2/2 Mapping Class Group and Curve Complex
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GeometrieKomplex <Algebra>GruppenkeimPhysikalische TheorieKlasse <Mathematik>Formation <Mathematik>Algebraische StrukturGeometrieKurveNumerische MathematikOrdnung <Mathematik>MengenlehreKategorie <Mathematik>FaserbündelOrthogonalitätÄquivalenzklasseGruppenoperationHyperbelHyperbolischer RaumKlassengruppeKomplex <Algebra>KreisringKreiszylinderMaßerweiterungMomentenproblemProjektive EbeneZeitzoneIsochoreGüte der AnpassungTeilbarkeitAbstandPunktSortierte LogikFlächentheorieKnotenmengeKreisbogenMultiplikationsoperatorRandwertZweiMinkowski-MetrikRechter WinkelMetrisches SystemKlasse <Mathematik>Vorlesung/Konferenz
07:23
KurveInvarianteEndlichkeitGruppenoperationIdealer PunktKomplex <Algebra>KreisringProjektive EbeneRestklasseSimplexverfahrenÜberlagerung <Mathematik>Gewicht <Ausgleichsrechnung>GammafunktionPunktPoisson-KlammerSortierte LogikProzess <Physik>Jensen-MaßGibbs-VerteilungKreisbogenMultiplikationsoperatorRandwertSchießverfahrenAuswahlaxiomKompaktifizierungKreisflächeFlächentheorieDifferenteElement <Gruppentheorie>Rechter WinkelVorlesung/Konferenz
13:12
Algebraische StrukturKurveOrdnung <Mathematik>ÄquivalenzklasseFundamentalgruppeGruppenoperationIdealer PunktKlassengruppeKomplex <Algebra>Metrisches SystemSimplexverfahrenGüte der AnpassungFamilie <Mathematik>Überlagerung <Mathematik>Helmholtz-ZerlegungGebundener ZustandKlasse <Mathematik>KreisflächeBeobachtungsstudieStabilitätstheorie <Logik>FlächentheorieIsotopie <Mathematik>Jensen-MaßObjekt <Kategorie>DickeKreisbogenRandwertMinkowski-MetrikRechter WinkelTransversalschwingungEinsKreisringLokales MinimumPunktLipschitz-StetigkeitMultiplikationsoperatorVorlesung/Konferenz
16:49
GraphKurveMathematikNumerische MathematikFinitismusGruppenoperationHochdruckKlassengruppeKomplex <Algebra>Lokales MinimumMereologieMetrisches SystemPunktrechnungIsochoreQuotientFaltungsoperatorStabilitätstheorie <Logik>Lipschitz-StetigkeitKnotenmengeObjekt <Kategorie>Element <Gruppentheorie>MultiplikationsoperatorStandardabweichungMinkowski-MetrikKlasse <Mathematik>Vorlesung/Konferenz
21:07
Einfach zusammenhängender RaumGruppenoperationPunktrechnungObjekt <Kategorie>MultiplikationsoperatorRechter WinkelFolge <Mathematik>Arithmetisches MittelGraphKurveNumerische MathematikZahlensystemProdukt <Mathematik>Kategorie <Mathematik>ProduktraumUngleichungAuswahlaxiomBetafunktionKomplex <Algebra>Projektive EbeneStichprobenfehlerAbstandGebundener ZustandPunktDifferenzkernWertevorratFlächentheorieJensen-MaßDifferenteKreisbogenRandwertMinkowski-MetrikAlgebraische StrukturOrdnung <Mathematik>RelativitätstheorieStatistische SchlussweiseTeilmengeWinkelTransitivitätFormation <Mathematik>SummierbarkeitInnerer PunktSortierte LogikPartielle DifferentiationTransversalschwingungEinsVorlesung/Konferenz
28:31
Folge <Mathematik>GeometrieKurveNumerische MathematikZahlensystemUniformer RaumUngleichungAuswahlaxiomBetafunktionEinheitskugelFundamentalkonstanteGeodätische LinieKomplex <Algebra>KugelLokales MinimumMomentenproblemProjektive EbeneQuantenzustandSimplexverfahrenTheoremVerschlingungStochastische AbhängigkeitAbstandZusammenhängender GraphGebundener ZustandDurchmesserPunktFlächentheorieJensen-MaßKnotenmengeObjekt <Kategorie>RandwertMinkowski-MetrikRechter WinkelMaß <Mathematik>GrundraumGüte der AnpassungVorlesung/Konferenz
35:24
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42:18
GeometrieKurveNumerische MathematikKategorie <Mathematik>Ganze ZahlBeweistheorieKomplex <Algebra>Projektive EbeneTheoremWinkelAbstandPunktSortierte LogikKreisbogenMultiplikationsoperatorRandwertGeodätische LinieReelle ZahlSummierbarkeitKörper <Algebra>FlächentheorieRechter WinkelVorlesung/Konferenz
47:30
ApproximationKurveNumerische MathematikProdukt <Mathematik>QuantorKoordinatenLokales MinimumMereologieProjektive EbeneTheoremWiderspruchsfreiheitKonstanteAbstandPunktRichtungNichtunterscheidbarkeitKonditionszahlRandwertBeweistheorieCW-KomplexKomplex <Algebra>Vorlesung/Konferenz
52:43
KurveNumerische MathematikGebäude <Mathematik>UngleichungBeweistheorieGruppenoperationHyperbolischer RaumKomplex <Algebra>MereologiePauli-PrinzipTheoremWiderspruchsfreiheitAbstandGewicht <Ausgleichsrechnung>PunktKonvexe HülleSortierte LogikRichtungFlächentheorieKonditionszahlMultiplikationsoperatorRandwertMinkowski-MetrikRechter WinkelKonvexe MengeStatistische SchlussweiseKnotenmengeEinsVorlesung/Konferenz
58:07
Numerische MathematikUngleichungMereologiePunktDifferenzkernFlächentheorieKnotenmengeRandwertZweiAlgebraische StrukturOrdnung <Mathematik>RelativitätstheorieGruppenoperationWinkelKonstanteTransitivitätAbstandParametersystemSortierte LogikPartielle DifferentiationObjekt <Kategorie>Vorlesung/Konferenz
01:03:31
Diagramm
Transkript: Englisch(automatisch erzeugt)
00:16
Let me try to go on to what I was going to do next.
00:23
So right, so what's the, so I want to talk in the second hour about the kind of subsurface structure and the actual course
00:41
geometry of the mapping class group. And I guess, let me just point out, this is related to a bunch of, this kind of modern versions of a bunch of this structure, which is worth mentioning. I mean, so these are just words for the moment.
01:05
But I guess you heard Francois talk about a cylindrical hyperbolicity. So this is, I think, Bodech, Osin, Domini, et cetera.
01:20
Actually, not so, I'm not sure how many names I should put on there, so I'll leave the parentheses open. This is, so the curve complex is a cylindrically hyperbolic. I think Francois mentioned it. So that's one kind of general category in which this discussion fits. There's another thing called, sort of more restrictive
01:45
situation called, I think, can I write bigger? I can. So this is a cylindrical hyperbolicity. I won't write that again, but there's a thing called projection complexes. I think that's what they're called,
02:00
which is a kind of a device worked out by Bestvina, Bromberg, and Fujiwara. Uh-huh. And the notion of hierarchically hyperbolic spaces,
02:24
groups and spaces. And this is Bierstock, Hagen, and Sisto. Why am I mentioning all this? So there are various features of these discussions, which
02:43
are kind of more general, which are closely related to the kind of structure that I want to talk about, and to some extent inspired by it. So that the mapping class group itself, I think as I said in the beginning, is kind of a good, rich setting in order to kind of get started understanding these somewhat more
03:00
general notions. So I'm staying in my own comfort zone of thinking about mapping class groups, but as you listen, if you've read any of these other things, you will recognize things you've seen. OK, so what's the story now? I want to tell you about, so we have these other curve
03:22
complexes. So I have to spend a couple of seconds adding a couple of technical points. So I think I talked about the fact that there are special cases for curve complexes of these surfaces. So I won't repeat that.
03:41
But there's one other thing you can do, is you can talk about arcs instead of curves. So actually, maybe for us, the simplest thing is to talk about the arc and curve complex. Say, maybe we'll call it AC of S or W, maybe. So think of W as a surface of boundary,
04:00
and now the arc and curve complex is, you look at the vertices are curves or arcs, where the arcs, so vertices, are curves or arcs. They should be essential, I'm not going to write everything down, but they should be essential, and they should be up to isotopes.
04:21
So the isotope, in general, involves moving the base point, moving the endpoints, as well as varying the arc. So think about keeping the base point in the boundary, but moving it. That isotope is the equivalence relation. So vertices are these, and edges are disjoint pairs, and so on, same setup. It turns out to be useful to think about that. One of the problems on the exercise set was to show that these are,
04:44
so AC of W is quasi-isometric to C of W. So in other words, if you allow yourself to use arcs as well as curves, you will only change the distance that you get
05:02
by some bounded factor. That's what that means, basically. And I didn't say this on the homework, but it's not quite true. So it's true except for the exceptional cases. So, except for the exceptions. So, okay, these are the exceptions, right?
05:22
These are, so you have to be, anyway, one could get bogged down in this, but it's better not to. So you can just work it out, what you have to do to make things work in the exceptional cases. But one exceptional case I want to think about a little more carefully is the annulus. I mentioned before, if you take an annulus,
05:41
so the curve complex of the annulus is empty because there's just the core of the annulus, and that's not, even that's not essential. It's parallel to the boundary. There are no curves that are non-parallel to the boundary and if you add arcs, then it's still kind of trivial because any two arcs are isotopic in the annulus.
06:02
So here I should take arcs up to homotopy with fixed endpoints. That's what I want to take. So, same discussion. Yes? Are those orthogonal at the boundary by definition?
06:21
Orthogonal to the boundary? No, I mean in the picture above. Well, there's no metric. It's just the topological picture, so there's no orthogonality. It's just topological. Here also. Okay, so if I take arcs up to homotopy with fixed endpoints, then they're not all the same.
06:40
In fact, well, now there's uncomfortably many of them, which is kind of unpleasant, but we're not going to worry about that. But the interesting thing that now happens is that if I twist an arc once around, this one and this one are no longer the same, right? You can't think about it. You can't isotope this arc to the other arc
07:00
while fixing the endpoints in the annulus. So in fact, what this complex does is it counts the number of times that an arc is twisted around, and even though there are uncountably many arcs because they're uncountably many points, this complex is still quasi-isometric to z,
07:22
where the z just measures the number of times, okay? So that's my device for thinking about annuli. You'll see in a minute why I have to do this. But the job of annuli is to measure twisting. Okay, so what do I want to do with all this? I want to talk about a sort of simple notion of projection.
07:41
Projection, so suppose I take w as a subsurface and alpha is a curve in s. It's one of our curves. And if alpha intersects w essentially,
08:04
so that means you cannot push alpha to be disjoint from w. If it has a non-trivial intersection, then we can produce, let's say it this way, alpha intersect w is a simplex
08:21
in the arc and curve complex of w, right? You take, maybe let me put like little brackets. I have to do something to this. So what do I have to do? Here's the picture. So here is w and here's the rest of s.
08:41
And if I have a curve alpha that does actually cut through w in a non-trivial way, well, I just want to intersect it and get an arc. But of course, there could be many, many arcs, right? It could come through many, many times, kind of unboundedly many times. And then come back this way maybe. So what do I mean? I mean this, up to,
09:02
so many of the arcs have to be parallel. Up to identifying parallel copies, there's only bound, I get a simplex, right? I get a finite number of arcs that are all disjoint, possibly curves, right? Alpha could have just been contained in w, then it would just be alpha itself. So this intersection, I can think of it as a simplex in here.
09:21
And in fact, so this produces almost a map from pi w, from the curve complex of s to the curve complex of w. It's not all of the curve complex of s, because I have to, I can't do it for the curves that are disjoint from w, okay? So maybe I should write, actually,
09:42
maybe I should write an actual map from this minus the alpha disjoint from w to c of w, okay? And if I want it to be an actual map, I should maybe pick a point in the simplex. So at the cost of making an arbitrary choice,
10:01
I can make this just a single point. I'm not gonna worry about that distinction. Okay, so that's what we call a subsurface projection. And of course, okay, and this works in all the cases, except for the case of the annulus. Because in the case of the annulus, what happens? So here's an annulus, w's an annulus,
10:23
alpha's running through somehow. Well, I get an arc, but the arc is not well defined. If I isotope alpha, I will get a different arc. And in fact, I could isotope alpha so it does this. After an isotope, I can make alpha go around a couple of times like this,
10:42
and then go back around this way, right? That curve is still isotopic to alpha, the first alpha, but now it intersects the annulus in a more complicated arc. So obviously, this is not a well-defined thing. So what is well-defined? And the way to get a well-defined thing
11:00
is to go to the universal cover, or actually to the annular cover. So, oh, we forgot to erase. So, okay, I'm just going to say this quickly, and then I'll try to pretend,
11:22
then I'll try to talk about annulis if there's no special case. So now, suppose the w's an annulus, then there's a kind of canonical annular cover. This is H2, the universal cover of S, modulo the group
11:41
generated by the core of w. Let me, so what do I mean by this? Oh, and then I have to take, and then I can take, shoot. Well, let me draw the picture. So I take H2. S has the universal cover, which is H2, if you want. It has a circular boundary.
12:00
If you lift w, you get some annulus. This is the lift of w. It's invariant by some element, which is just the core of w. It has two fixed points at infinity. Now, if I take H2, modulo this subgroup, I will get an annulus, a kind of canonical annulus,
12:23
with the real one inside. Okay, and then it actually has a compactification, which is canonical also. I take this thing, let's actually call it w hat, take this and compactify it by taking the boundary of H2 minus the fixed points of this group.
12:41
Maybe I'll give this a name, gamma, the fixed points of gamma. And then I mod out by gamma, right? So that has, that's these two arcs, modulo the action of gamma, it gives me two circles. So this is the compactified annulus, okay? Everybody clear?
13:00
So what do I do with this annulus? Now I have alpha, which is some curve in S that crosses w, okay? I can lift it, I can lift it to here. Of course, it has many lifts. This is the lift of alpha. But when I, if I put those lifts in here,
13:21
I can actually take all those lifts and just lift them not all the way to H2, but to this annulus, and I will get a bunch of arcs. They look like this. The essential ones, the ones that connect the boundaries, have well-defined endpoints. That's the whole point of this. So these endpoints, these are well-defined.
13:41
These are isotopy invariant endpoints. If I isotope alpha on the surface, right here is w. If I take alpha and I isotope it, then I isotope the lifts, but when I isotope the lifts, it's a bounded, it's some bounded length isotopy.
14:03
At infinity, it doesn't move at all, okay? So the endpoints are well-defined, so this defines, this gives a well-defined pi w of alpha. Okay?
14:20
So that's, hopefully I won't have to say this too many more times. Oh, good point, yes, it does. So in order to identify the surface with, give a hyperbolic metric on the surface, I have to choose a metric on the surface. So there could have been different ways of writing the universal cover as H2, right? But all of them identify the circle
14:42
in the same way to each other. There's a canonical, the circle in infinity is canonical, even if the different hyperbolic structures are not. The circle in infinity is naturally identified with the Gromov boundary of the fundamental group. And so, so it doesn't actually affect anything to change the metric.
15:02
That's a good question. Okay. Let's see. All right, so what are we gonna do with this?
15:21
Okay, I have, I have to introduce one more thing, one more kind of object on the surface. I wanna study the mapping class group itself, not just these different complexes. So to study the mapping class group geometrically, I want a geometric space that's a proxy for the mapping class group, a group that geometrically can be identified
15:42
with the mapping class group up to coarse equivalence. So that's, there's a million ways to do that. It's convenient to define the following objects. So in Anna's lectures, markings had their more kind of usual definition, which I won't repeat.
16:02
So this object is not a marking exactly the same way, but there's a kind of family resemblance between the objects. So what is a marking for the purpose of this discussion? It's just a drawing on the surface that kind of ties it down so it doesn't have any stabilizer in the mapping class group. So it's the following thing. So mu is a marking.
16:21
If it's, it's got two pieces. It's got the base and the transversals. I'm sorry, I wrote small, I apologize. Let me just draw the picture. I'll draw it for genus two. So the base is a Panse decomposition. It's a maximal simplex in a curve complex.
16:44
Or a Panse decomposition. So the base will be blue. Here is maybe the base. So this is an example of the base. And probably I should name the elements of the base. We'll call them maybe B1, B2, B3 or something.
17:00
Okay, and then that's the first part of the marking. Note that this already has the feature that the stabilizer of this object is one of our billion Daintwist groups. So it's got a relatively small stabilizer. But now, let's add one more thing which will tie things down. And that is, I'll add for every blue curve
17:22
a transversal curve, T1, T2, so T3 actually. The transversal curve intersects the blue curve that it's paired with, the minimal times it can, which is one in this case. And it's disjoint from all the other blue curves.
17:40
Okay, that's it. So that's the definition. So there's one more I have to draw and that's where, you'll see that there's a problem. First of all, in the complement of these two blue curves, the only kind of curve I can draw will intersect the blue curve here two or more times. So I have to draw something that intersects it exactly two times.
18:01
So here's an example, right? So this curve, let's call that T2, is disjoint from the other blue curves and intersects this blue curve the minimal number of times, which is two in this case. That's it, that's called a marking. Fine print is that the orange curves
18:21
are forced to intersect each other. This is a tragic fact about this picture because it's kind of messy. It makes some of the definitions messy. All right, anyway, that's a marking, it's this data. Okay, and what it's good for is that now this labeled object, its stabilizer is I think trivial
18:41
or certainly it's finite. Maybe there's like a hyper-elliptic involution that fixes things or something. Anyway, it's got a finite, is everybody clear that this should have a finite stabilizer in the mapping class group? Because if you fix all the blue curves then all you can do is twist, but the orange curves detect twisting, that's all.
19:04
Okay, so the stabilizer is at worst finite. And moreover, so that's, okay, that's a marking. Now we take, let's say M of S is the graph
19:21
whose vertices are markings and whose edges are elementary moves. And I don't want to talk about what these are exactly. They're not so important to our picture, except elementary moves need to be
19:42
some kind of standard little change in here. For example, twisting an orange curve around its blue curve, one twist is an elementary move. Okay, and there's a couple of other elementary moves that rearrange the picture. So I'll leave this as a kind of exercise to think about, but the features I want from this, I want this to be locally finite,
20:04
unlike the curve complex. I want it to be connected and I want, and it's obviously acted on by the mapping class group and I want the quotient to be finite.
20:25
And the action, oh, I didn't say action is with finite, sorry, locally finite MCG acts with finite stabilizers.
20:42
And all of this together implies that the mapping class group with its word metric on the Cayley graph is quasi-isometric to this thing. That's kind of a general fact. As soon as you have this kind of action on a nice space, so locally finite,
21:03
finite stabilizers connected with finite quotient, then the two, the group and the space are quasi-isometric. So I'm gonna kind of replace the group with this thing and work with this for the rest of the lecture. Now, of these things, the one that requires some work is this. Okay, I'm not gonna prove this, but think about, maybe think about this, right?
21:22
Draw two such objects, one of them enormously complicated relative to the other, they intersect 10 million times. And now you have to work it out, you have to work out these elementary moves so that you can then go from one to the other by a sequence of elementary moves. That's what connectivity means. Okay, so that tells you something about what you should make the elementary moves be
21:42
so that you can have a chance to make it connected. Okay, let's just leave that as it is. Oh, I was gonna... Okay, so the point of all that was to just have a space that we can use together with the curve complexes
22:00
to analyze what's going on. In particular, I leave... In particular, I get immediately a map from the marking graph into every curve complex. So pi w from the marking graph to any...
22:26
I guess I... Okay, I'm gonna cheat. I had this distinction between c of w and ac of w so that I could write the definition down. I claimed they are basically the same space and now I'm gonna pretend that they are the same space.
22:42
Okay, can you forgive me for that? So the final story is I have this map which makes sense as a map from a marking to the curve complex of a subsurface because every marking intersects every surface.
23:03
So there's always something I can map into intersect with the surface and get an image. Now, maybe I have to choose which of the curves of the marking I actually use, but that's not gonna make a big difference to this thing. I only care about this map up to bounded error anyway.
23:21
Okay, so this makes sense. I might have to make some choices and the choices only cost me bounded error. That's the way we're gonna work with this. Okay, so, and I claim that you can study these maps and learn things about the group with them. Okay, so let me, right, and in fact,
23:45
just to make it sound like a natural thing to do, I can put all these together, call it pi from here into the Cartesian product of all of these things.
24:03
Okay, right? Just use these as the coordinates of this map. So it looks like a math-y thing to do, right? I've mapped my group into some enormous product space. All I have to do is understand the product space and understand the image and I know something. Okay, so that's, all right, so the goal is to kind of understand this.
24:24
So let me describe some properties of this. I won't, this whole construction, I wanna describe some properties, say a little bit about what they mean, and then give you an actual example of using these properties to do something. That's kind of the goal of what I wanna accomplish.
24:43
So properties, well, here's a property, kind of a trivial one. If w is a subsurface inside v, which is another subsurface, then pi w composed with pi v
25:02
is basically the same as pi w. And what I mean when I write stuff like this is that the difference between them is bounded distance. These two, the images of these applied to anything are bounded distance in the image space. That's what I mean by this, where the bound is some uniform number.
25:21
Everybody okay with this? This is more or less obvious because here you intersect with w and here you first intersect with v and then intersect with w. That's obviously the same thing. The only thing that might mess this up and not make it an equality is that I made all these random choices like I took a bunch of arcs and maybe I only chose one of the arcs. And then maybe for w, I chose one arc and for v, I chose some other arc.
25:41
So maybe this is not exactly the same thing, but up to some bounded error. That's kind of a trivial property. There's two more properties that are more interesting. One of them is called bare stocks inequality. And I wanna spend a while talking about this one.
26:07
And it says the following. It says, instead of saying w is inside v, a more generic thing is that they intersect each other in some other way. So suppose that w intersects v,
26:21
but w is not in v and v is not in w, okay? So I usually, we have a shorthand for this. We write this w intersect v transversely. It's not really the same as transverse, but this is the notation I like to use. There's a shorthand for all this, okay? So you should think of this as,
26:41
typically the boundaries of w and v might intersect non-trivially, but they don't quite have to. All you need is the surfaces to intersect non-trivially. And when that's true, let me just point this out. So then pi w of boundary v and pi v of boundary w make sense.
27:03
That's not the property yet. That's just a fact because the boundaries have to meet each other's interiors. And the following thing happens now. If mu is any marking, then the following thing happen. You can do two things. You can take, actually, let me,
27:22
sorry, before I do this, let me do one more thing over here, one more notational shorthand that will keep me sane. I could take, if alpha and beta are two objects, I can, well, curves, let's just say curves, markings, or laminations.
27:43
I could have done all this with laminations. It's kind of important, actually. Laminations also can be intersected with surfaces and they get arcs, so that still works. Then d w of alpha beta is a shorthand for the distance in the image associated with w,
28:02
in the curve complex of w, of the projections of alpha and beta. Okay, so I just want this shorthand, which is convenient. All right, so then here's the inequality. The distance, I can compute the distance in w
28:22
between boundary v and mu, or mu at some point, and I can compute the distance in v between boundary w and mu. These are two numbers, okay? And the inequality says that at least one of them is small, so the minimum of these two numbers
28:43
is at most some constant, I'll just call it C1, where C1 is some universal constant, independent of anything here. So I have to explain to you what this means. The goal of this lecture, actually, is to explain to you what this means.
29:01
But let me, any questions on the actual notation? Yes? So if alpha is quite a bit, then the distance is better. Wait, if alpha equals beta? Yeah, and then, Then the distance should be zero, I guess, because then, if alpha is equal to beta,
29:21
then these are actually the same thing, so then. There are two markings, so you have average choice of the. Right, so this is, Well, the right answer is it's approximately zero, the distance is bounded. But if you imagine that you a priori made a choice
29:40
for every marking, then if alpha and beta are the same marking, then you made the same choice by definition. But it's, it's the kind of question that involves being careful with definitions, but it doesn't affect the way that the actual theorems work, so. All right, so, okay, so we have this funny looking
30:05
inequality, which I want to interpret for you. And then the third thing, what we call the bounded geodesic image theorem.
30:23
Oh, shoot, ran out of space.
30:42
So the bounded geodesic image theorem says the following, and, okay, so it says the following. So suppose that v is inside w. Suppose that g is a sequence of vertices,
31:01
say i, g i plus one, a geodesic in the curve complex of w, w could be all of s. Okay, I have a geodesic. So I think of this as just a sequence of curves
31:21
that are each one disjoint from its successor. But I need it to be a geodesic in the geometry of this curve, so an actual geodesic. And then the following holds. If pi v of g i is defined for all i,
31:44
then the diameter of the projection of the entire geodesic, all the vertices, oops, into v is again bounded by some a priori universal constant, okay?
32:03
So, and let me say just, okay, let me say something, I want to say something interpretive about this, and then maybe I'll leave this one alone and focus on the other one. So this is going to tell us a little bit
32:22
about how we think about what these projections mean. So let me make a couple of observations that will help a little bit. One is we want to think of pi v as a kind of visual projection in the following sense.
32:43
Suppose you're in a space, here you are, okay, and you're looking around, right? You're an observer of a space. Then if you want to, if you look at things, take any object you're looking at, what does it mean to look at it? It means that you look at the light ray
33:00
from that object to you, and you look at how it intersects your unit sphere, right, and so if you have some kind of an object here, you would project it to an object in your visual, this is the visual sphere, right, the unit sphere around your eye, okay? So what's the unit sphere? So what does that have to do with our discussion? Think of, so think of the boundary of v
33:21
is like a place inside here, okay? Roughly speaking, the boundary of v, it's a simplex in the curved complex of w. So maybe this space will be the curved complex of w, and then inside there I'll take an observer which, well, it might be a simplex and not a point, but I'm going to draw it as a point.
33:43
And what is the unit sphere around boundary v? It's just the link inside this bigger complex. In other words, take all the vertices adjacent to this one. So, for example,
34:01
let's just draw an example, it's good enough. If this is the boundary of v, just a curve, well, that means v is on this side and maybe there's some other component, u, on the other side, okay? The link of this curve is everybody disjoint from it, so take all the curves here and all the curves there. Take the whole curve complex
34:22
here and the curve complex here, and you also allow all the pairs, something here and something in there, or triples, a simplex in here and a simplex in there, join them together. So, this is really the join of the curve complex of, in this case, v and u. In general, there might be
34:42
more pieces, okay? So, the visual sphere of this observer is the curve complexes of a bunch of surfaces. And then, suppose I do exactly what I said, I take this object far away and I draw a geodesic
35:01
all the way to here, the boundary v, the last moment before it arrives, it's sitting in the link, and everywhere else it's distance two or more, and if you did one of the exercises, you'll see if to be distance two or more from a point, you need to intersect it. So, everybody from here
35:21
onward has a well-defined π sub v. So, π sub v makes sense for this whole geodesic, this whole light ray from the point to here, to the unit sphere, and the theorem says, I'm actually, so I'm fuzzing over the distinction between these two components, but if you, okay,
35:45
as soon as I get distance three away, I intersect both components and it doesn't matter. So, anyway, this whole thing has a kind of bounded image over here in the unit sphere. That's what this theorem says.
36:01
So, this theorem kind of tells you that, in particular, you can think of this projection as a visual projection, as just one observer looking out at the space, and what he sees on his unit sphere is this thing, okay? That gives you an interpretation, a philosophy for what we think of these many. It's uniform, it doesn't actually depend
36:21
on any of the surfaces. I think that's right, let's see. Ah, we didn't prove that it was uniform. Certainly, but I think Richard Webb proved that it was uniform, I think independent of the surface.
36:41
It's certainly independent of the subsurface. Okay, all right, so, okay, I need to give you
37:01
one more theorem, one more property which is really a theorem. Actually, it's different because, all right, all of these were kind of local properties. You can prove them about, well, I guess three is not quite local. Yeah, I was, okay, I'm gonna forget about three and talk about two, but first, let me tell you one more property
37:21
which is kind of relevant to the whole discussion, and that's, there's kind of a distance formula. I'm not gonna prove this for you, but you can start to imagine what the proof might look like after a while. But distance formula, it says the following. It says, so remember that the, if you look up there, we have the marking graph
37:41
being mapped to this product of curve complexes, okay? And I can measure the distance in the marking, marking graph's a metric space. I could try to put a reasonable metric on the product of curve complexes and then ask for this map to be a good map in terms of distances, and that's almost what the distance formula says, so let me write it. So mu, nu, any two points in the curve complex,
38:03
and then the distance in the curve, in the marking graph, sorry, two points in the marking graph, so it's like the mapping class group. The distance between these two points can be estimated, I have to tell you what this means, by a sum over all surfaces in S,
38:22
up to isotopy, of course, of these distances. So exactly, you project them in to the curve complex and compute their distance. This is still not right. I have to do one more thing, so I have to kind of, all right, so what does this little notation mean? It means this, if I take a number
38:41
and I put it in braces with an A, it means I just keep the number if it's bigger than A, let's say bigger than or equal to A, and I throw it away if it's less than A. So what this says is there's a little bit of noise I can't control, and the reason I can't control is I'm never careful about definitions,
39:01
so there's actually unavoidable noise in all these definitions. Once I throw away the noise and add up what's bigger, I get an estimate for this. Now this estimate just means up to multiplicative and additive error. So in other words, these sides differ
39:22
by multiplying by some constant and adding a constant, up to that. So that's like a quasi-isometric sort of thing, but it turns out this threshold thing is more trouble than it looks. It's really kind of, it makes a lot of things actually quite tricky, but as a statement, it's at least digestible.
39:41
So this tells you that that map up there is very much like a quasi-isometric embedding into an infinite product where you kind of want to think of the infinite product with its L1 metric. We just add up the distances and the factors, except for this little loophole where you throw away the noise. So there's something kind of difficult to deal with here.
40:04
Everybody okay with this? So this tells you that you have a chance of using the right-hand side to study the left-hand side and this theorem is proved by kind of putting together the other information among those properties and building paths in the marking graph using this data.
40:23
Right, so let me, yeah, I didn't get as far as I wanted to get with all this, but let me try. So let me try to tell you why Bierstock's inequality is true, what it means,
40:42
and try to give an example of using it, okay? So let's try that. So first of all, what does it mean? Actually, there's a little diagram that gives you the right point of view, whoops, on what Bierstock's inequality means. It's the following.
41:01
So what are the ingredients? There's two surfaces and some marking somewhere. So I think in terms of this discussion of visual blah, blah, blah, I just think of v and w as two observers in my space and mu is a third point.
41:21
And if I want to, so there's this kind of triangular picture. I can take, I can look from v and ask what is the angle between w and mu? What is the visual distance, right? So this angle is going to be dv of boundary w mu.
41:40
And this angle is dw of boundary v mu. And the inequality says that most one of them can be bigger than c1. So if I have, for example, a picture that looks like this where this is really big, then if I draw the other side,
42:04
it's going to be small. It's going to be less than or equal to c1, okay? That is certainly a property of Euclidean triangles, right, as soon as, because they add to 180. So, but here it's interesting. It's not quite that property. For example, this constant c1 is not a small number.
42:22
It's just some number. This big number here is not bounded by pi. It's an arbitrarily large real number or integer. So the geometry is a little more involved in that, but this is the sort of schematic interpretation, right? You can't have this picture. You can't have v, w, and then a really big angle here
42:44
and a big angle there, which then somehow managed to meet. Right, this can't be a picture where these are both geodesics. Everybody okay with that interpretation? That's okay. So, let's see, I don't know what I'm doing here.
43:05
Yeah, okay. So let me give you a quick proof. This is not Bearstock's original proof. This is a proof due to Chris Leininger. As in this whole kind of discussion, the sort of the original proofs are complicated
43:21
and then the kind of modern proofs are very simple. So proof, so here's the picture. Let's draw, okay, let's draw v. So here is v.
43:41
And then we'll draw the intersection of w with v. So we'll draw that in orange. Here's w. W is running through here in some way. Here's w. We don't quite know what it's doing, but maybe it's complicated. So all of this stuff is w. It goes around, comes back out.
44:03
That's w, okay? And now, somewhere else in this picture is mu, which I will draw, also complicated. Okay, and mu is running around somewhere, okay? So what I've drawn here is the, you can see from this, the boundary of w intersect v,
44:23
which is projection to w of boundary v. No, projection to v of boundary w is these arcs you see. And the projection to v of mu is the green arcs. So here's the first step. If the distance between, in v,
44:44
between boundary w and mu is bigger than a certain number, which you have to compute, but let's pretend that the number is 10. There's some number, like 10, so that if this distance is sufficiently large, then each arc of mu intersect v
45:06
meets each arc, well, just let's say, meets boundary w at least three times. So this is something you more or less proved in the exercises if you did this one.
45:22
If you intersect only three times, or less than three times, then your distance in the arc or curve complex is bounded above. So as soon as the distance is big enough, you must intersect a lot. In particular, you have to choose the number so you get at least three times. So if that's true, if mu intersects the boundary of w three times,
45:41
then there must be an arc that's contained in w. Here it is, right? Because it's three times. If you do three times, at some point you have to go in and then come out. Yes? So that's why you need three. Okay, so, but that implies that there exists an arc
46:03
of mu intersect w, there it is, which is disjoint from boundary v, right? Because it's inside v. So this implies that the distance in w
46:20
between boundary v and mu is at most one, yes? Because you have inside the orange surface, w, you have a green arc and you have the white arcs and they're disjoint. So the distance is at most one, okay?
46:43
That's the theorem, right? If this one is big, then this one is small. That's exactly the theorem. Okay, so that's the whole thing. All right, yeah, so what am I going to do with it?
47:02
Well, I think I have to stay. So this is a theorem I wanted to prove in these lectures, but let me at least say something about it. Oh, yuck, let's see.
47:35
So the question that this theorem answers is, what is the image of, what coarsely is the image of pi?
47:51
What? Oh, yeah, yeah, sorry.
48:00
I don't remember what's in there anymore. Oh, that's the proof. Yeah, what kind of a lecturer hides the proof? Is that visible? Okay, so now I want to ask, kind of finish up with the discussion of this question.
48:21
What is the image of this map pi up there which projects to all the curve complexes? So, and when I say coarsely, I mean, if you give me a point in the target, how can I tell if it's close to the image of pi in some bounded sense, okay? Now, they have to satisfy one and two
48:44
or some approximation of one and two, right? Any point that's close to the image has to satisfy some approximation of one and two because they're necessary facts about this. So the theorem, I think we call this the consistency theorem
49:03
but you could also call it the realization theorem. And this is, I guess, work with Jason Bierstock and Kleiner, Bruce Kleiner, V. Mosher. It's part of a longer project I'm not gonna talk about
49:21
but the theorem says that a point, let's write it as a tuple x, w in the product of all the Cw's, I need some quantifiers
49:40
but let me just write the statement first, is within, well, actually, let's just, sorry, given a point in here, there exists a point mu in M whose image is close to x.
50:01
So that means such that the distance, such that for all w, the distance in w between x, w and mu is bounded by some number which we have to talk about so let's call it B.
50:22
All right, this is the thing I want, right? I wanna know when is there a marking whose image is close to the points I chose for every coordinate. And that's true if and only if two conditions hold.
50:41
One is if v is inside w, then if you look at, I guess, the distance in v between x, v, oh, no, sorry. If, sorry, if the distance in w
51:01
between the boundary of v and x, w is large enough, we'll write c, I don't know what, zero, then the distance in v between boundary, between x, v and x, w is less than c, one. I'll explain these in a minute.
51:21
c, two is Bierstock's identity if v and w intersect without being nested, then the minimum of d, v, boundary w, x, v and d, w of boundary v, x, w is less than c, one.
51:45
Oh, and then now we need, how are the quantifiers in this statement? I guess, I think for all c, one, c, two, no, shoot, it's if and only if,
52:01
so my quantifier is gonna be messed up. In one direction, it's this. For all c, one, c, two, there exists a B such that the if condition holds. Let me, okay, so I allow, let me not fix this.
52:22
Okay, so, because there may be two versions of it for the two different directions. But the point is, there are suitable constants for which this is true, that the condition for being within a bounded number from the image of a marking is exactly the coarsified version of one and two up there. One is basically this kind of nested thing.
52:42
And this is the, the version of the nested thing that we have to deal with is we have to talk about the coordinates in each one and what they look like. So what is this saying? This is saying that this point in the curve complex of w needs to be far enough away from boundary v. First of all, it needs to be far away just so you can project it into v meaningfully, right?
53:03
If it's only distance one from boundary v, it might be disjoint from v. So it needs to be a little far away and the condition is that it needs, since we have kind of everything's coarsified, I need to be sufficiently far away. So if it's far enough away, then when I look at it from v, I see the same thing as I did
53:20
when I just took the point in v. So that's like number one. And number two is just like Berestock's inequality. Okay, so the theorem is that this is in fact what characterizes the image. And I won't, I think I won't get a chance to tell you how to use this, but it's just sort of a shame.
53:41
Question? C1 and C2 are C2 and C1? Yeah, yeah, thank you. All right, so I think in one direction for every C0 and C1 there's a B and in the other direction for every B there's a C0 and C1, that's it.
54:02
Okay, so that's the theorem. I wanna tell you a tiny bit about why it's true. So I have like four minutes, two minutes. What do I have? Yeah, five minutes, okay. Then I can do it. So first of all, I mean, okay,
54:22
the pep talk for why this kind of theorem should be useful is that hyperbolic spaces are good spaces for making various constructions in. You can build convex hulls, you can project from one thing to another, you can do a lot of work in a hyperbolic space that you can control. This theorem is kind of telling you that certain kinds of problems you can solve by solving them individually in each of the coordinates,
54:44
which is hyperbolic, and then kind of assembling them into something in the whole group by just checking this consistency condition. That's the way you use this theorem. Let me try in five minutes to say
55:03
something about the role of these inequalities in the proof, I guess. So let me try to say a few words about the proof, okay? So what are we doing? We have, so we're given, the hard part of the proof is we're given some xw that satisfies c1 and two.
55:24
We have to build mu, right? We need to build mu. So we will build mu kind of inductively. It's a marking, right? It's a bunch of curves in the surface. So we just have to find one curve that should live in mu. And then we find the right one,
55:42
and then in the complement of the surface, we find more curves, and we inductively work in each subsurface, and we build it up from the ground up, and we end up, after finding many steps, with something. So we have to explain how this might work. So that means that the first step should be finding a point in the curve complex to serve as the initial point of mu.
56:01
So here we are in the curve complex. And what do we have in the curve complex to work with? Well, we have xs, right? Which is the point we were given that is associated to the curve complex of the biggest surface, which is s itself.
56:21
And we want it to be close to the first point we choose from mu. But remember, everything was coarse. Maybe we chose a really terrible one, and maybe the right one would have been somewhere else. Right, let's call it mu zero. And we have to figure out where it is. We have to go from here to where we should be. And it's not obvious what we should do.
56:41
So let me, so what we'll point the way, yeah, it's a little, okay, let me try to explain this. So can I erase this? Who put it up? Okay, so remember that the mu that we are looking for
57:08
has more information than just xs. It has all the xw's in it, somehow encoded, right? So it's not enough to just know xs. So what could be in the way? What could be in the way is there could be an observer v,
57:21
which thinks that these are really different from each other. So we consider, so look at those sub-surfaces of s such that the distance in v between
57:41
the point we are starting with over an s and the information that really lives in v, which is xv, is very large, like let's say bigger than 100 times c1 or something, like some large number. These are sub-surfaces which are witnesses to the fact
58:03
that we can't use xs as our first vertex, because our first vertex has to look like mu in every sub-surface, and in particular, it should look like xv inside v. So if this number is big, it's bad. So the main point, which is that
58:22
Bierstock's inequality, which we now have a version of, I mean, not really Bierstock's inequality, but c2, which is Bierstock's inequality for this tuple x, right? So c2 helps to partially order this collection.
58:46
So let's call this, I don't know, J, just J. And let me, okay, let me draw,
59:00
I have negative 30 seconds, maybe. Actually, so, sorry. I claim, I'll explain, I'll draw one last picture that justifies this thing, but then what's the point? Bierstock's inequality is gonna partially order this collection of bad sub-surfaces, bad observers. And so I will get a picture that looks like this.
59:21
I'll get a bunch of surfaces, which are witnesses to the fact that I'm on the wrong side somehow of where I should be, that where I need to be is over here, and I've somehow chosen something so that these observers think that these are really far apart. The claim was that they're all partially ordered,
59:41
and then I'm gonna choose a minimal one. So choose, say, let's call it U, minimal in J, and let the boundary of U be the first part of mu.
01:00:00
and we do the induction. So I kind of, I look at all these bad ones, and I pick the one that's closest to the mystery place. That's gonna be, that's the one I choose by making it minimal in the partial order. And that's where I start. And I have to prove that that actually works. But I wanna draw a picture that tells you about the partial order, and then I'll stop. So, oh, maybe here.
01:00:22
So, let's see. It's more or less the following. So, it's a schematic of what we're looking for. But x is my tuple, and it kind of looks,
01:00:45
let's just first draw the following thing. Suppose I have two surfaces, v and w. Actually, let's just do this. What's the partial order? Here's v and here's w. I'm gonna say that v is less than w
01:01:01
if this angle is big, so it's closer to x. So that means that the distance in v between x, v, and boundary w is really big. Maybe 100, c1 or something. Okay, maybe I've defined the partial order.
01:01:21
So, Baire Stock's inequality, or number c2, implies if v is less than w, then w is not less than v, right? At most, one of these two angles is big. That's the schematic. And so the big one tells me which one, which of these surfaces is closer to my tuple x,
01:01:43
all the information in x, which is where I'm trying to get to. Okay, so maybe I'll, there's one more thing to check, which is, well, basically, transitivity. So, yeah, I think I don't wanna take more of my time, but this is a definition of something.
01:02:01
This, right, some kind of relation that's starting to be like a partial order because it's at least not symmetric, right? You can't, it's at least well-defined between pairs. And you have to choose, you have to prove some kind of transitivity property for this. And one of the problems with working with what, in the way that I've been working, I've been saying, well, you don't have to worry about things up to bounded distance,
01:02:22
and the definitions are only okay, you know, coarsely and so on. And that is, one of the things that immediately gets you is that in a definition like this is unstable because if you change things by a little bit, suddenly you were above the threshold and you go below the threshold, right? So, well, there's a transitivity law that applies,
01:02:46
but it's sort of a weak transitivity that if you have v less than w and w is less than z, then v is less than z, but with a smaller constant. And so you don't quite get transitivity. So you need a little more to actually turn this into a partial order,
01:03:01
and I'm not gonna explain how that goes, but this kind of, in some sense, this is a central idea of the whole thing, that you can take these inequalities, they really encode a kind of an ordering structure among all of the objects in this setup,
01:03:20
and that organizes kind of most of the, most of the arguments you try to make involve in some way using a kind of partial order of this type. Okay, so I'll stop. Sorry, I went over.
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