 . . . . . . . . . . . So this is not completely standard terminology, but I think it's becoming quite standard, let me nevertheless do it, so this is a standard boreal space equipped with a major class, so spaces do not come alone, they come with the proper class of morphisms between them, so let me elaborate on this, but before this, let me explain what a map is between a Lebesgue space, which typically I will denote x or y or z, and a standard boreal space, or a boreal space which I will call u. So if x is a Lebesgue space, and now u will be a boreal space, typically standard boreal space, by a map. From x to u, I will regard an equivalent, equivalence class of boreal functions, from x to u, define almost everywhere, and the equivalence again is up to almost everywhere equality. The space of all maps, I will denote it either maps, xu, or sometimes Lxu, those notations are standard, so again these are as usual spaces of equivalence classes, and of course if u is real or it has a field structure, then this becomes a vector space, in fact an algebra in general, not, but maybe let me mention that if x has a specific measure, then this becomes sometimes a metric space because I can integrate the metric, the difference says between functions, typically if x has a finite measure and u has a bounded diameter, then I can integrate the distance between images of maps and get a metric on this, this will become useful shortly. And now morphism or simply a map of Lebesgue spaces is a map between the underline or either the boreal spaces which preserves the measure class. So if I have x to y and here I have the measure mu and here I have phi and here I have the measure nu, sorry, x comma mu, y comma nu, then demand for this to be a morphism is that phi star mu is equivalent to nu. Now we study group actions, so we should, again typically for locally compact groups I will denote, I will use S or T, G and H are preserved for algebraic groups, so not to be confused, gamma usually is a lattice in an algebraic group, S, locally compact second countable group, then S has a standard boreal space structure, that actually holds for any Polish group, and a leg space structure, because it has the harm measure, in fact such groups are maybe, you can think of them as group objects in the category of boreal spaces, but you should notice that in the category of leg spaces there are no group objects, there is no meaning for having an identity for example, points are not visible. Nevertheless I can talk about actions and this is what I'll do, and S, leg space, X is a space and node with an action, so is a leg space set with homomorphism such that the map S times X to X, the action map is a morphism. Excuse me, it's equality, yes, it is written in equality, yes. If you read the brackets, you can write down equivalents if you wish, and just to remind, S acts on X, that's a symbol for action, is ergodic, if, and I gave you several equivalent definitions last time, they still hold, one of them is every S invariant map X to U is essentially constant for every standard boreal space U. That's the usual ergodicity, and now I want to discuss, it will be important for us to have a more, a stronger properties of ergodicity, so let me discuss several of them. So we'll fix an action, I'm defining several properties, it is called doubly ergodic, if the action of S, if the diagonal action of S on X times X is ergodic, maybe I'll mention, I say something about it already, you should note in the category of Lebesgue spaces, unlike the category of boreal spaces or others, there is no product, no categorical product, still we all know what the product of spaces and product measures are. One definition, the action is called metrical ergodic, if, now comes a bunch of parameters, if for every isometric action of S on U, U a separable metric space, usually I can assume it to be also complete, so it is a polished space, but the metric structure now is not implicit, it is important if for any isometric action of S, comma, every map, every S-equivalent, I will just write in the future S-map, S-equivalent map, X to U is constant. Constant means essentially constant by our definition of a map, constant and image that follows is S-fixed, the essential image which is a point is S-fixed, that's metrical ergodic, is the definition clear? The action is called weekly mixing, if for any ergodic PMP action, PMP, PMP means probability, now a measure, not a class. In our discussion of Lebesgue spaces, we have measure classes, sometimes in the class given on the Lebesgue space, there exist an actual invariant measure, maybe a finite measure, then I can always normalize it to be of total mass 1, then it is a probability measure, and this is a property of a Lebesgue action, the existence of such thing. Then if we have such, we call the action PMP, and typically if the action is ergodic by the way, then this probability, the invariant probability measure will be unique, because any two in the same class will be equivalent and the Radonikodim derivative of one with respect to another, both being invariant will be an invariant function into the reals, but such a function must be constant. And this, over a probability space, the integral of this constant must be 1, because both are a probability, so the constant must be 1, so these are the same. So this is just a remark, so ergodic PMP action is some special type of ergodic actions in our consideration. So an action, a Lebesgue action is called weakly mixing, if for every another action, the diagonal action on x times y is ergodic. And lastly 4, okay, this name is maybe stupid, has no compact factors, the action has no compact factors, if for every continuous homomorphism into, a compact group for any age in k closed, compact subgroup, k-mode age then could be equipped with the hard measure, which is an invariant measure here, under the k-action, but now through this homomorphism I have an s-action on k-mode age, for every map x to k-mode age, this is the compact factor of x. And now this is, I guess my notation doesn't show it explicitly, but it's clear from context, this should be a quotient map when this is with the hard measure. We have that age is k, that is this is just a single point. So you see, there are no compact factors, that's the property. So such a thing will be a compact factor of x, and the property is that we cannot find such in a non-trivial manner. So, well, don't claim, this property of this chain of implications, I will explain this, this is very easy, I will do it very briefly, but it is nice and it's a way for us to practice the definitions. So let me briefly explain what implies two. So I have a double ergodic action, I want to show that it is metrically ergodic. Or maybe before proving, sorry, I will make a remark. If x as any of the above, if s by this I mean an action, any of the above, y is a factor of x, by this I mean there exists s-map, no s-morphism, of the big spaces, of the big actions like that, then why has this property, the same property, that easy to see. And now I'll discuss what implies two. First observe, x is not countable, because, is there a question, Greg? So it's dominant, this is more or less by definition of a map for us, a map preserve the measure class. So the measure class on y is the same as the measure class on x, so you may call this dominant. I mean all maps have this property by definition, so you don't see points in y which are not in the image of x. Thank you. So first I want to observe that x cannot be countable, otherwise, then if it was countable, the measure would be atomic. Otherwise, the diagonal in x times x is an invariant subset where if x is not a single point, if x is not a single point, it is not everything and not null. Now the measure is atomic so everything is seen by the measure, if x is countable, invariant subset, hence x is a single term. So this is also x has no countable factors by this remark. So now I'm ready to prove one implies two. If I have a doubly ergodic x, it's not countable, then I'm sketching things, I will not write everything up, you tell me if this is too much for you. Now I'm assuming without writing that I have such a metric map, such a map from x into an isometric action u, while x is being a double ergodic, then I can take the double of the map x times x to u times u and here I have the metric which is defined and it is assumed to be invariant. So this map is invariant hence constant by ergodicity of the action of x on x times x and because here there is no action, this map is s invariant. So constant, let's call it alpha, the constant. You know x has countable factors, unit x has no countable factors? That may be the sentence just before the night. No? I was here, I didn't understand. Thank you. This will come useful in a second. Okay, now I'm assuming x is double ergodic and x has a map to u which is an isometric action and I claim that for every pair of points in x or almost every, I mean you have to deal with this if you want to write down the proof properly, but for almost every pair of points in x, the distance between these points is alpha. Now there are two options. If alpha is zero, then this map x to u is constant and we are done. This is what we wanted to prove. If alpha is not zero, then the image of x is a discrete set and here comes the assumption of separability on u which implies that it is countable, which I argue, I already explained that cannot be. So we are left with the first option, alpha is zero and the map is constant. That's one implies two, two implies three. Now I'm assuming having a metrically ergodic action and I want to explain that it is weekly mixing. So ergodicity is table for product with ergodic PMP actions. So again I will not write all the words, I will just give you the brief of the idea. Our enemy for ergodicity is having an asympt variant map on x times y. I want to show that this map is constant, this element f. The thing is that I can, out of a little f, I can cook another function, I will call it big f, from x to l infinity y. This is Fubini. This map f at x is a function, so at y it will be fx,y. Fubini tells me that I can do that. And now this space is not separable, this is an issue. I should have emphasized that separability is an issue here in this business. This space is not separable, but because I am assuming, and here this assumption is used, because I am assuming the measure is invariant on y, this space is a subspace of l2y. And now you see that I got myself an equivalent map from x into a separable metric space. So Me implies f is constant, so it's something in l2y, but moreover it will be an S invariant, I think I put it in parenthesis here. It will be an S invariant function in l2y, and it's a constant function in the y-parameter. And this means, if you follow the definition, that f is constant. So we have proved that every S invariant function on the double, on x times y is constant, so we have a good escape. As I told you, all the proof, I mean all these property implications are very easy to do, and just have to follow the definitions. So this is why I choose to give you the proof and maybe waste time on that. Okay, now I want to prove 4, I mean I want to assume weekly mixing, and to prove that there is no compact factor. If I had a compact factor, by this remark, I can actually assume that my x, I can assume that x is k mod h itself. And now I will take y to be k with the harm measure. And then I have x times y, k mod h times k. I have a natural map to k mod h. I'll take, I'll write this map, maybe I'll take x, k to k inverse x. This map is k invariant. So unless this space is just a point, I got myself an invariant function on x times y, contradicting ergodicity, which I'm assuming, that's weekly mixing. So this forces k mod h to be a point, and h to be k. Okay, so this is, everything here holds for every, I mean I proved the claim, all the implication holds for every Lebesgue action, but now I want to emphasize or to study the case where x itself is PMP. If x, or if s on x is PMP, then this is like more classical ergodic theory, and then it is well known that all these guys are the same. Let me prove this. I think 3 implies 1 is really very easy. Just take y to be x. Now x could be stand as y by assumption, and if I take y to be x, I'm just getting double ergodicity, and these shows that 1, 2, 3 are all the same. I just need to show that 4, which is implied by these, also implied them all. So I will explain, I will now explain why 4 implies 2, or maybe I will explain why not 2 implies not 4. So I want to assume now that 2 is not satisfied, so I do have some u and some map which is not constant, x to u, and I want to explain that by this I'm getting also a compact factor. Basically I will explain that if you have, if x is PMP, then I push the measure to u and I have a PMP action on u itself, and it is metric, then it must be a compact action. That's the point. And this is basically, I mean, I'm not sure how much I will do now and how much I will live as an exercise, let me see. So without loss of generality, as I said, u carries an ergodic PMP measure, just the measure that I push from x and can assume also, call it mu, mu is fully supported and u is complete, like it as an exercise. I will give you an hint. I claim that then u is compact. Why is u compact? Hint, u is showed that it is totally bounded. Given epsilon, consider, so maybe I'll choose a point u and u, and I will consider a maximal set of these joint balls of the form SU, I think in my note I wrote epsilon over 3 here. So take shifts, take the shifts of u under S. Remember that my measure is fully supported, so balls are all open balls, all carry a positive measure, and it is the same measure if the balls are just shifts of the same ball. So these are balls of positive measure, because my measure is bounded, I cannot have infinitely many of them, so the size of such a set of these joint balls of some positive measure is bounded by one over this measure, so I can choose a maximal thing like that and show that maybe S in some finite set inside S, show that f u is an epsilon net. That's the exercise. With this exercise I will explain I mean, then u is compact, and then Arcella Scully tells me that k, the isometry group of u is compact as well. And now I can consider the action of the isometry group on u, and this is the action of a compact group on a compact metric space. This is a household space, and this map is k invariant, so this map is constant, so the k action on u is transitive, so basically u is, as I told you, a compact factor. And we are now assuming that there aren't any compact factors, then u is trivial under all this reduction that I made, means that the original map I had, x to u, is constant. Did you say that 4 implied diagolic, let's show it for measure polynomial action. Excuse me? Did you say that 4 implied diagolic, I mean, I didn't say it, but is it true that 4 implied diagolic? I guess I'm assuming ergodicity here. If s on x is ergodic plus, thank you Francois, in all other definition ergodicity is implied, here this is an extra to ergodicity. Yes, otherwise I can take a bunch of things with no compact factor and glue them together. Okay, I mean, I guess I can reformulate it in a way that this is not needed, but nevertheless. So, we have all this 1 implies 2 implies 3 implies 4, and in case of a PMP action, then they are all equivalent. And classically, one use the term weekly mixing in this case, which we preserved for this property, which is the usual definition of weekly mixing, and maybe also this property could be thought of as a stable ergodic, not only the action, I mean, if the action of s on x is ergodic, it doesn't mean that when I multiply by another ergodic action, it stays ergodic under weekly mixing assumption, this is the case in the PMP category. Okay, that's about that. And all this is to emphasize the role of metric ergodicity, which is important for us in the PMP and in the non-PMP world. So, let me now discuss metric ergodicity. Let me start by a non-example. If I map s to g and g acts on g mod k, and actually g here, I mean, I said that I preserved g to be our algebraic group, but this is in general. If k is a compact subgroup of whatever group, locally compact group, then on g mod k I have a gene variant metric structure. So, this is not, I mean, this is not Me. Me means metric ergodic. Example, a very classical example is the action now a gamma countable, omega, a probability space, no action, and now I'll take the action of gamma by shift on omega to the gamma, this is the classical Bernoulli action, this is a PMP action, I take the product measure on, I mean this is omega times omega times omega, component indexed by the group gamma, and the action of gamma is by shifting the indices. It is well known to be ergodic action, in fact much more, it is a mixing action. In particular, I claim that it is weakly mixing. Let me explain why is it doubly ergodic. This is very, very simple, omega to the gamma, sometimes I use, by the way, power to the gamma to say fixed point. Here it's maps from gamma to omega, sorry for this, this is standard ambivalence that we all use. X times X is the same by the rules that we learned in high school to omega times gamma, omega to gamma, and again this is some probability space to the power of gamma, so this is ergodic, so this shows that this is ergodic and this is a weakly mixing thing. So this is an example, but the typical class of example that I will work with, let me keep this one, is the, now I want to discuss, to go back and to discuss algebraic oops, moreover classical ones. So now let me go back and assume that G is non-compact K-simple algebraic. I want to have this assumption and to discuss metric ergodicity in this context and to make the following claim. Ah, also, sorry, I will fix a lattice inside G and another non-compact closed subgroup. And the following claim will be important for us, both the action of gamma on G mod H and the action of H on G mod gamma RME. The reasons are somewhat similar, but not entirely the same. So let me explain why this, I will not prove it for you, but this is very classical, I mean the fact that here, this is a PMP action. So here we are in the realm of classical ergodic theory, or more classical ergodic theory, and this action is actually mixing, I decided not to discuss mixing here, I guess most of us know, for most of us this is anyway just a repetition, but nevertheless, I mean there is a theorem by Haomour that says that if G act on X, PMP, then this action is mixing. Again I will not give the definition, but mixing is something which defined usually using unitary representation, it implies ergodicity and it's stable under product, product of mixing actions is again mixing, in particular product of ergodic ones is ergodic, so Haomour theorem tells me that every action of G, every PMP action is weekly mixing, and also I should have said weekly mixing also passes to non-compact outputs. If G act mixingly and H is non-compact, then H act mixingly. So this shows that this action is mixing and weekly mixing. So in particular, since everything is PMP, it is ME, one implies two implies three implies four, equivalent. So this is supposed to explain this, again I did not give the proof, but I'm promising you that this is very classical, and this is, this follows from two facts. First the action of G on G mod H is ME. Okay, this is somewhat less classical, this is not PMP action, I mean it could have been proved by Haomour because it follows the same method, but reference to this is a paper I have with Tzachigelander, it's using the same kind of idea, but also another thing that is a general fact, if maybe I'll go back to neutral letters, if S act on X ME and gamma is a lattice, then also the action of gamma on X is ME. That's a general claim, maybe I'll explain this in one line, but obviously you see that blue and red here implies the white, maybe I'll very, very briefly explain, assume, so here in red I intend to explain the claim I made in red, if X is a gamma, I'll give context, X is an S space now, and it is S ME, I want to show it is gamma ME, so I'm taking a gamma metric space U, and I'm considering a gamma map to gamma metric space, let's call it phi, then it's easy to arrange a map from X to maps from S to U, which are gamma-equivariant, now as follow, big phi of X at S is phi of S X, and now I will not write any further word, I will just say, I mean, okay, you should check that this map here, that the image of X is gamma-equivariant map, with respect to what, to the conjugation, I mean to the gamma action on the left on S and on U. Moreover, this map is S-equivariant with respect to the action on X and the right action of S on S, I will not check it for you, but I mean there are two actions of, two side actions on S, so you should check it. So this is a G map, S map, sorry. Here I have an S structure. When I look at maps from S to U, I forget the S structure because they don't have on U, and I have just gamma structure on the other side, and I'm talking about the covariance with respect to this one, and now on this guy, we do have an invariant metric, because basically as I think I mentioned, I can assume, I can always assume that the metric on U is bounded, just change the metric, force a bound one on the metric as usual, I do it for the sake of integration in a minute, but it doesn't matter any of the fine structure, it doesn't change invariance of the metric by gamma, so I assume it is bounded, and now I can integrate the difference between two maps on the fundamental domain of gamma, and this gives me an invariant metric, or if you wish, if I take two maps from S to U and I take the difference, this difference is gamma invariant on S, so it is defined on S mod gamma. D alpha beta, if alpha and beta are mapped from S to U, D alpha beta is a map from S to zero infinity, so I'm taking the images in U, and then I apply the distance, and this is invariant under gamma, so I can integrate over this space and define a metric on this one, and by S ME, this should be constant, and you read what it means, you gain the fact that it is gamma, I mean the original map phi is gamma constant. Okay, sorry for being brief, but this... It's not possible to use gravity, and you have the blue thing, that H is a mixing even on G or a gamma. Does that mean that gamma cross H for the right and left action is metrically embedded on S? That gamma... You have the action of gamma cross H? Gamma cross H on G? Yes. And this one is metrically embedded, because looking at H, but looking at gamma gives you the other one? This is correct, I mean you can make it a proof, but it's basically the same proof, it requires some words, maybe it's a better explanation. Okay. Am I in time? Maybe now I will give So, so far, all this was a discussion of ergodic theoretical facts and properties. Ah, no, sorry, I have now yet another property that I want to discuss. For moving farther, I'll do it before the break, I will discuss very briefly the property of amenability just to have it on the board so I can use it later on. I'm in the context of an action, of a locally-compact group on a big space X and I want to define its amenability. There are many equivalent definitions, we all know this, there are many equivalent definitions for a group to be amenable and this is some sort of a generalization of this. What we discussed is sometimes called Zimmer amenability. I will give a somewhat different formulation just to have it on the board and then I will move on to a consequence. So, here is a formulation if there exists an S-equivariant conditional expectation L infinity S times X to L infinity X. I will not explain this. I mean, you might know it well, you might know other definition, I will emphasize that sometimes people call baby amenability which is the thing which is it's almost equivalent and this is what counts for us for every, that is going to be a fixed point property for every compact convex S space so I'm having here C which is subset of a topological vector space locally convex topological vector space on which S acts linearly and C preserve the C and C is convex and compact for topology. Typically C will be some subset some convex subset of a unit ball in a dual space or something or maybe weakly compact subset of a Banach space, et cetera, et cetera but we have the category of such things and this is just the thing I'm quantifying over for every such there exists an S-map so now I consider this as a braille space I forget all the fine structure and I'm just asking myself if I can map X in it and so this non-emptiness of this set is amenability or is a version of amenability implied by the classical amenability I'll give a bunch of examples and we'll go to break and after the break we will go back to discuss algebraic representations using these notions and terms that we discussed so examples the action of S on itself left regular action is amenable moreover the action of S on S mod H is amenable if H is amenable I mean this is the special case where H is trivial I mean you see with this notion I mean I don't care much about the notation up there it doesn't of interest for us but this is very easy to achieve if S is the action of S on itself just pick any point in C and take the orbit map if H is amenable and we have an S action on C just consider the H action on C and amenability implies an H fixed point take the orbit map of this H fixed point and gain a map from S mod H in an S experiment manner so these are the basic examples and also maybe I should say also for T in S closed if I restrict the action to T this is amenable I mean this is a general fact not just about this if you restrict an amenable action to a closed subgroup it stays amenable here is a fact which is very important for us for every locally compact second countable group S there exist X which is both amenable and ME and metrically ergodic I don't want to go into the theory but a way to construct a space is using what is called the notion of first person boundary first person boundary will have these properties and in fact a bit more but specifically I mean this is true in general but for classical groups if I combine this remark here with that remark there I'm getting some extra I mean this can never be amenable if gamma is a lattice in a classical group gamma is not amenable but this guy could if H is amenable and this is what I'm writing now example for G as there here gamma lattice H amenable but non-compact closed of course then this specific action is as is too nice somewhat complimentary property amenability and ME this is very important for us I mean this fact give some unknown space one can play with this gives me that space or such a space in a very concrete way in examples I care for so this is all very very useful that's enough about ergodic theory per se for today and after the break we'll take now a break and after the break we'll go back to actions on varieties thank you we are approaching the heart of this course and this is the definition of area algebraic representation of ergodic action so now I fix S again L, L C L C group and still fixed LB action moreover I fix K as I do always and G K group and I connect these two structures that I fixed by a continuous homomorphism from S to G if this S turns to gamma at some point as it is in my note please let me know so this is a fixed data again a data happening in the ergodic theoretical world a data happening in the algebraic geometric world connecting morphism and now I want I mean this is a fixed action but also I want to discuss actions of G algebraic actions and somehow represent X or the group A S acting on X in this algebraic world so and algebraic representation of S on X of this ergodic action maybe I'll write it assume S on X is ergodic I mean this is not necessarily for the definition but this is a typical assumption I will hold I will take an algebraic representation of this ergodic action with respect to rho is a K G variety V so something G act on it K morphically and of course then I denote V the underline the space of point on which I think with the K topology so it's a standard board space in particular with G action that we discussed and studied it's part of the varying data now V and an S equivalent map so I put marks like here to remind us that these are maps as discussed in the previous talk so a map from X to V from this Lebesgue space to this Borel space so in fact it is a class of map V which I'm assuming to be a covariant so I have the S action here the G action here and O in here and I want this to be commutative so this is what I abbreviate sometimes as an area part of the diagram so this is the basic object and having this object I want מופיזם in the category of areas so in fact if you you realize that the area is really a pair V I should have written here boldface V and of course implicitly there is the data of the G action on it and the map V and I assume I have another one U and C C is a K G map U let's bold boldify it V to U usually I will not boldify it just for once for the definition V to U which is S commutative so I have X goes to V goes to U and let me call this one α and I want a map α and again there is a distinction and I will write to emphasize it in writing there is a line here left side left side is ergodic theory right side is algebraic geometry the map α lives in the right side this is let me give it just regular map morphism let me emphasize it by choice of terminology so this is a morphism of varieties α and okay this line doesn't need to be here but you should have it in your mind so this is a morphism between representation usually I mean it's the same thing that we have or imitation of the same thing that we have many times in mathematics and let me give you I will give you some examples yes please S equivalent because here I have I mean it is I mean α should be G equivalent which is automatically if the image of S is a risky dance but I didn't say anything about it but this should be I mean this should be a commutative diagram such that such that this diagram is commutative better said this way let me give you the very basic example that this imitate consider the action of S on S mod T so homogenous action of S now how can I map S mod T S mod T into V we already discussed this in other examples we just take the orbit map of a T fixed point so every map is given by a pair a G-variate V and T fixed point in V and then then I get a map by the orbit by the orbit map based on that T fixed point I want you to notice that in fact if a point is fixed under T then it is fixed under the the risk closure of T so in fact what we get now I'll assume rho of T yes okay so I was just about to write an assumption now assume rho is a risky dance by this I mean that the image of S under rho is a risky dance then oh actually this is not not needed whatever we get a map actually I'm not using the risk density assumption but if we have a T a T bar fixed point then we get a map from G mod rho T Z 2V I mean the orbit map from of this point of the point fixed by the risk closure of T will give me a map from this so maybe let me call V0 this space maybe let me call rho T I can call it H0 so in fact I'm getting a map we get a map we get an area from X which is S mod T to G mod H0 and for every so let me call this maybe C0 F0 for every other F2V we have a morphism do it grammatically I have this one I have another and I just got alpha in fact I'm getting like this a unique such guy and also let me erase this I didn't use it I was confused now this here is basically the definition of initial object in a category observe G mod H0 is an initial object in the category of area S mod T so in a sense well H0 is the risky closure of T and what I'm saying here is basically I gave you a very fancy way to define the risky closure here what now I'm about to to sell is the fact that whenever the action of S on X is ergodic we are always we are all this time under this example title of homogeneous action of S on S mod T right I want to now say that this is general there is a philosophy in ergodic theory for many years yes no I am I am given I am given an algebraic representation V if there is no fixed point there is no algebraic representation I mean it is given I mean there is a correspondence between V's and fixed points and T fixed point inside them and algebraic representations of S mod T so yes please in this case is the inverse image of H the whole T in S sorry the inverse image of H H mod in S is it equality yes okay no sorry not necessarily maybe maybe it could be that the inverse image of H0 under under the map rho will be bigger than than T we will see examples H0 is defined to be the risk closure of the image of T under the given map rho so maybe I'll now take a theorem and then say something this is a very very basic theorem in this business so now let me emphasize I'm assuming S on X is ergodic and this is fixed in fact everything that appears here is fixed emphasizing then there exists an initial but now I'm not longer under the example title I mean this is general there exists an initial object in the category of areas associated so again what I started saying earlier there is a philosophy called philosophy of virtual groups whenever I have a transitive action like there it is given by a subgroup defined up to conjugacy T I mean the action of S on S mod T is given by T and of course it is ergodic it is homogeneous now I'm assuming having an ergodic action which is not necessarily homogeneous I think of it I may think of it as some sort of generalization of a subgroup in the world of locally compact second countable groups so this is sometimes called a virtual subgroup this is an old terminology I think due to MACI that people not use much but it's useful to have it in mind and as we observed before in the homogeneous setting the initial object that we had is just there's a basic closure of a subgroup now I claim that there exists such a generalization of this fact there is an initial object always and I will explain you will see that it is some G mod H0 always and this H0 could be thought maybe I'll add this as part of the theorem of the form X to G mod H0 P0 so the initial object will be a corset space of G and the stabilizing group defined again up to conjugacy could be thought of as there's a basic closure of the virtual group defined X we've seen last time I mean two days ago that in algebraic geometry in action of algebraic groups and algebraic varieties there is no interesting ergodic theory ergodic theory is just ergodic actions are just homogeneous they are just corset spaces so there are no virtual subgroups whenever I have a virtual subgroup there I mean there is a closure of it is a true group and basically this is what this theorem is telling us okay so now if there are no questions I will prove this theorem yes the map phi is where is the definition let me emphasize this is a Borel map almost ever defined so I consider V with the Borel structure defined up to equivalence of almost every equality of maps so this is thank you for making me clarifying it now I will prove this theorem and basically you are familiar with this line of thought that I am about to present I am using the same tricks over and over again I will look so I am looking for such a thing I will look at I will look on all options so I will consider the space of all age kLGB subgroup of G such that there exists an area from X to G mod H all candidates sitting here this is a non-empty collection as usual because G itself is here I always have the one point space and the trivial map so this is a non-empty collection I can again take a minimal element H0 and along with this minimal element I will make a choice of F0 defining the collection I just wanted the existence of such now I am choosing one and as we have seen before in different contexts this minimal element will actually be a least element or at least up to conjugacy and in certain sense this will be also canonical but this is in particular sense that is somewhat less trivial than appears a priori so now we will show this is an initial object by the way sorry for stopping in the middle of the proof but I want to remind you given a category initial object in a category is not uniquely defined two initial objects are naturally isomorphic in a unique way and this is very very important but in many categories that we are used to really we have just one canonical initial object this is not the case here we made some choices and this will become very very useful in the future so I am already emphasizing this fact okay but I should not stop in the middle of the proof and discuss things I should prove this so for this we consider another representation X 2V sorry by use now I am confused as a vanity property no no no no okay so let me emphasize this set here is a set of subgroups this existence of a map is a property of the subgroups I mean it is a way for me to distinguish some subgroups among others I am focusing on this is not a part of the data I am choosing a minimal object this is a very important question thank you for asking it I am just defining the collection of subgroups using it and on the collection of subgroups I have this net reality and I can choose a minimal object but once I did choose it I know the existence of such a thing and I further choose arbitrary of the many possibly options I have okay and I claim that actually and I I mentioned before this is not canonical but any choice will give me an initial object that we will see okay this clarifies things yeah but you have to accept life we make choices in fact I mean here you choose possibly a big set but manageable in a certain way I mean we will see it in the future okay so now I am considering another one and I will sketch a diagram that I hope will convince all that I can do things so now I am about to look at various representation of X I will not write X in the picture I will just write the spaces I am representing to so I was representing to V but actually I can take the diagonal representation into G times now I am I am taking now this I will take phi times phi 0 okay I will not write it not to spoil the diagram and I will have the projection to V so this will be the map phi in here and of course I will have the projection to G mod H0 so I have the map from X to this one but now I recall maybe I should have stated as a lemma but I mean this is now I am about to use a godicity this is the only place I am about to use a godicity this is very important the image of X inside this variety occupies just one single orbit so there is an orbit here G mod H1 now I will get some psi map too alright again this is a godicity this follows from the discussion we had later so in particular now I am getting this map okay now how come I get a map from G mod H1 G mod H0 from this I am getting that H1 is included in H0 up to conjugacy alright I mean if I have this is a G map this is a G map I mean all the maps over here are morphisms of varieties G morphisms of varieties so this is a G map between homogeneous G spaces and I am forced to have inclusions of stabilizers but now this psi over here is an area that I got out of this picture and H0 was supposed to be minimal object in this collection so by minimality of H0 I have that H0 H1 equals H0 and this map over here is actually an isomorphism and from this I can take the map to V so from G mod H0 I can climb up here and end up here I explained why I have such I mean remember this was in another discussion but this diagram represents what an initial object is I just showed that from G mod H0 I always have a map to V for an arbitrary V and in fact this map is unique I leave this as an easy exercise but everything here is G maps and transitivity tells you so we got a map from G mod H0 to V which I maybe I should call alpha which is unique and this is the end of the proof so we proved this claim and as I said this is an important one I here I didn't use the risky density or anything the only thing the only real assumption I had is ergodicity and it was used in choosing this thing here and so this is an important assumption but everything here is rather cheap now I want to give some examples because this discussion is somewhat mysterious okay I mean we have the first example S mod T which I already discussed in detail so then we understand everything and this is the motivating example maybe I'll say it if anyone knows the notion some of us do know the notion of the risky hull of a co-cycle that Zimmer was using in his theory of co-cycles I'm not sure maybe Mackie was using before then all these terms are inspired by this terminology and if you think about ideas there is nothing but these old ideas but it's somehow casted in a different language which is very useful I find okay so this is one example another example assume S on X is PMP probability measure preserving assume father that G is case simple so no normal subgroup no quotients and rho to G unbounded unbounded means that the image is not contained in a combat subgroup then the initial object the initial object is trivial so any rep of X is essentially constant essentially constant and the image is image is G fixed to say that initial object is trivial is to say that no I mean there are no representation which are not constant and these follows I will not write it down but it follows from the classification of measures of of PMP measures on algebraic variety that we discussed earlier we discussed last time we remember recall what I remember we remember that whenever I have an S invariant that was the statement I made last minutes of lecture 2 days ago any S invariant measure I think I used gamma back then every S invariant measure is coming from a pre-compact image of S in a quotient of G by some group N0 remember here I'm assuming there is no N0 is trivial no non-compact sorry I should have written this I'm assuming N0 is trivial so no compact quotient oh I guess this is it follows from unbounded I'm assuming that S as a subgroup of G or the image in G is unbounded so there are no no possible non-trivial images of X maybe on the same vein assume S on X is PMP and weekly mixing so it satisfies properties 1, 2, 3, 4 that we had before now maybe without this extra assumption of simplicity and unboundedness then again I0 is trivial again it follows from the same proof and the fact that part of definition of one of the equivalent definition of weekly mixing is that we don't have compact factors nevertheless let me I want to give you to sketch very briefly another proof of this fact so X weekly mixing and assume I have a map X to V X is PMP remember that if X is PMP in weekly mixing X times Y is also ergodic X times X is ergodic but now I can take X times X as Y so X times X times X is also ergodic so X to the N is always ergodic so I can look at the map from X to N to V to the N ah let me assume assume image I assume some dominance image is Z dense so I'm taking the support of the image and there's a risk closure and restrict instead of taking V I will take just that image as my target so if this is the case then image here is even but because this is ergodic I know that there is some G mod H here that takes this and I get that the dimension of V to the N is smaller than dimension of G mod H which is definitely smaller dimension of G now this is fixed but this goes for every N so I guess this is if you want N times dimension of V so dimension of V is here so this is just a finite set and this weekly mixing thing as I already told you that he doesn't have any countable quotient so he definitely cannot have a finite quotient unless this is just triviality that just one point so this is just a sketch alternative sketch for the following fact weekly mixing actions PMP ones are trivial are trivial from the point I cannot represent it represent them this ergodic strongly ergodic things inside the algebraic geometric algebraic geometric world questions about that okay another example in this list which I like a lot is something that we proved with Alex Thurman a ergodic advice if gamma is a lattice in G maybe I'll state it as a theorem elsewhere I'll give more respect to this example okay usually there is no much I can't tell about those initial object that we have by the theorem I just erased sometimes we do and this is when we have lattices in classical e groups so G say K simple non-compact gamma inside G a lattice I told you always that there is ergodic theoretical sides and algebraic geometric side now I'm about to confuse the two I'm about to put some algebraic data over here H inside G closed not necessarily algebraic closed non-compact subgroup I want to consider the gamma action on G mod H and that will be my X let's say K algebraic just to avoid taking the risk closure in a minute so this is this is a variety by itself but now I want to consider it as my X but I don't take it as a G space this was the example that we discussed before stating the theorem about initial object I'm taking it as a gamma space and I say this doesn't matter much and then then the identity map X to G mod H is the initial object I mean this would be clear if I was discussing this space as a G space and now I claim that this holds for of course rho being the inclusion of gamma inside G let me I will sketch the proof for this let me just illustrate let me give you an example which I like every Vorel map Rn to Rn which commutes with SLNZ is it's a scalar multiplication if I give you Rn and I take a borel map that commutes with all matrices application of all matrices all invertible matrices all SLN matrices it is very easy to see that these are just scalar multiplication I claim that it is enough to take this one and this is an easy application of this formalism and this theorem so I'm trying to say that this thing here has a known entirely trivial context content so let me sketch the proof of this claim so I'm taking G mod H and I have a map to V this is a gamma let's call it V take the projection natural map G to G mod H and consider this one so let me call this big thing the thing is that I want to discuss such maps and to say something about those gamma maps from G mod H to V where V is the G space so this map is gamma equivalent and age invariant on G I want to say that by a simple trick I can reverse the duality that one so I mentioned earlier I can reverse the role or invert the role of gamma and H this is what I'm about to do now so I define now Psi from G to V not Phi but Psi by taking just I wrote it so I will not get confused I'll take G inverse of Phi of G check Psi is gamma invariant on the right and age equivalent for the left action of age on G so actually I'm getting a map from G mod gamma age map so this is just an exercise with group symbolism but now we have age on G mod gamma and this is the PMP action and I already told you this is a weekly mixing PMP action so this map is actually constant if you wish I was just using this example because I actually got a representation P space into V and age representation so this is I mean this is an age representation of this ergodic action into V and it must end up in a fixed point is constant so this means that little V so G inverse Phi G is V and this means that Phi G is G V for every G and G and basically this says that the map Phi it follows that Phi is G is a G map so it's actually the original map was a G map and the image here is a fixed point and you can read out of it in fact this map is the famous map alpha that you can see here and everything is fine okay I was brief here because this is not my main interest but this is just to illustrate that sometimes we can't say what the initial objects are but in general this is a mystery and I'll answer in a minute but my next goal before releasing you for lunch is to say that sometimes we do have some control and In the last item on the launch is a gamma discreetness I use the fact that not discreetness really I use the fact that this is PMP yes but okay I mean if you ask under this assumption there are no ordinary subgroup of finite volume is already discreet so yes but it doesn't matter so much okay what I said so far I said that in the category of algebraic representation of ergodic actions there are initial objects and these guys are useful I said that sometimes under stronger they become trivial and in fact this is the case most of the time very rarely I can say something positive like in this statement over here and say exactly what are the initial objects but now I'm about to tell you that sometimes I can argue for the existence of non-trivial initial object without knowing what it is and this is very very important theorem in this business proposition because this is how it appears in my notes exist initial object the title assume this is amenable and also metacleargodic this is the assumption that I emphasize a pair of assumptions that I emphasize previously and assume g is case simple and rho unbounded so I mean a particular setting then there exists an initial object x to g mod h and I mean the existence we already know but the important fact is that age is a proper sub-woof so it's non-trivial amenability basically together with metacleargodicity buys us non-triviality of representation into algebraic varieties so this is a very transcendental assumption amenability or amenability and ME and somehow it gives us this miracle that we will see later is very useful for further elaborations and just this is the end this is the starter of an engine that we will build that is quite powerful and this theorem in my mind is very mysterious because this is the only place in a sense where transcendental assumptions on the ergodic actions come to play okay now in the remaining time I will give you the proof or sketch the proof so amenability now g is very classical it's a simple group or simple algebraic group so I can look at g mod p and p is a parabolic so this is a compact space and I can look at the probability measures on this compact space and this is again this is a compact convex set on which g act and since g act and we have a map s to g then s act and amenability gives me an s map so this is not an algebraic representation but it's almost it's a representation in a space which resembles an algebraic geometry and I told you that the action on this space is nice orbits are locally closed g orbits are locally closed not s orbits but g orbits are locally closed so I can find g mod a g in here let me call it h1 because this is not the age I'm looking for such that this map fuclos via this is by ergodicity of course plus locally closed locally closed orbits for this action for the g action here now h1 is not necessarily algebraic group but it's fairly close to be one h1 we know that it has it satisfies such a long exact sequence so it has h0 where this is algebraic normal sub of h1 and the quotient is co-compact quotient is co-compact we discussed what is the structure of stabilizers of measures on algebraic varieties we had this so h1 is not algebraic but h0 is algebraic and now various options algebraic and in fact maybe I'll say something this is the fixator of the support of the measure mu h1 is the stabilizer of a measure mu okay now there are various options maybe h1 is trivial but then h1 I claim this is not the case because then h1 is compact this map here and we get a map x to g mod k and this by m e is contradicted so h0 is not trivial I mean here this is the only use of m e metrically ergodic because this is a metric space ah no moreover it's not only m e this is a metric space so what I actually get by m e is that the map is is constant and this constant is s invariant and this means that s is contained in a conjugation of k so here I also use the assumption that unbounded these two assumptions were used here so now h is not e and also it is not g itself right okay this I didn't say let me put it here that's very important and I forgot to say h cannot be g itself if h was g itself then mu is maybe h0 is contained a fixator of the support of mu so h0 is contained in p up to conjugation right I mean it's in particular it fixes a point so it is in particular it cannot be g g cannot stabilize a measure h0 not normal now I'm using the fact that I gave that g is k simple again this is not a general tautological statement I'm using simplicity here so I can this means that h1 is contained in the normalizer of h0 let's call it n which is strict sub of g now h1 is not algebraic necessarily but h0 is so this is algebraic and let me finish the line here so I got x to g mod h1 and h1 is contained in n so I'm getting g mod n and this is maybe I should have called it h because oh no I'll keep it n so this is a non-trivial now I'm not claiming that this is the initial object the initial object is some subgroup h which is contained in n but necessarily a non-trivial and itself is non-trivial this is it so let me summarize because I made many many proofs and things get complicated but what are the essentials the first lecture today was about some ergodic theoretical stuff it was very standard but then we defined what an area is algebraic representation of ergodic actions and we learned that as in the homogeneous case we can always have some sort of a zirisic closure of an ergodic action in the algebraic world so an initial object in the category of such representations and this is of the form g mod h0 and in many classical cases there is no such theory no invariant essentially no invariant measures no PMP measures for algebraic actions so for x which is PMP maybe with assumptions but really no non-trivial algebraic representation every algebraic representation is fixed and this is very important but the theory is not empty because under this very strong assumption of amenability and ME we do have algebraic representations so we do have something to hold on to even though we don't know what it is I mean this is this proof that I gave is abstract it doesn't tell us what are the result but there exists something and moreover any group I told you has such an action amenable and ME so something is going on and we will see how to use it so next week I will explain some functoriality properties of this construction of an initial object so how it preserves symmetries from now on everything more or less is completely tautological, it's just playing with the tools we already studied so next week we will actually get some result and I will prove for you super agility in various context but things will be very very formal given the tools that we already made so no hard proofs I promise .