 Thank you very much. Welcome back to this last series of lectures. There will be very few proofs in these lectures. There will be some fairly honest summaries of proofs but not very many accurate proofs. The plan is to try and get to a fairly accurate statement of what's called geometricization theorem during the first hour and during the second hour I'll talk about some constructions and properties of hyperbolic manifolds which are a subclass. So I'll start with this rather mundane observation that was known to 19th century mathematicians that in dimension 2, real dimension 2, every surface s, by every I mean orientable and compact closed, is topologically classified by its genus. The genus is a number such that s is homeomorphic to a connected sum of that many copies of the torus. T2 is the torus and you do connected sum of torus with each other a number of times. You can also compute g from the fundamental group. It's the rank of the pi1 of s if you habilionize it and divide by 2. So that is all there is in the world of surfaces and also it admits a constant curvature metric modeled on one of the following, exactly one, the hyperbolic plane, the Euclidean plane R2, all the sphere. In fact there's just one example here the sphere, one example here the torus and one example in I mean as many examples as genera g are hyperbolic. Okay so the subject of this talk is what can we say in this direction about dimension 3 and of course it's much harder. First there is no such topological classification not nearly. But Poincaré still ventured a kind of a claim in this direction so that's called the Poincaré conjecture. If the fundamental group of your three manifolds, so I'm not going to call it s now but m as in manifold m3 for the dimension is trivial then m3 has to be homeomorphic to the three sphere. Again m is an orientable closed three manifold and as I was saying the a reasonable classification of such three manifolds is let's say hard but here are two statements. First connected sum is still our friend namely the there is a unique unique prime decomposition holds in dimension 3. So prime decomposition means connected sums into things that are no longer connected sums of other things. That is like for integer numbers so amounts. So it exists it's unique and that's already far from easy. You can have it by combining results of kneser in I think the 30s and later results by Milner. So in a sense it's enough to understand prime three manifolds and about prime manifolds here's what we can say a prime three manifold has a unique so that's isotopically unique minimal decomposition by incompressible tori. So those are tori t2 that find themselves injected in them in a way that's also injected at the pi one level. So if there are such tori then you can cut along them and what's left at the end is essentially unique into pieces that are either a toroidal meaning no incompressible t2 any longer or ciphered fiber. I'm going to say now what ciphered fibered manifolds are. This was the first class of three manifolds that can be said to have been well well understood. So ciphered fibered three manifolds were the first well understood class by ciphered in the 40s. Oh and by the way this this second statement about tori is due to jsj that's a very common way of calling it jco. I'm going to use it jco charlen and independently Johansson. I think it's one n double s but don't trust me. Okay so the ciphered fibered manifolds are the ones that are is ciphered fibered. If m comes with a partition into circles such that all but finitely many of these circles lie at the center of a solid torus. So that means that the nearby circles just travel along the same circle and close up at the same time it does are cores of s1 cross b2 the ball across the circle. I'm just thinking it as you have fibered your cylinder into segments and you close up. The remaining ones are the exceptional fibers being modeled on so s and I'm going to draw the picture first. You do the same vibration of the cylinder into segments but you close up with some fraction of a turn. So b2 cross 0 1 modulo x1 is the same as e to the i so I need to really keep track of the rational number that I put here k over n 2 pi x 0 so I have to subdivide it into n segments here and I identify the this point the point number 0 with point number k the color maybe. So as a result the fibers away from the core have to have to travel n times before they close up travel along the cylinder n times while the central fiber is travels only once. The sign does matter the sign of k over n so k is in z modulo nz okay so there is a full classification of those these are fully classified the space of fibers is a two-dimensional thing that's really an orbifold right if you if you travel inside the space of fibers there are two dimensions worth of ways of perturbing failure fiber and sometimes it becomes singular and that means the this two-dimensional space does something weird is in fact an orbifold is a two-dimensional orbifold namely it's modeled on r2 and r2 modulo and order n rotation and order n let's call it n sub s this n sub s being one of the the n that we see here so there's one n sub s for each singular fiber so the orbifold forgets about the k about the numerator but it knows how many times the fibers fold and so the this orbifold kind of forgets part of the data we still have to put back the data in and the the numerator data and there's a whole calculus of these things sometimes a space can have several orbifold realization sorry several cypher fibered realizations and the calculus of these things is well understood so when you cut by the incompressible torii you get something is boundary and is it part of the cypher is the cypher same with the boundary or not this torus here no it's just a local model so i'm just saying that there's an there's a an atlas whose charts would be just torii and with a sorry torii with a compatibility condition when they're good i'm talking about the pieces in this jfj state yes right so so there's there's a a version of all of this with boundary and when the when there's a a torus in the boundary in the jsj business this torus will be itself foliated by circles of the of the cypher vibration um all right uh this two-dimensional orbifold has a solid let's call it o with an orbifold oiler characteristic and there's a way of computing this it's just the oiler characteristic of the underlying surface of o if you forget that the singular points are singular you just have a topological surface in front of you call this well go between bars and you have to do something for each singular fiber namely you subtract 1 minus 1 over ns over the over all the singular fibers so this is a number that that that makes sense to measure it will show up a little bit later in the in the statement of geometrization now here are some examples of cypher fibered spaces it's of course perfectly okay to have no singular fibers like in the surface of genus g cross a circle or maybe the surface of genus g uh then take the unit tangent bundle t1 that's also fine and also you can look at the lens was called a lens space so l sub i put a fraction here lens space that's by definition s3 modulo so s3 seen as the unit sphere in in r4 modulo the action of uh r 2 pi over n rotation matrix uh sorry b let me use a 2 pi over a and r 2 b pi over a where r theta is the rotation matrix right you act on s3 by such a composition of rotations and the quotient is in fact a smooth manifold that comes with a cypher fiber structure in fact with many cypher fiber structures and uh there's a question about this in your exercise sheet now in um in 1980 person proposed so i'm going to uh give it as a definition a theorem and a conjecture i'm going to state all of these i'm going to take me a few blackboards first uh the definition the um a model geometry is a simply connected manifold since we work in three dimensions let's say a three manifold x with a smooth action transitive by a lee group g such that stabilizers g sub x are compact no larger group in the sense of inclusion um acts with compact stabilizers and also the uh there should exist so to rule out some some cases which are not of interest for us there should exist a compact quotient of the form uh gamma under x for gamma in g discreet well by different morphisms take the group and add the compact factor that's really a leaf oh right right that's uh thank you um now conspicuously there is no metric in this uh in this definition and that's that's on purpose namely um you could put a metric on x that's invariant under the action by compactness you can take a metric and then average it by compactness of the stabilizers x carries a gene variant metric possibly not unique so uh for example i'm going to give examples in fact the full list of model geometries in a moment but for example um uh if you take for x the what's called sl2 tilde which is also the unit tangent bundle uh of h2 well universal cover of that um then the the and g yes and g equals the same g equal so this is the x um so what i'm talking about is what if i want to put a metric on this x is how do i measure the the distance between two unit tangent vectors to the hyperbolic plane i want to travel um from this unit tangent vector to that unit tangent vector and let's say that traveling carrying it over along a segment by parallel transport costs me some price in euros proportional to the distance and uh so i add this up along the the path that i choose to travel and uh rotating as i go costs me some other amount of of but this time in dollars canadian dollars i guess and and what matters is the combined cost to define the the distance between the two points how much do i have to pay in the base and how much do i have to pay in terms of rotating my vector and the the actual geodesics will depend on the rate change between euros and dollars so the the geodesics of these metrics do not look the same uh so these are different metrics on the same manifold but for our definition it's still the same model geometry um okay now uh the theorem is that there exist exactly eight exactly eight model geometries uh namely so i'm going to start with the constant curvature one h3 so that this looks very similar to the the two-dimensional classification that we had at the at the beginning h3 r3 s3 um in those cases the dimension of g is always six isometry group of hyperbolic three space is six-dimensional and then there are the product geometries so h2 cross r s2 cross r so these are self-explanatory you let the the group of isometries of h2 act and you also allow yourself to travel along the the second factor so that's a four-dimensional group of isometries there's also the one i gave here sl2 twiddle and something called nil geometry about which i will say more towards the end of this lecture so for those dimension of g equals four and the last one called solve geometry the dimension of g equals three so the two that i have not really explained are the last two and there will be pictures and explanations about them at the end of this lecture if i'm lucky um and also for us a geometric structure on m is a defiomorphism to a space of the form gamma under x for gamma in one of those g's so now the conjecture so this was a conjecture of about 1980 and it was in fact proved is the geometrization theorem my parallel in about 90 2002 so it says the following if m is a oriented compact prime um then the jsj pieces mi of m have um all have finite volume geometric structures so over a gi xi where it may depend on the piece more precisely and i think i think i'm going to use a white blackboard for the more precisely part i mean the what's left when you remove those those tori those subsurface's those are submanifolds with boundaries and i mean yeah i mean i mean the interior you remove the boundaries um more precisely if so let's let's go if pi one of m is finite and just to be fancy i'm going to call this virtually trivial then xi the model equals s3 so you see how this generalizes the poincare conjecture that was here says the if pi one is trivial then x and then mi is the sphere s3 otherwise if pi one of m is virtually cyclic then the model has to be s2 cross r pi one of mi thank you and the corresponding model is called xi if not but it is still virtually abelian then we are are dealing with euclidean geometry so i think i called it r3 if not but it's still virtually nilpotent and we are dealing with the mysterious bill geometry if not that we are still virtually uh solvable then it has to be solved geometry and through five of them if not then the discussion becomes twofold so if there exists an exact sequence defining so the kernel is z the group the sequence defines is a finite index subgroup of pi one of m if mi so if if there virtually exists such a an exact sequence then two cases may arise if mi is compact meaning mi is in fact the only piece whenever i had to cut along some things since i had to remove the boundary that means the it's not not compact so if it's compact um then uh if the exact sequence splits uh then the model space has to be h2 cross r if not xi has to be the sl2 twiddle metric like on like on the unit tangent bundle of a surface if mi is not compact then both geometries actually can live on the on the manifold in question so that means both xi i don't have to choose both xi equal sl2 twiddle and h2 cross r are valid i'm going to say a a little bit more about how this can happen but the remaining case so if there is no such sequence mi was obtained from a compact manifold closed manifold by cutting it along some surfaces so if i had to cut at all then the pieces are not compact i've removed something if i have not then it is still compact so that's really that's really a matter of whether i had to cut or not and the splitting is up to finite index again or or just splitting uh the splitting is an actual splitting here um yeah and when you see both are valid it does mean both at the same time yes i can put two different geometries two different finite volume geometries on the same manifold and i'm going to say how this happens in a after i'm done with the statement so in the remaining case so if there is no such exact sequence in the remaining case which is also the case where um pi one of m well mi is irreducible a toroidal irreducible with pi uh with uh infinite fundamental group then xi is hyperbolic i'm going to add some more uh little things to this uh discussion one is that in those five cases automatically mi is compact namely n equals mi no no cutting uh and the other thing is that observation quotients of models other than h3 and sol are ciphered fibred and ciphered fibred means really well understood in fact if i do this in orange then there's a uh a discussion to be had in these cases the Euler characteristic of the in the the all before characteristic of the base of the ciphered fibred space is positive in the next two is zero in these two it's negative so all all of these six cases are really well classified they are ciphered the remaining case other than hyperbolic the sol case is is also also has a nice property which is okay what is the sol manifold the isometry group the trivial connected component of the isometry group of sol is just you act affinely on r3 by matrices of the form um e to the t e to the minus t in the first two coordinates and the translation should be by t in the third coordinate and here you put whatever you like right so this is uh this is really r semi-direct r2 where r acts on r2 by my compressing and and compressing one direction and stretching the other so the picture to be had in mind is you have let's say a little square in the first two directions and here's a square that's isometric to it looks like a rectangle up here and if i push it down there it looks like a rectangle with the other aspect ratio and everything in the way of horizontal translations is good so that that's a let's say description of your of the isometry group of them of sol and this should convince you hopefully that um mapping torii of an ossoff what's called an ossoff uh endomorphisms automorphisms phi of the torus are sol so what's a mapping torus is you take the torus cross interval and you glue the top to the bottom by an an sl2z identification and by assumption sl2 and the the identification phi is an ossoff meaning um two distinct real eigenvalues right so the way to put to realize such a such a quotient sorry such a mapping torus as a quotient of sol space is to rotate the the eigen directions of phi until they coincide with the vertical i mean with the first two axes the x axis and the y axis then your your lattice your the z2 by which you quotient r2 to get the torus looks like some sort of weird lattice in the horizontal plane right and it looks exactly the same up here after stretching and and compressing by by phi so uh that's an example of a sol of a manifold manifold with a sol geometry and theorem in fact all sol uh manifolds three manifolds are of this form are quotients thereof by uh by groups of order i think it's at most eight so really we have a good understanding of all three manifolds that do not fall into the last h3 class so i'm going to spend the remaining time discussing uh a little bit about the uh the remaining geometries especially nil but before that um another observation if you look at the new unit tangent bundle of genus g surface that's also the quotient of the unit tangent bundle of the hyperbolic plane and this has uh sl2 twiddle geometry as we mentioned sl2 acts on the hyperbolic plane while uh here's another circle bundle over the same surface while uh circle cross sg has h2 or this order r cross h2 h2 cross r geometry and these are different geometries however if i do this with a surface with boundary then as mentioned did the theorem the the the the the two uh collapse in a sense and that's because it's in fact the same bundle however um if i look at the unit tangent bundle of the tricepunched sphere and i'm drawing the tricepunched sphere as a hyperbolic surface uh then this is in fact homeomorphic to the trivial tangent bundle sorry the trivial circle bundle over the same surface because so these are homeomorphic and for the same reason it has both geometries and the reason they are homeomorphic is that um the tricepunched sphere has a field of directions so that's a trivialization of the of the unit tangent model and how do i do this you take a well there are many ways of doing it but basically you comb singularities of a of any field to the to the punctures here's a i guess we could do it that way and if you can find a field of directions with and then without singularities on the surface then that's a trivialization of the tangent bundle and so the it doesn't matter if you look at this circle bundle as the trivial or the tangent bundle it has both so that's okay that's why i'm cross the first s1 cross the surface yes so a a tangent vector on that surface is a tangent vector on the surface but it's also given by just go to a point and say how many radians away from the from the the orange direction you are so that's a trivial just a number at every point okay so that's that explains the phenomenon here now i'm going to say another it's not really a classification but a statement about spherical geometries because there are some questions about this in the exercise sheet so s3 in a sense the the first simplest case is also famously the double cover of s03 the orientation cover of the the group s03 the isometry group or the trivial component of the isometry group of s3 is given by multiplications by unit quaternions on both sides namely you take two unit quaternions you view the three spheres the unit sphere in the algebra quaternions and the way they these act on a on an element of s3 is just for some reason i like to write it like this b times u times a inverse so that gives you a very concrete handle on how to send the three sphere isometrically to itself and in fact for spherical manifolds s3 yeah there's an equivalence pi 1 of m is a billion if and only if it's cyclic if and only if it does not contain minus the identity of s3 so the antipodal map and equivalently it's a lens space so that's the first class of of spherical manifolds that they are and the remaining ones other spherical three manifolds are quotients of so i can write it s3 modulo plus or minus one namely s03 by subgroups so acting by right and left multiplication as above by subgroups that have finite index in fact that have index 1 2 or 3 in some product group gamma prime cross gamma second so you can put a gamma prime in s03 a gamma second in s03 let their product act by right and left multiplication on s03 and this will usually have some some usually not be a free action unless you're in one of those very special situations one of them so one of the two has to be cyclic and the other one could be icosahedron group or tetrahedron group or something now there's an exercise that does not really ask you to show this but with some weaker statements in the exercise sheet and okay i still have some time to draw some pictures of nil geometry which is the only one we don't have pictures for yet so nil geometry i'm going to draw it to use r3 as the model space the action on it will not be by affine transformations it will be by some sort of polynomial transformations and here's the picture isometries well there's an exact sequence the good news is you can write one r the isometries of nil a fiber over the isometries of the plane you view this as acting on r3 with three coordinates x y z and that acts on r2 and these here are translations along the z axis so over here let me draw i'm drawing tangent planes at every point so they are horizontal on the y axis and as i move along the x axis they they start to tilt they are sort of more vertical they get more and more vertical as i move to the right and over here they are more like they go down they flip and everything is invariant under on the z translations so above every isometry of r2 there is a there is a there's an isometry of nil space and the way it acts let let me show you how the stabilizer zero in r3 preserves the red quadric can i use red z equals x y so that's a quadric that looks something like goes down here here it goes up and it goes down on the other side it contains both axes and it's on the positive side when x and y are both the same sign and it's negative the rest of the time and yeah it's this saddle and you have to imagine that as you rotate in in the in the projection you rotate the points in the in the vertical x y coordinates and the vertical lines of nil have to do a little dance by going up and down and up and down along the quadric so the this map gets sent to that map by rotation so sorry this vertical line to the vertical line to the next now what can we say the pi inverse so if we call this i guess pi inverse of r2 translations is generated by yz translations x translations composed with a shear y z shears i guess what i'm trying to say here is you're allowed to translate the whole picture by in the yz plane but if you want to move into the x direction then you have to push down you have to basically send these squares to those squares it kind of bends your your space as you go in and you bend more and more as you move to the left so this description should show if you work it out a little bit carefully this shows two things one is that nil can be identified with one one so the heisenberg group i think you have to put it in this way so the heisenberg group but you have to to add things that are not just multiplications in the heisenberg group those are those are these funny rotations that i described it shows this and that the one one and zero mapping torus has nil geometry right namely you take you take a vertical square here in the yz direction as a fundamental domain for your torus you can identify opposite sides by translations which are which are in the group and then you push this in the x direction and do some shearing you push it by n units and you can do shearing with intensity n and that gives you a mapping torus with this monodromy okay i can stop here thank you