 Let me let me let me try to go on to what I was going to do next so Right, so so what's what's the So I want to talk in the second hour about the kind of subsurface structure And and the actual course geometry of the mapping class group and I guess Let me just point out this is this is related to a bunch of to this kind of Modern versions of a bunch of this structure, which is worth mentioning. I mean so there's So these are just words for the moment, but I guess you saw you heard from so I'll talk about a cylindrical hyperbolicity So this is I think Bodich Dominic, etc. I'm actually not so I'm not sure how many names I should put on there. I'll leave the parentheses open This is a so the curve complex is a cylindrical hyperbolic. I think from what I mentioned it So that's one Kind of general category in which this discussion Fits there's there's another thing called sort of more restrictive 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 Which is a kind of a device worked out by best Vina Bromberg Fujiwara and The notion of hierarchically hyperbolic spaces groups and spaces And this is bear stock Hagen Insisto Why am I mentioning all this so? All there are various features of these discussions Which are kind of more general which are closely related to the kind of structure that I want to talk about and some to some extent inspired by it It's so that 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 general notion so that so So I'm staying in my own comfort zone of thinking about mapping class groups But but as you listen you might if you've read any of these other things you will recognize Things you've seen okay, so So What's the story now? I want to tell you about so we have these Other curve 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 Of these surfaces So I won't repeat that But there's one other thing you you can do is you can talk about arcs instead of curves So actually maybe the for us the simplest thing is talk about the arc in curve complex Say maybe we'll call a c of s or w maybe so think of the w is a surface with boundary and now the arc in 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 isotope So so the isotope in general involves moving the base point moving the end points as well as varying the arc Okay, so think about keeping the base point in the value of it moving it that that isotope is the equivalence relation So vertices are these and edges are disjoint pairs and so on same setup It's it turns out to be useful to think about that one of the problems on the exercise set was to show that these are 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 by some bounded factor That means basically and I didn't say this on the on the on the homework, but It's not quite true. So it's true except for the exceptional cases except for the exceptions so Okay, these are the exceptions right these are so you have to be anyway, I'm One can get bogged down in this but it's better not to so you can ever 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 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 so here I should take arcs up to Homo topi with Fixed endpoints That's what I want to take. So same discussion. Yes Are talking to the boundary Well, there's no metric. It's just a topological picture. So there's no worth talking out. It's just topological here also Okay, so if I take arcs up to Homo topi with fixed endpoints Then then they're not all the same. In fact, well now there's uncountably many of them Which is kind of unpleasant, but we're not going to worry about that But but the interesting that now happens is that I can 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 while fixing the endpoints. Okay, so in the in the annulus so So in fact what this complex does is it counts the number of times that an arc is twisted around and it's not in even though There are uncountably many arcs because they're uncountably many points This complex is still quasi isometric to Z Where the Z just measures the number of times. Okay, so that's my device for thinking about annual You'll see in a minute why I have to do this But the job of annualized them as you're twisting, okay, so so what do I want to do with all this I want to talk about It's sort of simple notion of projection projection So suppose I take w is a subsurface and alpha is a curve In s one of our curves and if Alpha intersects W essentially so that means you cannot push alpha to be disjoint from w non-trivial intersection then we can produce Let's say this way alpha intersect w Is a simplex in the arc and curve complex of w right you take Maybe let me put like little brackets. I have to do something is what do I have to do? Let me here's the picture I mean So here is w and here is the rest of s 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 so many of the arcs have to be parallel up to identifying parallel copies There's only bound that 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 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 so maybe I should write actually maybe I should raise the actual map from this minus the alpha disjoint From w to c of w. Okay, and if I wanted to be an actual map I should maybe pick a point in the simplex So at the at the cost of making an arbitrary choice 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 is an annulus alpha is 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 I can make alpha after an isotope. I can make alpha go around a couple of times like this 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 Obviously, this is not a well-defined thing. So what is well-defined and The way to get a well-defined thing is to go to the universal cover Or actually to the annular cover so Oh, we forgot to erase So Okay, I'm just gonna say this quickly and and then I'll try to Pretend Then I'll try to talk about annualize if there's no special case But so so now suppose that w is an annulus and there's a kind of canonical annular cover this is H2 the universal cover of s modulo The group generated by the core of w. Let me so what do I mean by this? Oh, and then I have to take And I can take a 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 if you lift w you get some annulus This is 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 I will get an annulus kind of canonical annulus with the 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 compactified by taking the boundary of h2 Minus the fixed points of this group and maybe I'll give this a name gamma the fixed points of gamma And then I'm out out by gamma Right, so that has that's these two arcs Modular the action of gamma it gives me two circles. So this is the This is the compactified annulus. Okay, everybody clear So what do I do with this annulus now 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 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 these endpoints These are well-defined. These are isotopey variant 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 The it's a bounded it's some bounded length isotopey at infinity. It doesn't move at all Okay, so so the endpoints are well-defined. So this defines this gives a well-defined Pi W of alpha Okay So those 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 the height we 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 is h2 right but all of them identify the circle in the same way To each other. There's a canonical the circle in infinity is canonical Even if the the different hyperbolic structures are not the circle in infinity is naturally identified with the Gromov boundary of the fundamental group and so so So it doesn't actually affect anything to change the metric that's a good question, okay Let's see Alright, so what are we gonna do with this? Okay, I have I have to introduce one more thing one more kind of object on the surface I want to 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 map mapping class group a group that geometrically is can be identified with The mapping class group up to course 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 after I won't repeat so this this object is not a marking exactly the same way But there's a kind of family resemblance between the object 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 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, too So the base is a pants decomposition. It's a maximal simplex in a curved complex For a pants decomposition. So the base will be blue Here's 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 Okay, and then That's the first part of the marking note that it is already has the feature that the stabilizer of this object is One of our billion dain twist groups. So it's got a Relatively small stabilizer, but now Let's add one more thing which will which will tie things down and that is I'll add for every blue curve 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 Okay, that's it. So that's that's the definition So there's one more I have to draw and that's where that this is 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 at exactly two times. 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 a that's called a marking Fine print is that the orange curves are forced to intersect each other This is a tragic fact about this picture because it's kind of messing it makes some of the definitions messy All right, anyway, that's 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 or it certainly it's Certainly it's finite Maybe there's like a hyper elliptic involution that fixes things or something anyway It's got a finite state everybody clear that this should have a finite stabilizer in the mapping class group because If you fix all the blue curves and all you can do is twist but the orange curves detect twisting that's all Okay, so the stabilizers Is it worst finite and moreover so that that's okay, that's a marking now We take let's say M of s is the graph 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 important not so important to our picture except Elementary moves need to be some some kind of standard little change in here For example twisting an orange curve around its blue curve one one twist is an elementary move Okay, and there's a couple of other elementary moves that rearrange the picture so What so I'll leave this as a kind of exercise to think about but with the features I want from this I want this to be locally finite 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 and the action. Oh, and I didn't say action is with finite Sorry locally finite MCG acts With finite Stabilizers 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 finite stabilizers connected To find a quotient then the two the group in the space are quasi isometric So I'm going to 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 going to prove this but think about maybe think about this right draw two such objects one of them enormously complicated relative 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 where a sequence of elementary moves That's what connectivity means. Okay, so that right that tells you Something about what you should make the elementary moves be so that you can have a chance to make it connected Okay, let's just leave that as it is Okay, so so the point of all that was to just have a A space that we can use together with the curve complexes to analyze what's going on in particular In particular I get immediately a map from the the marking graph into every curve complex so so pi w from the marking graph to any I guess I Okay, I'm gonna cheat I I Had this this thing should be in C of w and a C of w so that I could write the definition down I claim they are basically the same space and I'm gonna pretend that they are the same space. Okay, cuz That can you forgive me for that? so So the final story is I have this This map which makes sense as a map from not from a marking to the curve complex of a subsurface because Every marking intersects every surface So there's always something I can map into Intersect with the surface and get an image now You 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 the to this thing. I only care about this map up to bounded error anyway Okay, so this makes sense. I might have to make some choices and the choices only cost me a bounded error That's the kind of that's 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 Okay, so let me Right, and in fact just to make it sound like it 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 Okay Right just use these as the coordinates of this map so it looks like a mathy thing to do right I have 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. That's so the goal is to kind of understand this So let me Describe some properties of this I won't This whole construction I want to describe some properties say a little bit about What they mean and it give you an actual example of using these properties to do something that's kind of the goal of what I want to accomplish so Properties well here's a property kind of a trivial one If if w is a subset subsurface inside v which is another subsurface then Pi w composed with Pi v 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 are 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 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. So 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 that are more interesting one of them is is Called Bearstocks inequality and I want to spend a while talking about this one 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 but But w is not in v and v is not in w. Okay So I usually a short that we have a shorthand for this this is we write this w Intersect v transversely. It's not really the same as transverse, but this is the notation I'd like to use is a shorthand for all this Okay, so you should think of this as typically the boundaries of w and v might intersect non-trivial But they don't quite have to all you need is the surfaces to intersect non-trivial II and when that's pot when that's true Let me just point this out. So then pi w of boundary v and Pi v of boundary w makes sense 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 happened you can do two things you can take Actually, let me 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 go curves. Let's just say curves Markings for laminations I could have done all this with laminations. It's kind of important actually Laminations also can be intersected with surfaces and I get arcs. So that still works Then D w of alpha beta is a shorthand for the distance In the image associated to W in the curve complex of W of the projections of alpha and Beta Okay, so I just want this shorthand which is convenient Okay All right, so then here's the inequality the the distance I Can compute the distance in W between boundary v and Mu or mu is some point and I can compute the distance in v between boundary w and mu There's two numbers Okay, and the inequality says that at least one of them is small. So the minimum of these two numbers Is it most some constant? I'll just call it C1 Where C1 is some uniform 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 but let me Any questions on the actual notation? Yes Wait if alpha equals beta Then the distance should be zero I guess because then if alpha is equal to beta then these are actually the same thing So then 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 show made a choice for every marking Then if alpha and beta are the same marking that 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 theorem All right, so Okay, so we have this Funny looking inequality, which I want to interpret for you and in the third thing What we call the Bounded geodesic image theorem. Oh shoot space so the boundary 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 Of geodec of vertices say I gi 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 is just a sequence of curves that are Each one disjoint from its successor just to but 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 gi is defined for all I then the diameter of The projection of the entire geodesic all the vertices whoops Into V is Again bounded by some a priori universal constant Okay, so And let me say just a okay. Let me let me say Something into 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 this is going to tell us a little bit 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 Suppose you're in a space Here you are Okay, and you're looking around right you're 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 from that object to you and you look at how it intersects your unit sphere Right, and it's so if you have some kind of an object here You would project it to an object on your visuals. 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 the boundary of V is like a place inside here Okay, roughly speaking the boundary of V It's a simplex in the curve complex of w so maybe this space will be a curve complex of w and then inside there I'll take an observer which well it might be it might be a simplex and not a point, but I'm going to draw it as a point and What is the 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 just draw an example it's good enough if this is the boundary of V just a curve Though that means V is on this side, and maybe there's some other component you on the other side, okay? The link of this Curve is everybody this joint from it so take all the curves here and all The curves there take the whole curve complex here and the curve complex here, and you also allow all the Pares something here and something in there or triples some some something a simplex in here and a simplex in there join them together so this is really the the join of the curve complex of in this case V and you In general there might be more pieces Okay, so the 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 I take this object far away, and I draw a geodesic all the way to here The boundary V the last moment before it arrives. It's sitting in the link And everywhere else its 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 onward has a well-defined Pi sub V So Pi sub V Makes sense for this whole geodesic this whole light ray from the point to to here to the 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 As soon as I get distance three away, I intersect both components, and it doesn't matter so Anyway, this whole thing has about kind of bounded image over here in the unit sphere That's what this theorem says So this theorem kind of tells you that in particular you can think of this projection as a visual projection It's just that one observer looking out at the space and what he sees on his unit sphere is this thing Okay, that gives you an interpretation for a philosophy for what we think of these meaning It's uniform. It doesn't actually depend on any of the surfaces Think that's right. Let's see. Ah We didn't prove that it was uniform. It's certainly but I think Richard Webb proved that it was uniform. I think independent of the surface. It's certainly independent of the subsurface Okay All right, so Okay, I need to give you 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, it was okay I'm going to forget about three and talk about two but first time I need let me tell you one more property Which is kind of relevant to the whole discussion and that's there's kind of a distance formula not going to 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 being mapped to this product of curve complexes Okay, and I can measure the distance in the marking marking graphs 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 only write it so mu nu any two points in the curve complex 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 up to isotope of course of These distances So exactly you project them in to the curve complex and computer distance This is still not right. I have to do one more thing. So I have to kind of All right, so what is this little notation mean? It's it means this if I take a number 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 equal to a and I just and I throw it away If it's less than a So what this says is I there's a little bit of noise I can't control and the reason I can't control is I've never careful about definitions So there's kind of an there's actually unavoidable noise in the 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 this means up to multiplicative and additive error so in other words These these sides differ by multiplying by some constant and adding a constant up to up to that So that's like a quasi-symmetric sort of thing, but this it turns out this Threshold thing is more 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 adjustable Okay, so this tells you that that map up there is very much like a quasi-symmetric embedding into an infinite product where 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. Everybody okay with this? so 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 Building paths in the marking graph using this data um Right, so let me Yeah, I didn't get as far as I wanted to get rid of all this, but let me Let me let me try so Let me try to tell you why bear stocks inequality is true what it means and Try to give an example of using it. Okay, so let's 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 bear stock inequality means. It's the following so what's what are the ingredients? There's there's there's a 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 and if I want to so This is 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 D V of boundary W mu and this angle is D W 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, 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 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 It's just some number this big number here is not bounded by pi is it's an arbitrarily large real number or integer So the geometry is a little more involved in that but this is this sort of Schematic interpretation right you can't have you can't have this picture. You can't have V W and then a really big angle here and a big angle there which then somehow managed to meet This can't be a picture where these are both geodesics They'll be okay with that interpretation. That's okay, so Let's see. I know what I'm doing here. 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 Is in this whole kind of discussion that sort of the original proofs are complicated and then the kind of modern proofs are Very simple So proof So so here's the picture Let's draw. Okay, let's draw V. So here is V and then We'll draw the intersection of bound of W with V. So that will draw that an 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. 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 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 here's the first step if the distance between In V 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 sufficiently large then each arc of Mu intersect V 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 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 and particularly 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 Then there must be an arc that's contained in W. Here it is Right because three times it's up if you do three times it at some point you have to go in and then come out Yes, so That's what you need three. Okay, so but that implies that there exists an arc of Mu intersect W there it is Which is this joint? from boundary V Right because it's inside V so this implies that the distance in W between Boundary V and Mu is at most one Yes Because they're just 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 the most one Okay, that's that's the theorem right? We if this one is big then this one is small. That's exactly the field Okay, so that's that's the whole thing. All right Yeah, so what am I gonna do with it? Well, I think I have to stay so this is a thing I wanted to prove in these Lectures, but let me at least Say something about it. Oh Yuck, let's see so the question that this theorem answers is what is the image of What? Corsely is the image of Pi What oh? Yeah, yeah, sorry. Remember what's in there anymore. Oh, that's the proof. Yeah, what kind of a lecturer hides the proof that visible Okay, so so now I want to ask kind of finish up with the the discussion of this question What is the image of this map pi up there? Which projects to the 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 or Some version 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 because they're necessary facts about this so the theorem. This is we we Think we call this the consistency theorem, but you could also call it the realization theorem And this is I guess work with Jason Bearstock and Kleiner was Kleiner the Mosher It's part of a longer project. I'm not going to talk about but the theorem says that a point Let's write it as a tuple xw in the product of all the Cw's I need some quantifiers, but let me just write the statement first is Within well actually let's just sorry given given a point in here There exists a point mu in M Whose image is close to x so that means such that The distance such that for all w the distance in w between xw and mu is Bounded by some number which we have to talk about so let's call it b All right, this is the thing I want right? I want to 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 um Two conditions hold 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 if the distance in w between the boundary of v and xw 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 c1 I'll explain these in a minute c2 is Bearstock's identity if v and w Intersect without being nested then the minimum of dv boundary w x v and Dw of boundary v x w is Less than c1. Oh, and I now we need to how are the quantifiers in the statement. I guess I think for all C1 c2 No shoot It's if and only if so my quantifiers are going to be messed up in one direction. It's this for all c1 c2 there exists a B Such that the Such that the if condition holds let me okay, so Allow let me not fix this Okay, so because there's maybe two versions of it for the two different directions But the point is there are suitable constants for which this is true that that the condition for being within a bounded number from the image of a marking is Exactly the corsified version of one and two up there right one is basically this kind of nested thing 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 this point in the curved 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 if it's only distance one from boundary v It might be disjoint from v So it needs to be put a little a little far away and the condition is that it needs Since since we have kind of everything's corsified. 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 when I just took the point in v So that's like number one and number two is just like Bearstock'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 cash in to tell you how to use this but Just sort of a shame 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 the C0 and C1 Okay, so that's the theorem. I want to tell you a tiny bit about why it's true I have like four minutes two minutes Five minutes, okay, then I can do First of all, I mean, okay, the the pep talk for why this kind of theorem should be useful is The hyperbolic spaces are 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 In each of the coordinates, which is hyperbolic and then kind of Assembling them into something in the whole group by just checking this consistency condition That's that's the way you use this theorem Let me try to let me try in five minutes to say something about the role of of these inequalities in In the proof I guess so so let me try to say a few words about the proof, okay, so So 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 2 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 we just we find the right one 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 so that means that the first step should be finding a point in the curve complex to serve as the initial point of mu so so here we are in the curve complex and And 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 of the biggest surface Which is s itself and we we wanted to be close to the first point we choose from you 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. Maybe 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 So let me so what will point the way? Yeah, it's a little okay, let me let me try to explain this so Can I erase this who put it up, okay, so remember that the The true the the mu that we are looking for has more information than just xs it has all the xw's in it somehow encoded right so 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 Which thinks that these are really different from each other? So so we consider so look at those subsurface of s such that the distance in v between the Point we are starting with over an s and the information that really lives in v, which is x v is Very large like let's say bigger than you know a hundred Times c1 or something like some large number These these are these are subsurface's which are witnesses to the fact that we can't use xs as our first vertex Because our first vertex has to look like mu in every subsurface and in particular it should look like xv inside v So if this number is big it's bad so So the main point which is is that bear stocks inequality which we now have a version of I mean Not really bear stocks in quality, but c2 which is bear stocks inequality for this Tupel x right so c2 Helps to Partially order this collection, so let's let's let's call this I don't know J Just J And let me let me okay. Let me let me draw. 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 what's the point? Bear stocks inequality is gonna partially order this collection of bad subsurface and bad observers And so I will get a picture that looks like this I'll get a bunch of of Surfaces which are witnesses to the fact that I'm 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 These the claim was that they're all partially ordered and then I'm gonna choose a minimal one so choose Say let's call it you minimal in J and let the boundary of you be the first part of Mu and then we do the induction so I kind of I look at all these bad ones and I picked the one That's closest to the mystery place That's gonna be that's the one I choose by making minimal in the partial order And that's where I start and I have to prove that that actually works But I want to draw a picture that tells you about the partial order and then I'll stop So I'll maybe here 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 and It kind of looks 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 if This angle is big so that's closer to x so that means that the distance in v between x v and Boundary w is really big. Maybe maybe a hundred C1 or something Okay, maybe I've I've defined the partial order the the so bear stocks 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 To my tuple x all the information and 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, right? So Yeah, I think I don't want to take more of my time but This is a definition of something this something 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 you know, 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 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 any 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 so things so trans in transit. Well There's a transitivity law that applies But it's sort of a weak transitivity that if you if you have you v less than w and w is less than z Then v is less than z but with a smaller constant And so you're you don't quite get transitivity So you need a little more to actually turn this into a partial order And I'm not going to explain how that goes but 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 In this setup and that organizes kind of most of the most of the items 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