The speaker discusses how to take a proper action on a Hilbert space by taking the regular representation and then creating a countable copy of it that acts on a countable copy of L2. They also use the folder condition to reformulate it, stating that there exist UK such that for any ball K, there exists UK where gUK is symmetric difference with SK is less than 1/2K.

The speaker explains that this can be done by defining a cycle as a direct sum of K Λg c k - CK, where CK is the same almost surely. They also mention that if G tends to infinity, the L2 norm is finite and the cycle is proper.

The speaker then asks if it's possible to have a geometrical explanation for how the orbit looks like in the corresponding median space. They also mention property T and refer to a reference by DoValet and BeaDoar, which is the standard reference for probability theory.

Additionally, they mention an alternative approach by Robertson and Stager from 1996, where they show that the original definition of property T can be proved directly using representation theory. However, this approach also quickly leads to the conclusion that everything can be found in their paper, suggesting that it might be a more difficult proof than expected.

Finally, the speaker discusses operator algebra terminology, noting that an operator is positive if it can be decomposed as a sum of T*T or similar operators. They mention that this terminology is used when saying that an operator is positive.

"WEBVTTKind: captionsLanguage: enso good morning everyone so the first Speaker today is Professor IND and she will continue with her uh mini course thank you very much so uh let me just recall what we discussed the first the first uh mini course that was my view of the world of finally generated groups and uh there is a quote by groof that says that any result that's true for any finally generated group is either trivial or false so uh so so we should keep that in mind when we when we try to classify finally generated groups the best we can do is you know put them in some type of bins and uh and then pick your favorite bin which could be a single group and uh and try to prove stuff and understand this particular bin but trying to understand the whole world of finally generated group is hopeless and uh Genevie was asking if anybody worked on the Zero divisor conjecture I spent quite a few years on the Zero divisor conjecture with no results uh and then I and then I think uh few famous mathematician then admitted to me that they actually see the same so there are some people who do that sometimes and not not it's yeah zero divisor conjecture typically is a dangerous dangerous one sorry hardest conure their hardest conjecture yes okay so this is my word of group I when I discussed it I made a mistake I told you I I I had put SL andz semidirect product in the wrong bin it actually has property t as long as n is n is greater or equal to three I would if FN is known for n greater equal to to four I don't know what happens for n equal to 3 I couldn't figure out I have a vague recollection that that that the answer is no or maybe it's still open so I don't really know um and uh we'll probably I'll leave that on the board so we'll probably come back to that so today I'm going to focus on those equivalences so one uh so I I already wrote it but I just rewrite the one I'm going to be focusing on today the equivalence that g has property T this is what we discussed with the felt topology the isolation and Etc uh and I understand that if you've never really worked with uh with unitary representation that was a bit rough but I hope that uh I hope that uh you started finding them trackable uh and there is so this is equivalent to any fine isometric action on a hilir space having a fixed point which is equivalent to any isometric action in a median space having a bounded orbit if you try to make a strong negation of this uh you get that there exists a proper action in a hrid space and there exists the proper action in a median space so this is uh what's also called uh also Hager property or at man ability uh and another a definition that one can give in terms of unitary action is that uh the trivial representation is weakly contained in a unitary repres entation but not any unitary representation a representation who has uh c0 coefficients so what's the coefficient of a representation if you take Pi a unitary representation and you take uh c a belonging to H you're going to define a map what do you call this map okay five C AA going from G to C you take an element and you map it to the inner product of Pi g c with AA and uh and you want this to be C 0 C 0 means that you go to zero at Infinity it means that each time you have Epsilon for every Epsilon greater than zero there exists a compact which depends on Epsilon such that uh this is so you land in C so you can compute the norm is less than Epsilon for any G outside uh this compact set and uh now that I've told you that uh we saw we saw also that um that Amino groups didn't have property T because of the regular representation we had the regular representation and you using fer sequence we found almost invariant vectors but if the group is infinite it has no invariant vectors so you could say maybe we with the rephrasing that meant that the trivial representation was weakly contained in the in the regular representation so let's see why the regular representation has c0 coefficients if you think of the regular representation so case uh right case of Lambda the regular representation so Lambda goes from G to the unitary operator of l2g all right so so at some point I'll mix G and Gamma G is equal to gamma it's a finally generated group for today yes I'm this defition is it saying exist oh um it's actually for all yeah so okay for for any two Vector you get a coefficient function and uh you want you want the c0 coefficient for any any c0 coefficient yeah yeah thank you so let's take it let's look at the let's look at the regular representation uh um yeah because of the definition recall that the definition of uh of like C and AA for the regular representation so it's l2g so it's the sum over X in G of TX AA X maybe a bar somewhere but anyway uh think think of C and A you could think of the direct mass at a point so it's the function that gives one at a given point and zero elsewhere and uh this this is an orthonormal basis right if you if C and eight are two direct masses at different points this inner product is going to be zero and if you take the inner product of a direct Mass with itself you get one so this is this is indeed an auton normal basis and now now why does this regular representation has c0 coefficient if you take if you pick some C or AA you know I like to describe them like this this is a kind of generalization of a of a direct Mass but you you you're very small at Infinity so that's C that say say say it's AA and then you're going to pick c c is another bunk but it's going to move around with G and once G gets big is going to Pi this is going to be moved off of the main mass of uh of ADA so so it's going to become more and more orthogonal so it's going to become more and more zero so so this is the reason why this has c0 coefficient okay uh right okay so uh amenability if you if you think of this definition amenability has is anti for for a straightforward reason where you just compute the the coefficients for the regular representation uh okay so there is there there are quite a few things I didn't discuss and I will not discuss and that we maybe discuss it uh next time so there's there there are some equivalent definition using conditionally negative definite kernels I chose not to talk about it we have some expert in the room about uh about that uh uh sub and Par go had a paper I mean know this really well uh so I didn't I didn't I didn't take this Viewpoint it's not very far from everything I discussed it was just uh I mean I made a choice um and uh and also there is a there is a new there is a new uh characterization so so not discuss that you might encountered is what we call CN D negative definite kernels which is just another way of describing hbert spaces um and also there is a very uh important characterization of Ozawa uh that tells you that you have property T if and only if uh the the laan which is the element of the group ring that's defined by one minus the average over say a finite generating set of just uh just the function as as you saw in in genev talk so this gives you an operator uh on uh on L2 and uh and this if you have it was known that if you have property T this operator gives you a positive operator with a spectral Gap in the full sister algebra and Ozawa uh proved that it's equivalent to this being positive in the group ring and the group ring is just finite stuff so immediately this is how property T became checkable by a computer because if you're in the full sister algebra it's hopeless you have to take soups and whatnot so computers cannot do that but now it became checkable with uh you know in a finite world and uh and this is how they proved the they proved that outn had Pro property T okay but uh today I'm going to say some few more words about median spaces and cader cubical complexes and then I'll also show you hopefully how uh I also hopefully construct some uh Hilbert space and median spaces on which some anti groups can act when I Was preparing I thought you know this starts looking like a political party and a political divide so I hope you're not taking it in that way okay so um right so recall that I have XD geodesic Matrix space and uh for geodesic metric space I have the interval between two points which is the set of T in X such that the distance between XT plus distance between Ty Y is equal to the distance between XY so if you're if you're uniquely geodesic that's going to be your geodesic otherwise it's all the jzx and uh right and X is median if for any three points uh the triple intersection of the interval is exactly one point um which is this is called M of XYZ it's the median of the three points and uh so we saw that uh the the main examples the first example the first obvious example is is r right um the second example was R2 but I'm putting the L1 which is the taxi cab metric and you can the it's the metric that you take the the difference of the coefficients of your points and and if you do that you can see that intervals look like this and if you add a third point then you can really see your triple intersection then uh another example are trees another example is trees uh again if you have three points the interval is a single geodesic because it's a tree so they cover each other and there is a a triple intersection here um C Zer cubical complexes head Zer cubical complexes are uh oh before I before I say that uh so if uh X1 if X1 D1 and X2 D2 are median so is X1 * X2 with the metric D1 + D2 with the just a sum of the metric uh if you did your homework you read the beginning of the bright and half lier so you know what CAD zero is CAD zero means that triangles are thinner than in ukian space if you have two cat zero space the product is CAD zero as well but you you can't put this metric it's not going to be CAD Zer with this metric is going to be cat Zer with the L2 combination of the metric the square sum of the of the metric but you know it's the same thing and and and here this is probably the first indication that uh that uh that those are those are related that median spaces and Hilbert spaces are related Hilbert space is the world where you take the square combination the square sum of your metric and in median is the world where you take the sum uh right so czer cubical complexes so what are those so you take a bunch of Cubes you you it's a it's a it's a it's a complex that you build out of cubes uh so first let's say first a cube what is a cube a cube is the interval 01 to the power n this is an N dimensional Cube you see many times people think people take minus one to one to the power n that's fine too the advantage of taking minus one one to the^ n is you automatically have a mid Cube which is in zero uh so that's an n-dimensional Cube so you have a Zero Dimensional cube a onedimensional cube that's a two-dimensional Cube that's a three-dimensional Cube and I'll stop here so my three yeah and uh what else yeah so that's a cube complex no so that's a cube Cube complex you glue cubes together isometrically along sub cubes so what's a subcube uh a subcube is uh is a cube of lower dimension in the lower dimensional skeleton of your Cube so for instance if I take this this uh this three-dimensional Cube and I split it somewhere here this is not a sub Cube this is so this this is not a sub Cube but this face here is going to be a subcube this is a subcube any vertex is a subcube so gluing isometrically together along subcubes means that uh means that uh you have the illegal gluings you have some illegal gluings you're not allowed to do that for instance and if you were to do that you would have to subdivide your your whole complex again if you have a cube complex you can always subdivide everything and then you start getting you know four cubes here and a bunch of Cubes here and Etc but uh yeah but this this this if you don't subdivide it's an illegal gluing and uh so here's some more illegal gluings you take you know you you you you shrink something uh or you shrink even more so that's those are all illegal gluings but it's fine for right all the rest is fine so and uh and then you want it to be CAD zero and this is where the real Miracle of uh of cubical complex arise yes so when you say isometrically what uh oh no so no on the cube uh on the cube I take the L2 metric yeah yeah here I take the L2 metric right so this is right so CAD zero is a triangles are thinner than in ukian space but then watch you have your Cube complex and how you going to check that it's CAD Zero by checking all the triangle that's tricky but this is where the the real Miracle happens is you can check it locally with the links um C zero condition is locally checkable on Links of vertices so what's the link of a Vertex uh so the link is the combinatorial structure of a sphere of radius a half of a sphere of radius 12 around the vertex and and uh right so so what you how do you compute the link uh for instance um let me just uh let me just draw here one of my favorite examples of as Z cubical complexes I should have done that ahead of time okay so if you if you kind of see what I'm doing I'm gluing a 3 Cube and then around this three Cube I'm going I'm gluing two two cubes and I keep going and if you're in the back and you don't have your glasses you might see a tri Veil and trees so that's why I don't go too far and uh so let's look at the link at this point so the link is a simpal complex a simpl complex where you put a Vertex for each Edge coming out of your of your vertex so so the vertices are edges out of your point V let's call this vertex V so here you're going to have one two three four and then you put an edge if uh two uh is if two vertices correspond to an edge to edges in a common square so here I have my three points and uh so they belong to a common square they uh belong to a common square I have this and then I have this that's it and then you keep going so since those three belong to a common Cube this is a three dimensional Cube you're going to you're going to fill your your your simplex and uh and you check your CAD Zer cubical condition by saying so links check Links of vertigo they should be uh no bygones and flag okay so no no bygones mean no bygones means you don't no bygones means this cannot happen and if you think about it what what's what how can you get such a link you get such a link if you take two squares and you glue them together isometrically along two adjacent sub cubes so you get something like this and this if you really think naively in terms of geometry you have a 90° angle if you go around you're going to see uh you're going to go around and you're going to have 180 degrees but if you're cader when you go around the point you need to have at least 360 that's definitely a non- cader piece that we're avoiding by saying there's no bygone flag flag means that each time each if you see if you see a complete graph of n vertices then you have an N minus one Simplex so here I saw I saw K K3 and I had a two Simplex you can remember that as saying every non sylex contains an on edge each time there's Simplex missing you you can't you can't have the full skeleton okay so uh that's a nice CAD zero Cube complex but it's not yet a median space because it's CAD zero and uh CAD zero spaces are median only when they're trees so but at zero spaces or median that's x uh C zero space is median if and only if x is a tree you can think of R2 R2 is really your toy example of a cad zero space uh so so so R2 with the ukian metric this is not median well because it's uniquely geodesic so there's no hope of having a triple intersection in in a triangle like like this so so why did I tell you that czer Cube complexes are are are examples of uh of median spaces because I ditched the cad Zer metric and I replaced it with the L1 metric on the cubes which coincides with the graph theoretical metric of the one skeleton yes question sorry oh yeah Simply Connected yes uh yeah so so that that's yeah thank you and uh a cube complex is CAD zero meaning that it admits a cad Zer metric if and only if it's simply connected and links are flags all links are flag are flag with no bone yeah and that's that's really a big difference if you take a simplicial complex you're not going to be able to check the kzero condition by by with just just with with links condition in fact if you take a simpal complex and you try to mimic that what you're going to get is systolic complexes uh for which you have a very rich theory that uh that has many similarities with uh with kzero Cube complexes okay so now yes yes yes an archery is a median space yeah exactly uh so so here I have a tree trees and are trees right trees are nice CER Cube complexes our trees are not Cube complexes anymore but they're still median um then what else uh yeah more median spaces what did they want to say more about median spaces yeah yeah it's it's going to yeah the link right so uh because links are not always simplical complexes with my definition uh yeah it's a yeah it's a cell complex because if you see if you do that the link is not going to be simplicial if you take my you're too uh you're too squared that you you glue like this the link you get is not simplicial anymore yeah so either yeah either you say the link has to be simplicial and flag or you say the link has no bygones and this flag yeah okay more questions uh I rule it out because it's simply connected uh all right do I need that so you want you want to do that that's my square like this oh okay so what should I say no Loop or should I just say that the link is uh is simp IAL okay no Loops all right do nothing funny and you'll be cero um okay so so but now if you want to be median what what you need to do uh if you take a c so x a ker Cube complex so you change the metric so you put on it instead of putting L2 Matrix on the cube in the total induced matric you're going to put L1 Matrix on the cubes because they are cubes you can you have a very nice L1 matric on it so L1 metric on the cubes and uh right and then the induced metric on the complex on the whole complex and uh and and that makes the that makes the the the metric then coincides with the coincides with the graph metric of the one skeleton so you can just now you can ditch all the meat you had on your on your uh on your cader Cube complex you just keep the one skeleton then you have a graph and uh so so with for this graph the zero skeleton is going to be a median space for the one skeleton okay uh so those are the first examples of median spaces let me just um look at more examples what's interesting in math is that make a theory you want few examples and you're happy you you get more examples but once you have too many examples uh it gets disappointing and this is what's going to happen with median spaces so uh yeah more examples so another thing so now you're going to take uh X mu measure space so it could be count counting measure it could be a discrete set with a counting measure um let's say it's nice enough I'm not going to tell you what reasonably nice so what's what's going to happen so for instance the first thing that's median is L1 of X mu that's not so surprising because it's like a generalization of R2 with the L1 metric trick uh and here you can see that uh if you take three function f GH in L1 so my I have my three functions f g and say h so the median is going to be uh the function that you get in the middle at each time you follow here here here hope I'm doing it correctly it's it's like I did before it's the median of each projection you know well okay I'm here assuming for the moment that I want function or continuous but what you do it you do it on continuous function you can find the median in that if your three functions are continuous then you find this this this function as a median so at each time you have you have a median and that's your function so that that's uh so this is a function so you kind of defined it as uh the median of f ofx g of x h of X and then another example now I keep my measure space and uh I'm going to look at uh Y which is the set of subsets in X such that they have finite measure finite measure subsets of X so in here you can define a pseudo distance between two sets just by the measure of the symmetric difference and uh with this shudo distance you get a distance you you can mod out by the by the elements of distance zero and uh you get a distance so so in order for for us to see why this is median we can look at the interval the interval between two sets I can erase that right yeah Ser is that so now I have sets and let's see why the measurable subsets finite measurable subset are form a median space um why is that so let's look at the interval between two sets so the interval between two sets is going to be the set of uh C and Y C and Y such that a intersect B is contained in C which is contained in a union B if your sets are not they don't have an intersection you're just going to grow uh from the empty set to to both of them and uh and then you can check you can check that it's indeed correct because uh I take here my set C so this C this is a this is B so let me check that the distance between a plus the distance between BC so if I like look at the distance no that's wrong c c is my element in the interval so the distance between a and c I said it's the symmetric difference so the distance between a and C is going to be the green part so the green part is is here so it's whatever is in a but not in C plus whatever is in C but not in a so this is what you get and then you look at the other piece and uh and what the other piece is whatever is in uh C but not in B and whatever is in B but not in C and uh and this is uh this is exactly the distance between A and B because it's the mass of whatever is in a but not in B and whatever is it B and not in C and then you can check as an exercise check that the median between three sets uh right so the median is going to be this pink part okay um and then um and then one can show that uh any median space embeds in a median space like this so this is where this is where the theory becomes kind of floppy because you know here you have like a bunch of really nice and concrete example but they all belong to that bigger class of example which is in some sense a bit too General I me which which is well which is very general um okay yeah so um median spaces were uh were around in the 50s already people were doing median algebras they've done quite quite a lot of work and then uh and then somehow it seemed to it didn't really die off it took off really in computer science and uh and in parts of logic but uh but it was uh it was uh was it was Martin rer who did kzer Cube complex and really popularized we did then a bunch of work on median algebra that uh that uh really revived the subject then he quit math so uh so he never published for instance his habilitat te that has a bunch of foundational theory on on median spaces and also people do you're doing median algebra in an algebraic way so it's less fun than drawing pictures right well I'll just say that M subject if not fun enough then maybe okay uh right so uh how do you go from median algebra to uh CER Cube complexes to Hilbert spaces um you use uh yeah you use walls so this is a this is a fact that uh median spaces have a wall structure which is a collection of partitions into uh so X is H Union H complement where H and H complement are convex for the metric now I have a median space and uh what I'm telling you which is not so so so the collection of walls is pretty easy to see on uh is pretty easy to see on a cad Zer Cube complex because in a cad Zer Cube complex uh you have walls in each Cube so here your wall you're so so so now I'm going to think just on the Zero skeleton of my complex it's like a discrete space and uh and if I if I if I cut if I cut every cube in two and I I I keep the so here for instance I cut my Cube but I keep going with my cuts I'm going to separate my space in two pieces and this is my collection of walls um on the tree the walls are pretty easy to see if you cut an edge in two pieces you're going to separate your your tree in two pieces that uh that this algorithm separate my cader Cube complex is a consequence of kader geometry and that uh that uh every median space has a wall structure so I didn't really so to put a wall structure in the median space uh what people had noticed already in the 50s is that so they had a notion of con they didn't have metrics but they still even without metrics they had intervals they had everything you wish for except that with a metric it's more visual um they had already shown that each time you take two points you can find uh a partition between two convex subsets that uh that separate those two points so the remark is that if you have for instance R2 with the L1 metric your walls are going to be given by all the partition along parallel to the x axis and all the partition parallel to the Y AIS but if you take anything else for instance that that's not going to be convex this is this is not convex because convex means that if I give you any two points the whole interval is contained in it and here if you take two points well the interval is the squared so it's not going to be contained in uh in uh in either either side so uh right so so now now we know that uh right so so once once okay you have your median space and now I'm telling you there is this whole collection of uh of walls and uh and last time I gave you the definition of a wall space so recall a wall space structure on X is a collection of partitions uh but if if you have no metric on X I'm not going to be able to tell you that the that set will be convex so for a wild space you don't need a metric all you want is a is a collection of partition uh such that for any X Y in X the and a measure on this and a measure on H such that the measure on the wall separating X from Y is finite and out of so out of this data you get you get you get a distance get a pseudo distance on X so just just imagine it's a distance where you mud out to get a distance so if you start with a median space you had a metric and if your collection of partitions are convex with with complement that's convex the metric you get is the same as the metric you started with so which uh so the wall matric if the walls are convex the wall metric corresponds to the original metric right and the the thing is that uh some spaces naturally have walls some spaces are naturally median so for instance for instance CAD Zer Cube complexes are kind of naturally median and then and then you have uh and then you produce walls but then you can ask if I only have walls will I recover a kader cube complexes or a median space and the answer is yes so uh that's this is basically um how you go from uh median spaces to space with walls yeah you don't have the so where yeah so the measure so it's not it's the measure of the set of walls between two points that is that is finite so if if your walls come from a cader cube complex you can just count the walls if you have a finite number of Walls between two points you can just count them if you have an infinite number you have to be smarter than that because you you need to you need to put a measure and uh you need to produce a measure right so so for instance for instance my L1 metric thing I have too many walls between two points but what I could do is decide that the measure of the number of Walls separating two points is just the distance because I have a distance and in fact this is how this is how you construct a measure on the set on the collection of walls in a in a in a median space you take all the walls and uh and uh so you you define your measure more or less like you define uh uh the Le measure on R you start by measuring intervals and then uh you extend your measure by with a bunch of sets with intersection and Etc yeah yeah you need to have a measure to you need to have a measure and once you have the measure you can you can produce a you can produce a a distance okay so more so now uh so yeah so from a wall space you can uh construct uh a median space so that's the that's the trippy tricky Park if if the if the median the if the wall space is discrete that was sagiv's work that's what sagiv did although he didn't call them wall space he called them could menion one subgroups uh so from from from the structure of wall space he has an algorithm is a way of constructing a czer cube complex which maybe I'll say a few words if I have time but that's not that's not totally clear uh but the the if you start with a wall space the cader cube complex or the median space you get can be quite different because uh right okay so let me just uh give you go back to uh to property T oh no let me just give you one more example the example is uh a RN now with a ukian metric so I told you that's not median but what so that definit so not median but you have a very nice wall structure if because you take okay let's say RN is R2 so now I'm going to take as wall structure all possible partitions along given by a line so that's a lot and uh and I'm going to construct a measure the measure I'm going to construct it's it's kind of here already between two points I have a UK fent distance so I'm going to say well the measure between of the of the number the measure of the set of walls separating X and Y is uh is the distance you define it like this so that gives me a measure wall space and uh and uh and as I told you that you can construct a median space so I'm going to have I embed this into a much bigger median space who's going to be something like uh L one of the in in the case of R2 you can really give the you can really give the embedding and this this now is a median space but but you can see that this in a sense is infinite dimensional and this was something very nice of Dimension two so wall spaces can potentially be much simpler than uh than than median spaces or czi or cube complexes and if you you were at Pier's talk yes yesterday he had a coxer group with a bunch of walls but if you try to construct the kader cube complex out of the wall space that P described you get something that's Dimension four so you couldn't draw it anymore on the board and the other example that I want to give is hn hyperbolic space this has naturally a bunch of hn minus 1es those are your walls so here you have a bunch of walls like this and you defined uh you define the measure in the same way and then you have to construct cat Zer Cube a median space out of that uh so th this so then then you get something which I we don't really un we don't really know what it is this is my work with Cornelia Dru and and uh so we get a median space which is some kind of thickening off your H2 or hn it's at finite hous D distance but you know it's the the walls from the median space correspond it just extends the wall you thicken it so that you can really act add your medians uh and I'm totally running out of time so let me just go back to property T and uh atmen ability with uh with the fact that uh the free group is at manable so if I want to show that the free group is aable either I'm happy with the definition of atmen ability that there is a proper action on a median space then here you have three groups are aable or anti um yeah so either you're happy with this action on a on yes yeah wall lines and if I take two point I have a set of lines so I need need to put a measure on this set of lines and I'm going to decide that this measure is the distance between X and Y exactly yeah and then I have to work to show no yeah I mean I can do it the other way if I if I if I only have a wall space and no distance but I have the measure on my my wall space I can Define the distance using the measure on the wall space but if I have if I have a distance and I want to define a measure on the wall space I'm going to use the the the the the distance to define the measure so um right so so here I have an action on a median space and how do I get an action on a Hilbert space out of this action so uh I get a proper action action on L2 of X CR X so X is the x is the zero skeleton here so as you remember maybe or not if you have an action Alpha action it had a unitary part and a cycle part so the unitary part so my Hilbert space is L2 of X cross X the unitary part I don't really have a choice right because I'm acting on it I'm going to permute the elements so that's a unitary action and I need to construct a cycle so the way to construct a cycle I'm going to define a cycle more or less as a co- boundary but a co- boundary of an element that's not in the right place so I'm going to Define that F0 minus G F0 where F0 so here I pick a base Point let's say zero and F0 is the maps that's you take two points X and Y and you're going to map it to one if uh if x y points to O and zero otherwise so here I have my tree here I have my base point and uh and I'm I'm going to Define fo is basically uh this map I have a pair of points if the pair of points are not adjacent then it's zero if they're adjacent and and you point to O then then you're going to to put one so this is clearly not L2 not in L2 of X CR X X but then uh when you do uh F0 minus gf0 so here you have zero and say here you have g0 everything cancels all the pairs of points that you're going to be left are the pairs of points that are on a geodesic between 0 and x0 all the other one pointing here they're going to whatever points to the geodesic is going to cancel out so you're only left with this and the geodesic meaning that this is indeed in L2 of X CR X and uh and it's a proper cycle because uh because B G tends to B of G if you compute the norm the L2 Norm of B of G is going to be proportional to the distance between o and go so the proper action on uh the proper action on the tree will give you that this tends to Infinity as G tends to infinity and you can do exactly the same on a cad zero Cube complex or a median space and you're going to be left uh BG is going to be some type of characteristic function on the half spaces separating o from go so again you have your your your Co cycle that gives you uh proper action on your Hilbert space uh so there's something that I didn't have the chance to do but you can ask me if you want uh is H how do you do for um for for for a for an amable group to construct the proper action uh oh this says that I still have three minutes is that true about five minutes okay well my my watch said I didn't have five minutes okay all right okay so okay so so let me just uh say a few more words so right so this is how you this is the tree case right this is how you get uh proper action on a tree so this goes back to Hagar I think Hagar really studied uh the free group he did a bunch of things he had a seminal paper on uh on the free on the free group where he did a bunch of stuff and uh he also among all the stuff he proved for for hyper for for for free group some of them gave uh the hauger property there is another property that stemmed out of this very same paper which is what I studied uh in my PhD which is the rapid Decay property and you can ask me later about that but it's here right uh but what I'd like to do in the few remaining minutes is uh show you how uh the case of amable groups so the case of amable group is kind of interesting because uh because aminal groups typically do not act on CER Cube complexes aminal groups if they're okay if they're Z to the power n they act on a very nice cader Cube complex which is the cube complex of ZN but otherwise they never do or not properly because you otherwise you're like polinomial growth you you're nil potent you get some Distortion and action on cader Cube complex don't take Distortion so in general they don't act on cader cubical complex uh so how do you get and and so and I I would like to understand the but this because the the this tells you that Amino groups act properly on a median space we like to understand the median space they act properly on but I don't and uh so but one thing we can we can say is the the Hilbert space they act on so G amable acts properly on a Hilbert space so how do you do that uh you take the regular representation and then you take a countable copy of the regular representation that's going to act on a countable copy of L2 and uh remember you had folder condition so you write gamma as a union of balls where K goes to infinity and folder tells you that for any ball uh there for for any K there exist UK you can reformulate foral condition by saying there exist UK such that g UK symmetric difference with SK is less than 1 over 2K so that's that's that's a reformulation of forer condition and that's for all g in a ball of radius k so then uh then you put CK which is the same almost invent Vector yes you said five minutes and then you check that if you define your cycle as a direct sum of K Lambda g c k minus CK uh you can check that this this is uh this is a cycle so the L2 Norm is finite and uh this is a proper cycle so tends to Infinity as G tends to infinity and uh the question that I'd like to understand is like can we can we have like a geometrical explanation on on how how the orbit looks like in the corresponding median space and I'm going to stop here thank you very much questions yes uh n yeah first you take it's weird direct sum the first one is the normal one and then twice and then three times and it's going to get bigger but it's still L2 because of the in fact it's easy to see that it's L2 because after a point those are those are less than 2 to the power K and so you so the sum is going to be converged but the part you left out is uh is going to be bigger and bigger and it's going to grow bigger depending on the size of G so that's why this thing is uh is is going to Infinity oh yeah then maybe I'd like to add a reference for property t uh so there is the so so there is a the do if you Google it you're going to find do valet and Bea doar valet which is the standard reference if you're interested in probity you should probably read that because it's important to to have the vocabulary that other people have there's also a more accessible uh reference exactly with this uh this the with this vocabulary is uh from Gander and uh and somebody else I forgot the name somebody Okay g the title is something like introduction to analytic uh group Theory I think uh Talco and and gallander not not to sure but anyway gallander introduction to analytic so and this is look like a class and uh it's it's it's not very thick the problem with the B is very thick so if you want to read it as an introduction is a bit harsh uh this one is a this one is a nice introduction sorry yeah yeah the thing is that if you once you if you have a if you have a if you have an action on a Hilbert space then your Co cycle in fact is a conditionally negative function and uh and if you have a yeah so in fact the way the way we did it with uh agun and Cornelia dutu when we were doing the equivalence between median spaces and wall spaces uh we we really were uh taking a median space funding the walls and Etc and that's and and and the equivalence between property T this is what how we wrote it up now I realize that in fact you could try to do it uh in a more direct way taking the definition of property T and uh with uh and going to conditionally negative definite kernels and uh trying to go directly and if you do that if you do that in fact you will you will quickly see that uh you will quickly see that uh everything is in the paper of Robertson Stager from 1996 way before we did so yeah if that that's that's thing that's something that would be nice to actually do is write the shortest possible proof of the the original definition of property T Dan's definition or the definition for at man ability with uh with the with the representation and uh and and show directly that it's equivalent to either a proper action in a median space or um no action in a median space if you try to do that you'll see that you very kind of quickly uh everything is in there yes yes yeah yeah so that's an operator algebra terminology when you say that an operator is positive if you can decompose it as a as tar t or a sum of T star T this is this is a positive operator so you're going to say that an operator is posit if it's a sum of this type of operators any other questions so let us thank the speaker againso good morning everyone so the first Speaker today is Professor IND and she will continue with her uh mini course thank you very much so uh let me just recall what we discussed the first the first uh mini course that was my view of the world of finally generated groups and uh there is a quote by groof that says that any result that's true for any finally generated group is either trivial or false so uh so so we should keep that in mind when we when we try to classify finally generated groups the best we can do is you know put them in some type of bins and uh and then pick your favorite bin which could be a single group and uh and try to prove stuff and understand this particular bin but trying to understand the whole world of finally generated group is hopeless and uh Genevie was asking if anybody worked on the Zero divisor conjecture I spent quite a few years on the Zero divisor conjecture with no results uh and then I and then I think uh few famous mathematician then admitted to me that they actually see the same so there are some people who do that sometimes and not not it's yeah zero divisor conjecture typically is a dangerous dangerous one sorry hardest conure their hardest conjecture yes okay so this is my word of group I when I discussed it I made a mistake I told you I I I had put SL andz semidirect product in the wrong bin it actually has property t as long as n is n is greater or equal to three I would if FN is known for n greater equal to to four I don't know what happens for n equal to 3 I couldn't figure out I have a vague recollection that that that the answer is no or maybe it's still open so I don't really know um and uh we'll probably I'll leave that on the board so we'll probably come back to that so today I'm going to focus on those equivalences so one uh so I I already wrote it but I just rewrite the one I'm going to be focusing on today the equivalence that g has property T this is what we discussed with the felt topology the isolation and Etc uh and I understand that if you've never really worked with uh with unitary representation that was a bit rough but I hope that uh I hope that uh you started finding them trackable uh and there is so this is equivalent to any fine isometric action on a hilir space having a fixed point which is equivalent to any isometric action in a median space having a bounded orbit if you try to make a strong negation of this uh you get that there exists a proper action in a hrid space and there exists the proper action in a median space so this is uh what's also called uh also Hager property or at man ability uh and another a definition that one can give in terms of unitary action is that uh the trivial representation is weakly contained in a unitary repres entation but not any unitary representation a representation who has uh c0 coefficients so what's the coefficient of a representation if you take Pi a unitary representation and you take uh c a belonging to H you're going to define a map what do you call this map okay five C AA going from G to C you take an element and you map it to the inner product of Pi g c with AA and uh and you want this to be C 0 C 0 means that you go to zero at Infinity it means that each time you have Epsilon for every Epsilon greater than zero there exists a compact which depends on Epsilon such that uh this is so you land in C so you can compute the norm is less than Epsilon for any G outside uh this compact set and uh now that I've told you that uh we saw we saw also that um that Amino groups didn't have property T because of the regular representation we had the regular representation and you using fer sequence we found almost invariant vectors but if the group is infinite it has no invariant vectors so you could say maybe we with the rephrasing that meant that the trivial representation was weakly contained in the in the regular representation so let's see why the regular representation has c0 coefficients if you think of the regular representation so case uh right case of Lambda the regular representation so Lambda goes from G to the unitary operator of l2g all right so so at some point I'll mix G and Gamma G is equal to gamma it's a finally generated group for today yes I'm this defition is it saying exist oh um it's actually for all yeah so okay for for any two Vector you get a coefficient function and uh you want you want the c0 coefficient for any any c0 coefficient yeah yeah thank you so let's take it let's look at the let's look at the regular representation uh um yeah because of the definition recall that the definition of uh of like C and AA for the regular representation so it's l2g so it's the sum over X in G of TX AA X maybe a bar somewhere but anyway uh think think of C and A you could think of the direct mass at a point so it's the function that gives one at a given point and zero elsewhere and uh this this is an orthonormal basis right if you if C and eight are two direct masses at different points this inner product is going to be zero and if you take the inner product of a direct Mass with itself you get one so this is this is indeed an auton normal basis and now now why does this regular representation has c0 coefficient if you take if you pick some C or AA you know I like to describe them like this this is a kind of generalization of a of a direct Mass but you you you're very small at Infinity so that's C that say say say it's AA and then you're going to pick c c is another bunk but it's going to move around with G and once G gets big is going to Pi this is going to be moved off of the main mass of uh of ADA so so it's going to become more and more orthogonal so it's going to become more and more zero so so this is the reason why this has c0 coefficient okay uh right okay so uh amenability if you if you think of this definition amenability has is anti for for a straightforward reason where you just compute the the coefficients for the regular representation uh okay so there is there there are quite a few things I didn't discuss and I will not discuss and that we maybe discuss it uh next time so there's there there are some equivalent definition using conditionally negative definite kernels I chose not to talk about it we have some expert in the room about uh about that uh uh sub and Par go had a paper I mean know this really well uh so I didn't I didn't I didn't take this Viewpoint it's not very far from everything I discussed it was just uh I mean I made a choice um and uh and also there is a there is a new there is a new uh characterization so so not discuss that you might encountered is what we call CN D negative definite kernels which is just another way of describing hbert spaces um and also there is a very uh important characterization of Ozawa uh that tells you that you have property T if and only if uh the the laan which is the element of the group ring that's defined by one minus the average over say a finite generating set of just uh just the function as as you saw in in genev talk so this gives you an operator uh on uh on L2 and uh and this if you have it was known that if you have property T this operator gives you a positive operator with a spectral Gap in the full sister algebra and Ozawa uh proved that it's equivalent to this being positive in the group ring and the group ring is just finite stuff so immediately this is how property T became checkable by a computer because if you're in the full sister algebra it's hopeless you have to take soups and whatnot so computers cannot do that but now it became checkable with uh you know in a finite world and uh and this is how they proved the they proved that outn had Pro property T okay but uh today I'm going to say some few more words about median spaces and cader cubical complexes and then I'll also show you hopefully how uh I also hopefully construct some uh Hilbert space and median spaces on which some anti groups can act when I Was preparing I thought you know this starts looking like a political party and a political divide so I hope you're not taking it in that way okay so um right so recall that I have XD geodesic Matrix space and uh for geodesic metric space I have the interval between two points which is the set of T in X such that the distance between XT plus distance between Ty Y is equal to the distance between XY so if you're if you're uniquely geodesic that's going to be your geodesic otherwise it's all the jzx and uh right and X is median if for any three points uh the triple intersection of the interval is exactly one point um which is this is called M of XYZ it's the median of the three points and uh so we saw that uh the the main examples the first example the first obvious example is is r right um the second example was R2 but I'm putting the L1 which is the taxi cab metric and you can the it's the metric that you take the the difference of the coefficients of your points and and if you do that you can see that intervals look like this and if you add a third point then you can really see your triple intersection then uh another example are trees another example is trees uh again if you have three points the interval is a single geodesic because it's a tree so they cover each other and there is a a triple intersection here um C Zer cubical complexes head Zer cubical complexes are uh oh before I before I say that uh so if uh X1 if X1 D1 and X2 D2 are median so is X1 * X2 with the metric D1 + D2 with the just a sum of the metric uh if you did your homework you read the beginning of the bright and half lier so you know what CAD zero is CAD zero means that triangles are thinner than in ukian space if you have two cat zero space the product is CAD zero as well but you you can't put this metric it's not going to be CAD Zer with this metric is going to be cat Zer with the L2 combination of the metric the square sum of the of the metric but you know it's the same thing and and and here this is probably the first indication that uh that uh that those are those are related that median spaces and Hilbert spaces are related Hilbert space is the world where you take the square combination the square sum of your metric and in median is the world where you take the sum uh right so czer cubical complexes so what are those so you take a bunch of Cubes you you it's a it's a it's a it's a complex that you build out of cubes uh so first let's say first a cube what is a cube a cube is the interval 01 to the power n this is an N dimensional Cube you see many times people think people take minus one to one to the power n that's fine too the advantage of taking minus one one to the^ n is you automatically have a mid Cube which is in zero uh so that's an n-dimensional Cube so you have a Zero Dimensional cube a onedimensional cube that's a two-dimensional Cube that's a three-dimensional Cube and I'll stop here so my three yeah and uh what else yeah so that's a cube complex no so that's a cube Cube complex you glue cubes together isometrically along sub cubes so what's a subcube uh a subcube is uh is a cube of lower dimension in the lower dimensional skeleton of your Cube so for instance if I take this this uh this three-dimensional Cube and I split it somewhere here this is not a sub Cube this is so this this is not a sub Cube but this face here is going to be a subcube this is a subcube any vertex is a subcube so gluing isometrically together along subcubes means that uh means that uh you have the illegal gluings you have some illegal gluings you're not allowed to do that for instance and if you were to do that you would have to subdivide your your whole complex again if you have a cube complex you can always subdivide everything and then you start getting you know four cubes here and a bunch of Cubes here and Etc but uh yeah but this this this if you don't subdivide it's an illegal gluing and uh so here's some more illegal gluings you take you know you you you you shrink something uh or you shrink even more so that's those are all illegal gluings but it's fine for right all the rest is fine so and uh and then you want it to be CAD zero and this is where the real Miracle of uh of cubical complex arise yes so when you say isometrically what uh oh no so no on the cube uh on the cube I take the L2 metric yeah yeah here I take the L2 metric right so this is right so CAD zero is a triangles are thinner than in ukian space but then watch you have your Cube complex and how you going to check that it's CAD Zero by checking all the triangle that's tricky but this is where the the real Miracle happens is you can check it locally with the links um C zero condition is locally checkable on Links of vertices so what's the link of a Vertex uh so the link is the combinatorial structure of a sphere of radius a half of a sphere of radius 12 around the vertex and and uh right so so what you how do you compute the link uh for instance um let me just uh let me just draw here one of my favorite examples of as Z cubical complexes I should have done that ahead of time okay so if you if you kind of see what I'm doing I'm gluing a 3 Cube and then around this three Cube I'm going I'm gluing two two cubes and I keep going and if you're in the back and you don't have your glasses you might see a tri Veil and trees so that's why I don't go too far and uh so let's look at the link at this point so the link is a simpal complex a simpl complex where you put a Vertex for each Edge coming out of your of your vertex so so the vertices are edges out of your point V let's call this vertex V so here you're going to have one two three four and then you put an edge if uh two uh is if two vertices correspond to an edge to edges in a common square so here I have my three points and uh so they belong to a common square they uh belong to a common square I have this and then I have this that's it and then you keep going so since those three belong to a common Cube this is a three dimensional Cube you're going to you're going to fill your your your simplex and uh and you check your CAD Zer cubical condition by saying so links check Links of vertigo they should be uh no bygones and flag okay so no no bygones mean no bygones means you don't no bygones means this cannot happen and if you think about it what what's what how can you get such a link you get such a link if you take two squares and you glue them together isometrically along two adjacent sub cubes so you get something like this and this if you really think naively in terms of geometry you have a 90° angle if you go around you're going to see uh you're going to go around and you're going to have 180 degrees but if you're cader when you go around the point you need to have at least 360 that's definitely a non- cader piece that we're avoiding by saying there's no bygone flag flag means that each time each if you see if you see a complete graph of n vertices then you have an N minus one Simplex so here I saw I saw K K3 and I had a two Simplex you can remember that as saying every non sylex contains an on edge each time there's Simplex missing you you can't you can't have the full skeleton okay so uh that's a nice CAD zero Cube complex but it's not yet a median space because it's CAD zero and uh CAD zero spaces are median only when they're trees so but at zero spaces or median that's x uh C zero space is median if and only if x is a tree you can think of R2 R2 is really your toy example of a cad zero space uh so so so R2 with the ukian metric this is not median well because it's uniquely geodesic so there's no hope of having a triple intersection in in a triangle like like this so so why did I tell you that czer Cube complexes are are are examples of uh of median spaces because I ditched the cad Zer metric and I replaced it with the L1 metric on the cubes which coincides with the graph theoretical metric of the one skeleton yes question sorry oh yeah Simply Connected yes uh yeah so so that that's yeah thank you and uh a cube complex is CAD zero meaning that it admits a cad Zer metric if and only if it's simply connected and links are flags all links are flag are flag with no bone yeah and that's that's really a big difference if you take a simplicial complex you're not going to be able to check the kzero condition by by with just just with with links condition in fact if you take a simpal complex and you try to mimic that what you're going to get is systolic complexes uh for which you have a very rich theory that uh that has many similarities with uh with kzero Cube complexes okay so now yes yes yes an archery is a median space yeah exactly uh so so here I have a tree trees and are trees right trees are nice CER Cube complexes our trees are not Cube complexes anymore but they're still median um then what else uh yeah more median spaces what did they want to say more about median spaces yeah yeah it's it's going to yeah the link right so uh because links are not always simplical complexes with my definition uh yeah it's a yeah it's a cell complex because if you see if you do that the link is not going to be simplicial if you take my you're too uh you're too squared that you you glue like this the link you get is not simplicial anymore yeah so either yeah either you say the link has to be simplicial and flag or you say the link has no bygones and this flag yeah okay more questions uh I rule it out because it's simply connected uh all right do I need that so you want you want to do that that's my square like this oh okay so what should I say no Loop or should I just say that the link is uh is simp IAL okay no Loops all right do nothing funny and you'll be cero um okay so so but now if you want to be median what what you need to do uh if you take a c so x a ker Cube complex so you change the metric so you put on it instead of putting L2 Matrix on the cube in the total induced matric you're going to put L1 Matrix on the cubes because they are cubes you can you have a very nice L1 matric on it so L1 metric on the cubes and uh right and then the induced metric on the complex on the whole complex and uh and and that makes the that makes the the the metric then coincides with the coincides with the graph metric of the one skeleton so you can just now you can ditch all the meat you had on your on your uh on your cader Cube complex you just keep the one skeleton then you have a graph and uh so so with for this graph the zero skeleton is going to be a median space for the one skeleton okay uh so those are the first examples of median spaces let me just um look at more examples what's interesting in math is that make a theory you want few examples and you're happy you you get more examples but once you have too many examples uh it gets disappointing and this is what's going to happen with median spaces so uh yeah more examples so another thing so now you're going to take uh X mu measure space so it could be count counting measure it could be a discrete set with a counting measure um let's say it's nice enough I'm not going to tell you what reasonably nice so what's what's going to happen so for instance the first thing that's median is L1 of X mu that's not so surprising because it's like a generalization of R2 with the L1 metric trick uh and here you can see that uh if you take three function f GH in L1 so my I have my three functions f g and say h so the median is going to be uh the function that you get in the middle at each time you follow here here here hope I'm doing it correctly it's it's like I did before it's the median of each projection you know well okay I'm here assuming for the moment that I want function or continuous but what you do it you do it on continuous function you can find the median in that if your three functions are continuous then you find this this this function as a median so at each time you have you have a median and that's your function so that that's uh so this is a function so you kind of defined it as uh the median of f ofx g of x h of X and then another example now I keep my measure space and uh I'm going to look at uh Y which is the set of subsets in X such that they have finite measure finite measure subsets of X so in here you can define a pseudo distance between two sets just by the measure of the symmetric difference and uh with this shudo distance you get a distance you you can mod out by the by the elements of distance zero and uh you get a distance so so in order for for us to see why this is median we can look at the interval the interval between two sets I can erase that right yeah Ser is that so now I have sets and let's see why the measurable subsets finite measurable subset are form a median space um why is that so let's look at the interval between two sets so the interval between two sets is going to be the set of uh C and Y C and Y such that a intersect B is contained in C which is contained in a union B if your sets are not they don't have an intersection you're just going to grow uh from the empty set to to both of them and uh and then you can check you can check that it's indeed correct because uh I take here my set C so this C this is a this is B so let me check that the distance between a plus the distance between BC so if I like look at the distance no that's wrong c c is my element in the interval so the distance between a and c I said it's the symmetric difference so the distance between a and C is going to be the green part so the green part is is here so it's whatever is in a but not in C plus whatever is in C but not in a so this is what you get and then you look at the other piece and uh and what the other piece is whatever is in uh C but not in B and whatever is in B but not in C and uh and this is uh this is exactly the distance between A and B because it's the mass of whatever is in a but not in B and whatever is it B and not in C and then you can check as an exercise check that the median between three sets uh right so the median is going to be this pink part okay um and then um and then one can show that uh any median space embeds in a median space like this so this is where this is where the theory becomes kind of floppy because you know here you have like a bunch of really nice and concrete example but they all belong to that bigger class of example which is in some sense a bit too General I me which which is well which is very general um okay yeah so um median spaces were uh were around in the 50s already people were doing median algebras they've done quite quite a lot of work and then uh and then somehow it seemed to it didn't really die off it took off really in computer science and uh and in parts of logic but uh but it was uh it was uh was it was Martin rer who did kzer Cube complex and really popularized we did then a bunch of work on median algebra that uh that uh really revived the subject then he quit math so uh so he never published for instance his habilitat te that has a bunch of foundational theory on on median spaces and also people do you're doing median algebra in an algebraic way so it's less fun than drawing pictures right well I'll just say that M subject if not fun enough then maybe okay uh right so uh how do you go from median algebra to uh CER Cube complexes to Hilbert spaces um you use uh yeah you use walls so this is a this is a fact that uh median spaces have a wall structure which is a collection of partitions into uh so X is H Union H complement where H and H complement are convex for the metric now I have a median space and uh what I'm telling you which is not so so so the collection of walls is pretty easy to see on uh is pretty easy to see on a cad Zer Cube complex because in a cad Zer Cube complex uh you have walls in each Cube so here your wall you're so so so now I'm going to think just on the Zero skeleton of my complex it's like a discrete space and uh and if I if I if I cut if I cut every cube in two and I I I keep the so here for instance I cut my Cube but I keep going with my cuts I'm going to separate my space in two pieces and this is my collection of walls um on the tree the walls are pretty easy to see if you cut an edge in two pieces you're going to separate your your tree in two pieces that uh that this algorithm separate my cader Cube complex is a consequence of kader geometry and that uh that uh every median space has a wall structure so I didn't really so to put a wall structure in the median space uh what people had noticed already in the 50s is that so they had a notion of con they didn't have metrics but they still even without metrics they had intervals they had everything you wish for except that with a metric it's more visual um they had already shown that each time you take two points you can find uh a partition between two convex subsets that uh that separate those two points so the remark is that if you have for instance R2 with the L1 metric your walls are going to be given by all the partition along parallel to the x axis and all the partition parallel to the Y AIS but if you take anything else for instance that that's not going to be convex this is this is not convex because convex means that if I give you any two points the whole interval is contained in it and here if you take two points well the interval is the squared so it's not going to be contained in uh in uh in either either side so uh right so so now now we know that uh right so so once once okay you have your median space and now I'm telling you there is this whole collection of uh of walls and uh and last time I gave you the definition of a wall space so recall a wall space structure on X is a collection of partitions uh but if if you have no metric on X I'm not going to be able to tell you that the that set will be convex so for a wild space you don't need a metric all you want is a is a collection of partition uh such that for any X Y in X the and a measure on this and a measure on H such that the measure on the wall separating X from Y is finite and out of so out of this data you get you get you get a distance get a pseudo distance on X so just just imagine it's a distance where you mud out to get a distance so if you start with a median space you had a metric and if your collection of partitions are convex with with complement that's convex the metric you get is the same as the metric you started with so which uh so the wall matric if the walls are convex the wall metric corresponds to the original metric right and the the thing is that uh some spaces naturally have walls some spaces are naturally median so for instance for instance CAD Zer Cube complexes are kind of naturally median and then and then you have uh and then you produce walls but then you can ask if I only have walls will I recover a kader cube complexes or a median space and the answer is yes so uh that's this is basically um how you go from uh median spaces to space with walls yeah you don't have the so where yeah so the measure so it's not it's the measure of the set of walls between two points that is that is finite so if if your walls come from a cader cube complex you can just count the walls if you have a finite number of Walls between two points you can just count them if you have an infinite number you have to be smarter than that because you you need to you need to put a measure and uh you need to produce a measure right so so for instance for instance my L1 metric thing I have too many walls between two points but what I could do is decide that the measure of the number of Walls separating two points is just the distance because I have a distance and in fact this is how this is how you construct a measure on the set on the collection of walls in a in a in a median space you take all the walls and uh and uh so you you define your measure more or less like you define uh uh the Le measure on R you start by measuring intervals and then uh you extend your measure by with a bunch of sets with intersection and Etc yeah yeah you need to have a measure to you need to have a measure and once you have the measure you can you can produce a you can produce a a distance okay so more so now uh so yeah so from a wall space you can uh construct uh a median space so that's the that's the trippy tricky Park if if the if the median the if the wall space is discrete that was sagiv's work that's what sagiv did although he didn't call them wall space he called them could menion one subgroups uh so from from from the structure of wall space he has an algorithm is a way of constructing a czer cube complex which maybe I'll say a few words if I have time but that's not that's not totally clear uh but the the if you start with a wall space the cader cube complex or the median space you get can be quite different because uh right okay so let me just uh give you go back to uh to property T oh no let me just give you one more example the example is uh a RN now with a ukian metric so I told you that's not median but what so that definit so not median but you have a very nice wall structure if because you take okay let's say RN is R2 so now I'm going to take as wall structure all possible partitions along given by a line so that's a lot and uh and I'm going to construct a measure the measure I'm going to construct it's it's kind of here already between two points I have a UK fent distance so I'm going to say well the measure between of the of the number the measure of the set of walls separating X and Y is uh is the distance you define it like this so that gives me a measure wall space and uh and uh and as I told you that you can construct a median space so I'm going to have I embed this into a much bigger median space who's going to be something like uh L one of the in in the case of R2 you can really give the you can really give the embedding and this this now is a median space but but you can see that this in a sense is infinite dimensional and this was something very nice of Dimension two so wall spaces can potentially be much simpler than uh than than median spaces or czi or cube complexes and if you you were at Pier's talk yes yesterday he had a coxer group with a bunch of walls but if you try to construct the kader cube complex out of the wall space that P described you get something that's Dimension four so you couldn't draw it anymore on the board and the other example that I want to give is hn hyperbolic space this has naturally a bunch of hn minus 1es those are your walls so here you have a bunch of walls like this and you defined uh you define the measure in the same way and then you have to construct cat Zer Cube a median space out of that uh so th this so then then you get something which I we don't really un we don't really know what it is this is my work with Cornelia Dru and and uh so we get a median space which is some kind of thickening off your H2 or hn it's at finite hous D distance but you know it's the the walls from the median space correspond it just extends the wall you thicken it so that you can really act add your medians uh and I'm totally running out of time so let me just go back to property T and uh atmen ability with uh with the fact that uh the free group is at manable so if I want to show that the free group is aable either I'm happy with the definition of atmen ability that there is a proper action on a median space then here you have three groups are aable or anti um yeah so either you're happy with this action on a on yes yeah wall lines and if I take two point I have a set of lines so I need need to put a measure on this set of lines and I'm going to decide that this measure is the distance between X and Y exactly yeah and then I have to work to show no yeah I mean I can do it the other way if I if I if I only have a wall space and no distance but I have the measure on my my wall space I can Define the distance using the measure on the wall space but if I have if I have a distance and I want to define a measure on the wall space I'm going to use the the the the the distance to define the measure so um right so so here I have an action on a median space and how do I get an action on a Hilbert space out of this action so uh I get a proper action action on L2 of X CR X so X is the x is the zero skeleton here so as you remember maybe or not if you have an action Alpha action it had a unitary part and a cycle part so the unitary part so my Hilbert space is L2 of X cross X the unitary part I don't really have a choice right because I'm acting on it I'm going to permute the elements so that's a unitary action and I need to construct a cycle so the way to construct a cycle I'm going to define a cycle more or less as a co- boundary but a co- boundary of an element that's not in the right place so I'm going to Define that F0 minus G F0 where F0 so here I pick a base Point let's say zero and F0 is the maps that's you take two points X and Y and you're going to map it to one if uh if x y points to O and zero otherwise so here I have my tree here I have my base point and uh and I'm I'm going to Define fo is basically uh this map I have a pair of points if the pair of points are not adjacent then it's zero if they're adjacent and and you point to O then then you're going to to put one so this is clearly not L2 not in L2 of X CR X X but then uh when you do uh F0 minus gf0 so here you have zero and say here you have g0 everything cancels all the pairs of points that you're going to be left are the pairs of points that are on a geodesic between 0 and x0 all the other one pointing here they're going to whatever points to the geodesic is going to cancel out so you're only left with this and the geodesic meaning that this is indeed in L2 of X CR X and uh and it's a proper cycle because uh because B G tends to B of G if you compute the norm the L2 Norm of B of G is going to be proportional to the distance between o and go so the proper action on uh the proper action on the tree will give you that this tends to Infinity as G tends to infinity and you can do exactly the same on a cad zero Cube complex or a median space and you're going to be left uh BG is going to be some type of characteristic function on the half spaces separating o from go so again you have your your your Co cycle that gives you uh proper action on your Hilbert space uh so there's something that I didn't have the chance to do but you can ask me if you want uh is H how do you do for um for for for a for an amable group to construct the proper action uh oh this says that I still have three minutes is that true about five minutes okay well my my watch said I didn't have five minutes okay all right okay so okay so so let me just uh say a few more words so right so this is how you this is the tree case right this is how you get uh proper action on a tree so this goes back to Hagar I think Hagar really studied uh the free group he did a bunch of things he had a seminal paper on uh on the free on the free group where he did a bunch of stuff and uh he also among all the stuff he proved for for hyper for for for free group some of them gave uh the hauger property there is another property that stemmed out of this very same paper which is what I studied uh in my PhD which is the rapid Decay property and you can ask me later about that but it's here right uh but what I'd like to do in the few remaining minutes is uh show you how uh the case of amable groups so the case of amable group is kind of interesting because uh because aminal groups typically do not act on CER Cube complexes aminal groups if they're okay if they're Z to the power n they act on a very nice cader Cube complex which is the cube complex of ZN but otherwise they never do or not properly because you otherwise you're like polinomial growth you you're nil potent you get some Distortion and action on cader Cube complex don't take Distortion so in general they don't act on cader cubical complex uh so how do you get and and so and I I would like to understand the but this because the the this tells you that Amino groups act properly on a median space we like to understand the median space they act properly on but I don't and uh so but one thing we can we can say is the the Hilbert space they act on so G amable acts properly on a Hilbert space so how do you do that uh you take the regular representation and then you take a countable copy of the regular representation that's going to act on a countable copy of L2 and uh remember you had folder condition so you write gamma as a union of balls where K goes to infinity and folder tells you that for any ball uh there for for any K there exist UK you can reformulate foral condition by saying there exist UK such that g UK symmetric difference with SK is less than 1 over 2K so that's that's that's a reformulation of forer condition and that's for all g in a ball of radius k so then uh then you put CK which is the same almost invent Vector yes you said five minutes and then you check that if you define your cycle as a direct sum of K Lambda g c k minus CK uh you can check that this this is uh this is a cycle so the L2 Norm is finite and uh this is a proper cycle so tends to Infinity as G tends to infinity and uh the question that I'd like to understand is like can we can we have like a geometrical explanation on on how how the orbit looks like in the corresponding median space and I'm going to stop here thank you very much questions yes uh n yeah first you take it's weird direct sum the first one is the normal one and then twice and then three times and it's going to get bigger but it's still L2 because of the in fact it's easy to see that it's L2 because after a point those are those are less than 2 to the power K and so you so the sum is going to be converged but the part you left out is uh is going to be bigger and bigger and it's going to grow bigger depending on the size of G so that's why this thing is uh is is going to Infinity oh yeah then maybe I'd like to add a reference for property t uh so there is the so so there is a the do if you Google it you're going to find do valet and Bea doar valet which is the standard reference if you're interested in probity you should probably read that because it's important to to have the vocabulary that other people have there's also a more accessible uh reference exactly with this uh this the with this vocabulary is uh from Gander and uh and somebody else I forgot the name somebody Okay g the title is something like introduction to analytic uh group Theory I think uh Talco and and gallander not not to sure but anyway gallander introduction to analytic so and this is look like a class and uh it's it's it's not very thick the problem with the B is very thick so if you want to read it as an introduction is a bit harsh uh this one is a this one is a nice introduction sorry yeah yeah the thing is that if you once you if you have a if you have a if you have an action on a Hilbert space then your Co cycle in fact is a conditionally negative function and uh and if you have a yeah so in fact the way the way we did it with uh agun and Cornelia dutu when we were doing the equivalence between median spaces and wall spaces uh we we really were uh taking a median space funding the walls and Etc and that's and and and the equivalence between property T this is what how we wrote it up now I realize that in fact you could try to do it uh in a more direct way taking the definition of property T and uh with uh and going to conditionally negative definite kernels and uh trying to go directly and if you do that if you do that in fact you will you will quickly see that uh you will quickly see that uh everything is in the paper of Robertson Stager from 1996 way before we did so yeah if that that's that's thing that's something that would be nice to actually do is write the shortest possible proof of the the original definition of property T Dan's definition or the definition for at man ability with uh with the with the representation and uh and and show directly that it's equivalent to either a proper action in a median space or um no action in a median space if you try to do that you'll see that you very kind of quickly uh everything is in there yes yes yeah yeah so that's an operator algebra terminology when you say that an operator is positive if you can decompose it as a as tar t or a sum of T star T this is this is a positive operator so you're going to say that an operator is posit if it's a sum of this type of operators any other questions so let us thank the speaker again\n"