The Wonderful World of Caster Cube Complexes and Hilbert Spaces

Caster Cube Complexes are a fascinating topic that requires some background knowledge to understand. To begin, we need to clarify what a Caster Cube complex is. A Caster Cube complex is essentially a collection of cubes glued together in a specific way, without any smashing or cutting, to create a unique and intricate object. The key aspect of a Caster Cube complex is that it has zero curvature, meaning there are no "bends" or "corners" in the space.

The process of creating a Caster Cube complex involves taking an edge of the cube and repeatedly cutting across squares and cubes perpendicularly, while ensuring that the procedure does not allow for positive curvature. This means that each cut must be made in such a way that it separates the space into two distinct regions, without allowing for any "bends" or "corners". The magic of Caster geometry forces this procedure to always separate the space into two pieces, making it possible to navigate through the complex without ever cutting myself again.

One of the most interesting examples of a Caster Cube complex is taking a tree and cutting an edge. This will inevitably cut the tree in two, illustrating the concept of walls separating different regions of space. When we have multiple points in our Caster Cube complex, we need to ensure that there are only finitely many walls separating any two points. This is where Hilbert spaces come into play.

A Hilbert space is a type of metric space that has certain properties, including being complete and having a well-defined distance between points. In the context of Caster Cube complexes, Hilbert spaces provide a way to measure the distance between points in the complex. The walls of a Hilbert space are given by the kernel of a linear functional, which means that they are defined by a set of hyperplanes that partition the space into different regions.

The concept of a wall space is closely related to Caster Cube complexes and Hilbert spaces. A wall space is essentially a metric space with a specific type of "wall" separating different regions of the space. The key aspect of a wall space is that it has codimension one, meaning that the walls are defined by a linear functional. In other words, the walls are given by the kernel of a linear transformation.

When we have a wall space, we can use the concept of relative probability to define an invariant measure. This involves taking an almost-invariant vector and using it to construct a representation with almost-invariant properties. The result is that we get an sl3 R-invariant vector, which provides a way to measure the distance between points in the complex.

The relationship between Caster Cube complexes, Hilbert spaces, and wall spaces is more than just superficial. In fact, these concepts are closely linked, and understanding one can provide insight into the others. The concept of relative probability plays a crucial role in establishing this connection, allowing us to translate properties of one space into another.

In particular, when we have a representation with almost-invariant properties, we can use the LMA (Lovasz-Milgram-Arndt theorem) to establish sl3 R-invariance. This theorem provides a powerful tool for understanding the relationship between different types of spaces and their corresponding invariant measures. By using relative probability to construct representations with almost-invariant properties, we can gain insight into the structure of wall spaces and their relationships to Caster Cube complexes.

The study of Caster Cube complexes and Hilbert spaces is an active area of research, and there is still much to be discovered about these fascinating topics. By exploring the connections between these concepts, we can gain a deeper understanding of the underlying mathematics and develop new tools for analyzing complex systems.

In conclusion, the world of Caster Cube complexes and Hilbert spaces is a rich and fascinating one, full of intricate relationships and powerful mathematical tools. Whether you're an experienced mathematician or just starting to explore these topics, there's always something new to learn and discover.

"WEBVTTKind: captionsLanguage: enthank you very much um so oh there's like a am I allowed to go all the way here okay I'm not going here because I want to repeat the advertisement for Thursday evening there's a play you should all come we have fantastic actors it's a CO if you if you want to know it's a comedy it's not a drama or something it should be more or less an hour but it's a comedy it's nothing heavy and hopefully nothing boring hopefully easier to follow than a MTH do okay so um let me well recall that I was talking about property T and uh and uh let me just recall the following are equivalent so TFA means the following are equivalent and I'm going to make here G discret of course G is always a group uh and I'm going to give you a list of equivalent definition of property T um so this is the original one is that the trivial representation is isolated in Jihad um which there is a slight uh reformulation in terms of uh so this is the felt topology that we discussed last time uh any unitary representation with almost invariant vectors has an invariant Vector um there exists s Epsilon where s contained in G compact Epsilon greater than zero such that so okay uh such that any for any representation either you have an invariant vector or for any G in s the displacement of the unitary representation of any unit vector or any Vector is greater than Epsilon so that's must be obviously a nonzero Vector in your hbert space uh there is an equivalent definition which says that uh any finite dimensional representation is isolated in Jihad now in the space of yeah gu okay um and any a fine isometric action on a hbert space has a fixed point and so if you're if you're in a Hilbert space if you find a bounded orbit you're going to find a fix point so that's the same thing to saying that has a bounded orbit because hbert spaces being cat zero if you have a bounded orbit you can find a barer center or your B bounded set and that's going to be your fix point and uh any isometric action on a median space has a bounded orbit but that doesn't necessarily give you a fix point or yeah okay um and given the questions uh that I had I thought that I will uh say a few more words on the fact that if you're a meable and have property T then you're a finite group but before I do that let me just make uh a comment oh yeah first the list of groups with property t uh let me just list of groups with T so the one that were the motivation of kdan are the lesses in G where uh so this is a lattice so you mean that gamma is discret finite volume and uh and G is a higher rank a simple Le group so Le groups uh yeah simple and semi- simple Le groups are Le groups whose Le algebra either simple or semi- simple but the point is that they're classified so there is a list of those so you can theoretically be lazy and not learn the definition because you could learn the list uh so in the list there most of them are Matrix groups like slnr spnr um so you can think think of slnr for n greater or equal to three and the rank so there's an algebraic definition of the rank but there is a geometric definition of the rank which is the largest RN you can isometrically embed in your group so so higher rank or slnr for n greater or equal to three means that you're going to find a flat copy a copy of R2 within your group in particular you cannot be word hyperbolic because you have this flat piece so they're never hyperbolic and uh in the classification of Le group you also have a list of rank one Le group so lank one Le group and the list is s o N1 s N1 s pn1 and F4 minus 20 and uh those are the those are the the isometries of the real hyperbolic space of Dimension n and uh so as so so that that that guy you might have seen already it's a space with constant curvature minus one and if you if you if you take a co- compact lattice you're going to have a nice hyperbolic manifold and uh your your Lattis is going to be a hyperbolic group uh so but if you have a non-compact lattice still has finite volume it's a relatively hyperbolic group meaning that you'll start finding Flats in it but they're very isolated ones from each other so this one is the isometry group of the complex hyperbolic plane uh and uh okay there's I'm not going to really go into the definition I'm going to wave my hands to describe it it's uh uh it's it's a hyperbolic plane that you get it's it's a hyperbolic space those are hyperbolic spaces that you get so they get Dimension 2 N because it's over the complex uh but the thing is that the curvature is not constant negative one anymore it varies a little bit between minus a half and minus one but it's still a very nice uh hyperbolic space and the co- compact ltis is are word hyperbolic groups and this is the isometry group of the hyperbolic space over the querian the Hamilton querian so you know you get C out of making R plus I then you get the querian out of making C plus I C and you know there's some difficulty that's follow for instance it's it looks like a field but it's no longer commutative and uh and if you iterate the procedure uh you get a hyperbolic plane over the octonian but then there's only then uh there there's you can only do one the octonian are no longer associative so so it's it's a very really an exotic thing so this they don't have property T but those have property T I'm not going to give the proof I'm going to explain why those one don't have you're going to see why those ones don't have probability but I'm not going to say anything about why those have probabil except that now you know that you have a nice class of hyperbolic groups with property T and uh what does what do so okay so so so it's good to know that we have a bunch of hyperb groups that can have property t uh but what can you do with hyperbolic groups with property T hyperbolic groups all have this property which is say SQ Universal and uh which means that SQ universality means that uh if I choose a hyperbolic group and if any of you choose the worst possible group you can think of I can produce a quotient of my hyperbolic group in which your horrible group embeds so hyperbolic groups are SQ universal meaning that uh uh any for any gamma hyperbolic any uh any H finally generated uh H embed in a quotient of of of gamma so it's it's pretty clear if you start with the free group you have a finally generated group is a quotient of the free group but so so so as a result as a result you see that any group embeds in a property T group so being a subgroup of a property group T group is no information okay any finaly generated group embeds in a property T group all right so this is really to say that being a quotient uh being a subgroup of a property T group is no information okay and I I find it pretty interesting that the original motivation for uh for kajan was to prove something using property T but then uh most of the other result is you can't prove anything property now there's a few thing you can prove I'll tell you a few more thing you can prove using property T but in many in many uh in many questions we ask oursel geometric group Theory many times propert is telling you that's not going to happen because because this this definition any isometric action on a median space has a bounded orbit it's the same it implies that any isometric action on a cader cube complex has a fixed point so it means that you cannot use any cader cubical geometry to understand the property T group uh okay right so so so so back to the amable case um right so so recall that gamma amable means that for any F containing gamma finite for any Epsilon greater than zero there exists U which depends on on F and Epsilon such that f with u is smaller than Epsilon so um yeah so so this really means that if so f a finite set you can think of when I it is a set of moves in your group so really this means that u u is uh Epsilon moved by F compared to the size of U right and we had the regular representation and let me make a comment about L2 of G so we saw in Genevie talk that L2 of G is a L2 of gamma is a Hilbert space but you might remember that you saw in some class or somebody told you that there are not that many Hilbert spaces there is there is C to the power n and then there's L2 to the power n so why the hell are we saying L2 of gamma and we're changing the gamma because we know that this in fact is L2 of n the reason why we write L2 of gamma because and we don't write L2 of n even though we know it's the same thing is because when you write L2 of gamma you have your Hilbert space L2 of n but with it you have a bunch of unitary operators so L2 of gamma is really a hilber space with a bunch of unitary operators and uh because because if you have L2 of gamma or L2 of something that has some structure the structure will give you a bunch of unit you you will give you a bunch of op operators and uh right so uh so the the action so if you have C in L2 of gamma uh the action Lambda g c and now I'm going to to say x given by C of G inverse X is just so L2 of gamma is this I always represent it like this because because if your space is infinite you need to go to zero at Infinity so uh so so so Lambda G will displace this function and uh and so an invariant Vector C is invariant is is is an invariant vector means that CX is C of G inverse X for all G and that will force C to be constant but now now a constant function that uh that L2 can only be zero otherwise it's if it has a little Mass constant Mass somewhere it will blow up um okay right uh yeah I'm just coming back to and then and then the and then you can check that uh the almost invariant vectors are uh almost inent vectors are going to be given by one the characteristic function of U divided by a square root of U and this is going to be your almost invariant vectors if I'm here do we get a anyway so so this are going to be your almost invariant vectors so that shows that as soon as your group is infinite with and amable and infinite you get a sequence of almost invent vectors but no invent vector and uh and this is how you disprove properity this is actually the right so um so this show kind of shows that so this Al all implies that if you have a quotient that that that is amable oh yeah this is maybe something that I wanted to say um which is uh something that I will uh leave as an exercise I'm going say proposition but the the proof can be done as an exercise so if you have a short exact sequence of groups um and Q and both groups on each side have t then G has t as well but the converse is not true okay so uh the convert is true for a direct product but it's not true in general but not true in general uh and in fact in general you have C theorem which uh says that uh if G finally generated uh so that the center of G which I denote by zg like Centrum I think it comes from the German such that the Cent the center uh the G mod the center has t and the abelianization is finite then uh then G has property t as well all right so how do we how why is that why is that result interesting uh for us or for geometric group theorists uh it's because of the following coroller and here I'm really telling you a bit of a story that you just can can take it as a story I'm not claiming that you should really understand the math that I'm telling you uh after that after the little story I'll come back to to Really explaining some math so this you can take it as a story so uh the coroller is that uh if you take a co- compact lattice in spn1 twiddle which part of the story that I'm telling you is uh is uh spn1 is not uh this is not Simply Connected it has a fundament it has a it has a unique Universal cover and uh s t but in fact this uh this gamma fits into a short exact sequence where this is q and this this is a CO compact lce in spn1 so uh this is we said it was a hyperbolic group and uh and this is a central extension and Central extensions of hyperbolic groups are given by bounded two cycle Central intensions in general are given by an element in each2 in the second chology group of your group uh in the case of hyperbolic groups all H2 is bounded so the extension is given by a bounded Tuco cycle and hence well this is again an exercise that one can do uh gamma is quasi is quasi isometric to z z cross q and this shows that property T is not a quasi isometry invariant because you have a group that does not have properity t since it has a quotient which is Zed and you have a group here that has property T because of those results all right so so that was um um right that was the few comments I wanted to make um right I want also wanted to say a few words about so more groups with property T list of group with property T So this this is not t right um but then uh yeah so once you gromov also showed that if you take a hyperbolic group and you take a ction by hyperb of this hyperbolic group using a really really really long word then you get something that's still hyperbolic so out of out of those hyperbolic group with probability T you get a bunch of hyperbolic groups with property T that are not necessarily those nice and theoretically well- behaved subgroup of a Matrix Group once you take those weird coent you can get stuff that are consider that I don't really understand but that are considerably more difficult the other interesting point that I want to mention is that uh higher rank simple e groups they there is a huge there is like a really big Theory going with it but uh you can associate to to them what's called a symmetric space so they have a maximal compact subgroup if you mud out you have a very nice very nice had zero space so a manifold with non-positive curvature so all those property T group groups act very nicely in on some CAD zero spaces but they can never act on a kader cube complex so the kader spaces that come from those those those Le groups are very far from cadero Cube complexes this is also to show that there is a wide range of CAD zero spaces that are I mean CAD Z CAD Zer n doesn't tell you anything about property T even though cubical CAD Zer n tells you that you cannot have properity if you act on it okay um right so I'm not allowed to do go until the end can I okay so here okay um right so let me say a few word about why why sl3 R has probability and it's a sketch so the first step is a a lce uh so gamma has t if and only if G has t and uh so this direction right I'm going to say a few words about this direction and uh in this direction is surprisingly is more difficult and arguably not that useful but uh yeah so this direction is is interestingly difficult kajan didn't discuss it the first proof I saw I think is in mari's book in the other direction what you will do is uh the idea is you you start with a representation of gamma induce it up to representation of G and then if this has almost inent vectors show that this will have almost inent vectors as well find an inent Vector go back to the inent to to here the difficulty is if you start with a representation here and you restrict it since one is something continuous and something discrete you might not see much of what you're doing okay so uh so to to do this direction you need induced representations you need in use representations and uh there is a okay I'm not going to so so you need to induce your presentation there's a formula for it uh okay but I'm not going to do finally because I want to do something else um yeah so so you you start with pi a representation of gamma so the induced representation from gamma to G of Pi is a representation of G and all almost invariant vectors are also uh found in the induced representation and so T will give t for G will give an invariant Vector which is also I invariant okay so all right no so that that was yl3 Z has probabil t so uh first first you show that probability T goes down to Lattis is and then what you show is and then and then you have to see why uh so and then you have to study sl3 R so how do you study s sl3 r first there is um you need to study the following subgroup of s sl3 r you're going to look at H which is SL2 R semder product R2 so there is an action of SL there is the obvious action of SL2 r on R2 and then you can put do the semidirect product but what's interesting is that this semidirect product kind of sits in sl3 R and how does it sit in a c3r all right just do do it here um yeah so the embedding you can embed uh so SL2 r semidirect product with R2 you can think of you can put you put SL2 R here you put R2 here one0 0 and that's that's your isomorphism this is how you see your group and so if you look at this and and then you can look at this subgroup s which has 1 1 0 0 1 0 0 you can check that this is indeed a subgroup and the point is that if if you take this is a this is an important Lemma that if you take a representation of sl3 r with s invariant vectors so I want to point out that s is equal to R2 so it's pretty small in sl3 R and if you have S and varent vectors then those vectors in fact it has sl3 r and uh so and I think that this this fact is a is a particularly neat fact uh the the proof is not very deep is really moving around vectors I mean it's not it it's not very involved ma it's really a matter of moving around stuff but it it's also it relies on the fact that if you take any of those if you if you take any of the elementary matrices they normally generate a C3 r see because you take an elementary Matrix it has a one somewhere you just conjugate you're going to put the one everywhere so you're going to generate everything uh and uh and if you have an action if you have a group action and you have an element that normally generates your group or subgroup that normally generates your group and the subgroup acts trivially where it will force the whole thing to act trivially all right so uh so once you so so but this is this is really an this is an important comment if you have a group that's normally generated by something and do something act trivially somewhere the whole thing actually really and uh right so and to finish to finish the proof uh they're using is a definition of relative property T you have a subgroup g and you say that the pair GH has probity this is called also relative probility um well so it's it's kind of you mimic the definition so you're going to say that if if any representation with almost invariant vectors has invariant vectors has each invariant vectors so so it means that uh if you have a pair with itself has t that's the same thing as saying that g has t and also G neutral element always as T and then so to to conclude to conclude prop properity t for l3r uh so then uh then it remains to show that SL2 R semidirect product with R2 relative to R2 uh has relative and uh and to do that uh to do that you make SL2 R act on its unitary Jewel so since SL2 acts on R2 it will act on its unitary dual you look at the action of SL2 r on R2 hat which is R2 and you translate uh having almost invariant vectors in a statement about measure and uh and uh um invariant Vector Right comes from uh from the fact that uh that uh that the direct the direct mass at zero in in R2 hat is the only SL2 R invariant measure you look at SL2 are acting on uh on R2 uh what does the action do right the action of s so2r is just going to rotate stuff but SL2 R is going to smash things so so it's kind of believable that the only the only measure that's that's going to be invariant is the direct Mass is at at zero because that's the only Point that's not really moved okay so uh so that was uh right so that that that was yeah another list of group was property t uh okay so uh so here we have sl3 Z as a lattice in sl3 R and another important class of group with property T is sln QP or uh and uh and Gamma acting co- compactly on X the rits building uh Associated to SL nqp so this I'm just mentioning because uh the way you prove that is using the paper I'd like to read uh next week and the paper I'd like to read next week is so so so it uses it uses the action on the build mixing properties um yeah and this is so so let me just mention maybe the the full results yeah no I mean not not not not a not an intelligent one uh not not one that doesn't go through I have property te and I'm going to follow that that step yeah uh but but yeah so so um but yeah that's really something that I would like to understand uh is if we can get a geometric proof for sl3 R having property T out of this rather combinatorial proof of sl3 QP having property T it's it's it's it's a bit of an Audi many many results that people have for for sl3 r or sl3 QP they mimic in a way what's happening Ju Just just like similarly to when we try to do something for hyperbolic spaces we do we we understand what we're doing on a tree and then we try to imitate what we're doing on a tree on the hyperbolic space just just with this idea we should be able because math is is like a perfect theory if we understand it we should be able to kind of mimic what we're doing here uh on a building with with a manifold but but there's results that explain you how to not do it so any but you know okay uh but since I can see that people want to see actions uh what I'm going to explain now is uh how you get how you get here and here from property T how you get to action and then and that's that's brings me to the definition of so this is and G has t and so if you want to have anti which you also heard as Hagar property or at man ability uh then uh you also um going to skip on that but I'm going to just go here and say there exist a proper action a find isometric action on a Hilbert space and then you don't do the rest and then there exist a proper action so now we're going to see how from a unitary representation you go to an action so let me just make this comment again on terminology because I could see that the first the first uh the first uh terminology about property T was calling property T cash down groups and uh while I understand the need to to honor the people who do great math or who exist I don't know well I I think that when you give some when you give a name to something it should be a name that says something about what you're naming so if you say property te even if you understood nothing about property during my lectures you might remember that there's a trivial representation isolated somewhere if I had say kdan group and you understood nothing of my lecture you get out with nothing and so if if I say anti or at man ability you might get out something even if you understand nothing of the map if I say Hager property you really end up with nothing so anyway that's my speech about naming things not with names of people whom you like to impress but but names of really things that show you that that say something about what you're naming and also you know naming something about after somebody is a way of saying well this stuff that this guy and that I don't really understand so you might as well give a good name okay uh yeah so how do you go from unitary representations to to actions so uh yeah what is an aine isometric action so uh it's G acting on a Hilbert space and this is this is where I think the terminology is kind of misleading because uh it's not a linear action you're not going to find it's it's actually an a fine Hilbert space so it's a Hilbert space but you you ditch the zero and you just think of it as a set with an action of itself that you now think as a Vector space so uh if you have an aine action aine isometric action on a hbit space is what is a map from alpha is a map that I call Alpha from G now G is a discrete group I'm going to ditch locally now it's just finally generated to it's just a map to the isometry group of H that's what that's what an action is it's a group homomorphism from your group to the isometry group of the place you want to act on by isometries so uh but then it happens so that just in the case of RN or CN the isometry group of H decomposes as a semi product of H with uh so so each time you have an action so H with uh well it's a sem product it's a nice group and you have a map which now this is a group homomorphism because it's just a projection so if you compose if you compose your Alpha with this this just standard projection uh here you're going to get Pi that's the linear Part of Your Action it's the unitary representation that we've been talking about this yeah this is the linear part of the action but then it's a ceric product so you have a section meaning that you have a map okay you have a map that goes to h which is not a group homomorphism it's still there and uh and this composition here is another part of interest is uh B so this is B is not a homomorphism but uh but a cycle meaning that I'm not going to go into the the theory but I'll tell you the just the definition meaning that it satisfies B of GH is equal to B of G plus pi g b of H so B takes an element G and Maps it to your Hilbert space and it satisfy this this equation this equation is a r reformulation that Alpha is an action if you if you write if you write out the fact that Al Alpha is an action and that uh in this map here this is the equation you you get now what happens if the action has a fix point if the action has a fixed Point C in h so it means that Alpha g c is equal to C for all G and G then well then C is equal to Pi g c plus b of G oh yeah I didn't tell you that uh yeah this the alpha the action is given by pi g c plus b of G this is really out of spelling out whatever is in here Pi is a unitary representation yeah so if you have an action because if you have an action because the action is a homomorphism into isometry group of H and because H decomposes the isometric group of AG decomposes like this well if you compose your action with the canonical projection you get a unitary operator yeah here you get a unitary operator which is which is a unitary representation yeah you get your unitary representation out of taking your action and composing with the projection on the unitary part of the action and uh and then and that will decompose your action as Alpha G plus pi G plus b G which is which is right your cycle yeah yeah exactly exactly yeah yeah exactly you you get you get this because since it's an action you need to have uh you get this equation you get it you get this equation out of alpha GH c equal to Alpha G of alpha h of C when you write out everything that's that's the equation that's the equation you get yeah from PI and B you recover Alpha yes from PI and B you recover Alpha like this yeah if you have a unitary oper if you have a unitary if you have a unitary representation and a cycle then you can you can stick them together to get an action yeah so it's really it's really the same thing uh okay so so now we have our action and we we di we have a unitary part and the cycle part and let's see what happens if you have a fixed point in the 15 last minute that's good uh so what happens if you have a fix point then C is equal to Pi g c plus BG so BG is C minus Pi GX and uh this is what we called a co- boundary and uh yeah so that uh that brings me yeah that bring me to uh to another right okay so and and and uh so sometimes uh you see uh five five rephrased as um H1 of G Pi is equal to zero so this is exactly saying that any aine isometric action has a fixed point because because this is this is the space of Cycles so it's called and this is the space of co- boundary and if you have a fixed Point your cycle is in fact a co- boundary um okay great so um okay so how now do we go from isometric actions to median spaces to hbert spaces to median spaces oh yeah just before I do that uh let me since I discussed anti I can say that uh yeah the action Alpha is proper proper means that uh if a sequence goes to Infinity in the group meaning that it leaves every finite set it will go to Infinity in the Hilbert space the action uh Alpha is proper meaning that uh um for any c in h and any sequence GN going to Infinity then then Alpha GN C which is an element in the Hilbert space so you can compute the norm in h so that's a that's a sequence that goes to Infinity in R but what does it mean to go to Infinity in if you have if you have an action that's decomposed like this this is never going to move because it's a unitary action so this doesn't do anything else than having Norm one if you want to go to Infinity you need your cycle to go to Infinity uh so this means that for any sequence GN going to Infinity in G then the norm of B of GN goes to Infinity in H oh this is an R right and uh so that's that's how you can go from uh that's how you go from uh from unary actions with uh with almost invent Vector so okay I I I was I kind of this is how you go from unitary operators to uh to to to actions and of course I skipped under the rug how you prove that H one is zero if you have a representation with almost invent vectors and uh you force it to have if you the idea is like you have a representation with almost invar vectors and uh you get a bunch of uh co- boundaries that tend to zero and out of that you get your you get your uh you get your you you you get your your you get your Co you get your co- boundary out of your out of your your your fixed vectors anyway so let's how how do you go from uh um yeah so again it's much it seems much easier to show that uh that uh that something is proper rather than show that everything is bounded because if you want to show that you have you're at manable or anti you just have to find this one proper cycle if you want to show property T you have to look at all unitary representations uh so to go to go from uh uh to go from from uh from um Hilbert spaces to median spaces go from bu spaces to median spaces and back I guess you use spaces with walls I mean this is this is how uh Drew to Haun and I did it when we did it uh there is actually a shorter way using other stuff that I might discuss briefly um so right so x a space with walls wall space is a set with a collection of partitions X is H Union H complement and uh H is in H H and H complement oh uh H is not a Hilbert space anymore okay let me just call it h like this uh this is the set of of half spaces so you have a collection of half spaces and you want it closed closed under complement and uh you're going to have walls which are pair of half spaces and uh you want that uh for for any X Y in X you're going to denote by wxy the set of pairs the set of uh pairs just call this W such that X is an h and y is an H complement and uh and you also want mu a measure on W such that for any X Y in X the measure of the set of uh walls separating X from Y is is finite right so this is the definition of a space with walls so you have need to have a bunch of partitions and uh so that's in it's you know it's a partition so you can say when a partition separates two points if you're one side and the other and uh you want a finite measure on the set of points so the class iCal example is uh X is the vertices of czer cube complex so a c zero cubical complex I'll come back to that next time uh I guess if you don't know what a cader cubical complex is you should ask me by next time and I'll explain to you uh but I'll I'll I'll recall what is a cader cube complex next time but you might need to take a head start if you want to enjoy it but anyway so czer Cube complex I think they're very nice oh it's it's time to oh it's time to rub it up okay so cader Cube compx are very nice object you you take cubes like the cubes you would give to a child like wooden cubes or something and you glue them together you glue them together without doing in in the in the most naive possible way so you don't smash anything and Etc you look at the C the the the ukian metric on the cubes and you want it to be CAD zero meaning that you you're not allowed to create positive curvature and so when you have a cube complex you can uh you take an edge and you start cutting the edge along along the cube and you keep cutting across across you keep cutting perpendicularly across squares and cubes and the magic of kader geometry forces this procedure to to separate your your space into two pieces you're not you're not going to come back and uh and cut yourself again so the the I guess the the most the easiest example is you take no I guess the easiest example you take a tree if you take a tree you cut an edge it will cut your tree in two pieces uh so that's that's an example of space with walls and you see that if you have two points now you need to take only one wall per Edge if you have two points you have a finite jasic so you have finitely many walls that separate any two points so uh Hilbert spaces have a bunch of walls because right so and uh and the thing is that a space with wall out of a space with wall you can recover a median space you can build a median space and uh and a hill bit space Hur spaces have walls guess I can write it here and uh okay no so Hilbert spaces have walls so what are your walls walls need to be a co-dimension one Hilbert Subspace what's a codimension one Hilbert Subspace it's the kernel of a linear functional so you take a linear functional in your Hill space take the current you're going to get uh you're going to get a partition and uh and uh all right so I'm not going to write anything because I'm left with one minute exactly so uh but that I repeat next time so your linear functional gives you uh gives you a bunch of gives you a a partition the partition is is given by a hyperplane and the both sides if you have a you have a metric because you're Hilbert space both sides are convex you have one side that's convex the other side is also convex so when you have a space with walls you can get a metric or pseudometric first out of deciding that that the distance is the measure of the number of Walls separating two point that gives you a pseudometric and uh if the walls you had or convex with convex complement the metric you get is the same you started with and um and in fact uh in fact Hilbert just as cader cubical complexes and uh and wall spaces are in fact the same thing ilbert spaces and measured wall spaces are the same thing as well and I'm going to talk about that next time thank thank you very much yeah oh yeah so it was it's normal you didn't follow nobody should have followed it was too fast so okay uh so why is it so if you have right if you have relative probability if you have relative probility T then what do you do you take an almost invar sl3 invariant Vector it's going to be almost SL2 R semidirect product R2 invariant as well because this is a subgroup and because of relative probity it's going to be R2 invariant and because of this fantastic Lemma which I think this is this is one of the best Lemma in Matrix groups uh then you have an sl3 R inent Vector so start with a representation with almost inant Vector restricted to the semidirect product use relative T to get R2 invariance and uh and then use this LMA to get sl3 R invariance yeah so relative T this is really the part I didn't I didn't explain uh it there is a lot there is quite a bit of analysis involved into understanding how you can translate uh relative property into understanding what are the invariant measure of the SL2 action on R2 hat there it's a it's a it's a bit of analysis but for analysts is relatively well relatively standard are there any other questions commentsthank you very much um so oh there's like a am I allowed to go all the way here okay I'm not going here because I want to repeat the advertisement for Thursday evening there's a play you should all come we have fantastic actors it's a CO if you if you want to know it's a comedy it's not a drama or something it should be more or less an hour but it's a comedy it's nothing heavy and hopefully nothing boring hopefully easier to follow than a MTH do okay so um let me well recall that I was talking about property T and uh and uh let me just recall the following are equivalent so TFA means the following are equivalent and I'm going to make here G discret of course G is always a group uh and I'm going to give you a list of equivalent definition of property T um so this is the original one is that the trivial representation is isolated in Jihad um which there is a slight uh reformulation in terms of uh so this is the felt topology that we discussed last time uh any unitary representation with almost invariant vectors has an invariant Vector um there exists s Epsilon where s contained in G compact Epsilon greater than zero such that so okay uh such that any for any representation either you have an invariant vector or for any G in s the displacement of the unitary representation of any unit vector or any Vector is greater than Epsilon so that's must be obviously a nonzero Vector in your hbert space uh there is an equivalent definition which says that uh any finite dimensional representation is isolated in Jihad now in the space of yeah gu okay um and any a fine isometric action on a hbert space has a fixed point and so if you're if you're in a Hilbert space if you find a bounded orbit you're going to find a fix point so that's the same thing to saying that has a bounded orbit because hbert spaces being cat zero if you have a bounded orbit you can find a barer center or your B bounded set and that's going to be your fix point and uh any isometric action on a median space has a bounded orbit but that doesn't necessarily give you a fix point or yeah okay um and given the questions uh that I had I thought that I will uh say a few more words on the fact that if you're a meable and have property T then you're a finite group but before I do that let me just make uh a comment oh yeah first the list of groups with property t uh let me just list of groups with T so the one that were the motivation of kdan are the lesses in G where uh so this is a lattice so you mean that gamma is discret finite volume and uh and G is a higher rank a simple Le group so Le groups uh yeah simple and semi- simple Le groups are Le groups whose Le algebra either simple or semi- simple but the point is that they're classified so there is a list of those so you can theoretically be lazy and not learn the definition because you could learn the list uh so in the list there most of them are Matrix groups like slnr spnr um so you can think think of slnr for n greater or equal to three and the rank so there's an algebraic definition of the rank but there is a geometric definition of the rank which is the largest RN you can isometrically embed in your group so so higher rank or slnr for n greater or equal to three means that you're going to find a flat copy a copy of R2 within your group in particular you cannot be word hyperbolic because you have this flat piece so they're never hyperbolic and uh in the classification of Le group you also have a list of rank one Le group so lank one Le group and the list is s o N1 s N1 s pn1 and F4 minus 20 and uh those are the those are the the isometries of the real hyperbolic space of Dimension n and uh so as so so that that that guy you might have seen already it's a space with constant curvature minus one and if you if you if you take a co- compact lattice you're going to have a nice hyperbolic manifold and uh your your Lattis is going to be a hyperbolic group uh so but if you have a non-compact lattice still has finite volume it's a relatively hyperbolic group meaning that you'll start finding Flats in it but they're very isolated ones from each other so this one is the isometry group of the complex hyperbolic plane uh and uh okay there's I'm not going to really go into the definition I'm going to wave my hands to describe it it's uh uh it's it's a hyperbolic plane that you get it's it's a hyperbolic space those are hyperbolic spaces that you get so they get Dimension 2 N because it's over the complex uh but the thing is that the curvature is not constant negative one anymore it varies a little bit between minus a half and minus one but it's still a very nice uh hyperbolic space and the co- compact ltis is are word hyperbolic groups and this is the isometry group of the hyperbolic space over the querian the Hamilton querian so you know you get C out of making R plus I then you get the querian out of making C plus I C and you know there's some difficulty that's follow for instance it's it looks like a field but it's no longer commutative and uh and if you iterate the procedure uh you get a hyperbolic plane over the octonian but then there's only then uh there there's you can only do one the octonian are no longer associative so so it's it's a very really an exotic thing so this they don't have property T but those have property T I'm not going to give the proof I'm going to explain why those one don't have you're going to see why those ones don't have probability but I'm not going to say anything about why those have probabil except that now you know that you have a nice class of hyperbolic groups with property T and uh what does what do so okay so so so it's good to know that we have a bunch of hyperb groups that can have property t uh but what can you do with hyperbolic groups with property T hyperbolic groups all have this property which is say SQ Universal and uh which means that SQ universality means that uh if I choose a hyperbolic group and if any of you choose the worst possible group you can think of I can produce a quotient of my hyperbolic group in which your horrible group embeds so hyperbolic groups are SQ universal meaning that uh uh any for any gamma hyperbolic any uh any H finally generated uh H embed in a quotient of of of gamma so it's it's pretty clear if you start with the free group you have a finally generated group is a quotient of the free group but so so so as a result as a result you see that any group embeds in a property T group so being a subgroup of a property group T group is no information okay any finaly generated group embeds in a property T group all right so this is really to say that being a quotient uh being a subgroup of a property T group is no information okay and I I find it pretty interesting that the original motivation for uh for kajan was to prove something using property T but then uh most of the other result is you can't prove anything property now there's a few thing you can prove I'll tell you a few more thing you can prove using property T but in many in many uh in many questions we ask oursel geometric group Theory many times propert is telling you that's not going to happen because because this this definition any isometric action on a median space has a bounded orbit it's the same it implies that any isometric action on a cader cube complex has a fixed point so it means that you cannot use any cader cubical geometry to understand the property T group uh okay right so so so so back to the amable case um right so so recall that gamma amable means that for any F containing gamma finite for any Epsilon greater than zero there exists U which depends on on F and Epsilon such that f with u is smaller than Epsilon so um yeah so so this really means that if so f a finite set you can think of when I it is a set of moves in your group so really this means that u u is uh Epsilon moved by F compared to the size of U right and we had the regular representation and let me make a comment about L2 of G so we saw in Genevie talk that L2 of G is a L2 of gamma is a Hilbert space but you might remember that you saw in some class or somebody told you that there are not that many Hilbert spaces there is there is C to the power n and then there's L2 to the power n so why the hell are we saying L2 of gamma and we're changing the gamma because we know that this in fact is L2 of n the reason why we write L2 of gamma because and we don't write L2 of n even though we know it's the same thing is because when you write L2 of gamma you have your Hilbert space L2 of n but with it you have a bunch of unitary operators so L2 of gamma is really a hilber space with a bunch of unitary operators and uh because because if you have L2 of gamma or L2 of something that has some structure the structure will give you a bunch of unit you you will give you a bunch of op operators and uh right so uh so the the action so if you have C in L2 of gamma uh the action Lambda g c and now I'm going to to say x given by C of G inverse X is just so L2 of gamma is this I always represent it like this because because if your space is infinite you need to go to zero at Infinity so uh so so so Lambda G will displace this function and uh and so an invariant Vector C is invariant is is is an invariant vector means that CX is C of G inverse X for all G and that will force C to be constant but now now a constant function that uh that L2 can only be zero otherwise it's if it has a little Mass constant Mass somewhere it will blow up um okay right uh yeah I'm just coming back to and then and then the and then you can check that uh the almost invariant vectors are uh almost inent vectors are going to be given by one the characteristic function of U divided by a square root of U and this is going to be your almost invariant vectors if I'm here do we get a anyway so so this are going to be your almost invariant vectors so that shows that as soon as your group is infinite with and amable and infinite you get a sequence of almost invent vectors but no invent vector and uh and this is how you disprove properity this is actually the right so um so this show kind of shows that so this Al all implies that if you have a quotient that that that is amable oh yeah this is maybe something that I wanted to say um which is uh something that I will uh leave as an exercise I'm going say proposition but the the proof can be done as an exercise so if you have a short exact sequence of groups um and Q and both groups on each side have t then G has t as well but the converse is not true okay so uh the convert is true for a direct product but it's not true in general but not true in general uh and in fact in general you have C theorem which uh says that uh if G finally generated uh so that the center of G which I denote by zg like Centrum I think it comes from the German such that the Cent the center uh the G mod the center has t and the abelianization is finite then uh then G has property t as well all right so how do we how why is that why is that result interesting uh for us or for geometric group theorists uh it's because of the following coroller and here I'm really telling you a bit of a story that you just can can take it as a story I'm not claiming that you should really understand the math that I'm telling you uh after that after the little story I'll come back to to Really explaining some math so this you can take it as a story so uh the coroller is that uh if you take a co- compact lattice in spn1 twiddle which part of the story that I'm telling you is uh is uh spn1 is not uh this is not Simply Connected it has a fundament it has a it has a unique Universal cover and uh s t but in fact this uh this gamma fits into a short exact sequence where this is q and this this is a CO compact lce in spn1 so uh this is we said it was a hyperbolic group and uh and this is a central extension and Central extensions of hyperbolic groups are given by bounded two cycle Central intensions in general are given by an element in each2 in the second chology group of your group uh in the case of hyperbolic groups all H2 is bounded so the extension is given by a bounded Tuco cycle and hence well this is again an exercise that one can do uh gamma is quasi is quasi isometric to z z cross q and this shows that property T is not a quasi isometry invariant because you have a group that does not have properity t since it has a quotient which is Zed and you have a group here that has property T because of those results all right so so that was um um right that was the few comments I wanted to make um right I want also wanted to say a few words about so more groups with property T list of group with property T So this this is not t right um but then uh yeah so once you gromov also showed that if you take a hyperbolic group and you take a ction by hyperb of this hyperbolic group using a really really really long word then you get something that's still hyperbolic so out of out of those hyperbolic group with probability T you get a bunch of hyperbolic groups with property T that are not necessarily those nice and theoretically well- behaved subgroup of a Matrix Group once you take those weird coent you can get stuff that are consider that I don't really understand but that are considerably more difficult the other interesting point that I want to mention is that uh higher rank simple e groups they there is a huge there is like a really big Theory going with it but uh you can associate to to them what's called a symmetric space so they have a maximal compact subgroup if you mud out you have a very nice very nice had zero space so a manifold with non-positive curvature so all those property T group groups act very nicely in on some CAD zero spaces but they can never act on a kader cube complex so the kader spaces that come from those those those Le groups are very far from cadero Cube complexes this is also to show that there is a wide range of CAD zero spaces that are I mean CAD Z CAD Zer n doesn't tell you anything about property T even though cubical CAD Zer n tells you that you cannot have properity if you act on it okay um right so I'm not allowed to do go until the end can I okay so here okay um right so let me say a few word about why why sl3 R has probability and it's a sketch so the first step is a a lce uh so gamma has t if and only if G has t and uh so this direction right I'm going to say a few words about this direction and uh in this direction is surprisingly is more difficult and arguably not that useful but uh yeah so this direction is is interestingly difficult kajan didn't discuss it the first proof I saw I think is in mari's book in the other direction what you will do is uh the idea is you you start with a representation of gamma induce it up to representation of G and then if this has almost inent vectors show that this will have almost inent vectors as well find an inent Vector go back to the inent to to here the difficulty is if you start with a representation here and you restrict it since one is something continuous and something discrete you might not see much of what you're doing okay so uh so to to do this direction you need induced representations you need in use representations and uh there is a okay I'm not going to so so you need to induce your presentation there's a formula for it uh okay but I'm not going to do finally because I want to do something else um yeah so so you you start with pi a representation of gamma so the induced representation from gamma to G of Pi is a representation of G and all almost invariant vectors are also uh found in the induced representation and so T will give t for G will give an invariant Vector which is also I invariant okay so all right no so that that was yl3 Z has probabil t so uh first first you show that probability T goes down to Lattis is and then what you show is and then and then you have to see why uh so and then you have to study sl3 R so how do you study s sl3 r first there is um you need to study the following subgroup of s sl3 r you're going to look at H which is SL2 R semder product R2 so there is an action of SL there is the obvious action of SL2 r on R2 and then you can put do the semidirect product but what's interesting is that this semidirect product kind of sits in sl3 R and how does it sit in a c3r all right just do do it here um yeah so the embedding you can embed uh so SL2 r semidirect product with R2 you can think of you can put you put SL2 R here you put R2 here one0 0 and that's that's your isomorphism this is how you see your group and so if you look at this and and then you can look at this subgroup s which has 1 1 0 0 1 0 0 you can check that this is indeed a subgroup and the point is that if if you take this is a this is an important Lemma that if you take a representation of sl3 r with s invariant vectors so I want to point out that s is equal to R2 so it's pretty small in sl3 R and if you have S and varent vectors then those vectors in fact it has sl3 r and uh so and I think that this this fact is a is a particularly neat fact uh the the proof is not very deep is really moving around vectors I mean it's not it it's not very involved ma it's really a matter of moving around stuff but it it's also it relies on the fact that if you take any of those if you if you take any of the elementary matrices they normally generate a C3 r see because you take an elementary Matrix it has a one somewhere you just conjugate you're going to put the one everywhere so you're going to generate everything uh and uh and if you have an action if you have a group action and you have an element that normally generates your group or subgroup that normally generates your group and the subgroup acts trivially where it will force the whole thing to act trivially all right so uh so once you so so but this is this is really an this is an important comment if you have a group that's normally generated by something and do something act trivially somewhere the whole thing actually really and uh right so and to finish to finish the proof uh they're using is a definition of relative property T you have a subgroup g and you say that the pair GH has probity this is called also relative probility um well so it's it's kind of you mimic the definition so you're going to say that if if any representation with almost invariant vectors has invariant vectors has each invariant vectors so so it means that uh if you have a pair with itself has t that's the same thing as saying that g has t and also G neutral element always as T and then so to to conclude to conclude prop properity t for l3r uh so then uh then it remains to show that SL2 R semidirect product with R2 relative to R2 uh has relative and uh and to do that uh to do that you make SL2 R act on its unitary Jewel so since SL2 acts on R2 it will act on its unitary dual you look at the action of SL2 r on R2 hat which is R2 and you translate uh having almost invariant vectors in a statement about measure and uh and uh um invariant Vector Right comes from uh from the fact that uh that uh that the direct the direct mass at zero in in R2 hat is the only SL2 R invariant measure you look at SL2 are acting on uh on R2 uh what does the action do right the action of s so2r is just going to rotate stuff but SL2 R is going to smash things so so it's kind of believable that the only the only measure that's that's going to be invariant is the direct Mass is at at zero because that's the only Point that's not really moved okay so uh so that was uh right so that that that was yeah another list of group was property t uh okay so uh so here we have sl3 Z as a lattice in sl3 R and another important class of group with property T is sln QP or uh and uh and Gamma acting co- compactly on X the rits building uh Associated to SL nqp so this I'm just mentioning because uh the way you prove that is using the paper I'd like to read uh next week and the paper I'd like to read next week is so so so it uses it uses the action on the build mixing properties um yeah and this is so so let me just mention maybe the the full results yeah no I mean not not not not a not an intelligent one uh not not one that doesn't go through I have property te and I'm going to follow that that step yeah uh but but yeah so so um but yeah that's really something that I would like to understand uh is if we can get a geometric proof for sl3 R having property T out of this rather combinatorial proof of sl3 QP having property T it's it's it's it's a bit of an Audi many many results that people have for for sl3 r or sl3 QP they mimic in a way what's happening Ju Just just like similarly to when we try to do something for hyperbolic spaces we do we we understand what we're doing on a tree and then we try to imitate what we're doing on a tree on the hyperbolic space just just with this idea we should be able because math is is like a perfect theory if we understand it we should be able to kind of mimic what we're doing here uh on a building with with a manifold but but there's results that explain you how to not do it so any but you know okay uh but since I can see that people want to see actions uh what I'm going to explain now is uh how you get how you get here and here from property T how you get to action and then and that's that's brings me to the definition of so this is and G has t and so if you want to have anti which you also heard as Hagar property or at man ability uh then uh you also um going to skip on that but I'm going to just go here and say there exist a proper action a find isometric action on a Hilbert space and then you don't do the rest and then there exist a proper action so now we're going to see how from a unitary representation you go to an action so let me just make this comment again on terminology because I could see that the first the first uh the first uh terminology about property T was calling property T cash down groups and uh while I understand the need to to honor the people who do great math or who exist I don't know well I I think that when you give some when you give a name to something it should be a name that says something about what you're naming so if you say property te even if you understood nothing about property during my lectures you might remember that there's a trivial representation isolated somewhere if I had say kdan group and you understood nothing of my lecture you get out with nothing and so if if I say anti or at man ability you might get out something even if you understand nothing of the map if I say Hager property you really end up with nothing so anyway that's my speech about naming things not with names of people whom you like to impress but but names of really things that show you that that say something about what you're naming and also you know naming something about after somebody is a way of saying well this stuff that this guy and that I don't really understand so you might as well give a good name okay uh yeah so how do you go from unitary representations to to actions so uh yeah what is an aine isometric action so uh it's G acting on a Hilbert space and this is this is where I think the terminology is kind of misleading because uh it's not a linear action you're not going to find it's it's actually an a fine Hilbert space so it's a Hilbert space but you you ditch the zero and you just think of it as a set with an action of itself that you now think as a Vector space so uh if you have an aine action aine isometric action on a hbit space is what is a map from alpha is a map that I call Alpha from G now G is a discrete group I'm going to ditch locally now it's just finally generated to it's just a map to the isometry group of H that's what that's what an action is it's a group homomorphism from your group to the isometry group of the place you want to act on by isometries so uh but then it happens so that just in the case of RN or CN the isometry group of H decomposes as a semi product of H with uh so so each time you have an action so H with uh well it's a sem product it's a nice group and you have a map which now this is a group homomorphism because it's just a projection so if you compose if you compose your Alpha with this this just standard projection uh here you're going to get Pi that's the linear Part of Your Action it's the unitary representation that we've been talking about this yeah this is the linear part of the action but then it's a ceric product so you have a section meaning that you have a map okay you have a map that goes to h which is not a group homomorphism it's still there and uh and this composition here is another part of interest is uh B so this is B is not a homomorphism but uh but a cycle meaning that I'm not going to go into the the theory but I'll tell you the just the definition meaning that it satisfies B of GH is equal to B of G plus pi g b of H so B takes an element G and Maps it to your Hilbert space and it satisfy this this equation this equation is a r reformulation that Alpha is an action if you if you write if you write out the fact that Al Alpha is an action and that uh in this map here this is the equation you you get now what happens if the action has a fix point if the action has a fixed Point C in h so it means that Alpha g c is equal to C for all G and G then well then C is equal to Pi g c plus b of G oh yeah I didn't tell you that uh yeah this the alpha the action is given by pi g c plus b of G this is really out of spelling out whatever is in here Pi is a unitary representation yeah so if you have an action because if you have an action because the action is a homomorphism into isometry group of H and because H decomposes the isometric group of AG decomposes like this well if you compose your action with the canonical projection you get a unitary operator yeah here you get a unitary operator which is which is a unitary representation yeah you get your unitary representation out of taking your action and composing with the projection on the unitary part of the action and uh and then and that will decompose your action as Alpha G plus pi G plus b G which is which is right your cycle yeah yeah exactly exactly yeah yeah exactly you you get you get this because since it's an action you need to have uh you get this equation you get it you get this equation out of alpha GH c equal to Alpha G of alpha h of C when you write out everything that's that's the equation that's the equation you get yeah from PI and B you recover Alpha yes from PI and B you recover Alpha like this yeah if you have a unitary oper if you have a unitary if you have a unitary representation and a cycle then you can you can stick them together to get an action yeah so it's really it's really the same thing uh okay so so now we have our action and we we di we have a unitary part and the cycle part and let's see what happens if you have a fixed point in the 15 last minute that's good uh so what happens if you have a fix point then C is equal to Pi g c plus BG so BG is C minus Pi GX and uh this is what we called a co- boundary and uh yeah so that uh that brings me yeah that bring me to uh to another right okay so and and and uh so sometimes uh you see uh five five rephrased as um H1 of G Pi is equal to zero so this is exactly saying that any aine isometric action has a fixed point because because this is this is the space of Cycles so it's called and this is the space of co- boundary and if you have a fixed Point your cycle is in fact a co- boundary um okay great so um okay so how now do we go from isometric actions to median spaces to hbert spaces to median spaces oh yeah just before I do that uh let me since I discussed anti I can say that uh yeah the action Alpha is proper proper means that uh if a sequence goes to Infinity in the group meaning that it leaves every finite set it will go to Infinity in the Hilbert space the action uh Alpha is proper meaning that uh um for any c in h and any sequence GN going to Infinity then then Alpha GN C which is an element in the Hilbert space so you can compute the norm in h so that's a that's a sequence that goes to Infinity in R but what does it mean to go to Infinity in if you have if you have an action that's decomposed like this this is never going to move because it's a unitary action so this doesn't do anything else than having Norm one if you want to go to Infinity you need your cycle to go to Infinity uh so this means that for any sequence GN going to Infinity in G then the norm of B of GN goes to Infinity in H oh this is an R right and uh so that's that's how you can go from uh that's how you go from uh from unary actions with uh with almost invent Vector so okay I I I was I kind of this is how you go from unitary operators to uh to to to actions and of course I skipped under the rug how you prove that H one is zero if you have a representation with almost invent vectors and uh you force it to have if you the idea is like you have a representation with almost invar vectors and uh you get a bunch of uh co- boundaries that tend to zero and out of that you get your you get your uh you get your you you you get your your you get your Co you get your co- boundary out of your out of your your your fixed vectors anyway so let's how how do you go from uh um yeah so again it's much it seems much easier to show that uh that uh that something is proper rather than show that everything is bounded because if you want to show that you have you're at manable or anti you just have to find this one proper cycle if you want to show property T you have to look at all unitary representations uh so to go to go from uh uh to go from from uh from um Hilbert spaces to median spaces go from bu spaces to median spaces and back I guess you use spaces with walls I mean this is this is how uh Drew to Haun and I did it when we did it uh there is actually a shorter way using other stuff that I might discuss briefly um so right so x a space with walls wall space is a set with a collection of partitions X is H Union H complement and uh H is in H H and H complement oh uh H is not a Hilbert space anymore okay let me just call it h like this uh this is the set of of half spaces so you have a collection of half spaces and you want it closed closed under complement and uh you're going to have walls which are pair of half spaces and uh you want that uh for for any X Y in X you're going to denote by wxy the set of pairs the set of uh pairs just call this W such that X is an h and y is an H complement and uh and you also want mu a measure on W such that for any X Y in X the measure of the set of uh walls separating X from Y is is finite right so this is the definition of a space with walls so you have need to have a bunch of partitions and uh so that's in it's you know it's a partition so you can say when a partition separates two points if you're one side and the other and uh you want a finite measure on the set of points so the class iCal example is uh X is the vertices of czer cube complex so a c zero cubical complex I'll come back to that next time uh I guess if you don't know what a cader cubical complex is you should ask me by next time and I'll explain to you uh but I'll I'll I'll recall what is a cader cube complex next time but you might need to take a head start if you want to enjoy it but anyway so czer Cube complex I think they're very nice oh it's it's time to oh it's time to rub it up okay so cader Cube compx are very nice object you you take cubes like the cubes you would give to a child like wooden cubes or something and you glue them together you glue them together without doing in in the in the most naive possible way so you don't smash anything and Etc you look at the C the the the ukian metric on the cubes and you want it to be CAD zero meaning that you you're not allowed to create positive curvature and so when you have a cube complex you can uh you take an edge and you start cutting the edge along along the cube and you keep cutting across across you keep cutting perpendicularly across squares and cubes and the magic of kader geometry forces this procedure to to separate your your space into two pieces you're not you're not going to come back and uh and cut yourself again so the the I guess the the most the easiest example is you take no I guess the easiest example you take a tree if you take a tree you cut an edge it will cut your tree in two pieces uh so that's that's an example of space with walls and you see that if you have two points now you need to take only one wall per Edge if you have two points you have a finite jasic so you have finitely many walls that separate any two points so uh Hilbert spaces have a bunch of walls because right so and uh and the thing is that a space with wall out of a space with wall you can recover a median space you can build a median space and uh and a hill bit space Hur spaces have walls guess I can write it here and uh okay no so Hilbert spaces have walls so what are your walls walls need to be a co-dimension one Hilbert Subspace what's a codimension one Hilbert Subspace it's the kernel of a linear functional so you take a linear functional in your Hill space take the current you're going to get uh you're going to get a partition and uh and uh all right so I'm not going to write anything because I'm left with one minute exactly so uh but that I repeat next time so your linear functional gives you uh gives you a bunch of gives you a a partition the partition is is given by a hyperplane and the both sides if you have a you have a metric because you're Hilbert space both sides are convex you have one side that's convex the other side is also convex so when you have a space with walls you can get a metric or pseudometric first out of deciding that that the distance is the measure of the number of Walls separating two point that gives you a pseudometric and uh if the walls you had or convex with convex complement the metric you get is the same you started with and um and in fact uh in fact Hilbert just as cader cubical complexes and uh and wall spaces are in fact the same thing ilbert spaces and measured wall spaces are the same thing as well and I'm going to talk about that next time thank thank you very much yeah oh yeah so it was it's normal you didn't follow nobody should have followed it was too fast so okay uh so why is it so if you have right if you have relative probability if you have relative probility T then what do you do you take an almost invar sl3 invariant Vector it's going to be almost SL2 R semidirect product R2 invariant as well because this is a subgroup and because of relative probity it's going to be R2 invariant and because of this fantastic Lemma which I think this is this is one of the best Lemma in Matrix groups uh then you have an sl3 R inent Vector so start with a representation with almost inant Vector restricted to the semidirect product use relative T to get R2 invariance and uh and then use this LMA to get sl3 R invariance yeah so relative T this is really the part I didn't I didn't explain uh it there is a lot there is quite a bit of analysis involved into understanding how you can translate uh relative property into understanding what are the invariant measure of the SL2 action on R2 hat there it's a it's a it's a bit of analysis but for analysts is relatively well relatively standard are there any other questions commentsthank you very much um so oh there's like a am I allowed to go all the way here okay I'm not going here because I want to repeat the advertisement for Thursday evening there's a play you should all come we have fantastic actors it's a CO if you if you want to know it's a comedy it's not a drama or something it should be more or less an hour but it's a comedy it's nothing heavy and hopefully nothing boring hopefully easier to follow than a MTH do okay so um let me well recall that I was talking about property T and uh and uh let me just recall the following are equivalent so TFA means the following are equivalent and I'm going to make here G discret of course G is always a group uh and I'm going to give you a list of equivalent definition of property T um so this is the original one is that the trivial representation is isolated in Jihad um which there is a slight uh reformulation in terms of uh so this is the felt topology that we discussed last time uh any unitary representation with almost invariant vectors has an invariant Vector um there exists s Epsilon where s contained in G compact Epsilon greater than zero such that so okay uh such that any for any representation either you have an invariant vector or for any G in s the displacement of the unitary representation of any unit vector or any Vector is greater than Epsilon so that's must be obviously a nonzero Vector in your hbert space uh there is an equivalent definition which says that uh any finite dimensional representation is isolated in Jihad now in the space of yeah gu okay um and any a fine isometric action on a hbert space has a fixed point and so if you're if you're in a Hilbert space if you find a bounded orbit you're going to find a fix point so that's the same thing to saying that has a bounded orbit because hbert spaces being cat zero if you have a bounded orbit you can find a barer center or your B bounded set and that's going to be your fix point and uh any isometric action on a median space has a bounded orbit but that doesn't necessarily give you a fix point or yeah okay um and given the questions uh that I had I thought that I will uh say a few more words on the fact that if you're a meable and have property T then you're a finite group but before I do that let me just make uh a comment oh yeah first the list of groups with property t uh let me just list of groups with T so the one that were the motivation of kdan are the lesses in G where uh so this is a lattice so you mean that gamma is discret finite volume and uh and G is a higher rank a simple Le group so Le groups uh yeah simple and semi- simple Le groups are Le groups whose Le algebra either simple or semi- simple but the point is that they're classified so there is a list of those so you can theoretically be lazy and not learn the definition because you could learn the list uh so in the list there most of them are Matrix groups like slnr spnr um so you can think think of slnr for n greater or equal to three and the rank so there's an algebraic definition of the rank but there is a geometric definition of the rank which is the largest RN you can isometrically embed in your group so so higher rank or slnr for n greater or equal to three means that you're going to find a flat copy a copy of R2 within your group in particular you cannot be word hyperbolic because you have this flat piece so they're never hyperbolic and uh in the classification of Le group you also have a list of rank one Le group so lank one Le group and the list is s o N1 s N1 s pn1 and F4 minus 20 and uh those are the those are the the isometries of the real hyperbolic space of Dimension n and uh so as so so that that that guy you might have seen already it's a space with constant curvature minus one and if you if you if you take a co- compact lattice you're going to have a nice hyperbolic manifold and uh your your Lattis is going to be a hyperbolic group uh so but if you have a non-compact lattice still has finite volume it's a relatively hyperbolic group meaning that you'll start finding Flats in it but they're very isolated ones from each other so this one is the isometry group of the complex hyperbolic plane uh and uh okay there's I'm not going to really go into the definition I'm going to wave my hands to describe it it's uh uh it's it's a hyperbolic plane that you get it's it's a hyperbolic space those are hyperbolic spaces that you get so they get Dimension 2 N because it's over the complex uh but the thing is that the curvature is not constant negative one anymore it varies a little bit between minus a half and minus one but it's still a very nice uh hyperbolic space and the co- compact ltis is are word hyperbolic groups and this is the isometry group of the hyperbolic space over the querian the Hamilton querian so you know you get C out of making R plus I then you get the querian out of making C plus I C and you know there's some difficulty that's follow for instance it's it looks like a field but it's no longer commutative and uh and if you iterate the procedure uh you get a hyperbolic plane over the octonian but then there's only then uh there there's you can only do one the octonian are no longer associative so so it's it's a very really an exotic thing so this they don't have property T but those have property T I'm not going to give the proof I'm going to explain why those one don't have you're going to see why those ones don't have probability but I'm not going to say anything about why those have probabil except that now you know that you have a nice class of hyperbolic groups with property T and uh what does what do so okay so so so it's good to know that we have a bunch of hyperb groups that can have property t uh but what can you do with hyperbolic groups with property T hyperbolic groups all have this property which is say SQ Universal and uh which means that SQ universality means that uh if I choose a hyperbolic group and if any of you choose the worst possible group you can think of I can produce a quotient of my hyperbolic group in which your horrible group embeds so hyperbolic groups are SQ universal meaning that uh uh any for any gamma hyperbolic any uh any H finally generated uh H embed in a quotient of of of gamma so it's it's pretty clear if you start with the free group you have a finally generated group is a quotient of the free group but so so so as a result as a result you see that any group embeds in a property T group so being a subgroup of a property group T group is no information okay any finaly generated group embeds in a property T group all right so this is really to say that being a quotient uh being a subgroup of a property T group is no information okay and I I find it pretty interesting that the original motivation for uh for kajan was to prove something using property T but then uh most of the other result is you can't prove anything property now there's a few thing you can prove I'll tell you a few more thing you can prove using property T but in many in many uh in many questions we ask oursel geometric group Theory many times propert is telling you that's not going to happen because because this this definition any isometric action on a median space has a bounded orbit it's the same it implies that any isometric action on a cader cube complex has a fixed point so it means that you cannot use any cader cubical geometry to understand the property T group uh okay right so so so so back to the amable case um right so so recall that gamma amable means that for any F containing gamma finite for any Epsilon greater than zero there exists U which depends on on F and Epsilon such that f with u is smaller than Epsilon so um yeah so so this really means that if so f a finite set you can think of when I it is a set of moves in your group so really this means that u u is uh Epsilon moved by F compared to the size of U right and we had the regular representation and let me make a comment about L2 of G so we saw in Genevie talk that L2 of G is a L2 of gamma is a Hilbert space but you might remember that you saw in some class or somebody told you that there are not that many Hilbert spaces there is there is C to the power n and then there's L2 to the power n so why the hell are we saying L2 of gamma and we're changing the gamma because we know that this in fact is L2 of n the reason why we write L2 of gamma because and we don't write L2 of n even though we know it's the same thing is because when you write L2 of gamma you have your Hilbert space L2 of n but with it you have a bunch of unitary operators so L2 of gamma is really a hilber space with a bunch of unitary operators and uh because because if you have L2 of gamma or L2 of something that has some structure the structure will give you a bunch of unit you you will give you a bunch of op operators and uh right so uh so the the action so if you have C in L2 of gamma uh the action Lambda g c and now I'm going to to say x given by C of G inverse X is just so L2 of gamma is this I always represent it like this because because if your space is infinite you need to go to zero at Infinity so uh so so so Lambda G will displace this function and uh and so an invariant Vector C is invariant is is is an invariant vector means that CX is C of G inverse X for all G and that will force C to be constant but now now a constant function that uh that L2 can only be zero otherwise it's if it has a little Mass constant Mass somewhere it will blow up um okay right uh yeah I'm just coming back to and then and then the and then you can check that uh the almost invariant vectors are uh almost inent vectors are going to be given by one the characteristic function of U divided by a square root of U and this is going to be your almost invariant vectors if I'm here do we get a anyway so so this are going to be your almost invariant vectors so that shows that as soon as your group is infinite with and amable and infinite you get a sequence of almost invent vectors but no invent vector and uh and this is how you disprove properity this is actually the right so um so this show kind of shows that so this Al all implies that if you have a quotient that that that is amable oh yeah this is maybe something that I wanted to say um which is uh something that I will uh leave as an exercise I'm going say proposition but the the proof can be done as an exercise so if you have a short exact sequence of groups um and Q and both groups on each side have t then G has t as well but the converse is not true okay so uh the convert is true for a direct product but it's not true in general but not true in general uh and in fact in general you have C theorem which uh says that uh if G finally generated uh so that the center of G which I denote by zg like Centrum I think it comes from the German such that the Cent the center uh the G mod the center has t and the abelianization is finite then uh then G has property t as well all right so how do we how why is that why is that result interesting uh for us or for geometric group theorists uh it's because of the following coroller and here I'm really telling you a bit of a story that you just can can take it as a story I'm not claiming that you should really understand the math that I'm telling you uh after that after the little story I'll come back to to Really explaining some math so this you can take it as a story so uh the coroller is that uh if you take a co- compact lattice in spn1 twiddle which part of the story that I'm telling you is uh is uh spn1 is not uh this is not Simply Connected it has a fundament it has a it has a unique Universal cover and uh s t but in fact this uh this gamma fits into a short exact sequence where this is q and this this is a CO compact lce in spn1 so uh this is we said it was a hyperbolic group and uh and this is a central extension and Central extensions of hyperbolic groups are given by bounded two cycle Central intensions in general are given by an element in each2 in the second chology group of your group uh in the case of hyperbolic groups all H2 is bounded so the extension is given by a bounded Tuco cycle and hence well this is again an exercise that one can do uh gamma is quasi is quasi isometric to z z cross q and this shows that property T is not a quasi isometry invariant because you have a group that does not have properity t since it has a quotient which is Zed and you have a group here that has property T because of those results all right so so that was um um right that was the few comments I wanted to make um right I want also wanted to say a few words about so more groups with property T list of group with property T So this this is not t right um but then uh yeah so once you gromov also showed that if you take a hyperbolic group and you take a ction by hyperb of this hyperbolic group using a really really really long word then you get something that's still hyperbolic so out of out of those hyperbolic group with probability T you get a bunch of hyperbolic groups with property T that are not necessarily those nice and theoretically well- behaved subgroup of a Matrix Group once you take those weird coent you can get stuff that are consider that I don't really understand but that are considerably more difficult the other interesting point that I want to mention is that uh higher rank simple e groups they there is a huge there is like a really big Theory going with it but uh you can associate to to them what's called a symmetric space so they have a maximal compact subgroup if you mud out you have a very nice very nice had zero space so a manifold with non-positive curvature so all those property T group groups act very nicely in on some CAD zero spaces but they can never act on a kader cube complex so the kader spaces that come from those those those Le groups are very far from cadero Cube complexes this is also to show that there is a wide range of CAD zero spaces that are I mean CAD Z CAD Zer n doesn't tell you anything about property T even though cubical CAD Zer n tells you that you cannot have properity if you act on it okay um right so I'm not allowed to do go until the end can I okay so here okay um right so let me say a few word about why why sl3 R has probability and it's a sketch so the first step is a a lce uh so gamma has t if and only if G has t and uh so this direction right I'm going to say a few words about this direction and uh in this direction is surprisingly is more difficult and arguably not that useful but uh yeah so this direction is is interestingly difficult kajan didn't discuss it the first proof I saw I think is in mari's book in the other direction what you will do is uh the idea is you you start with a representation of gamma induce it up to representation of G and then if this has almost inent vectors show that this will have almost inent vectors as well find an inent Vector go back to the inent to to here the difficulty is if you start with a representation here and you restrict it since one is something continuous and something discrete you might not see much of what you're doing okay so uh so to to do this direction you need induced representations you need in use representations and uh there is a okay I'm not going to so so you need to induce your presentation there's a formula for it uh okay but I'm not going to do finally because I want to do something else um yeah so so you you start with pi a representation of gamma so the induced representation from gamma to G of Pi is a representation of G and all almost invariant vectors are also uh found in the induced representation and so T will give t for G will give an invariant Vector which is also I invariant okay so all right no so that that was yl3 Z has probabil t so uh first first you show that probability T goes down to Lattis is and then what you show is and then and then you have to see why uh so and then you have to study sl3 R so how do you study s sl3 r first there is um you need to study the following subgroup of s sl3 r you're going to look at H which is SL2 R semder product R2 so there is an action of SL there is the obvious action of SL2 r on R2 and then you can put do the semidirect product but what's interesting is that this semidirect product kind of sits in sl3 R and how does it sit in a c3r all right just do do it here um yeah so the embedding you can embed uh so SL2 r semidirect product with R2 you can think of you can put you put SL2 R here you put R2 here one0 0 and that's that's your isomorphism this is how you see your group and so if you look at this and and then you can look at this subgroup s which has 1 1 0 0 1 0 0 you can check that this is indeed a subgroup and the point is that if if you take this is a this is an important Lemma that if you take a representation of sl3 r with s invariant vectors so I want to point out that s is equal to R2 so it's pretty small in sl3 R and if you have S and varent vectors then those vectors in fact it has sl3 r and uh so and I think that this this fact is a is a particularly neat fact uh the the proof is not very deep is really moving around vectors I mean it's not it it's not very involved ma it's really a matter of moving around stuff but it it's also it relies on the fact that if you take any of those if you if you take any of the elementary matrices they normally generate a C3 r see because you take an elementary Matrix it has a one somewhere you just conjugate you're going to put the one everywhere so you're going to generate everything uh and uh and if you have an action if you have a group action and you have an element that normally generates your group or subgroup that normally generates your group and the subgroup acts trivially where it will force the whole thing to act trivially all right so uh so once you so so but this is this is really an this is an important comment if you have a group that's normally generated by something and do something act trivially somewhere the whole thing actually really and uh right so and to finish to finish the proof uh they're using is a definition of relative property T you have a subgroup g and you say that the pair GH has probity this is called also relative probility um well so it's it's kind of you mimic the definition so you're going to say that if if any representation with almost invariant vectors has invariant vectors has each invariant vectors so so it means that uh if you have a pair with itself has t that's the same thing as saying that g has t and also G neutral element always as T and then so to to conclude to conclude prop properity t for l3r uh so then uh then it remains to show that SL2 R semidirect product with R2 relative to R2 uh has relative and uh and to do that uh to do that you make SL2 R act on its unitary Jewel so since SL2 acts on R2 it will act on its unitary dual you look at the action of SL2 r on R2 hat which is R2 and you translate uh having almost invariant vectors in a statement about measure and uh and uh um invariant Vector Right comes from uh from the fact that uh that uh that the direct the direct mass at zero in in R2 hat is the only SL2 R invariant measure you look at SL2 are acting on uh on R2 uh what does the action do right the action of s so2r is just going to rotate stuff but SL2 R is going to smash things so so it's kind of believable that the only the only measure that's that's going to be invariant is the direct Mass is at at zero because that's the only Point that's not really moved okay so uh so that was uh right so that that that was yeah another list of group was property t uh okay so uh so here we have sl3 Z as a lattice in sl3 R and another important class of group with property T is sln QP or uh and uh and Gamma acting co- compactly on X the rits building uh Associated to SL nqp so this I'm just mentioning because uh the way you prove that is using the paper I'd like to read uh next week and the paper I'd like to read next week is so so so it uses it uses the action on the build mixing properties um yeah and this is so so let me just mention maybe the the full results yeah no I mean not not not not a not an intelligent one uh not not one that doesn't go through I have property te and I'm going to follow that that step yeah uh but but yeah so so um but yeah that's really something that I would like to understand uh is if we can get a geometric proof for sl3 R having property T out of this rather combinatorial proof of sl3 QP having property T it's it's it's it's a bit of an Audi many many results that people have for for sl3 r or sl3 QP they mimic in a way what's happening Ju Just just like similarly to when we try to do something for hyperbolic spaces we do we we understand what we're doing on a tree and then we try to imitate what we're doing on a tree on the hyperbolic space just just with this idea we should be able because math is is like a perfect theory if we understand it we should be able to kind of mimic what we're doing here uh on a building with with a manifold but but there's results that explain you how to not do it so any but you know okay uh but since I can see that people want to see actions uh what I'm going to explain now is uh how you get how you get here and here from property T how you get to action and then and that's that's brings me to the definition of so this is and G has t and so if you want to have anti which you also heard as Hagar property or at man ability uh then uh you also um going to skip on that but I'm going to just go here and say there exist a proper action a find isometric action on a Hilbert space and then you don't do the rest and then there exist a proper action so now we're going to see how from a unitary representation you go to an action so let me just make this comment again on terminology because I could see that the first the first uh the first uh terminology about property T was calling property T cash down groups and uh while I understand the need to to honor the people who do great math or who exist I don't know well I I think that when you give some when you give a name to something it should be a name that says something about what you're naming so if you say property te even if you understood nothing about property during my lectures you might remember that there's a trivial representation isolated somewhere if I had say kdan group and you understood nothing of my lecture you get out with nothing and so if if I say anti or at man ability you might get out something even if you understand nothing of the map if I say Hager property you really end up with nothing so anyway that's my speech about naming things not with names of people whom you like to impress but but names of really things that show you that that say something about what you're naming and also you know naming something about after somebody is a way of saying well this stuff that this guy and that I don't really understand so you might as well give a good name okay uh yeah so how do you go from unitary representations to to actions so uh yeah what is an aine isometric action so uh it's G acting on a Hilbert space and this is this is where I think the terminology is kind of misleading because uh it's not a linear action you're not going to find it's it's actually an a fine Hilbert space so it's a Hilbert space but you you ditch the zero and you just think of it as a set with an action of itself that you now think as a Vector space so uh if you have an aine action aine isometric action on a hbit space is what is a map from alpha is a map that I call Alpha from G now G is a discrete group I'm going to ditch locally now it's just finally generated to it's just a map to the isometry group of H that's what that's what an action is it's a group homomorphism from your group to the isometry group of the place you want to act on by isometries so uh but then it happens so that just in the case of RN or CN the isometry group of H decomposes as a semi product of H with uh so so each time you have an action so H with uh well it's a sem product it's a nice group and you have a map which now this is a group homomorphism because it's just a projection so if you compose if you compose your Alpha with this this just standard projection uh here you're going to get Pi that's the linear Part of Your Action it's the unitary representation that we've been talking about this yeah this is the linear part of the action but then it's a ceric product so you have a section meaning that you have a map okay you have a map that goes to h which is not a group homomorphism it's still there and uh and this composition here is another part of interest is uh B so this is B is not a homomorphism but uh but a cycle meaning that I'm not going to go into the the theory but I'll tell you the just the definition meaning that it satisfies B of GH is equal to B of G plus pi g b of H so B takes an element G and Maps it to your Hilbert space and it satisfy this this equation this equation is a r reformulation that Alpha is an action if you if you write if you write out the fact that Al Alpha is an action and that uh in this map here this is the equation you you get now what happens if the action has a fix point if the action has a fixed Point C in h so it means that Alpha g c is equal to C for all G and G then well then C is equal to Pi g c plus b of G oh yeah I didn't tell you that uh yeah this the alpha the action is given by pi g c plus b of G this is really out of spelling out whatever is in here Pi is a unitary representation yeah so if you have an action because if you have an action because the action is a homomorphism into isometry group of H and because H decomposes the isometric group of AG decomposes like this well if you compose your action with the canonical projection you get a unitary operator yeah here you get a unitary operator which is which is a unitary representation yeah you get your unitary representation out of taking your action and composing with the projection on the unitary part of the action and uh and then and that will decompose your action as Alpha G plus pi G plus b G which is which is right your cycle yeah yeah exactly exactly yeah yeah exactly you you get you get this because since it's an action you need to have uh you get this equation you get it you get this equation out of alpha GH c equal to Alpha G of alpha h of C when you write out everything that's that's the equation that's the equation you get yeah from PI and B you recover Alpha yes from PI and B you recover Alpha like this yeah if you have a unitary oper if you have a unitary if you have a unitary representation and a cycle then you can you can stick them together to get an action yeah so it's really it's really the same thing uh okay so so now we have our action and we we di we have a unitary part and the cycle part and let's see what happens if you have a fixed point in the 15 last minute that's good uh so what happens if you have a fix point then C is equal to Pi g c plus BG so BG is C minus Pi GX and uh this is what we called a co- boundary and uh yeah so that uh that brings me yeah that bring me to uh to another right okay so and and and uh so sometimes uh you see uh five five rephrased as um H1 of G Pi is equal to zero so this is exactly saying that any aine isometric action has a fixed point because because this is this is the space of Cycles so it's called and this is the space of co- boundary and if you have a fixed Point your cycle is in fact a co- boundary um okay great so um okay so how now do we go from isometric actions to median spaces to hbert spaces to median spaces oh yeah just before I do that uh let me since I discussed anti I can say that uh yeah the action Alpha is proper proper means that uh if a sequence goes to Infinity in the group meaning that it leaves every finite set it will go to Infinity in the Hilbert space the action uh Alpha is proper meaning that uh um for any c in h and any sequence GN going to Infinity then then Alpha GN C which is an element in the Hilbert space so you can compute the norm in h so that's a that's a sequence that goes to Infinity in R but what does it mean to go to Infinity in if you have if you have an action that's decomposed like this this is never going to move because it's a unitary action so this doesn't do anything else than having Norm one if you want to go to Infinity you need your cycle to go to Infinity uh so this means that for any sequence GN going to Infinity in G then the norm of B of GN goes to Infinity in H oh this is an R right and uh so that's that's how you can go from uh that's how you go from uh from unary actions with uh with almost invent Vector so okay I I I was I kind of this is how you go from unitary operators to uh to to to actions and of course I skipped under the rug how you prove that H one is zero if you have a representation with almost invent vectors and uh you force it to have if you the idea is like you have a representation with almost invar vectors and uh you get a bunch of uh co- boundaries that tend to zero and out of that you get your you get your uh you get your you you you get your your you get your Co you get your co- boundary out of your out of your your your fixed vectors anyway so let's how how do you go from uh um yeah so again it's much it seems much easier to show that uh that uh that something is proper rather than show that everything is bounded because if you want to show that you have you're at manable or anti you just have to find this one proper cycle if you want to show property T you have to look at all unitary representations uh so to go to go from uh uh to go from from uh from um Hilbert spaces to median spaces go from bu spaces to median spaces and back I guess you use spaces with walls I mean this is this is how uh Drew to Haun and I did it when we did it uh there is actually a shorter way using other stuff that I might discuss briefly um so right so x a space with walls wall space is a set with a collection of partitions X is H Union H complement and uh H is in H H and H complement oh uh H is not a Hilbert space anymore okay let me just call it h like this uh this is the set of of half spaces so you have a collection of half spaces and you want it closed closed under complement and uh you're going to have walls which are pair of half spaces and uh you want that uh for for any X Y in X you're going to denote by wxy the set of pairs the set of uh pairs just call this W such that X is an h and y is an H complement and uh and you also want mu a measure on W such that for any X Y in X the measure of the set of uh walls separating X from Y is is finite right so this is the definition of a space with walls so you have need to have a bunch of partitions and uh so that's in it's you know it's a partition so you can say when a partition separates two points if you're one side and the other and uh you want a finite measure on the set of points so the class iCal example is uh X is the vertices of czer cube complex so a c zero cubical complex I'll come back to that next time uh I guess if you don't know what a cader cubical complex is you should ask me by next time and I'll explain to you uh but I'll I'll I'll recall what is a cader cube complex next time but you might need to take a head start if you want to enjoy it but anyway so czer Cube complex I think they're very nice oh it's it's time to oh it's time to rub it up okay so cader Cube compx are very nice object you you take cubes like the cubes you would give to a child like wooden cubes or something and you glue them together you glue them together without doing in in the in the most naive possible way so you don't smash anything and Etc you look at the C the the the ukian metric on the cubes and you want it to be CAD zero meaning that you you're not allowed to create positive curvature and so when you have a cube complex you can uh you take an edge and you start cutting the edge along along the cube and you keep cutting across across you keep cutting perpendicularly across squares and cubes and the magic of kader geometry forces this procedure to to separate your your space into two pieces you're not you're not going to come back and uh and cut yourself again so the the I guess the the most the easiest example is you take no I guess the easiest example you take a tree if you take a tree you cut an edge it will cut your tree in two pieces uh so that's that's an example of space with walls and you see that if you have two points now you need to take only one wall per Edge if you have two points you have a finite jasic so you have finitely many walls that separate any two points so uh Hilbert spaces have a bunch of walls because right so and uh and the thing is that a space with wall out of a space with wall you can recover a median space you can build a median space and uh and a hill bit space Hur spaces have walls guess I can write it here and uh okay no so Hilbert spaces have walls so what are your walls walls need to be a co-dimension one Hilbert Subspace what's a codimension one Hilbert Subspace it's the kernel of a linear functional so you take a linear functional in your Hill space take the current you're going to get uh you're going to get a partition and uh and uh all right so I'm not going to write anything because I'm left with one minute exactly so uh but that I repeat next time so your linear functional gives you uh gives you a bunch of gives you a a partition the partition is is given by a hyperplane and the both sides if you have a you have a metric because you're Hilbert space both sides are convex you have one side that's convex the other side is also convex so when you have a space with walls you can get a metric or pseudometric first out of deciding that that the distance is the measure of the number of Walls separating two point that gives you a pseudometric and uh if the walls you had or convex with convex complement the metric you get is the same you started with and um and in fact uh in fact Hilbert just as cader cubical complexes and uh and wall spaces are in fact the same thing ilbert spaces and measured wall spaces are the same thing as well and I'm going to talk about that next time thank thank you very much yeah oh yeah so it was it's normal you didn't follow nobody should have followed it was too fast so okay uh so why is it so if you have right if you have relative probability if you have relative probility T then what do you do you take an almost invar sl3 invariant Vector it's going to be almost SL2 R semidirect product R2 invariant as well because this is a subgroup and because of relative probity it's going to be R2 invariant and because of this fantastic Lemma which I think this is this is one of the best Lemma in Matrix groups uh then you have an sl3 R inent Vector so start with a representation with almost inant Vector restricted to the semidirect product use relative T to get R2 invariance and uh and then use this LMA to get sl3 R invariance yeah so relative T this is really the part I didn't I didn't explain uh it there is a lot there is quite a bit of analysis involved into understanding how you can translate uh relative property into understanding what are the invariant measure of the SL2 action on R2 hat there it's a it's a it's a bit of analysis but for analysts is relatively well relatively standard are there any other questions comments\n"