The geometry of 3 - manifolds before and after Perelman
Hamenstädt, Ursula
The rank of a hyperbolic manifold is the smallest number of generators of its fundamental group. McMullen conjectured that for all $k\geq 2$, the pointwise injectivity radius of a closed hyperbolic 3-manifold of rank at most k is uniformly bounded from above. We explain some methods which were introduced before and after the foundational work of Perelman to study these manifolds, and we show how these methods can be used to prove McMullen's conjecture in many cases including random 3-manifolds.
 Well, let me thank you for inviting me here. It's a great honor and pleasure. I met Marcel in 1981 in fall when I came to Paris as an exchange student, being completely naive and ignorant, I have to say. But due to not only Marcel, but the atmosphere that he created around him and his young people. So when I came, I had some vague idea that maybe geometry was something that I could like. And after one year, it was kind of clear that I would never escape geometry again. I was completely captured by and fascinated by the subject, and in particular also by Marcel's way to discuss mathematics and introduce concrete problems that had difficult, but perhaps concrete answers. So I hope I can give you today a little bit of lecture in the spirit of this philosophy that you might be interested in concrete questions. There's been a lot of progress ever since, but somehow maybe the progress leads now to us being able in the hopefully near future to solve problems that were not reachable back then, but maybe there isn't reach right now. Let me start with something so very simple related to Bruce talk. So Bruce said, now that we understood geometrization, we can ask a sort of more advanced questions that follow up. And let me just go back to something that is familiar to all of you. Let's look at a closed surface where somehow you could think of everything is understood. And for me, everything is oriented. I don't write this down. All my manifolds will be oriented. And let's take a closed surface of genus G bigger equal than 2. And we all know that this surface has a hyperbolic structure. That means a constant curvature, a metric, a constant curvature of minus 1. And furthermore, there exists a 6G minus 6 dimensional parameter space of such metrics. Of curvature minus 1. And we in addition have Gauss-Boné theorem. Which says that for each such metric, the integral or the volume of the surface is equal to 2 pi times the Euler characteristic, the absolute value of the Euler characteristic. So it's a fixed number. And this says a kind of very easy consequence that for all x and s. So now I think of it as equipped with such a metric. The injectivity radius at x is less or equal to some constant times logarithm of G. But does this come from the fact that if you have a disk in the hyperbolic plane of a radius r, then the volume is roughly e to the power r. It's not quite correct, but up to some constant. And so if we had a large injectivity radius around a point, the volume would be very large. But the volume is not very large because the volume is uniformly bounded. Of course, we can have points that the injectivity radius is arbitrarily small. We can produce hyperbolic metrics on any of these surfaces such that the system, that means the length of the shortest closed geodesic is arbitrarily small. That's not a problem at all. So now, all these sort of things related, we can look at the rank of the fundamental group of s. What is this? This is a minimal number of generators of pi 1s. That's a number, and this is actually the number that we can easily compute. It's 2G. So what we can say is that the injectivity radius here point wise is bounded from above by a function which only depends on the rank. That's a kind of very satisfactory result if you look at understanding the geometry of the surfaces. And you immediately observe that there's nothing much to do with having a hyperbolic metric here. We could have a negatively curved, pinched metric, and we would get not the same constant, but we would get very similar statements here. So there's no problem with that. Let me give you just to make an advertisement for my student, Bram Beatri, give you a kind of nice result that he proved last year. So we can have actually, this is sort of sharp in a way, a strong sense, there exists a sequence of surfaces, hyperbolic surfaces, GI of genus GI going to infinity, such that s, say si, GI has a systolic pen composition. So what is the systolic pen composition? Pense decomposition is something that I might wish to draw on the right hand side here. So you decompose your surface into pairs of pens, a pair of pens is just, for example, like this. A pair of pens is just a three-fold sphere, and now you can decompose your surface into such three-fold spheres. And now what he proved is that there's a pencil composition of this particular surface, which is systolic, that means each of these pens curves from the decomposition has, the length is equal to the smallest length of any closed geodesic on the surface, so the metric is hyperbolic, and the length of each of these curved systols is bigger equal to 4 over 3 times the logarithm of GI minus some constant. So in other words, you can produce things where, I mean, you have really have your sharp estimate, I showed you here the point-wise injectivity radius is less or equal than C times log G, and now he produced something where you actually have a pencil composition and all these pens, these curves here are systolic, so that means injectivity radius on each of these curves is what you get from the number, and you have plenty of them. So no, no, it came out of the proof. Yeah, it's just a number. I mean here, so it's, I mean, it has to do with the construction, but I mean, you may or may not be able to improve this, I don't know. So now the question that arises is, what about three manifolds? Now we know that due to the work of Paramand there is geometrization. So what we know in particular is that if M is closed, again my manifolds will be oriented for simplicity, so I don't write this down as closed, three manifolds, what, a spherical, meaning that the second fundamental group of M is trivial and a to order, meaning that so that their pi 1M, so I'm slightly informal here, does not contain Z cross Z, then Paramand showed, so this is one of his consequences of his work, so this statement is weaker than geometrization, but it comes out of Paramand's work on geometrization that M admits a hyperbolic metric with curvature minus 1. So now, most of the rigidity as Bruce explained to you in the last talk, most of the rigidity implies that this metric is unique up to isometry, we are now in 3D, so we're not in the two-dimensional case anymore, but of course the diffeomorphism group acts on this metric and so it looks, if you try to describe this metric, it's maybe slightly complicated due to the fact that the diffeomorphism group acts, but up to isometry this metric is unique. Now once we have understood this, we somehow can ask now, we find questions as you see here that are known for surfaces, so one of the refined questions would be that was a question that was raised by McMullen actually in the early 90s, so the conjecture by McMullen says, let's M be closed, hyperbolic, that means, and again I mean this has a friend, this conjecture has a friend, so this conjecture was stated before geometrization, so now in the view of geometrization, this conjecture should have a friend for variable negative curvature and it says if for all k bigger equal than 2, there exists a constant u of k bigger than 0, such that if the rank of pi 1 of M is lesser equal than k, rank in this sense, so you look at the minimal number of generators of the fundamental group, then the injectivity radius of x is lesser equal than c of k for all x, so meaning that although there are definitely examples with arbitrarily small injectivity radius at some point, we require that at every point, the injectivity radius is bounded from above, so and again if this were true due to geometrization, we assume that the same thing is true also for variable negative curvature provided that you have some pinching assumption. Now this would shed a light on the geometry of those three manifolds which would lead to a better understanding, let me now give you some overview of what's known towards this conjecture. So first of all there is a C-O by white, unfortunately lots of people whose name is white, so and I forgot this first, I never met this person, he's a student of McCullough, so he did in the early 2000s, he proved that for all such m the axis, the axis x and m such that the injectivity radius at x is lesser equal than say c prime of k, so the global injectivity radius for any such manifold whose rank of the fundamental group is lesser equal than k, it can be bounded from above by a uniform number, but of course this is very far away from saying that the point was injectivity radius is bounded, that's much much stronger. So there is a serum which somehow solves the conjecture which is due to a Birner and Zutow and they show the proof the conjecture, so this is 2017, this is maybe 2005, this is 2017, that proves the conjecture for manifolds with injectivity radius at x bigger equal, some constant a nu bigger than 0, nu is universal plus some small additional assumptions. I confess that I did not read this paper, this paper is available on the archive and it's 170 pages long, so it's a very difficult technical work, but I mean that proves this conjecture under these assumptions. So now let me give you a sort of weaker form of this conjecture of McMullen that is maybe geometrically more accessible than which I will discuss in the for the remainder of my talk, so the rank is controlled by something else which is called the Higa genus and let me explain what that is. So here's the fact, any clues, and this is easy to prove, this has various ways to do it, I mean you take it for, if you believe that manifolds can be triangulated, just take a triangulation and you take two polyneboids of one skeleton, you get going, that's an easy result, any closed stream manifold can be glued from two handle bodies with specific amorphous of the boundary. So what does that mean, so I in some sense draw already here a picture of a handle body, so handle body is a neighborhood of a bouquet of G circles in R3, so you think of it as this thing here, this surface you built, filled in the interior, now you can take a point here and you draw a bouquet of in this case we circled and you flatten this up and you get this handle body as we draw here, and now what's assumed this is, if you take a closed stream manifold then you can find some G, so the genus of the handle body is the genus of the boundary surface, such that you obtain this stream manifold by taking the boundary surface here, taking another copy and gluing it with a different morphism. So now the smallest G, smallest genus, such that this is possible, that means such that we can glue our manifold from these two handle bodies of genus G, it's called the HECA genus. Now the HECA genus, understanding the HECA genus is a bit more friendly if it comes to understanding, trying to understand the topology of such a manifold, so what we know is that if the HECA genus is of M is less than equal to G than the rank of pi 1 of M is less than equal to G, and this comes from the fact that the fundamental group of such a handle body is a free group into NG generators. That's an easy statement, but now if you replace in McMullan's conjecture the rank by the HECA genus and it becomes a little easier. So let me however state that the rank is not equal to the HECA genus because there's a theorem by Tau Li, it was conjectured to be the same by Waldhausen but this is false, there is a theorem by Tau Li such that the HECA genus of M minus the rank of pi 1 of M can be arbitrary in the order. So some care has to be taken into account. So now if you think about however using the HECA genus to understand our manifold then it reduces to understanding how the gluing works and the fact is that the manifold only depends on, so as a topological manifold depends on a differentiable manifold on the element of the gluing map or the class of the gluing map. In what Bruce already introduced it was a mapping class group, mod S, so this is a group of different morphisms of the boundary surface S divided out by the different morphisms which are isotopic to the identity. So in other words if you take two different morphisms that are isotopic you get the same manifold back and actually it depends on much less on a double core set but this will not be important. The reason being that an element if you have a different morphism of the boundary surface which extends to the three manifold then we are actually in the setup of Bruce's theorem that we deal with pulling everything back by different morphisms of the three manifold and so this is not what we will be interested in. So now the mapping class group here, this is a nice group, it's finally presented, it's finally generated, was proven by Lickorich finding sort of recovering of an old theorem by Dain and Nielsen and then it's finally presented, this was initiated by Hedges-Resson the same Hedges as before. And then there's work by Weinrub, so this was done in the early 80s, everything was done in the early 80s, Lickorich's earlier. So now what we can do is we can choose now a probability measure nu on mod s. So for the sake of sort of making things simple let's assume that is finally supported and that the support generates mod s. Well for example you could take a simple probability measure which just takes a final generating set and gives each of these generators an equal weight such that the weight sum up to 1 and then you get such a probability measure and then you get a random walk on mod s. And what you can ask now is I mean what is a random manifold which random manifolds in some sense may be easier to study than other manifolds with other properties. Now I have the challenge here that I have to somehow maneuver the blackboards. Alright so what I mean a random walk, so what we do is I mean we have a probability space so the probability to go from the identity to one of my generators is exactly what is prescribed by this measure so the total mass of this generator and then the probability so we go here say from the identity to some point g with probability say epsilon and so now we start our process again so we have this weight epsilon and we go to the next step so we attach a generator here and then we ask ourselves what is the probability to go from the g epsilon to the next element which is of the form g epsilon times s where s is one of these generating things and this will be given according to the weight of this s and then we have to sum up overall ways to get to this point in order to get the probability but now we can ask ourselves what is a random property. Let me define this formulae a property p holds for a random three manifold of rank g say if the probability for p tends to one with s the step length of the random walk tends to infinity so now what does that mean we fix the genus so we fix the genus of this handle body now we look at all elements in this mapping class group and we ask ourselves I mean we produce now using the elements in the mapping class group we produce three manifolds by gluing them together with this corresponding different morphism or in different morphism we are presenting this class so we glue together to this manifold and we ask ourselves what are the expected properties of such a manifold as a step length goes to infinity now I assume which was proven by Dunfield and Zürsten in maybe 2000 about 2000 and it says a random so random three manifold in the sense of rank g so it's not clear that the rank is actually equal to g if you start this process a random three manifold of rank g has rank g so the rank is actually equal to g of the fundamental group and not small not bigger has Hager genus equal to g so the Hager genus is actually this g and not anything smaller so you could have do some sort of cancellation here that produces for example the sphere and the Hager genus of the of the of the sphere is definitely zero in the way you constructed so this is actually the random manifold is actually Hager genus g and and it is hyperbolic so now let me state a theorem that I proved in joint work with my student W. L. Ibiachi so what we showed is that a random so we cannot prove McMullan's conjecture but we can prove the following so a random for a random sorry for a random three manifold of rank g McMullan's theorem holds two and we showed something else which is related to some other quantity that you can re-emit the quantities that you associate to such a manifold and maybe show that for random three manifold so McMullan's conjecture holds two meaning that there is a constant only depending on the rank here that a random three manifold has satisfies McMullan's conjecture for this particular constant and for random three manifold we have that of rank g in the sense we have that lambda one of m the first eigenvalue of m is contained in and now comes a uniform constant b over the volume of m square and a constant depending on g over the volume of m square so we somehow understand at least part of the spectrum that means we understand the smallest eigenvalue for such a manifold. Now let me add some remarks on this lambda one of m so where does this estimate come from this comes from an old theorem of which chain but he proved in I have to check the date for an 82 but he proved is that regardless for all closed hyperbolic three manifolds over under some suitable curvature conditions so if say n let me give a different symbol so make sure that we don't we are not in the realm of hyperbolic manifolds anymore but n of uniformly bounded curvature uniformly bounded curvature closed say dimension I mean this theorem works for all dimensions but we have looked at dimensions three so the constants depend on the dimension and the curvature then lambda one of m is bigger equal than some constant b and again this constant depends on the data that I was discussing so what is so b is so in the case of their pinch negative curvature b is actually explicit that's nice now this was generalized to finite volume benefits of non-prose of negative curvature per random and let's see Randall sorry and that joke in 86 so let me discuss briefly what is known for the upper bound so here what we say is for random three manifold this is essentially shop so this is essentially what we what you expect of course is c of g is not really explicit unfortunately but I mean this is all what we can say let me discuss a little bit where upper bounds for the first eigenvalue come from because it will somehow show you how things are a little bit related so the first upper bound comes from the celebrated Chica constant so what is this is this is something that is kind of very easy to write down which makes it so attractive so the Chica constant is the infimum probably all I've seen this before over all set e so submanifold e that are hyper surfaces dividing my manifold into two pieces and now we look at the volume or the surface area if in our case it will be a two dimensional manifold then we can look at the surface area of this e divided over the minimum over the volume of a and volume of b where these are the two components that I obtained by cutting along e so in other words I have a disjoint union of b is equal to m minus e this is a way to write it down so this is a celebrated Chica constant and now what the Chica inequality says at lambda okay oh yeah that was funny so the Chica constant is nowadays is so very celebrated I'm very sort of honored that now Jeff was just ended up so lambda so the Chica constant this lambda one is bigger equal to an h2s square over m over 4 alright and so this gives a lower bound for lambda one this looks the same as here but I mean now we also have under some curvature assumptions and dimension restrictions we have an upper bound that comes from the work of Boozer and Boozer showed that lambda one of m is lesser equal and this constant is now dependent on the dimension and the curvature but I mean here everything is bounded lesser equal than b times h of m plus h square of m so this gives and so this was proven in the 80s this gives an upper bound for lambda one of m and let me now sort of quote a theorem that allows us to use the upper bound oops now this was to use the upper bound coming from the Chica constant to give us in the case we are interested in to give us a bound on the on lambda one and this is work of Mark Lackenby so Mark Lackenby proved in 2006 the following beautiful theorem so it's very explicit and he showed that this Chica constant of m so now we are look at m a closed hyperbolic sphere manifold so this is lesser equal than 4 pi times and so in order to formulate it in the sharp form I don't use a Hager genus I use a Hager Euler characteristic so this is given by 2 Hager genus minus 2 and you divide out over the volume of m and now if you put this together you get immediately that lambda one of m is contained in universal constant b over volume of m square and some constant over the volume of m and this is always holds true in the case of hyperbolic sphere manifolds using this theorem of Lackenby and what we saw before now this turned out to be conjecturally somehow not sharp in our case the upper bound the lower bound is as I showed you up to constant but conjecturally so he's a conjecture if the bank so let G be the bank of pi one of m then lambda one of m should be less than constant over the volume of m and now comes an explicit exponent 2 to the power 2G minus 2 over 2 to the power 2G minus 2 minus 1 that looks a bit funny so the exponent is 2 to the power 2G minus 2 over 2 to the power 2G minus 1 sorry 2G minus 2 minus 1 so this is a little smaller this one so you have a little higher exponent so this is a conjecture that this upper bound should be okay and this is essentially is this true and sharp for manifolds which fiber so hyperbolic sphere manifolds which fiber over the circle and I guess that the same should hold true in general but I mean we are very far away from from proving this so this is due to myself and during work with Hungold Beik and Ilya Gekhtman and independently due to Lensen and Zutow so that's that's a picture and now let me spend the last maybe 10 minutes or so giving you some idea on how to prove the theorem that I wrote down for the random three manifold and a little bit what goes what kind of tools go in so the tools is actually a combination of things that were discovered somehow after Pellman and and and tools that were developed before Pellman's work and the Ricci flow does not enter so Pellman does not enter so what the starting point is the starting point is foundational work of Jaya Minsky and then collaboration of Jaya Minsky together with Jeff Brock and Dick Canary and so what Jaya proved is this is an amazing film and then so Brock Canary Minsky so they proved the following so for M closed hyperbolic as before I don't write this down for the sake of time saving and fibring over the circle so if I go for closed manifold which fibers over the circle then this admits an infinite cyclic covering whose fundamental group is the fundamental group of a closed surface so m hat the infinite cyclic cover so pi 1 of m hat is equal to pi 1 of some surface as G of some C genus the existing explicit model manifold so that means a manifold that you can write down by hand where you can write down the geometry by hand explicitly and the curvature is a bounded curvature the curvature bounds to depend on the G so G is fixed a bounded curvature negative curvature this can be arranged and such that this model metric is by lipchitz uniformly by lipchitz to the metric the hyperbolic metric of course is model manifold satisfies McBrown's conjecture that's something amazing I mean so we discussed I discussed at the beginning of this lecture how you can describe surfaces explicitly so that's in the case of surface is no big deal but here we can write down for those particular manifolds we can write down explicitly how the geometry looks like that we recall also a theorem, a celebrated theorem of Egan Egel, he proved that any m so m closed hyperbolic admits a final cover fibering over the circle now you might think that you are set with these two results but you are not very far from being set simply because the degree of this cover cannot be controlled so I mean once you cannot control the degree the injectivity radius is going to grow up, blow up and so we don't know we cannot argue using the injectivity radius but now so the goal is somehow to try to understand somehow how you obtain an arbitrary manifold using the information or at least a random manifold using the information that you get from this theorem now what the idea is the idea is very simple and then comes some tools that were sort of paper element which I am going to explain to you so the idea for random three manifold and let me now give you the idea for the control on lambda one because this is a little easier, the idea for random three manifold is somehow you have information on the gluing map you don't have information on the geometry but you have information on the gluing map and now how you would like to think about the random three manifold is as follows so you have your two handle bodies and you glue them together with an element in this mapping class group which is very large let me do a picture here you have a handle body now what you think of it this is sort of a precise piece here you have a handle body and now just picture what you think of as if we didn't glue them directly together along the boundary but what we fit in is a large collar and this collar will be the surface a boundary surface cause some enormous interval say minus kk and now you think of this manifold being decomposed into three pieces you have these pieces this piece that is I mean different morphic to s-quasers interval you have here the bottom and the top you have here handle body where you know nothing much about and now you try to use mince case model for this piece somehow you think of this piece as being part of this manifold or piece of the manifold that you obtain by taking a manifold fibering over the circle by taking this infinitesimally cover and you like to import this piece somehow into such a manifold this is actually what you can do in the for the random manifold in the following sense you can identify gluing regions where you produce the following picture so it's possible to decompose this manifold m into three components say v1 v0 v1 v2 and these components are as follows so v0 is stiff morphic to a handle body v1 is stiff morphic to the surface cross say an interval let me take an open interval and v2 is stiff morphic to handle body now in addition in this case of so you use some information on the random manifold to do this in such a way that you can equip these pieces with some hyperbolic metric suitable and then you glue the pieces so the metrics on the pieces glue them to a along the overlap so you think of it as these pieces overlap these are open submanifold they overlap somehow along v0 intersected v1 and v1 intersected v2 we will have v1 v0 and v1 v2 which is joined we glue them along this intersection components to metrics of bounded negative gravitation now we don't quite know what's going on on v0 but on v1 but we can produce this model geometry on v1 where we have a lot of control the model geometry is imported from the work of of Jaj Minsky and his collaborators so we can arrange we can assure that a large piece a large portion of v1 is uniformly by lipchits embedded in the hyperbolic manifold so what does it mean is we have this metric which has bounded curvature and we have this this large region where we have essential perfect perfect control and we can assure that this large region so we have basically the identity map which maps this region into the hyperbolic manifold this original manifold has a hyperbolic metric this is a uniform by lipchits map so it's a different geometry is different but nevertheless we have a uniform by lipchits map in between and now we could use the control on this v1 if we knew somehow that we have some control on v0 and v2 as well and this is a lot more difficult because this work of Minsky and his collaborators is not valid in this case but we however can assure also that we can assure for the metric of verbal curvature the volume of v0 union v1 v2 sorry is much much less than the volume of v1 so almost all the volume is eaten up by this region where we have control but now what we are going to do so what comes now what we like to say is this geometry here is not too far away from the geometry that we are aiming at namely the hyperbolic geometry so I the entire summer I stared at this problem and try to use a Ricci flow to get this settled and I never succeeded I asked everyone that I knew who works on the Ricci flow to help me out and I didn't get any answer and then eventually I remembered that there was another fantastic tool and that's called the BCG map all right and I'm very happy now to say that this was to my came to my rescue the work of Besant Kutva and Galovic states is this pre-Perelman so what that proof is that now we are in a situation where we can apply it so what that proof is and so this is a some souped up version of their work is that in the case at hand there exists a map now we are talking about the same underlying manifolds but the geometry is different between the variable curvature and here comes to the constant curvature and this essentially so I mean I'm slightly cheating because I'm running out of time it's essentially decreasing volume it's a point wise decreasing volume essentially so point wise it's not literally true but I mean so let me cheat a little bit point wise decreasing volume and now we use this and so now we analyze this map what it does on the part that we have a perfect control and we realize there are nothing much happens so the volume essentially remains the same this uses precise information on this piece in the middle and on this on this piece is here v0 and v1 where we do not have much control it decreases the volume so now we use a volume information to get lambda 1 going very easily so this allows you to show to deduce that the volume of this v0 and union v1 for the hyperbolic metric is not bigger not bigger is not really true there is a small constant but this is I mean we had arranged this for the variable curvature metric that is volume of this union is much much smaller it's not bigger than for the variable curvature metric now even more this this approach by Besson-Court-Var and Galou is so amazing that it can be souped up improved so I don't use exactly what they do but I use a slight variation of what they do and it can be souped up to also prove McMullan's conjecture and the situation where I have to analyze a little bit what these great minds did in their work and then using my souped up version that get me going so that's my time is over thank you very much I'm not sure it makes much sense but could one hope to get some average value in this Ago theorem you quoted and could one hope this number to be one then get a proof you want here's a theorem that we can prove we meaning me and seven of those are paper that I wrote with seven other courses one would name all of them but so what we what you proved is that for the random three so you fix a number any number of ten thousand for the random three manifold you will not find a cover of degree ten thousand which does a job so you have to go a higher degree yeah so this I mean all this can be sort of made a quite a so this isn't the sort of the power of random method that you can sure make sure that you don't try to do something that's not going to work if there are any geometrical understanding about the bone on the run could you get in the case sorry what you get these C of K yes around if there are any geometrical understanding of these concepts yeah I mean you can basically you this can be given an explicit interpretation actually the C of K can be more or less calculated more or less explicitly so you have to do a lot of refined analysis depends a little bit on the random model however but you have to visit the refined analysis has to do with dynamics of the mapping class group actually that can be made quite explicit okay no more questions so thank you again
 Well, let me thank you for inviting me here.  It's a great honor and pleasure.  I met Marcel in 1981 in fall when I came to Paris as an exchange student, being completely  naive and ignorant, I have to say.  But due to not only Marcel, but the atmosphere that he created around him and his young people.  So when I came, I had some vague idea that maybe geometry was something that I could like.  And after one year, it was kind of clear that I would never escape geometry again.  I was completely captured by and fascinated by the subject, and in particular also by  Marcel's way to discuss mathematics and introduce concrete problems that had difficult, but  perhaps concrete answers.  So I hope I can give you today a little bit of lecture in the spirit of this philosophy  that you might be interested in concrete questions.  There's been a lot of progress ever since, but somehow maybe the progress leads now to  us being able in the hopefully near future to solve problems that were not reachable  back then, but maybe there isn't reach right now.  Let me start with something so very simple related to Bruce talk.  So Bruce said, now that we understood geometrization, we can ask a sort of more advanced questions  that follow up.  And let me just go back to something that is familiar to all of you.  Let's look at a closed surface where somehow you could think of everything is understood.  And for me, everything is oriented.  I don't write this down.  All my manifolds will be oriented.  And let's take a closed surface of genus G bigger equal than 2.  And we all know that this surface has a hyperbolic structure.  That means a constant curvature, a metric, a constant curvature of minus 1.  And furthermore, there exists a 6G minus 6 dimensional parameter space of such metrics.  Of curvature minus 1.  And we in addition have Gauss-Boné theorem.  Which says that for each such metric, the integral or the volume of the surface is equal  to 2 pi times the Euler characteristic, the absolute value of the Euler characteristic.  So it's a fixed number.  And this says a kind of very easy consequence that for all x and s.  So now I think of it as equipped with such a metric.  The injectivity radius at x is less or equal to some constant times logarithm of G.  But does this come from the fact that if you have a disk in the hyperbolic plane of a radius  r, then the volume is roughly e to the power r.  It's not quite correct, but up to some constant.  And so if we had a large injectivity radius around a point, the volume would be very large.  But the volume is not very large because the volume is uniformly bounded.  Of course, we can have points that the injectivity radius is arbitrarily small.  We can produce hyperbolic metrics on any of these surfaces such that the system, that  means the length of the shortest closed geodesic is arbitrarily small.  That's not a problem at all.  So now, all these sort of things related, we can look at the rank of the fundamental  group of s.  What is this?  This is a minimal number of generators of pi 1s.  That's a number, and this is actually the number that we can easily compute.  It's 2G.  So what we can say is that the injectivity radius here point wise is bounded from above  by a function which only depends on the rank.  That's a kind of very satisfactory result if you look at understanding the geometry  of the surfaces.  And you immediately observe that there's nothing much to do with having a hyperbolic  metric here.  We could have a negatively curved, pinched metric, and we would get not the same constant,  but we would get very similar statements here.  So there's no problem with that.  Let me give you just to make an advertisement for my student, Bram Beatri, give you a kind  of nice result that he proved last year.  So we can have actually, this is sort of sharp in a way, a strong sense, there exists a sequence  of surfaces, hyperbolic surfaces, GI of genus GI going to infinity, such that s, say si,  GI has a systolic pen composition.  So what is the systolic pen composition?  Pense decomposition is something that I might wish to draw on the right hand side here.  So you decompose your surface into pairs of pens, a pair of pens is just, for example,  like this.  A pair of pens is just a three-fold sphere, and now you can decompose your surface into  such three-fold spheres.  And now what he proved is that there's a pencil composition of this particular surface,  which is systolic, that means each of these pens curves from the decomposition has, the  length is equal to the smallest length of any closed geodesic on the surface, so the  metric is hyperbolic, and the length of each of these curved systols is bigger equal to  4 over 3 times the logarithm of GI minus some constant.  So in other words, you can produce things where, I mean, you have really have your sharp  estimate, I showed you here the point-wise injectivity radius is less or equal than  C times log G, and now he produced something where you actually have a pencil composition  and all these pens, these curves here are systolic, so that means injectivity radius  on each of these curves is what you get from the number, and you have plenty of them.  So no, no, it came out of the proof.  Yeah, it's just a number.  I mean here, so it's, I mean, it has to do with the construction, but I mean, you may  or may not be able to improve this, I don't know.  So now the question that arises is, what about three manifolds?  Now we know that due to the work of Paramand there is geometrization.  So what we know in particular is that if M is closed, again my manifolds will be oriented  for simplicity, so I don't write this down as closed, three manifolds, what, a spherical,  meaning that the second fundamental group of M is trivial and a to order, meaning that  so that their pi 1M, so I'm slightly informal here, does not contain Z cross Z, then Paramand  showed, so this is one of his consequences of his work, so this statement is weaker than  geometrization, but it comes out of Paramand's work on geometrization that M admits a hyperbolic  metric with curvature minus 1.  So now, most of the rigidity as Bruce explained to you in the last talk, most of the rigidity  implies that this metric is unique up to isometry, we are now in 3D, so we're not in the two-dimensional  case anymore, but of course the diffeomorphism group acts on this metric and so it looks,  if you try to describe this metric, it's maybe slightly complicated due to the fact that the  diffeomorphism group acts, but up to isometry this metric is unique.  Now once we have understood this, we somehow can ask now, we find questions as you see  here that are known for surfaces, so one of the refined questions would be that was a  question that was raised by McMullen actually in the early 90s, so the conjecture by McMullen  says, let's M be closed, hyperbolic, that means, and again I mean this has a friend,  this conjecture has a friend, so this conjecture was stated before geometrization, so now in  the view of geometrization, this conjecture should have a friend for variable negative  curvature and it says if for all k bigger equal than 2, there exists a constant u of  k bigger than 0, such that if the rank of pi 1 of M is lesser equal than k, rank in this  sense, so you look at the minimal number of generators of the fundamental group, then the  injectivity radius of x is lesser equal than c of k for all x, so meaning that although  there are definitely examples with arbitrarily small injectivity radius at some point, we  require that at every point, the injectivity radius is bounded from above, so and again  if this were true due to geometrization, we assume that the same thing is true also for  variable negative curvature provided that you have some pinching assumption.  Now this would shed a light on the geometry of those three manifolds which would lead to  a better understanding, let me now give you some overview of what's known towards this  conjecture.  So first of all there is a C-O by white, unfortunately lots of people whose name is  white, so and I forgot this first, I never met this person, he's a student of McCullough,  so he did in the early 2000s, he proved that for all such m the axis, the axis x and m  such that the injectivity radius at x is lesser equal than say c prime of k, so the global  injectivity radius for any such manifold whose rank of the fundamental group is lesser equal  than k, it can be bounded from above by a uniform number, but of course this is very  far away from saying that the point was injectivity radius is bounded, that's much much stronger.  So there is a serum which somehow solves the conjecture which is due to a Birner and Zutow  and they show the proof the conjecture, so this is 2017, this is maybe 2005, this is  2017, that proves the conjecture for manifolds with injectivity radius at x bigger equal,  some constant a nu bigger than 0, nu is universal plus some small additional assumptions.  I confess that I did not read this paper, this paper is available on the archive and  it's 170 pages long, so it's a very difficult technical work, but I mean that proves this  conjecture under these assumptions.  So now let me give you a sort of weaker form of this conjecture of McMullen that is maybe  geometrically more accessible than which I will discuss in the for the remainder of  my talk, so the rank is controlled by something else which is called the Higa genus and let  me explain what that is.  So here's the fact, any clues, and this is easy to prove, this has various ways to do  it, I mean you take it for, if you believe that manifolds can be triangulated, just take  a triangulation and you take two polyneboids of one skeleton, you get going, that's an  easy result, any closed stream manifold can be glued from two handle bodies with specific  amorphous of the boundary.  So what does that mean, so I in some sense draw already here a picture of a handle body,  so handle body is a neighborhood of a bouquet of G circles in R3, so you think of it as  this thing here, this surface you built, filled in the interior, now you can take a point  here and you draw a bouquet of in this case we circled and you flatten this up and you  get this handle body as we draw here, and now what's assumed this is, if you take a  closed stream manifold then you can find some G, so the genus of the handle body is the  genus of the boundary surface, such that you obtain this stream manifold by taking the  boundary surface here, taking another copy and gluing it with a different morphism.  So now the smallest G, smallest genus, such that this is possible, that means such that  we can glue our manifold from these two handle bodies of genus G, it's called the HECA genus.  Now the HECA genus, understanding the HECA genus is a bit more friendly if it comes to  understanding, trying to understand the topology of such a manifold, so what we know is that  if the HECA genus is of M is less than equal to G than the rank of pi 1 of M is less than  equal to G, and this comes from the fact that the fundamental group of such a handle body  is a free group into NG generators.  That's an easy statement, but now if you replace in McMullan's conjecture the rank  by the HECA genus and it becomes a little easier.  So let me however state that the rank is not equal to the HECA genus because there's a  theorem by Tau Li, it was conjectured to be the same by Waldhausen but this is false,  there is a theorem by Tau Li such that the HECA genus of M minus the rank of pi 1 of  M can be arbitrary in the order.  So some care has to be taken into account.  So now if you think about however using the HECA genus to understand our manifold then  it reduces to understanding how the gluing works and the fact is that the manifold only  depends on, so as a topological manifold depends on a differentiable manifold on the element  of the gluing map or the class of the gluing map.  In what Bruce already introduced it was a mapping class group, mod S, so this is a group  of different morphisms of the boundary surface S divided out by the different morphisms which  are isotopic to the identity.  So in other words if you take two different morphisms that are isotopic you get the same  manifold back and actually it depends on much less on a double core set but this will not  be important.  The reason being that an element if you have a different morphism of the boundary surface  which extends to the three manifold then we are actually in the setup of Bruce's theorem  that we deal with pulling everything back by different morphisms of the three manifold  and so this is not what we will be interested in.  So now the mapping class group here, this is a nice group, it's finally presented, it's  finally generated, was proven by Lickorich finding sort of recovering of an old theorem  by Dain and Nielsen and then it's finally presented, this was initiated by Hedges-Resson  the same Hedges as before.  And then there's work by Weinrub, so this was done in the early 80s, everything was done  in the early 80s, Lickorich's earlier.  So now what we can do is we can choose now a probability measure nu on mod s.  So for the sake of sort of making things simple let's assume that is finally supported  and that the support generates mod s.  Well for example you could take a simple probability measure which just takes a final generating  set and gives each of these generators an equal weight such that the weight sum up to  1 and then you get such a probability measure and then you get a random walk on mod s.  And what you can ask now is I mean what is a random manifold which random manifolds in  some sense may be easier to study than other manifolds with other properties.  Now I have the challenge here that I have to somehow maneuver the blackboards.  Alright so what I mean a random walk, so what we do is I mean we have a probability space  so the probability to go from the identity to one of my generators is exactly what is  prescribed by this measure so the total mass of this generator and then the probability  so we go here say from the identity to some point g with probability say epsilon and so  now we start our process again so we have this weight epsilon and we go to the next  step so we attach a generator here and then we ask ourselves what is the probability to  go from the g epsilon to the next element which is of the form g epsilon times s where  s is one of these generating things and this will be given according to the weight of this  s and then we have to sum up overall ways to get to this point in order to get the probability  but now we can ask ourselves what is a random property.  Let me define this formulae a property p holds for a random three manifold of rank g say  if the probability for p tends to one with s the step length of the random walk  tends to infinity so now what does that mean we fix the genus so we fix the genus of this  handle body now we look at all elements in this mapping class group and we ask ourselves  I mean we produce now using the elements in the mapping class group we produce three  manifolds by gluing them together with this corresponding different morphism or in different  morphism we are presenting this class so we glue together to this manifold and we ask  ourselves what are the expected properties of such a manifold as a step length goes to  infinity now I assume which was proven by Dunfield and Zürsten in maybe 2000 about 2000  and it says a random so random three manifold in the sense of rank g so it's not clear that  the rank is actually equal to g if you start this process a random three manifold of rank  g has rank g so the rank is actually equal to g of the fundamental group and not small  not bigger has Hager genus equal to g so the Hager genus is actually this g and not anything  smaller so you could have do some sort of cancellation here that produces for example  the sphere and the Hager genus of the of the of the sphere is definitely zero in the way  you constructed so this is actually the random manifold is actually Hager genus g and and  it is hyperbolic so now let me state a theorem that I proved in joint work with my student  W. L. Ibiachi so what we showed is that a random so we cannot prove McMullan's conjecture  but we can prove the following so a random for a random sorry for a random three manifold  of rank g McMullan's theorem holds two  and we showed something else which is related to some other quantity that you can re-emit  the quantities that you associate to such a manifold and maybe show that for random  three manifold so McMullan's conjecture holds two meaning that there is a constant only  depending on the rank here that a random three manifold has satisfies McMullan's conjecture  for this particular constant and for random three manifold we have that of rank g in the  sense we have that lambda one of m the first eigenvalue of m is contained in and now comes  a uniform constant b over the volume of m square and a constant depending on g over  the volume of m square so we somehow understand at least part of the spectrum that means we  understand the smallest eigenvalue for such a manifold. Now let me add some remarks on  this lambda one of m so where does this estimate come from this comes from an old theorem of  which chain but he proved in I have to check the date for an 82 but he proved is that regardless  for all closed hyperbolic three manifolds over under some suitable curvature conditions  so if say n let me give a different symbol so make sure that we don't we are not in  the realm of hyperbolic manifolds anymore but n of uniformly bounded curvature uniformly  bounded curvature closed say dimension I mean this theorem works for all dimensions  but we have looked at dimensions three so the constants depend on the dimension and the  curvature then lambda one of m is bigger equal than some constant b and again this constant  depends on the data that I was discussing so what is so b is so in the case of their  pinch negative curvature b is actually explicit that's nice now this was generalized to finite  volume benefits of non-prose of negative curvature per random and let's see Randall  sorry and that joke in 86 so let me discuss briefly what is known for the upper bound so  here what we say is for random three manifold this is essentially shop so this is essentially  what we what you expect of course is c of g is not really explicit unfortunately but  I mean this is all what we can say let me discuss a little bit where upper bounds for  the first eigenvalue come from because it will somehow show you how things are a little  bit related so the first upper bound comes from the celebrated Chica constant so what  is this is this is something that is kind of very easy to write down which makes it  so attractive so the Chica constant is the infimum probably all I've seen this before  over all set e so submanifold e that are hyper surfaces dividing my manifold into two pieces  and now we look at the volume or the surface area if in our case it will be a two dimensional  manifold then we can look at the surface area of this e divided over the minimum over the  volume of a and volume of b where these are the two components that I obtained by cutting  along e so in other words I have a disjoint union of b is equal to m minus e this is a  way to write it down so this is a celebrated Chica constant and now what the Chica inequality  says at lambda okay oh yeah that was funny so the Chica constant is nowadays is so very  celebrated I'm very sort of honored that now Jeff was just ended up so lambda so the Chica  constant this lambda one is bigger equal to an h2s square over m over 4 alright and so  this gives a lower bound for lambda one this looks the same as here but I mean now we also  have under some curvature assumptions and dimension restrictions we have an upper bound  that comes from the work of Boozer and Boozer showed that lambda one of m is lesser equal  and this constant is now dependent on the dimension and the curvature but I mean here  everything is bounded lesser equal than b times h of m plus h square of m so this gives  and so this was proven in the 80s this gives an upper bound for lambda one of m and let  me now sort of quote a theorem that allows us to use the upper bound oops now this was  to use the upper bound coming from the Chica constant to give us in the case we are interested  in to give us a bound on the on lambda one and this is work of Mark Lackenby so Mark  Lackenby proved in 2006 the following beautiful theorem so it's very explicit and he showed  that this Chica constant of m so now we are look at m a closed hyperbolic sphere manifold  so this is lesser equal than 4 pi times and so in order to formulate it in the sharp form  I don't use a Hager genus I use a Hager Euler characteristic so this is given by 2 Hager  genus minus 2 and you divide out over the volume of m and now if you put this together  you get immediately that lambda one of m is contained in universal constant b over volume  of m square and some constant over the volume of m and this is always holds true in the  case of hyperbolic sphere manifolds using this theorem of Lackenby and what we saw before  now this turned out to be conjecturally somehow not sharp in our case the upper bound the  lower bound is as I showed you up to constant but conjecturally so he's a conjecture if  the bank so let G be the bank of pi one of m then lambda one of m should be less than  constant over the volume of m and now comes an explicit exponent 2 to the power 2G minus  2 over 2 to the power 2G minus 2 minus 1 that looks a bit funny so the exponent is 2 to  the power 2G minus 2 over 2 to the power 2G minus 1 sorry 2G minus 2 minus 1 so this is  a little smaller this one so you have a little higher exponent so this is a conjecture that  this upper bound should be okay and this is essentially is this true and sharp for manifolds  which fiber so hyperbolic sphere manifolds which fiber over the circle and I guess that  the same should hold true in general but I mean we are very far away from from proving  this so this is due to myself and during work with Hungold Beik and Ilya Gekhtman and independently  due to Lensen and Zutow so that's that's a picture and now let me spend the last maybe  10 minutes or so giving you some idea on how to prove the theorem that I wrote down for  the random three manifold and a little bit what goes what kind of tools go in so the  tools is actually a combination of things that were discovered somehow after Pellman  and and and tools that were developed before Pellman's work and the Ricci flow does not  enter so Pellman does not enter so what the starting point is the starting point is foundational  work of Jaya Minsky and then collaboration of Jaya Minsky together with Jeff Brock and  Dick Canary and so what Jaya proved is this is an amazing film and then so Brock Canary  Minsky so they proved the following so for M closed hyperbolic as before I don't write  this down for the sake of time saving and fibring over the circle so if I go for closed  manifold which fibers over the circle then this admits an infinite cyclic covering whose  fundamental group is the fundamental group of a closed surface so m hat the infinite  cyclic cover so pi 1 of m hat is equal to pi 1 of some surface as G of some C genus  the existing explicit model manifold so that means a manifold that you can write down by  hand where you can write down the geometry by hand explicitly and the curvature is a  bounded curvature the curvature bounds to depend on the G so G is fixed a bounded curvature  negative curvature this can be arranged and such that this model metric is by lipchitz  uniformly by lipchitz to the metric the hyperbolic metric  of course is model manifold satisfies McBrown's conjecture that's something amazing I mean  so we discussed I discussed at the beginning of this lecture how you can describe surfaces  explicitly so that's in the case of surface is no big deal but here we can write down  for those particular manifolds we can write down explicitly how the geometry looks like  that we recall also a theorem, a celebrated theorem of Egan Egel, he proved that any m  so m closed hyperbolic admits a final cover fibering over the circle  now you might think that you are set with these two results but you are not very far  from being set simply because the degree of this cover cannot be controlled so I mean  once you cannot control the degree the injectivity radius is going to grow up, blow up and so  we don't know we cannot argue using the injectivity radius but now so the goal is somehow to try  to understand somehow how you obtain an arbitrary manifold using the information or at least  a random manifold using the information that you get from this theorem now what the idea  is the idea is very simple and then comes some tools that were sort of paper element  which I am going to explain to you so the idea for random three manifold and let me  now give you the idea for the control on lambda one because this is a little easier, the idea  for random three manifold is somehow you have information on the gluing map you don't have  information on the geometry but you have information on the gluing map and now how you would like  to think about the random three manifold is as follows so you have your two handle bodies  and you glue them together with an element in this mapping class group which is very  large let me do a picture here you have a handle body now what you think of it this  is sort of a precise piece here you have a handle body and now just picture what you  think of as if we didn't glue them directly together along the boundary but what we fit  in is a large collar and this collar will be the surface a boundary surface cause some  enormous interval say minus kk and now you think of this manifold being decomposed into  three pieces you have these pieces this piece that is I mean different morphic to s-quasers  interval you have here the bottom and the top you have here handle body where you know  nothing much about and now you try to use mince case model for this piece somehow you  think of this piece as being part of this manifold or piece of the manifold that you  obtain by taking a manifold fibering over the circle by taking this infinitesimally  cover and you like to import this piece somehow into such a manifold this is actually what  you can do in the for the random manifold in the following sense you can identify gluing  regions where you produce the following picture so it's possible to decompose this manifold  m into three components say v1 v0 v1 v2 and these components are as follows so v0 is stiff  morphic to a handle body v1 is stiff morphic to the surface cross say an interval let me  take an open interval and v2 is stiff morphic to handle body now in addition in this case  of so you use some information on the random manifold to do this in such a way that you  can equip these pieces with some hyperbolic metric suitable and then you glue the pieces  so the metrics on the pieces glue them to a along the overlap so you think of it as  these pieces overlap these are open submanifold they overlap somehow along v0 intersected v1  and v1 intersected v2 we will have v1 v0 and v1 v2 which is joined we glue them along this  intersection components to metrics of bounded negative gravitation now we don't quite know  what's going on on v0 but on v1 but we can produce this model geometry on v1 where we  have a lot of control the model geometry is imported from the work of of Jaj Minsky and  his collaborators so we can arrange we can assure that a large piece a large portion  of v1 is uniformly by lipchits embedded in the hyperbolic manifold so what does it mean  is we have this metric which has bounded curvature and we have this this large region where we  have essential perfect perfect control and we can assure that this large region so we  have basically the identity map which maps this region into the hyperbolic manifold this  original manifold has a hyperbolic metric this is a uniform by lipchits map so it's  a different geometry is different but nevertheless we have a uniform by lipchits map in between  and now we could use the control on this v1 if we knew somehow that we have some control  on v0 and v2 as well and this is a lot more difficult because this work of Minsky and  his collaborators is not valid in this case but we however can assure also that we can  assure for the metric of verbal curvature the volume of v0 union v1 v2 sorry is much  much less than the volume of v1 so almost all the volume is eaten up by this region  where we have control but now what we are going to do so what comes now what we like  to say is this geometry here is not too far away from the geometry that we are aiming  at namely the hyperbolic geometry so I the entire summer I stared at this problem and  try to use a Ricci flow to get this settled and I never succeeded I asked everyone that  I knew who works on the Ricci flow to help me out and I didn't get any answer and then  eventually I remembered that there was another fantastic tool and that's called the BCG map  all right and I'm very happy now to say that this was to my came to my rescue the work  of Besant Kutva and Galovic states is this pre-Perelman so what that proof is that now  we are in a situation where we can apply it so what that proof is and so this is a some  souped up version of their work is that in the case at hand there exists a map now we  are talking about the same underlying manifolds but the geometry is different between the  variable curvature and here comes to the constant curvature  and this essentially so I mean I'm slightly cheating because I'm running out of time it's  essentially decreasing volume it's a point wise decreasing volume essentially so point  wise it's not literally true but I mean so let me cheat a little bit point wise decreasing  volume and now we use this and so now we analyze this map what it does on the part that we  have a perfect control and we realize there are nothing much happens so the volume essentially  remains the same this uses precise information on this piece in the middle and on this on  this piece is here v0 and v1 where we do not have much control it decreases the volume  so now we use a volume information to get lambda 1 going very easily so this allows  you to show to deduce that the volume of this v0 and union v1 for the hyperbolic metric  is not bigger not bigger is not really true there is a small constant but this is I mean  we had arranged this for the variable curvature metric that is volume of this union is much  much smaller it's not bigger than for the variable curvature metric  now even more this this approach by Besson-Court-Var and Galou is so amazing that it can be souped  up improved so I don't use exactly what they do but I use a slight variation of what they  do and it can be souped up to also prove McMullan's conjecture and the situation where I have  to analyze a little bit what these great minds did in their work and then using my souped  up version that get me going so that's my time is over thank you very much  I'm not sure it makes much sense but could one hope to get some average value in this  Ago theorem you quoted and could one hope this number to be one then get a proof you  want here's a theorem that we can prove we meaning me and seven of those are paper that  I wrote with seven other courses one would name all of them but so what we what you proved  is that for the random three so you fix a number any number of ten thousand for the random three  manifold you will not find a cover of degree ten thousand which does a job so you have  to go a higher degree yeah so this I mean all this can be sort of made a quite a so  this isn't the sort of the power of random method that you can sure make sure that you  don't try to do something that's not going to work  if there are any geometrical understanding about the bone on the run could you get in  the case sorry what you get these C of K yes around if there are any geometrical understanding  of these concepts yeah I mean you can basically you this can be given an explicit interpretation  actually the C of K can be more or less calculated more or less explicitly so you have to do a  lot of refined analysis depends a little bit on the random model however but you have to  visit the refined analysis has to do with dynamics of the mapping class group actually  that can be made quite explicit  okay no more questions so thank you again
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Well, let me thank you for inviting me here. It's a great honor and pleasure. I met Marcel in 1981 in fall when I came to Paris as an exchange student, being completely naive and ignorant, I have to say. But due to not only Marcel, but the atmosphere that he created around him and his young people. So when I came, I had some vague idea that maybe geometry was something that I could like. And after one year, it was kind of clear that I would never escape geometry again. I was completely captured by and fascinated by the subject, and in particular also by Marcel's way to discuss mathematics and introduce concrete problems that had difficult, but perhaps concrete answers. So I hope I can give you today a little bit of lecture in the spirit of this philosophy that you might be interested in concrete questions. There's been a lot of progress ever since, but somehow maybe the progress leads now to us being able in the hopefully near future to solve problems that were not reachable back then, but maybe there isn't reach right now. Let me start with something so very simple related to Bruce talk. So Bruce said, now that we understood geometrization, we can ask a sort of more advanced questions that follow up. And let me just go back to something that is familiar to all of you. Let's look at a closed surface where somehow you could think of everything is understood. And for me, everything is oriented. I don't write this down. All my manifolds will be oriented. And let's take a closed surface of genus G bigger equal than 2. And we all know that this surface has a hyperbolic structure. That means a constant curvature, a metric, a constant curvature of minus 1. And furthermore, there exists a 6G minus 6 dimensional parameter space of such metrics. Of curvature minus 1. And we in addition have Gauss-Boné theorem. Which says that for each such metric, the integral or the volume of the surface is equal to 2 pi times the Euler characteristic, the absolute value of the Euler characteristic. So it's a fixed number. And this says a kind of very easy consequence that for all x and s. So now I think of it as equipped with such a metric. The injectivity radius at x is less or equal to some constant times logarithm of G. But does this come from the fact that if you have a disk in the hyperbolic plane of a radius r, then the volume is roughly e to the power r. It's not quite correct, but up to some constant. And so if we had a large injectivity radius around a point, the volume would be very large. But the volume is not very large because the volume is uniformly bounded. Of course, we can have points that the injectivity radius is arbitrarily small. We can produce hyperbolic metrics on any of these surfaces such that the system, that means the length of the shortest closed geodesic is arbitrarily small. That's not a problem at all. So now, all these sort of things related, we can look at the rank of the fundamental group of s. What is this? This is a minimal number of generators of pi 1s. That's a number, and this is actually the number that we can easily compute. It's 2G. So what we can say is that the injectivity radius here point wise is bounded from above by a function which only depends on the rank. That's a kind of very satisfactory result if you look at understanding the geometry of the surfaces. And you immediately observe that there's nothing much to do with having a hyperbolic metric here. We could have a negatively curved, pinched metric, and we would get not the same constant, but we would get very similar statements here. So there's no problem with that. Let me give you just to make an advertisement for my student, Bram Beatri, give you a kind of nice result that he proved last year. So we can have actually, this is sort of sharp in a way, a strong sense, there exists a sequence of surfaces, hyperbolic surfaces, GI of genus GI going to infinity, such that s, say si, GI has a systolic pen composition. So what is the systolic pen composition? Pense decomposition is something that I might wish to draw on the right hand side here. So you decompose your surface into pairs of pens, a pair of pens is just, for example, like this. A pair of pens is just a three-fold sphere, and now you can decompose your surface into such three-fold spheres. And now what he proved is that there's a pencil composition of this particular surface, which is systolic, that means each of these pens curves from the decomposition has, the length is equal to the smallest length of any closed geodesic on the surface, so the metric is hyperbolic, and the length of each of these curved systols is bigger equal to 4 over 3 times the logarithm of GI minus some constant. So in other words, you can produce things where, I mean, you have really have your sharp estimate, I showed you here the point-wise injectivity radius is less or equal than C times log G, and now he produced something where you actually have a pencil composition and all these pens, these curves here are systolic, so that means injectivity radius on each of these curves is what you get from the number, and you have plenty of them. So no, no, it came out of the proof. Yeah, it's just a number. I mean here, so it's, I mean, it has to do with the construction, but I mean, you may or may not be able to improve this, I don't know. So now the question that arises is, what about three manifolds? Now we know that due to the work of Paramand there is geometrization. So what we know in particular is that if M is closed, again my manifolds will be oriented for simplicity, so I don't write this down as closed, three manifolds, what, a spherical, meaning that the second fundamental group of M is trivial and a to order, meaning that so that their pi 1M, so I'm slightly informal here, does not contain Z cross Z, then Paramand showed, so this is one of his consequences of his work, so this statement is weaker than geometrization, but it comes out of Paramand's work on geometrization that M admits a hyperbolic metric with curvature minus 1. So now, most of the rigidity as Bruce explained to you in the last talk, most of the rigidity implies that this metric is unique up to isometry, we are now in 3D, so we're not in the two-dimensional case anymore, but of course the diffeomorphism group acts on this metric and so it looks, if you try to describe this metric, it's maybe slightly complicated due to the fact that the diffeomorphism group acts, but up to isometry this metric is unique. Now once we have understood this, we somehow can ask now, we find questions as you see here that are known for surfaces, so one of the refined questions would be that was a question that was raised by McMullen actually in the early 90s, so the conjecture by McMullen says, let's M be closed, hyperbolic, that means, and again I mean this has a friend, this conjecture has a friend, so this conjecture was stated before geometrization, so now in the view of geometrization, this conjecture should have a friend for variable negative curvature and it says if for all k bigger equal than 2, there exists a constant u of k bigger than 0, such that if the rank of pi 1 of M is lesser equal than k, rank in this sense, so you look at the minimal number of generators of the fundamental group, then the injectivity radius of x is lesser equal than c of k for all x, so meaning that although there are definitely examples with arbitrarily small injectivity radius at some point, we require that at every point, the injectivity radius is bounded from above, so and again if this were true due to geometrization, we assume that the same thing is true also for variable negative curvature provided that you have some pinching assumption. Now this would shed a light on the geometry of those three manifolds which would lead to a better understanding, let me now give you some overview of what's known towards this conjecture. So first of all there is a C-O by white, unfortunately lots of people whose name is white, so and I forgot this first, I never met this person, he's a student of McCullough, so he did in the early 2000s, he proved that for all such m the axis, the axis x and m such that the injectivity radius at x is lesser equal than say c prime of k, so the global injectivity radius for any such manifold whose rank of the fundamental group is lesser equal than k, it can be bounded from above by a uniform number, but of course this is very far away from saying that the point was injectivity radius is bounded, that's much much stronger. So there is a serum which somehow solves the conjecture which is due to a Birner and Zutow and they show the proof the conjecture, so this is 2017, this is maybe 2005, this is 2017, that proves the conjecture for manifolds with injectivity radius at x bigger equal, some constant a nu bigger than 0, nu is universal plus some small additional assumptions. I confess that I did not read this paper, this paper is available on the archive and it's 170 pages long, so it's a very difficult technical work, but I mean that proves this conjecture under these assumptions. So now let me give you a sort of weaker form of this conjecture of McMullen that is maybe geometrically more accessible than which I will discuss in the for the remainder of my talk, so the rank is controlled by something else which is called the Higa genus and let me explain what that is. So here's the fact, any clues, and this is easy to prove, this has various ways to do it, I mean you take it for, if you believe that manifolds can be triangulated, just take a triangulation and you take two polyneboids of one skeleton, you get going, that's an easy result, any closed stream manifold can be glued from two handle bodies with specific amorphous of the boundary. So what does that mean, so I in some sense draw already here a picture of a handle body, so handle body is a neighborhood of a bouquet of G circles in R3, so you think of it as this thing here, this surface you built, filled in the interior, now you can take a point here and you draw a bouquet of in this case we circled and you flatten this up and you get this handle body as we draw here, and now what's assumed this is, if you take a closed stream manifold then you can find some G, so the genus of the handle body is the genus of the boundary surface, such that you obtain this stream manifold by taking the boundary surface here, taking another copy and gluing it with a different morphism. So now the smallest G, smallest genus, such that this is possible, that means such that we can glue our manifold from these two handle bodies of genus G, it's called the HECA genus. Now the HECA genus, understanding the HECA genus is a bit more friendly if it comes to understanding, trying to understand the topology of such a manifold, so what we know is that if the HECA genus is of M is less than equal to G than the rank of pi 1 of M is less than equal to G, and this comes from the fact that the fundamental group of such a handle body is a free group into NG generators. That's an easy statement, but now if you replace in McMullan's conjecture the rank by the HECA genus and it becomes a little easier. So let me however state that the rank is not equal to the HECA genus because there's a theorem by Tau Li, it was conjectured to be the same by Waldhausen but this is false, there is a theorem by Tau Li such that the HECA genus of M minus the rank of pi 1 of M can be arbitrary in the order. So some care has to be taken into account. So now if you think about however using the HECA genus to understand our manifold then it reduces to understanding how the gluing works and the fact is that the manifold only depends on, so as a topological manifold depends on a differentiable manifold on the element of the gluing map or the class of the gluing map. In what Bruce already introduced it was a mapping class group, mod S, so this is a group of different morphisms of the boundary surface S divided out by the different morphisms which are isotopic to the identity. So in other words if you take two different morphisms that are isotopic you get the same manifold back and actually it depends on much less on a double core set but this will not be important. The reason being that an element if you have a different morphism of the boundary surface which extends to the three manifold then we are actually in the setup of Bruce's theorem that we deal with pulling everything back by different morphisms of the three manifold and so this is not what we will be interested in. So now the mapping class group here, this is a nice group, it's finally presented, it's finally generated, was proven by Lickorich finding sort of recovering of an old theorem by Dain and Nielsen and then it's finally presented, this was initiated by Hedges-Resson the same Hedges as before. And then there's work by Weinrub, so this was done in the early 80s, everything was done in the early 80s, Lickorich's earlier. So now what we can do is we can choose now a probability measure nu on mod s. So for the sake of sort of making things simple let's assume that is finally supported and that the support generates mod s. Well for example you could take a simple probability measure which just takes a final generating set and gives each of these generators an equal weight such that the weight sum up to 1 and then you get such a probability measure and then you get a random walk on mod s. And what you can ask now is I mean what is a random manifold which random manifolds in some sense may be easier to study than other manifolds with other properties. Now I have the challenge here that I have to somehow maneuver the blackboards. Alright so what I mean a random walk, so what we do is I mean we have a probability space so the probability to go from the identity to one of my generators is exactly what is prescribed by this measure so the total mass of this generator and then the probability so we go here say from the identity to some point g with probability say epsilon and so now we start our process again so we have this weight epsilon and we go to the next step so we attach a generator here and then we ask ourselves what is the probability to go from the g epsilon to the next element which is of the form g epsilon times s where s is one of these generating things and this will be given according to the weight of this s and then we have to sum up overall ways to get to this point in order to get the probability but now we can ask ourselves what is a random property. Let me define this formulae a property p holds for a random three manifold of rank g say if the probability for p tends to one with s the step length of the random walk tends to infinity so now what does that mean we fix the genus so we fix the genus of this handle body now we look at all elements in this mapping class group and we ask ourselves I mean we produce now using the elements in the mapping class group we produce three manifolds by gluing them together with this corresponding different morphism or in different morphism we are presenting this class so we glue together to this manifold and we ask ourselves what are the expected properties of such a manifold as a step length goes to infinity now I assume which was proven by Dunfield and Zürsten in maybe 2000 about 2000 and it says a random so random three manifold in the sense of rank g so it's not clear that the rank is actually equal to g if you start this process a random three manifold of rank g has rank g so the rank is actually equal to g of the fundamental group and not small not bigger has Hager genus equal to g so the Hager genus is actually this g and not anything smaller so you could have do some sort of cancellation here that produces for example the sphere and the Hager genus of the of the of the sphere is definitely zero in the way you constructed so this is actually the random manifold is actually Hager genus g and and it is hyperbolic so now let me state a theorem that I proved in joint work with my student W. L. Ibiachi so what we showed is that a random so we cannot prove McMullan's conjecture but we can prove the following so a random for a random sorry for a random three manifold of rank g McMullan's theorem holds two and we showed something else which is related to some other quantity that you can re-emit the quantities that you associate to such a manifold and maybe show that for random three manifold so McMullan's conjecture holds two meaning that there is a constant only depending on the rank here that a random three manifold has satisfies McMullan's conjecture for this particular constant and for random three manifold we have that of rank g in the sense we have that lambda one of m the first eigenvalue of m is contained in and now comes a uniform constant b over the volume of m square and a constant depending on g over the volume of m square so we somehow understand at least part of the spectrum that means we understand the smallest eigenvalue for such a manifold. Now let me add some remarks on this lambda one of m so where does this estimate come from this comes from an old theorem of which chain but he proved in I have to check the date for an 82 but he proved is that regardless for all closed hyperbolic three manifolds over under some suitable curvature conditions so if say n let me give a different symbol so make sure that we don't we are not in the realm of hyperbolic manifolds anymore but n of uniformly bounded curvature uniformly bounded curvature closed say dimension I mean this theorem works for all dimensions but we have looked at dimensions three so the constants depend on the dimension and the curvature then lambda one of m is bigger equal than some constant b and again this constant depends on the data that I was discussing so what is so b is so in the case of their pinch negative curvature b is actually explicit that's nice now this was generalized to finite volume benefits of non-prose of negative curvature per random and let's see Randall sorry and that joke in 86 so let me discuss briefly what is known for the upper bound so here what we say is for random three manifold this is essentially shop so this is essentially what we what you expect of course is c of g is not really explicit unfortunately but I mean this is all what we can say let me discuss a little bit where upper bounds for the first eigenvalue come from because it will somehow show you how things are a little bit related so the first upper bound comes from the celebrated Chica constant so what is this is this is something that is kind of very easy to write down which makes it so attractive so the Chica constant is the infimum probably all I've seen this before over all set e so submanifold e that are hyper surfaces dividing my manifold into two pieces and now we look at the volume or the surface area if in our case it will be a two dimensional manifold then we can look at the surface area of this e divided over the minimum over the volume of a and volume of b where these are the two components that I obtained by cutting along e so in other words I have a disjoint union of b is equal to m minus e this is a way to write it down so this is a celebrated Chica constant and now what the Chica inequality says at lambda okay oh yeah that was funny so the Chica constant is nowadays is so very celebrated I'm very sort of honored that now Jeff was just ended up so lambda so the Chica constant this lambda one is bigger equal to an h2s square over m over 4 alright and so this gives a lower bound for lambda one this looks the same as here but I mean now we also have under some curvature assumptions and dimension restrictions we have an upper bound that comes from the work of Boozer and Boozer showed that lambda one of m is lesser equal and this constant is now dependent on the dimension and the curvature but I mean here everything is bounded lesser equal than b times h of m plus h square of m so this gives and so this was proven in the 80s this gives an upper bound for lambda one of m and let me now sort of quote a theorem that allows us to use the upper bound oops now this was to use the upper bound coming from the Chica constant to give us in the case we are interested in to give us a bound on the on lambda one and this is work of Mark Lackenby so Mark Lackenby proved in 2006 the following beautiful theorem so it's very explicit and he showed that this Chica constant of m so now we are look at m a closed hyperbolic sphere manifold so this is lesser equal than 4 pi times and so in order to formulate it in the sharp form I don't use a Hager genus I use a Hager Euler characteristic so this is given by 2 Hager genus minus 2 and you divide out over the volume of m and now if you put this together you get immediately that lambda one of m is contained in universal constant b over volume of m square and some constant over the volume of m and this is always holds true in the case of hyperbolic sphere manifolds using this theorem of Lackenby and what we saw before now this turned out to be conjecturally somehow not sharp in our case the upper bound the lower bound is as I showed you up to constant but conjecturally so he's a conjecture if the bank so let G be the bank of pi one of m then lambda one of m should be less than constant over the volume of m and now comes an explicit exponent 2 to the power 2G minus 2 over 2 to the power 2G minus 2 minus 1 that looks a bit funny so the exponent is 2 to the power 2G minus 2 over 2 to the power 2G minus 1 sorry 2G minus 2 minus 1 so this is a little smaller this one so you have a little higher exponent so this is a conjecture that this upper bound should be okay and this is essentially is this true and sharp for manifolds which fiber so hyperbolic sphere manifolds which fiber over the circle and I guess that the same should hold true in general but I mean we are very far away from from proving this so this is due to myself and during work with Hungold Beik and Ilya Gekhtman and independently due to Lensen and Zutow so that's that's a picture and now let me spend the last maybe 10 minutes or so giving you some idea on how to prove the theorem that I wrote down for the random three manifold and a little bit what goes what kind of tools go in so the tools is actually a combination of things that were discovered somehow after Pellman and and and tools that were developed before Pellman's work and the Ricci flow does not enter so Pellman does not enter so what the starting point is the starting point is foundational work of Jaya Minsky and then collaboration of Jaya Minsky together with Jeff Brock and Dick Canary and so what Jaya proved is this is an amazing film and then so Brock Canary Minsky so they proved the following so for M closed hyperbolic as before I don't write this down for the sake of time saving and fibring over the circle so if I go for closed manifold which fibers over the circle then this admits an infinite cyclic covering whose fundamental group is the fundamental group of a closed surface so m hat the infinite cyclic cover so pi 1 of m hat is equal to pi 1 of some surface as G of some C genus the existing explicit model manifold so that means a manifold that you can write down by hand where you can write down the geometry by hand explicitly and the curvature is a bounded curvature the curvature bounds to depend on the G so G is fixed a bounded curvature negative curvature this can be arranged and such that this model metric is by lipchitz uniformly by lipchitz to the metric the hyperbolic metric of course is model manifold satisfies McBrown's conjecture that's something amazing I mean so we discussed I discussed at the beginning of this lecture how you can describe surfaces explicitly so that's in the case of surface is no big deal but here we can write down for those particular manifolds we can write down explicitly how the geometry looks like that we recall also a theorem, a celebrated theorem of Egan Egel, he proved that any m so m closed hyperbolic admits a final cover fibering over the circle now you might think that you are set with these two results but you are not very far from being set simply because the degree of this cover cannot be controlled so I mean once you cannot control the degree the injectivity radius is going to grow up, blow up and so we don't know we cannot argue using the injectivity radius but now so the goal is somehow to try to understand somehow how you obtain an arbitrary manifold using the information or at least a random manifold using the information that you get from this theorem now what the idea is the idea is very simple and then comes some tools that were sort of paper element which I am going to explain to you so the idea for random three manifold and let me now give you the idea for the control on lambda one because this is a little easier, the idea for random three manifold is somehow you have information on the gluing map you don't have information on the geometry but you have information on the gluing map and now how you would like to think about the random three manifold is as follows so you have your two handle bodies and you glue them together with an element in this mapping class group which is very large let me do a picture here you have a handle body now what you think of it this is sort of a precise piece here you have a handle body and now just picture what you think of as if we didn't glue them directly together along the boundary but what we fit in is a large collar and this collar will be the surface a boundary surface cause some enormous interval say minus kk and now you think of this manifold being decomposed into three pieces you have these pieces this piece that is I mean different morphic to s-quasers interval you have here the bottom and the top you have here handle body where you know nothing much about and now you try to use mince case model for this piece somehow you think of this piece as being part of this manifold or piece of the manifold that you obtain by taking a manifold fibering over the circle by taking this infinitesimally cover and you like to import this piece somehow into such a manifold this is actually what you can do in the for the random manifold in the following sense you can identify gluing regions where you produce the following picture so it's possible to decompose this manifold m into three components say v1 v0 v1 v2 and these components are as follows so v0 is stiff morphic to a handle body v1 is stiff morphic to the surface cross say an interval let me take an open interval and v2 is stiff morphic to handle body now in addition in this case of so you use some information on the random manifold to do this in such a way that you can equip these pieces with some hyperbolic metric suitable and then you glue the pieces so the metrics on the pieces glue them to a along the overlap so you think of it as these pieces overlap these are open submanifold they overlap somehow along v0 intersected v1 and v1 intersected v2 we will have v1 v0 and v1 v2 which is joined we glue them along this intersection components to metrics of bounded negative gravitation now we don't quite know what's going on on v0 but on v1 but we can produce this model geometry on v1 where we have a lot of control the model geometry is imported from the work of of Jaj Minsky and his collaborators so we can arrange we can assure that a large piece a large portion of v1 is uniformly by lipchits embedded in the hyperbolic manifold so what does it mean is we have this metric which has bounded curvature and we have this this large region where we have essential perfect perfect control and we can assure that this large region so we have basically the identity map which maps this region into the hyperbolic manifold this original manifold has a hyperbolic metric this is a uniform by lipchits map so it's a different geometry is different but nevertheless we have a uniform by lipchits map in between and now we could use the control on this v1 if we knew somehow that we have some control on v0 and v2 as well and this is a lot more difficult because this work of Minsky and his collaborators is not valid in this case but we however can assure also that we can assure for the metric of verbal curvature the volume of v0 union v1 v2 sorry is much much less than the volume of v1 so almost all the volume is eaten up by this region where we have control but now what we are going to do so what comes now what we like to say is this geometry here is not too far away from the geometry that we are aiming at namely the hyperbolic geometry so I the entire summer I stared at this problem and try to use a Ricci flow to get this settled and I never succeeded I asked everyone that I knew who works on the Ricci flow to help me out and I didn't get any answer and then eventually I remembered that there was another fantastic tool and that's called the BCG map all right and I'm very happy now to say that this was to my came to my rescue the work of Besant Kutva and Galovic states is this pre-Perelman so what that proof is that now we are in a situation where we can apply it so what that proof is and so this is a some souped up version of their work is that in the case at hand there exists a map now we are talking about the same underlying manifolds but the geometry is different between the variable curvature and here comes to the constant curvature and this essentially so I mean I'm slightly cheating because I'm running out of time it's essentially decreasing volume it's a point wise decreasing volume essentially so point wise it's not literally true but I mean so let me cheat a little bit point wise decreasing volume and now we use this and so now we analyze this map what it does on the part that we have a perfect control and we realize there are nothing much happens so the volume essentially remains the same this uses precise information on this piece in the middle and on this on this piece is here v0 and v1 where we do not have much control it decreases the volume so now we use a volume information to get lambda 1 going very easily so this allows you to show to deduce that the volume of this v0 and union v1 for the hyperbolic metric is not bigger not bigger is not really true there is a small constant but this is I mean we had arranged this for the variable curvature metric that is volume of this union is much much smaller it's not bigger than for the variable curvature metric now even more this this approach by Besson-Court-Var and Galou is so amazing that it can be souped up improved so I don't use exactly what they do but I use a slight variation of what they do and it can be souped up to also prove McMullan's conjecture and the situation where I have to analyze a little bit what these great minds did in their work and then using my souped up version that get me going so that's my time is over thank you very much I'm not sure it makes much sense but could one hope to get some average value in this Ago theorem you quoted and could one hope this number to be one then get a proof you want here's a theorem that we can prove we meaning me and seven of those are paper that I wrote with seven other courses one would name all of them but so what we what you proved is that for the random three so you fix a number any number of ten thousand for the random three manifold you will not find a cover of degree ten thousand which does a job so you have to go a higher degree yeah so this I mean all this can be sort of made a quite a so this isn't the sort of the power of random method that you can sure make sure that you don't try to do something that's not going to work if there are any geometrical understanding about the bone on the run could you get in the case sorry what you get these C of K yes around if there are any geometrical understanding of these concepts yeah I mean you can basically you this can be given an explicit interpretation actually the C of K can be more or less calculated more or less explicitly so you have to do a lot of refined analysis depends a little bit on the random model however but you have to visit the refined analysis has to do with dynamics of the mapping class group actually that can be made quite explicit okay no more questions so thank you again
volume is uniformly bounded. Of course, we can things related, we can look at the rank of the you look at the minimal number of generators of radius of x is lesser equal than c of k for all point, the injectivity radius is bounded from be bounded from above by a uniform number, but of random manifold which random manifolds in some you some idea on how to prove the theorem that I and and tools that were developed before point is the starting point is foundational work fundamental group of a closed surface so m hat