 CAT(o) Cube Complexes
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GeometryComplex (psychology)CubeGroup theoryTheorySubgroupCubeField of setsKodimensionDuality (mathematics)Group actionSpacetimeFinitismusInfinityPrice indexCategory of beingLecture/Conference
02:34
InfinitySubgroupConjugacy classNetwork topologyHeegaard splittingField of setsGroup actionKodimensionPositional notationCubeGroup theoryVertex (graph theory)Duality (mathematics)FundamentalgruppeProduct (business)Extension (kinesiology)Stability theoryState of matterTranslation (relic)Connected spaceMany-sorted logicPhysical systemDimensional analysisPrice indexDifferent (Kate Ryan album)Right angleFree productHeuristicGraph coloringLecture/Conference
10:19
SubgroupGraph (mathematics)Group actionAxiom of choiceCubeFinitismusKodimensionHyperbolische GruppePartition (number theory)TheoremExtension (kinesiology)Dimensional analysisDuality (mathematics)Right angleCompact spaceLecture/Conference
12:29
Group actionCompact spaceElement (mathematics)Lecture/Conference
13:15
Set theoryDuality (mathematics)Many-sorted logicGroup actionPartition (number theory)SpacetimeField of setsRight angleCubeLecture/Conference
15:12
Group actionTranslation (relic)Point (geometry)Identical particlesOrbitElement (mathematics)SpacetimeMultiplication signPower (physics)Axiom of choiceIdentifiabilityLecture/Conference
17:44
Element (mathematics)Right angleMetrischer RaumSpacetimeLimit of a functionCartesian coordinate systemAlgebraic structureLecture/Conference
19:02
Game theoryElement (mathematics)Duality (mathematics)DistanceGroup actionRight angleField of setsCondition numberMany-sorted logicOrbitNumerical analysisSpacetimeCubeLinearizationCategory of beingMoment (mathematics)Flow separationMetrischer RaumIsometrie <Mathematik>Logical constantGroup theoryGraph (mathematics)Cartesian coordinate systemEinbettung <Mathematik>Point (geometry)Metric systemLecture/Conference
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Cartesian coordinate systemGroup actionDistanceHyperbolischer RaumStability theoryInfinityNumerical analysisFlow separationLinearizationGraph (mathematics)CubeOrbitFinitismusPoint (geometry)Field of setsRight angleSpacetimeCategory of beingAnnihilator (ring theory)Dimensional analysisDuality (mathematics)SubgroupMany-sorted logicLimit of a functionMetric systemEinbettung <Mathematik>Arithmetic meanAsymmetryLecture/Conference
31:49
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38:35
Field of setsCovering spaceSurfaceCubeBoundary value problemFundamentalgruppePoint (geometry)Plane (geometry)Group actionHyperbolische GruppeHyperbolische MannigfaltigkeitPoint at infinityCircleHyperbolischer RaumMultiplication signCompact spaceFilm editingSubgroupRight angleManifoldParameter (computer programming)InfinityFinitismusSpacetimeGoodness of fitDimensional analysisTheoremTranslation (relic)Lecture/Conference
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SubgroupCubePrice indexField of setsFinitismusGroup actionState of matterRight angleCompact spaceKontraktion <Mathematik>Object (grammar)ManifoldCovering spaceHelmholtz decompositionReflection (mathematics)Torsion (mechanics)Hyperbolische GruppeGraph (mathematics)Closed setSpacetimeComplex numberIsometrie <Mathematik>Multiplication signCategory of beingFree productMany-sorted logicPosition operatorFundamentalgruppeLattice (group)Protein foldingCombinatory logicBlock (periodic table)MathematicsDimensional analysisGroup theoryGoodness of fitPrime idealMusical ensembleSet theorySign (mathematics)CurveRandom matrixSurfaceAmalgam (chemistry)Lecture/Conference
Transcript: English(auto-generated)
00:17
subgroup of a finitely generated subgroup of F2
00:31
is codimension one if and only if, this is a free group,
00:42
if and only if it's infinite index. OK. So you can do that exercise while I do my exercise.
01:06
And sorry, I forgot about this. I got punished. Is there one more?
01:26
We'll leave that. We'll leave that for me to take a short break during the talk. I want to talk a little bit about finiteness, properties,
01:51
the dual. So well, we had a group.
02:05
We found some codimension one subgroups. We made a wall space. We created a dual cube complex. And now we have a group acting on this cat 0 cube complex. But often, the dual arising from the construction
02:45
is locally infinite, is infinite dimensional,
03:03
and it has cubes of arbitrarily high dimension in it. And among the exercises, if you have three pairwise crossing walls, there's a three cube. If you have four pairwise crossing walls, you'd expect to see a four cube and so forth.
03:21
So you'll be forced to look at that a little bit and consider that a little bit. And actually, we heard about Thompson's group.
03:40
So let's see. Locally infinite may be the simplest example would be like a Bass-Ser tree. That's really one of the simplest versions
04:04
of this dual cube complex, is really the Bass-Ser tree. And it's actually the case. It's also an exercise, lemma. What's the difference? Theorem? Everything is just an exercise, I think. Right?
04:22
It's an example. All right, I see. Yeah, ex means example. If you have an amalgamated product,
04:42
then the amalgamated subgroup has co-dimension one. And likewise, provided that we're going to assume that c is not equal to a or b.
05:04
So we're assuming that it's a genuine splitting. Likewise, if you look at an HNN extension, then the edge group is co-dimension one.
05:27
And really, the Bass-Ser tree ends up being, it really is just a dual to a wall system that you're creating with this co-dimension one subgroup. That's really all that's happening. That's why you're getting a tree.
05:41
And it's a very nice type of co-dimension one subgroup. It's a co-dimension one subgroup, which it's a very nice type of co-dimension one subgroup. It's a co-dimension one subgroup. It's a little bit heuristic, but this is fairly close to the truth. If you had your group and you found a co-dimension one subgroup, giving you walls that don't cross their translates.
06:07
And so the dual is going to be a tree. That's really the connection with the Bass-Ser tree. But unfortunately, typically what's going to happen
06:24
is that the group is not going to act freely on this dual cube complex, this Bass-Ser tree. And people were really, really very concerned with the stabilizers of the vertices. And we know that in this case, they're conjugates of A and B,
06:43
conjugates of the vertex groups. They're called vertex groups. They're conjugates of A and B, or the conjugates of A in the HNN extension case. What does this notation mean, A star CT equals D? This is sometimes notation for an HNN extension.
07:04
So have you heard of the notion of HNN extension? So then you'll learn about that and you'll find out. It's a generalization of an inaugmented product. But if you like co-dimension one subgroups, then you can first do co-dimension one subgroups and then work your way down to the case where you get a one-dimensional cube complex.
07:21
And then you'll know about Bass-Ser trees. You could skip it, more or less. Is it possible that both C and D are proper subgroups or one of them? It's not necessary. In this case, it's not necessary for them to be proper subgroups. Ah, you can just have any both groups? Sure. Sure. OK.
07:40
So I'm not going to use this notation very much as we proceed, so I'll just continue. Does it work backwards too, if you get the dual complex as a tree and you get a splitting? If the dual complex is a tree, that's the most important way that it works. If the dual complex is a tree, then you get an action on a tree. And Ser's book is, if you've got an action on a tree,
08:02
then it's a splitting. You can write a book about it. So we saw another, so I just wanted to mention the connection to the Bass-Ser tree, just not to be remiss. And we saw briefly another.
08:22
So this is sort of the classical case where you have locally infinite behavior, because usually the action on the Bass-Ser tree, well, the Bass-Ser tree usually has locally infinite vertices. Unless the vertex groups were finite, or the edge groups more generally
08:40
were finite indexed in the vertex groups, that doesn't happen very often. Usually the edge groups are not finite indexed in the vertex groups, and so the vertices in the Bass-Ser tree are locally infinite, which is one sort of pathology to a certain extent.
09:00
A more interesting case, it could be infinite dimensional, so there are some tubular groups that exhibit that behavior. But what's more exciting, locally finite,
09:29
yet but infinite dimensional kyova, am I correct? For Thompson's group.
09:40
And I think the cube complex that you described, which was described not using this dual cube complex construction, it actually is a different way of getting to cube complexes, which is more closely related to what people call state complexes.
10:02
So there are lots of pathological examples and pathological behaviors around. I am interested in, as you know from the special cube complexes, I'm interested in the group not acting on a cube complex in an arbitrary way, but actually being the fundamental group of a cube complex.
10:20
So I want it to be acting freely. And even though I said that I'm open minded and not everything has to be compact, truth is that I prefer compactness. So we're looking for theorems that are going to give us control over the action of the group on the dual.
10:41
So I want to describe a few to give you some sense of what's out there. The first thing that he did, and to a certain extent the last that he did in this topic until he came back to the topic some years later,
11:02
interested in the Krapphalle conjecture, which is something a little bit esoteric, he proved the following. Let G be hyperbolic and H be a codimension one subgroup
11:31
that is also quasi-convex.
11:41
Then the dual, so there's more things to do over here. You have to choose a Cayley graph. Pick your walls, choose what your partitions are going to be. But it doesn't matter what choices you make. The dual is G co-compact.
12:04
In particular, it's finite dimensional. If it's G co-compact, the group is acting, permuting the cubes around. There's a bound to the dimension of a cube over here.
12:24
Now, here's another theorem that gives us some sense. Suppose that for each, it will give us
12:42
some sense towards freeness of the action. This is towards co-compactness of the action. Suppose that for each non-trivial element, little g, there exists a wall that cuts g, in a sense
13:07
that I'm not going to make totally precise. I'll draw you a picture. Then g acts freely on the dual.
13:31
So this is now a setting. Let me maybe clarify. So what's the setting? I better specify that I'm sorry. So the setting over here, let g act on a wall space.
14:03
So I have a wall space. There's a group which is acting on it. So the g is acting on the set. And it's permuting the partitions around. It's permuting the walls around as it acts. And then, as you can imagine, there's an induced action of g on the dual-cube complex.
14:21
And we're interested in, OK, what sort of thing could guarantee that the group will act freely on the dual to this wall space? And this is one of the criteria that comes in handy quite often. So what do I mean by cuts?
14:50
In the sense that, let's see. Usually, and there's a few ways of thinking about it.
15:10
Usually, how about this? How about this? Let's choose a point in the wall space.
15:23
Let's do it like that. So we have the identity times a point. So let s be some fixed point. And what we'll do is we'll look at all translates of that point by powers of g.
15:57
And we want there to be some wall that separates them
16:11
into the translates which are negative and the translates which are positive.
16:22
It's a little more general than that. We'd be OK with an orange wall that did that. And so eventually, the sufficiently positive
16:46
translates are going to be on one side of the wall and sufficiently negative translates are going to be on the other side of the wall. So maybe I'll say that. Here s is a choice we want or a fixed point. Yeah, just choose some point just so that we can identify group elements with just we're
17:03
looking at an orbit. We'll just choose some orbit. So this definition won't depend upon the choice. OK, so in practice, our wall space often s
18:01
has extra structure, e.g., s might be a geodesic metric space, as is the case when s is epsilon.
18:24
It's a nice geodesic metric space. And a nice way of thinking about this is so cutting w cutting g, it's w cutting g.
18:47
So let me say it like this. Then we would g maybe has an axis, so maybe a nice geodesic, and that the element g is stabilizing.
19:08
And the picture that we really like is something like this, where our wall cuts a nice geodesic axis.
19:21
And the axis goes far away from the wall on each side. And that's really the motivating case. And it's not very difficult to prove that when every element is cut by a wall,
19:43
then it is impossible for a zero cube to get stabilized by a group element. Because you try to imagine a zero cube which is stabilized, and then you reach some type of contradiction in the definitions of the way a zero cube behaves.
20:01
I'm not going to get lost in it right now. Instead, I want to convey one more thing. And I think together they will give you a pretty good intuition about when a group will act freely. So here's another condition.
20:23
Suppose that we're still in this setting that g is acting on a wall space, but we're going to assume that S is a metric space.
20:42
S is a nice metric space, like the Cayley graph is. Let's suppose that the number of walls separating two points, so this is measured inside of the wall space of course, is greater than or equal to, I don't know,
21:04
a distance between p and q minus b for some a and b. Suppose for some constants a and b,
21:20
we had this property which I like to call linear separation. Linear separation. Then actually you get, not only does g act freely,
21:46
then the action of g on the dual cube complex has a quasi-symmetrically embedded qi-embedded orbit.
22:12
So let me maybe say it like this. So there's a map from a Cayley graph of g to the dual cube complex.
22:23
There exists a qi-embedding from the Cayley graph to the dual cube complex. It's not a quasi-symmetry between them.
22:41
The cube complex is probably much, much bigger than the group because of all of the surprise zero cubes that showed up. I mean, in my mind, there's a, I'll draw a picture of that in a moment.
23:09
Talking about freeness now, what you should absorb from this is somehow that the,
23:25
so the intuition. So in the theorem, what is the assumption on g? g, let's let g be a finally generated group
23:41
so that we can give it some nice metric. You know, the culture over here, by the way, is we usually just care about finally generated groups. You know, maybe countable, but usually just finally generated.
24:01
So g is a finally generated group and it's acting on a wall space and so that we can compare the number of walls with a metric. There's a metric on the wall space, right?
24:21
So the picture for that criterion, maybe I'll reuse this picture over here. The picture for that, I'll draw a new picture. So the picture for that criterion is that we've got p and q and there's a certain distance between p and q in our wall space
24:45
and we want the number of walls that are separating p and q to be roughly proportional to the distance between them.
25:02
Now, it's a tricky, I mean, this is giving you, this is saying that it's lower bounded by the distance between them, but the way the finiteness properties go, you can never expect more than a linear number of walls as candidates.
25:23
Okay, so this is the best, this is the best you can hope for. Is it f by eigenvalues? Okay, so does what act by asymmetries? g act on the cube complex, it's acting by asymmetries. You're on the wall space with metric structure.
25:42
So, yes, thank you. But you need some facefulness to embed the Cayley graph in c, you need something about...
26:01
Okay, so all of this is, you know, the sort of generalizations of the group acting on its Cayley graph. Okay, and you're saying that if it didn't act faithfully over here, then you wouldn't expect that the walls are going to lead
26:20
to faithful actions on the cube complex. Well, might, I don't know. But I'm going to assume that it's acting. Usually what we do is we act, we usually act properly.
26:42
So, we will often add a little bit of, a little bit of, there could be a little bit of finite stabilizers in this situation that we normally are considering, okay? But you don't have to make, when you're first learning about this, don't make everything so difficult by thinking about grand generalities. Think of two cases.
27:02
One is that you have a group acting on a Cayley graph, and the other case that you should be thinking of is a group acting on hyperbolic space, right? And you're going to have walls in hyperbolic space coming from codimensional subgroups in hyperbolic space. That's the other type of situation that you should be thinking about over here.
27:22
Now, if you just require that the number of walls separating p and q is going to infinity as the distance between p and q goes to infinity, you'll still get something. So, maybe I'll just say that over here.
27:42
If the number of walls separating p and q goes to infinity as the distance between p and q goes to infinity, so as two points are getting farther and farther away from each other, the number of walls separating them is increasing,
28:07
then that will be enough to know that the map from epsilon to c,
28:27
or if you prefer the orbit instead of being quasi-estimetrically embedded, it will be metrically properly embedded. So, the orbit of the cube complex
28:42
is going to be sufficiently spread out inside of the orbit of g, and the cube complex is going to be sufficiently spread out. This is a minimal amount that we want, but that's probably the one that you should be looking at. The intuition is really, I'm getting a little bit out from where I wanted to focus,
29:02
the intuition is that if the walls cut the wall space very well into small pieces,
29:32
then the group will act nicely,
29:42
meaning freely, properly, with finite stabilizers on the dual cube complex.
30:07
And that's what both of these statements are really telling you. And cutting it up means what you think it does.
30:28
So that's the intuition. But you have to be very, very careful. Unfortunately, this is a soft statement,
30:41
so nobody can criticize me about this, because it doesn't mean anything. Unfortunately, it's hard to tell.
31:03
So, you can be fooled, easily fooled. Why? Well, you have these points P and Q, and it's looking good.
31:26
Looks like there are lots of walls separating them, right? But maybe it was all the same wall.
31:42
Hey, they're not even separated. So this is a pitfall that often occurs when we're studying this, when we're trying to recognize that we've found enough walls
32:03
to actually have an interesting action of G on the cube complex. We're looking to see that we've cut it up many, many ways, but you can get into some trouble. And these statements are all just trying to pinpoint what's happening. So that the intuition, which should be guiding you,
32:23
isn't going to give you too much guidance and send you in the wrong direction. So, well, we would like to see that there really are actual walls separating them. It shouldn't just look like you're chopping it up into little pieces. Or this is enough as well, to ensure it.
32:56
Let's look at an exciting thing that happened a few years ago,
33:05
that increased the interest in the topic for people who were not yet believers. So, Kahn and Markovich proved that the fundamental group
33:36
of a hyperbolic manifold contains many, many quasi-convex
33:53
Codimension 1, that has to be the case, closed surface subgroups.
34:06
So the setting over here is let M be a closed hyperbolic manifold.
34:27
Now, it's a general fact that, actually, another example is that if,
34:41
I think I stated this already, that an n-dimensional, an orientable n-dimensional manifold subgroup, a closed orientable n-dimensional manifold subgroup inside of an orientable n plus 1-dimensional manifold. Did I state it last hour? No, I didn't.
35:03
So, if you have an inclusion of fundamental groups
35:36
of closed orientable manifolds, an n-manifold fundamental group
35:42
in an n plus 1 manifold fundamental group, then you get a Codimension 1 subgroup, which is the motivation for the name over here, for the terminology.
36:06
Although, during the break, someone was complaining that the trivial subgroup is a Codimension 1 subgroup in a cyclic group, which is a Codimension 1 subgroup in a free group of rank 2. So, you don't want to get too attached to notions of dimension here,
36:28
but this is the motivation for the language. So, these are all Codimension 1. So, the dimension didn't drop here, but then it dropped over there.
36:45
All right. So, Kahn and Markovitch gave us, answering a long-standing problem about hyperbolic 3-manifolds, they gave us a quasi-convex surface subgroups,
37:06
which of course are Codimension 1. It's a general all-purpose fact. So, in fact, there's various ways to think about why this is the case, but we then can deduce, so, using sufficiently many,
37:30
maybe two, three, maybe ten of them, of these surface subgroups, and applying the dual-cube complex construction,
37:47
we obtain a free co-compact action of the fundamental group of this 3-manifold
38:04
on a cat-zero-cube complex. So, this is a corollary to their discovery.
38:26
The co-compactness of the action is just coming from Sagiv's observation. The freeness of the action is coming from the fact that there are many.
38:48
You could choose so many surface subgroups, you could choose so many surface subgroups, that when you look at the hyperbolic 3-space,
39:06
in fact, what they showed was that for any two points in the boundary of hyperbolic 3-space, they found a closed surface subgroup,
39:25
so this is the universal cover of our manifold, Kahn and Markovitch found that for any two points, you could find a surface, probably very hygienist,
39:43
whose universal cover maps pi-1-injectively into the manifold, but when you look at the induced map between the universal covers,
40:01
so this is the universal cover of my surface over here, and it has a nice yellow boundary at infinity, I guess, you could think of it as a hyperbolic plane, and it very nearly, so if you choose a circle
40:24
that separates the two red points on the boundary, Kahn and Markovitch can choose this surface S so that its universal cover comes so close
40:43
to this orange hyperbolic plane over here that actually its boundary kind of looks like that, just kind of travels right along nearby it, and this yellow boundary circle of the universal cover
41:02
of our surface, which you're looking at it, you're saying that's going to be a very good wall, this surface manages to separate these two points at infinity.
41:21
With that, you can actually choose sufficiently many of these, and by some compactness argument, you can actually, and of course you can use all of their translates, as we discussed, maybe you've chosen seven of these, plus all of their translates, you actually get into a situation
41:43
where you're going to cut this up sufficiently well, and so you know that the action is also going to be free, it's not just co-compact, which is a generality from Sagiv's theorem, but it's actually going to be free,
42:01
because you've cut hyperbolic space up sufficiently well, and you could do it like that as well. So we know that our three-manifold is actually the fundamental group of a compact cube complex,
42:33
and from there, of course, we hope that we could show that that cube complex has a special finite cover,
42:44
which is what we were talking about yesterday, and that's what happened in the end. That was the plan.
43:13
So one would like to hope that every closed hyperbolic manifold
43:38
is pi1 of a compact, non-positively curved cube complex.
43:49
But the main thing that we are missing, and we're probably very far away from that, is to understand where the walls are going to come from.
44:02
People knew to be interested in finding these walls even before they knew what to do with them. They wanted to find these surface subgroups, very useful to cut a three-manifold along a surface subgroup, and there obviously hasn't been as much interest yet in finding ways of cutting four-manifolds along.
44:25
It won't be along three-manifolds. It will be along something, along some quasi-convex codimensional subgroups, and to find enough of them so that your four-manifold, its universal cover is going to give you a nice wall space,
44:43
and you could do the same thing. So your hyperbolic four-manifold is also going to be the fundamental group of a non-positively curved cube complex. This is all conjectural. This is what I believe mostly because I want it to be true, because it means that cube complexes are the glorious answer to many things.
45:02
But that's what I've felt for about 20 years, just because I didn't know how to understand anything else. Now, for three-manifolds, by the way, maybe we can spend a little time talking about that,
45:23
because we heard about three-manifolds yesterday. For three-manifolds, so theorem,
45:41
almost all compact three-manifolds have the property that pi 1 of M is the fundamental group of a non-positively curved cube complex,
46:06
but often not compact. So some exceptions,
46:31
well, nil and sol,
46:43
and the psl2 tilde, I think, I haven't thought about that in a long time. The more serious exceptions are, which are a little bit interesting,
47:02
certain closed graph manifolds. But if you, for instance, took a few hyperbolic three-manifolds,
47:24
and you glued them together along their tori, so I'll just write h for hyperbolic, these are little hyperbolic pieces, if the JSJ decomposition were non-trivial,
47:44
and, for instance, all of the pieces, were they called pieces yesterday in the JSJ decomposition, or blocks pieces, I think, yeah? Well, if all the pieces in the JSJ decomposition are hyperbolic, then this is actually the fundamental group of a non-positively curved cube complex.
48:06
Not necessarily compact, but it's still virtually special. So the specialness is really working very nicely together with hypervelicity over here.
48:24
I should probably spend a few minutes mentioning, listing some other examples. We heard about Coxeter groups. So every Coxeter group, finally generated Coxeter group,
48:51
is equal to the fundamental group of X, where X is a non-positively curved cube complex.
49:01
This is Niblo and Reeves. And also, if I'm not mistaken, Pierre Pansu didn't publish it, and they were using the wall space construction,
49:22
I'm not sure if they had the words wall, they knew of the terminology wall space already, but it was pretty clear what the essence of Sagiv's construction was, and they used the reflection walls that I spoke about, that I mentioned just briefly before.
49:43
However, X is not compact. Oh, and I said something wrong. This is incorrect. A Coxeter group has torsion, of course,
50:01
and a non-positively curved cube complex, as Ruth mentioned, its universal cover is contractible, but if it's compact, you're certainly not going to have, these are finite dimensional, finite dimensional.
50:20
So let me state this correctly. So I'll say, is virtually, so it has a finite index subgroup, which is the fundamental group of such a thing, and let me state it a little bit more accurately, let's call the group G. G acts properly on a finite dimensional cat zero cube complex.
51:00
So that's a little bit more correct, and actually here too, there exists G' contained in G, such that G' acts freely. Let's call this cat zero cube complex X tilde.
51:21
There exists G' contained in G, so that G' acts freely, and in fact, X tilde mod G' is special. So all Coxeter groups using Sagiv's construction on the walls, you get an action on a cat zero cube complex,
51:45
which the dimension goes up quite a bit, but it's something organized that you can understand, and then you can pass to a finite index subgroup, and see that your Coxeter group is virtually inside of a rag, like this.
52:00
This is the sort of continuing theme. If you've heard of hyperbolic arithmetic lattices of simple type,
52:25
they also fall to this approach, which also act on cat zero cube complex and are virtually special.
52:57
So there were discussions of arithmetic lattices at the beginning of last week,
53:04
is that correct? But you were probably talking about bad lattices that had property T. Is that right? Yeah, those are dead already. I have about three or four minutes to wrap things up.
53:24
So maybe I'll describe one more suggestive example, which is really, my interests are to convey the intuition of, when this procedure of using Sagiv's construction is going to work.
53:51
And maybe it's a good idea also to mention an example in regards to the Coxeter groups.
54:07
Cat zero two complexes were discussed here last week. The week before last week, maybe. Not really? Okay.
54:32
Let me then change the, I won't try to do that, I'll change the topic entirely, and close with another thing of a different flavor. So let me talk about a combination theorem.
54:47
This is with Tim Shue. If G is equal to an amalgamated product,
55:03
where G is hyperbolic, and C is quasi-convex, and A and B are virtually fundamental groups
55:37
of compact special cube complexes,
55:51
then G is a fundamental group of a non,
56:01
where X is a non-positively curved cube complex that is compact. So sometimes you can take two non-positively curved cube complexes, and you can glue them together. That's maybe the simplest case. You might just have cube complex number one,
56:24
and cube complex number two, and then create an amalgamated product of them. And well, if you glue them together, we know that if you're gluing them together along local isometries, then maybe you might be lucky and you can actually,
56:42
the object that you get might already be a non-positively curved cube complex. But usually it's not. Usually it's a huge mess. But in a very good situation where X1 and X2 are special,
57:03
and the amalgamated subgroup C, this is A and this is B, when C is quasi-convex, you can actually get this whole group, the fundamental group of this new space, to be acting on a cat-zero-cube complex.
57:22
So how would you do that? Huh? What? You'd find walls, because there isn't another way of doing it. You'd somehow find walls, and so you'd look inside of this object,
57:41
and you'd say, and you're kind of looking for, it's almost like you're looking for a three-manifold, right? Just for a three-manifold, what you'd be doing is you'd say, okay, there must be a surface in here somewhere, and you hope to find one, and the same thing is happening over here. You're looking for kind of immersed walls inside.
58:07
Maybe you found one. You have to be very careful about it. Hopefully it's going to correspond to something which in the universal cover of this object is going to give you an honest wall, and hopefully it'll be quasi-convex. And then maybe you'll have to find another one.
58:24
And perhaps you'll observe that, oh, there's already one waiting for you right over here, because that edge group right over here is going to be a nice wall. And then you have the universal cover.
58:41
You have the picture of all these walls cutting it up, and that's how you'll get an action of the amalgam on a cube complex. And this is how the subject has been progressing, and there's lots of things that we don't know. There's lots of things as we drop hyperbolicity,
59:02
then it's trying to generalize this statement over here. It has a generalization to a relatively hyperbolic setting, but as you increase the sort of group that you want to apply this to, because hyperbolic groups are amazing,
59:22
but those are the easy groups that are easy to understand. What about the difficult groups? So then as you stretch this, things get messier and more complicated, and you try to extend what we've done in the hyperbolic case, and it's going to get more and more complicated,
59:41
as that's the way math goes. So thank you very much for your attention.