 Thank you very much. So first of all I would of course like to also say Gelukkige verjaardag to Alain Valet. So as Claire said yesterday, I think Alain is like one of the best examples of how to guide PhD students and postdocs. I have seen the effects by hiring some of them afterwards in Leuven. So I think he does an exemplary job in that way. But not only that, I think I learned a lot from Alain's lecture series and talks where he manages in one way or another to have the entire audience leaving the room motivated to start working on something. So I think that that is a wonderful thing that he accomplishes all the time. So I will be talking about von Neumann algebras today and as Bashir already said in the previous talks. So very quickly von Neumann algebras are algebras of operators on a Hilbert space closed in the weak topology and we will be mainly concerned with factors. So factors being the algebras where the center is reduced to the multiples of one. And very importantly there is a notion of a man ability in this world and a man ability for groups has been mentioned already a few times and a man ability of a group gamma would be the same as a man ability of its group von Neumann algebra. But a man ability has an intrinsic definition in terms of von Neumann algebras. It's called injectivity and as introduced by Alain Korn there is also a version that refers to what in the group language would be the existence of an invariant mean on the group. There is something like that. One can formulate given of von Neumann algebra M living somewhere in B of H the existence of a specific functional on B of H playing the role of the mean and being called a hyper trace. So we will be restricting most of the time to so-called type two one factors namely those that admit a tracial state. So a state on the factor that satisfies the trace property and the very fundamental theorem for the entire talk is Alain Korn's theorem from 1976 saying that up to isomorphism there is only one amenable type two one factor and it's a very concrete one. It's the so-called hyperfinite two one factor that was the very first example of a two one factor introduced by Marie and von Neumann and it's very easy to construct this two one factor. You start by putting the two by two matrices in the four by four in the eight by eight by every time putting matrices along the diagonal you take a direct limit that gives you a star algebra and there is a unique way to complete that into a two one factor. So that's the basic object for today the hyperfinite two one factor. So from point of view of classification there is nothing more to say than I mean there is just one I mean there is the unique amenable type two one factor. But on the other hand when you consider sub-algebras of the hyperfinite one factor there is already a very first remark to make sub-algebras inherit amenability. So again up to isomorphism the sub-algebras well they could be like finite dimensional and they could be some center but essentially it's again just the same algebra that that can arise but the way they sit inside can encode extremely rich structures. So something that I won't be talking today at all but that you may know or have heard of is Jones's theory of sub-factors which is like a whole world of considering inclusions of the hyperfinite one factor sitting in one way or another inside the hyperfinite one factor and with all the connections to to all kinds of fields. But that will not be today's topic but it shows how just an inclusion of one in the other can already encode a lot of structure. The topic today rather will be sub-algebras that behave differently. So we will be looking at sub-algebras that satisfy a certain say normality property we call them regular or they were called regular by I think Dix-Mierre and regularity means that the normalizer of the sub-algebra and that consists of all unitaries in the algebra that normalize after conjugacy in this way the sub-algebra. This normalizer should be sufficiently large in the sense that it generates the entire algebra in the von Neumann algebra sense of the word. So that's regularity and then secondly there is a kind of maximality type of assumption in the sense that the only things that can commute with the sub-algebra B inside R should be restricted to the center of B. Now this may sound as a kind of weird set of assumptions and kind of weird why to consider this but I will show you the two extreme cases that will convince you that it's very natural to look at this setting. So the two extreme cases are the cases where B is a billion then this condition reduces to saying that the relative commentant of B inside R is equal to B itself so it's saying that B is a maximal a billion sub-algebra and the other extreme case that I that is very well understood is the case where B is factorial where the center of B is the multiples of one and in that case these are the so-called irreducible regular sub-algebras. So these two cases I will just quickly introduce them they are extremely well understood and then the general case becomes much more intricate. So this is the case where B is a billion and then this definition or this notion amounts to what is called the carton sub-algebra. So a carton sub-algebra B of a two-unfactor is such a regular sub-algebra satisfying the relative commentant condition but now with B a billion so it means that the relative commentant of B is just B itself. Now why is it so natural to look at carton sub-algebras in two-unfactors is because then by the work of Feldman and Moore one can actually realize the two-unfactor as the algebra associated with some other data namely one can realize the two-unfactor M from an equivalence relation. So now I should take some time to explain the concepts so Feldman and Moore showed that any such carton inclusion arises from the following data there is some probability space somewhere x mu x mu arises by writing B the billion algebra as L infinity of a probability space so that gives you x mu then on x mu there is an equivalence relation but it's not just an equivalence relation R is also Borel in the sense that as a subset of x times x it's a Borel set it has countable equivalence classes and then there is a way to express that this equivalence relation preserves the measure so one way to imagine what this means is that if you take a transformation of the space with the graph in the equivalence relation then this transformation should preserve the measure mu and then there is a second ingredient there is a scalar tucco cycle so there is a map from say equivalent triples to the circle somehow making an obstruction to being just just multiplicative and then together so this equivalence relation and the scalar tucco cycle you can define with data to one factor well the factor would come from ergodicity so you can define with data to one factor it contains naturally L infinity of x as a maximum abelian subalgebra and any carton inclusion comes in this way from an equivalence relation so equivalence relation plus tucco cycle goes hand in hand with carton subalgebras what do you mean by countable equivalence relation that that every equivalence class is countable okay so this is what I summarized here now the second fundamental theorem is again a uniqueness theorem by corn Feldman and weiss saying that again if you restrict to the amenable world the word amenable went out well there is a unique ergodic amenable countable probability measure preserving equivalence relation so in in the amenable world there's again a unique object r and not only there is a unique one it has no two homology so here there is a a scalar tucco cycle that appears but in the amenable case the tucco cycle is automatically a co boundary so there is only one equivalence relation that can appear there is no tucco cycle that can appear conclusion just saying the same theorem in other words is that this hyperfina to one factor has a unique carton subalgebra this is up to of course automorphisms of of the algebra no not necessarily inner arbitrary automorphisms okay so again one could say that there is not no extra structure nothing to classify there is just a unique carton subalgebra but let them look at the other extreme case where b is a factor and things already get more interesting so when b is a factor so that here is again the definition we look at regular subalgebras of a two one factor now satisfying the irreducibility assumption so b is a factor and the relative commutant is equally reduced to multiples of one then again there is a way a canonical way to write your algebra not as an algebra of an equivalence relation but from some other data this time and it is as a so-called a co-cycle cross product and what is this group there is a countable group that appears and that's very natural that there is a countable group appearing because the regularity means that the normalizer of b this is a group of unitaries inside m is large enough generates m now an obvious part of the normalizer of b is the unitaries in b itself of course yeah like an inner normalizer if you wish the quotient group in this context where this inclusion is irreducible it's a normal subgroup always and the quotient is a countable group so out of the data comes a countable group and then there is obviously somewhere a two co-cycle or a two comology obstruction because you have this quotient group gamma and you want to associate to every element of gamma a unitary a normalizer here but of course there is no unique choice there is a two co-cycle obstruction that appears extremely naturally and the whole data is then like tautologically say summarized as given by this countable group gamma and a so-called free co-cycle action of the group on the factor so for every group element we have an automorphism alpha g it's not an action it's only an action up to a two co-cycle u and that's just the same as looking at irreducible regular inclusions okay what happens now in the amenable case so the amenable case so m being amenable m being the hyperfinate to one factor amounts exactly to both b being amenable and gamma amenable yeah so in the amenable case when both b and gamma are amenable okneano proved the fundamental results first very surprisingly although there is a wealth of amenable groups the two co-cycle is always a co-boundary I mean there is no way to see this like in a very explicit way but by some miracle if gamma is an amenable group there is always a lift back from the quotient to to the normalizer which is very remarkable and secondly there is essentially only one action as well so essentially there is only one way a count an amenable group gamma can act on the amenable to one factor hyperfinate to one factor b up to the the right notion of conjugacy which in this case would be called co-cycle conjugacy so now there is already something more there is not just uniqueness there is this group gamma that pops up but again there is a complete classification I mean the the regular irreducible subalgebras of the hyperfinate to one factor they are exactly described by this group that's the precise invariant the group in itself up to isomorphism so two such inclusions are isomorphic even only if these quotient groups are the same and this is just a translation of Oknian's theorem of course okay so the the first thing I want to present today is this work that I have been doing recently with Sorin Popa and Dimash Liatenko and it's a very classical topic because let's now just look at the two together so you consider all regular subalgebras satisfying the relative commutant condition and again there is a translation exercise to make because the way the inclusion is written as coming from an equivalence relation in the abelian case or as a cross product in the factorial case I mean all that is kind of straightforward with nowadays techniques so here again there is a straightforward way to summarize what happens but of course you have to put on the same footing an equivalence relation and a group well we know how to put that on the same footing it's by looking at group poets so what will arise is a group poet and I will immediately give some more definitions not to be z of b the center of b is at infinity of x I'm sorry yes I'm sorry the center of b z of b thank you of course so the center of b is now abelian of course it's abelian the center of b is l infinity of a probability space that space will be the space of units of the group poet so in the extreme case where we are looking at the reducible inclusions it's one point so that we will get an ordinary group in the other extreme case we will turn to an equivalence relation yeah so that pops up very naturally a group poet with unit space given here should be z of b then of course since b z of b is the center of b your b is written as a direct integral of of two one factors so above every point lifts above above every point of x lives a two one factor b of x and then there is an action of the group poet on on this and this is not very difficult to understand so maybe I should give you one notation so the group poet has units being this space x so this means that there is a source map and there is a target map and we can compose if things match yeah the source of g is the target of h and so what is then an action of such a group poet on on this field of two one factors b well for every element in the group poet you get an isomorphism from the algebra in the source to the algebra in the target and then very naturally it's not exactly an action there is a tucco cycle obstruction so the composition of the two is up to an inner alpha of gh and the use satisfy the obvious tucco cycle relation so this is just a translation exercise well they need in this generality they need not be all isomorphic but they will be isomorphic along the orbits here is no tucco cycle obstruction no no that would be if you would that's because there is already already the algebra it's like really choosing a lift from the group poet to the normalizer say so yeah and then the theorem is as you could expect that what is the full classification of these regular sub algebras it's exactly determined up to isomorphism by the group poet so in the equivalence relation case there was a unique equivalence relation in the in the group case it's classified by the group and in general it's classified by the group poet now if you it's a type of b you mean either type one or type two exactly because you have to be careful that there could be the silly carton case yeah and when you say associated group in itself yes as an abstract I a discrete measured group poet and again in the amenable case of course now this is something like a theorem that should belong actually to the time when when they proved all the other theorems in the 1980s now actually an important thing had been done actually namely the fact that these amenable group poets can act in a unique way that is a theorem from back then proven by sutherland and dakazaki but it's the tucco cycle problem that is the tricky one that was not on so the the question is how to prove that every tucco cycle is a is a co-boundary so now you may for a moment forget about these cross products and and and so on because now I will look at this tucco cycle problem in its own right so you only remember what is a group poet and then a group poet acts on on a field of factors in the sense that from the source algebra to the target algebra you have these and then there is this the inner perturbation and the thing is that tucco cycles vanish actually for amenable group poets acting on any field of two one factors so you don't even need to assume that the field of factors on which you're acting is amenable but it's essential of course to assume that the group poet that is acting is is amenable otherwise this is very much false so whenever you have an amenable group poet for any free co-cycle action on a field of factors like there the the tucco cycle is a co-boundary now first of all we have to look at the case where where the the group poet is just a group so we have just an amenable group and for and the amenable group is having a co-cycle action on on a two one factor b and then actually it's a very recent result of Soren Popa of this summer that in that case the tucco cycle is always a co-boundary and his proof is of course I will not give the proof but I will give the the main point in the proof because it's an extremely remarkable proof because the the main step in the proof is so the existence of a perturbation of the action so one should realize that actually perturbing alpha g with an inner automorphism is something you have always to to allow because that that's kind of random yeah okay so you allow this inner perturbation of the action so after an inner perturbation of the action there all of a sudden exists a copy of the hyperfine a two one factor that's globally preserved by the action what about the yeah and you have to control that as well exactly and you can and then you apply Oknianus theorem because you land you also get the tucco cycle with values in r you have to control everything but you he can and and and you go back to Oknianus theorem for actions on the hyperfine a two one factor and moreover the construction of this globally invariant copy of the hyperfine a two one factor goes through his methods from sub-factor already how do you construct r I mean it's yeah but it could be very small no no but you do construct it in a kind of inductive way by refining refining refining the the partitions and and and then all the time you control that they commute less and less with a pre-given bigger and bigger set of elements yeah what do you see here it's your arbitrary two one factor on which the group is acting and this is even a characterization of a man ability in the sense that being able to always find the copy of the hyperfine a two one factor that's globally invariant can be done even only if the group is amenable so it's it's a sharp result okay so for amenable groups we can annihilate two co cycles all two co cycles are co boundaries I mean in the case where the actual is free then it's equivalent to that so you don't have a co-cycle isa yes in the equivalence relation case you mean yes in the and now I will immediately show because there and then we come indeed to the crux of the matter immediately so here is again the notion of of group poet that I wrote there on the blackboard as well so the group poet actually so let's now consider what is a general group poet that is amenable well first of all there are the isotropic groups so for every for every unit x you have this countable group that's the isotropic group at x and secondly if your group poet is amenable then actually it's not so complicated as an arbitrary group poet in the sense that it's always a semi direct product of this field of amenable groups on which the amenable equivalence relation is acting now what does it mean that the amenable equivalence relation is acting so on top of the field of isotropic groups you have your amenable equivalence relation so you have an equivalence relation on the space x and then for every pair in the equivalence relation for every x i in r there is a an isomorphism between the fiber gamma y and the fiber gamma x and all this just mixes together into a semi direct product which is this amenable group poet and because amenable equivalence relation have no two comologies as Alan was saying there is no more general amenable discrete group poet than such a semi direct product okay so amenable group poets are such a semi direct product of a field of amenable groups and then amenable equivalence relation okay now we look back to the so they're just resuming what I was saying we go back to the co-cycle actions so the group poet is acting on your field of two one factors and now by Soren's theorem we already know that in each fiber you can solve the two co-cycle so for each isotropic group the two co-cycle is co-boundary so say let's not bother about what the notations mean but you can imagine something it's it's co-boundary and you can do that measurably that's not not issue fiber by fiber for each isotropic group you solve the two co-cycle you write it as a co-boundary but now in order to prove that the entire u is a co-boundary you should be able to choose the the the the cycles here in an equivalent way with respect to this equivalence relation that is acting everywhere and a priori you would say that this is impossible so a priori you say that there is an obstruction but actually of course why is there an obstruction if these vx would be unique then because everything is canonical and so on that this would be automatically co-equivariant but they are not unique they are intrinsically non-unique they are only unique up to a one co-cycle of course that is the way co-homology is working so there seems to be an obstruction except that there is a very general lemma and there is a very general lemma so again you can now for a while forget even about group oh it's now we go even some kind of higher level of obstruction just consider the following data some standard probability space and we have a measurable field of groups polish groups metric complete separable acting on polish spaces in a continuous way think of something like a unitary group acting on a two-on-factor by inner automorphisms or an automorphism group yeah well that's the context that will appear later yeah okay now on top of that we have an equivalence relation on x and this is amenable and this equivalence relation acts by what I would call conjugacies of my data in the sense that when x y is in the equivalence relation then say alpha of x y would be a homeomorphism from py to px and delta of x y would be a homeomorphic group isomorphism I mean preserving all the structure present yeah well if the equivalence relation is amenable there is an equivariant choice up to a co-cycle twist that you should always allow but if the if the the amenable the equivalence relation is amenable you can choose such a section that is equivariant with respect to this kind of thing but with one big if if the actions have dense orbits and it's the density of the orbits together with so here is there is one thing to say if the equivalence relation would have finite orbits then of course you can choose equivariantly your your sections now the equivalence relation is only a union of more and more finite sub equivalence relations so you have to do this and you have to make this converge and that you can exactly do if the orbits are dense so this I have to say is kind of abstractization and generalization of several co-homology lemmas that have been used in the 1980s but it's it's put in this very general context now what does this have to do with our topic today there was this obstruction in in solving the two-core cycle where we needed this equivariant choice of cycles solving the the two-core cycles and we can finish the proof if we can verify this kind of hypothesis which will amount and I will explain you in a minute what this means if we can show that the one-core cycles are approximately co-boundaries that is what this density would mean so what we really need as an ingredient now is that for amenable groups one-core cycles are approximately co-boundaries so now we leave the world of group poets you can for a while forget about that and we now just go back to amenable groups acting on two-one factors so the setting is now again very straightforward we have just an amenable group and we have an action by automorphisms of a two-one factor so what is a one-core cycle a one-core cycle is just a family of unitary satisfying the one-core cycle relation what is a co-boundary it's a one-core cycle that looks like this and what is an approximate co-boundary it would mean that instead of exactly writing vg like this there is a sequence of unitaries such that with convergence say in the in the strong topology vg is such a limit of co-boundaries and the theorem that we proved in the paper is that for amenable groups every one-core cycle is approximately co-boundary and even better this is a characterization of amenability so a group is amenable even only if whenever the group is acting on a two-one factor a one-core cycle is approximately a co-boundary now it may be tempting to think that this is something that has to do with just taking a kind of mean over felner sequences but we actually believed for quite a while that this is maybe not true for the following reason think of the much more known case of homology for a unitary representation so if if we have a unitary representation of your group gamma then there is a notion of a one-core cycle which is nothing else than a map from gamma to the Hilbert space with this one-core cycle relation now a linear one and this is something with the Hayru property and so on the Dalin likes a lot and knows a lot here it's not at all true that for an amenable group one-core cycles are approximately inner by no means so it's and it's not true that you could take them the the mean of the cg over felner sequences and so on it doesn't work at all and so we believed somehow that this should be viewed as the the linear term of that kind of more complicated setting so that maybe there is even an obstruction to have approximate internals of one-core cycles but it's not so and it has to do with another characterization of amenability namely the characterization saying that outractions of the group gamma of an amenable group are never strongly ergodic and this means that there are although there are no not necessarily invariant elements there are always unitaries that are approximately invariant and so here again you may be tempted to say okay that's easy let's just take the mean over a felner sequence but typically that would completely kill the make very small l infinity norm so this is not the the way to go to to prove this yeah okay so how can it be be proven this kind of results so I will just give you a hint of a few of the implications so I will show you how to go from the fact that there is never strong ergodicity to solving or approximately solving one-core cycles and I will also show you how to go from amenability to the fact that that there is no strong ergodicity okay so assume that you have an action of your amenable group on a factor and the tool that we will be using is because we will be looking at approximately invariant elements so we will be looking at sequences of unitaries wn sequences of unitaries that are approximately invariant now in order to express this in a much more convenient way you can view the sequence as an element of the so-called ultra power algebra I will not introduce the ultra power to one factor but there is a way to to to view to put a two-one factor structure on the space of sequences modded out by those that converge to zero okay now in that case strong ergodicity would mean that if you have the group then act on the ultra power that there are that there is nothing except the scalars that there is invariant in the ultra power algebra okay so the first step and this is really actually a method of Alain Cohen if let's assume a little more than just the non-triviality of these fixed points if this is a factor then one-core cycles are approximately co-boundary and that's really Alain's method because what you do is the following assume that you have somewhere a one-core cycle then you build a two by two matrices over b and on the two by two matrices over b you have this action that Alain wrote and then you notice that there are two projections in this algebra that there are two projections in this algebra namely this one and this one they are invariant by construction under bg now our assumption is saying that the fixed points in such a ultra power algebra is always a factor so these two projections must be equivalent now these projections being equivalent in this factor produces you a sequence of unitary sitting in in that position there and it exactly solves you approximately the one-core cycle so this is say classical by now okay so that's for that part then we have to go from the assumption that the group is amenable to producing almost invariant unitaries and this is much more tricky than you would expect because you cannot just take the mean over over a Felner sequence so take an amenable group and let it act on a two-one factor and I will produce for you a sequence of unitaries that's approximately invariant under the group action so because the action is assumed to be by outer automorphisms if you make the cross product the relative commutant of b in this cross product will be reduced to multiples of one and now there is another very interesting result of Sorin Popa in general if you have this kind of situation a subalgebra with trivial relative commutants then you can find in the ultra power of this subalgebra a unitary element that is free with respect to the entire algebra free in the sense of free probability theory if you make all kinds of words between new and elements in the larger algebra you never get one you never get three zero and so on and so forth and and you see that the relative commutant condition is essential because if there would be something in b cross gamma commuting with b yet then it would commute with everything in the ultra power you can never get freeness so it's but whenever the relative commutant is trivial you find in the ultra power of b an element that's completely free with respect to b cross gamma now if you take this element b you now build inside the ultra power algebra you look at this subalgebra and all the ways the group acts on it now if you translate freeness you can you probably have no real idea what this exactly means but what you then find is inside the ultra power b omega you find a free family of copies of b all free with each other indexed by gamma and the way the group is acting the way the group is acting on b so it's acting on b omega and the way that it's acting is just by shifting the the free product around so you find the free Bernoulli shift if you wish inside b omega automatically always and this is the big advantage that this free this is a very concrete thing this is a free Bernoulli shift you can very concretely now construct almost the invariant elements in the for the free Bernoulli shift and and therefore you find it in b omega and therefore you have it in the in the two one factor okay this by now very recently actually it has been shown to be true as well in the type three case so at least for state preserving actions so the similar reasoning works for state preserving actions on arbitrary factors by now that's a result of amin marachi the group is amenable but the factor need not be amenable here it's not supposed to be amenable so what is then the conclusion while we can then put everything together one co cycles are approximately inner that gives you enough of a covariant choice so that you can solve the two co cycle problem and then you have the the classification result that I said for for regular sub algebras but this also poses actually a lot of questions because you could now wonder as serene did in his paper what is this class of groups satisfying two co-mology vanishing for any action on a two one factor with the with the two co cycle so what what serene proved in the group case is that amenable groups have this co-mology vanishing property for obvious reasons if you make a free product then of course if you can solve the two co cycle on gamma one and on gamma two you can do it on the free product so that's so it's stable under free products it's actually even stable under you can amalgamate over finite sub groups that's not not a problem and there are other things that serene could prove for instance property t groups are never inside yes that's the next they always have the high group approximation property they cannot be property t there is no room for commuting things inside so you can in such a group you have no room for a direct product of an infinite group and an amenable group that cannot happen so all this leads to a wild guess that I want to discuss in the final minutes of my talk say you have examples of groups which are not because they could still I mean high group groups could still be such direct products so yeah so say z times f2 would be have the high group property but not be exactly but the guess rather is that it could be the so-called treeable groups and that's a notion that I want to to discuss so this class of groups satisfying co-mology vanishing could be exactly the class of treeable groups so what is a treeable group so countable group is said to be treeable if you can have it act somewhere freely and preserving a probability measure in such a way that the associated orbit equivalence relation so the equivalence relation of lying in the same orbit is treeable now I have to tell you what is a treeable equivalence relation but let me first say what is strongly treeable it would be the same but requiring that for every action free action preserving a probability the relation is is treeable but what is a treeable equivalence relation well this is again not a real definition but still you can put on the orbit structure so you can choose you can put on every tree sorry every orbit for the moment is just a set of points countable set of points you could say okay this countable set of points has actually the structure of a tree that you can always do you just choose a tree in the orbit but then you have to do it measurably throughout being able to do that measurably throughout is what is called treeability but maybe you want to see a much more concrete definition of three that I don't think so I mean I will the concrete definition will will show you what it really means it means that you find that's another definition of treeability a family of elements phi i in the so-called full pseudo group I mean these are partial transformations yes of x with grav in the relation so treeability means that you can find such a family such that no word unless the you make a trivial composition say is the identity somewhere the identity transformation so it's it's really that you can find a free family and generating phi i generate the relation but the group acting freely on a tree would be just a free group it's okay no I see you're right absolutely yeah so everything is up to measure 0 yes everything is up to measure 0 yes so that's the notion of treeability for an equivalence relation and then for a group treeable and and strongly treeable by the way it's a huge open problem whether every treeable group is automatically strongly treeable so that's it would be very interesting I think there should be counter examples but that's a difficult problem now of course in the same way as groups can act group poets can act and there is an obvious notion of treeability for for group poets and there is a similar obvious guess that group poets satisfy to co homology vanishing even only if the group poet is vanishing the even only if the group poet is treeable now there is a little nuance now let's get back to equivalence relations I just defined you treeability of an equivalence relation but I also defined you or gave a hint what is treeability of a group poet so meaning that the group poet is acting somewhere such that the orbit relation is treeable now I have to reinterpret this what this would mean for an equivalence relation I have now two definitions of treeability for an equivalence relation this one and that one now what is the second one meaning um so this was the original definition now if you viewed this in the group poet definition an equivalence relation would be treeable say in the group poet way if there happens something else if there is this orbit relation which would be another equivalence relation it sits higher up there is a treeable equivalence relation s on a space living higher up on a space y and then a factor map measure preserving from y on to x that happens to be bijective on the orbits of the relation now this is just a map that is a subjective and measure preserving yeah and essentially subjective up to measure zero okay now this is weaker so you could call it I mean I don't know whether this is a good notion but let's call it weak treeability just for one minute and now there is an equally open problem whether a weakly treeable equivalence relation is is automatically treeable and this is a question about equivalence relation if you restrict this question to equivalence relation coming from group actions then it's the previous question of course and that's in the yes exactly that's in the and that the sub relation of an equivalent a treeable equivalence relation is treeable which is a theorem of of them yeah and now the guess was that a group poet satisfies two-core muller g vanishing even only if it is treeable so an equivalence relation satisfies two-core muller g vanishing even only if it is treeable now there is some indication but that's something that I don't have time to discuss today instead of just looking at group poets and equivalence relations and groups acting on two one factors you can also have them act on carton inclusions that has to do with normal sub equivalence relations and so on that's another story and there actually you show that if two-core muller g vanishes in that category then automatically the group poet must be weakly treeable so there is a beginning of indication that this could be the the right notion but I cannot go deeper into that so that's all thank you the problem is even that there is no true cosmology theory so but we didn't think of it but it why not no that's right yeah