 I'm truly honoured to speak at Tomah's Fest and as same as many other people, I've benefited a lot from his creativity and wisdom and both influenced me a lot. So let me now open my slides and I should say that I really wanted to be there in person, but France has classified Amber by UK and my daughter is due to start school in a few days, so it was really practically impossible. Otherwise I would have been there definitely and I'm really envious of all the people who are in the room and capable to experience a conference in sort of normal conditions. Now, so let's see, I should share my screen, which means slides, okay, right, so this is it and I noticed that Zoom talks are not the most exciting and for this reason I got in the habit of starting a Zoom talk with some quotations, sort of inspirational and this time I've selected a hue that reminded me of Tomah in fact, so I apologise if the sort of recognised names of mathematicians will disturb him a bit, but really sometimes this reminded me of what he sometimes said and was very useful. So the first one is from Dalambe, who apparently said to his students, so that helped especially for some projects we seem to never end. The second one is from William Thurston is one of my favourites, mathematics is not about numbers equations, computations or algorithms, it is about understanding and the third one is from Hardy, sometimes one has to say difficult things but one ought to say them as simply as one knows how. Now to get back to the subject of what compactability one may find between the hyperbolic geometry so all the various degrees of hyperbolic geometry that a group may have and median geometry, I will begin with sort of the strongest and the most striking, cubulabel. So a group is cubulable if that's properly discontinues to co-compact on a cat zero cube complex and fundamental groups of hyperbolic three manifolds are cubulable and using this result and the theory of special cube complexes of Haddington wise, Jan Egol was able to prove the virtual hack and conjecture. So that was an important ingredient. So why am I mentioning cat zero cube complexes in relation with median geometry? I apologize of course to the experts who already know why but I thought it may be worth reminding why because of a theorem of Chapoen-Gerasimov stating that a graph is the one skeleton of the cat zero cube complex if and only if the set of vertices with a simplicial distance is median. So if we are given a cube complex, we can decide if it is a cat zero cube complex by checking if the set of vertices is median. Again, let me also remind what a median space is so it's a metric space such that every triple of points admits a median point meaning a point that is in between any two of them in this sense. So the sum of the distances is the distance between any of those three points x1, x2, x3. And as usual, the best way to check to understand that is to also look at other examples. So sort of the key example is the set of vertices of a cat zero cube complexes. Other examples are obviously a few trees, one can easily check the tripod, any tripod has a center, Rn with a norm L1 or the space L1 itself. And in a sense, the space L1 is universal because that's what prove that every median space embeds isometrically into an L1 space. So L1 space is played in this, in this case, the role of those universal real trees into each any real tree embeds. And another result that may tell you things about this, about this geometry is that in the realm of normed spaces, median means linearly isometric to an L1 space. So really L1 space as soon as we, as we work with normed spaces complete. So the best way to think of median spaces is as nondescript versions of, of zero skeletal cat zero cube complexes, in the same way in which we can think of real trees as nondescript versions of simply short trees. So that is the sort of the image one has to have in mind. And there is also a cat zero geometry lurking behind a median space, at least in a certain case for the so-called finite rank median spaces, because Bowditch proved that if we consider a median space that is complete and connected, and finite rank, one can always deform the metric in a bilibschitz way, so that we get a cat zero metric. And the set of collection convex subset space is the same. So what is the rank of a median metric space? It is the supremum, the supremum of the dimensions of the cubes, again, discrete cubes. So we just look at the set of vertices that can be isometrically embedded inside that metric space. So it's some sort of notion equivalent to the notion of dimension for cat zero cube complexes. And to simplify, we will always assume that median spaces are complete and connected, which also implies that they are geodesic. So why is the median geometry interesting? Because it relates to two important properties. So we proved with Indira and Frederick that a locally compact second countable group has property t, if and only if any continuous action by isometries on a median space has bounded orbits. While at minable is equivalent to the fact that it admits a proper continuous action by isometries on a median space. Another reason why this geometry is interesting is that a number of groups and spaces have median geometry asymptotically. That simply means that asymptotic cones are median with a certain standard deformation of the metric, obviously, because at least when we talk about finitely generated groups with a word metric, we have a range of metrics that are by-legitimate equivalent. So there's no choice, but there is a way of a standard way of deforming the induced metric on the asymptotic cone that will give the median metric. So why would we be interested in that, among other things, because the geometry of asymptotic cones is key in the study of compactifications of spaces of representations. So that means that for all the groups that have median geometry, the boundary of the space of representations will have representations by isometries acting on median spaces. So mapping class groups of surfaces with word metrics are median. We proved this with Jason Bierstock and Mark Sapir. Taishmiller spaces with a wild peterson metric, this is a result of Brian Baldic. And last but not least, in the theory of graphs, median, especially finite graphs this time, median graphs are relevant in optimization theory and computer science. So I did mention that there are various degrees of compatibility with the median geometry, and I mentioned the strongest, tubulable, proper discontinuous co-compact action on a cat zero cube complex. The next one, let's call it strongly medianizable, proper discontinuous co-compact action on a median space of finite track. So this is sort of the next best thing. And medianizable or weakly medianizable, the same assumption only we allow for any kind of rank. So proper discontinuous co-compact action on a median space of infinite track. Well, and there's also weaker, yeah, but we're just looking at co-compact actions. Well, well, obviously in the last two cases, strongly medianizable or medianizable, that means that actually we are looking at a median space that is locally compact. And since we are working with that assumption that median spaces are complete, then it is also JDZ. So in the last two cases, the median space that is acted on is locally compact, JDZ, and therefore proper and complete. So therefore proper by Hopchino. So obviously that's it goes from stronger to weaker. So one other reason why these various degrees of compatibility with the median geometry are interesting. Is the connection with various degrees of amenability. So I mentioned already one connection. So at amenability is equivalent to a proper action on a median space. I should also mention this other result that for the strongest version of compatibility, there are three cubulable groups. Goodner and Hickson prove that they are always weakly amenable with cowling haggard of constant one. I'm not an expert of that, but I'm going to recall the definitions. So first, the easiest definition amenable, the one we are all familiar with, there is a reformulation of it in this terms. So amenable is equivalent to the fact that there exists a sequence of positive, definite, compactly supported functions on that group converging to one uniformly on compact subsets. Amenable is equivalent to again the existence of a sequence of positive definite. Sorry, in the first definition also we have continuous if we're working with a topological group. Positive definite function instead of compactly supported, we have vanishing that infinity. And weak amenability, now for this one, we are no longer asking for positive definite, we're just asking for a sequence of continuous compactly supported functions on G converging uniformly. Obviously amenable implies amenable and amenable implies weakly amenable. The question that is still not completely clear is what is the connection between 80 amenable and weakly amenable? Right, and I should also say what is the Kowloon-Hagorov constant? It is the best possible upper bound for this norm and this norm is, so it is described below. So again, if the group is kubelable, then it is also weakly amenable. So kubelable implies weakly amenable. Now 80 amenable is equivalent to a proper action on a median space. So in a way this question that is still not completely clear, which is what is the connection between 80 amenable and weakly amenable, might be transferred in terms of median geometry too. What is the connection between a proper action or maybe a co-compact action on a median space and a co-compact action on a casual cube complex? So this is exactly what we're asking. So as I mentioned, the question of connection between 80 amenable and weakly amenable becomes a question of connection between medianizable, let's say, and kubelable. Now in between kubelable and medianizable, there was also the strongly medianizable, the case in which we assume moreover that the median space has finite rank. For this one, one may sort of relatively safely say that maybe they are equivalent. So maybe kubelable is precisely equivalent to co-compact action on a median space, a finite rank. So possibly we have this. The evidence for that is that until now, both classes of groups share the same properties. So they both share the t-saltanative and the superdigidity. So Papras and Saghev proved it for kubelable and Elia Fioravanti for strongly medianizable. And for instance, there is also some incompatibility, irreducible uniform lattices of isometes of products of hyperbolic spaces with at least two factors cannot act on casual kub complexes. If they act on a finite dimensional casual kub complex, they must have a fixed point. And Elia Fioravanti proved that the same is true for these lattices if we replace casual kub complex finite dimensional with median spaces of finite rank. Now what we proved with Indira is that these lattices, so the lattices here are at least medianizable. So they cannot be kubelable, they cannot be strongly medianizable but they are medianizable. So they act co-compactly on median spaces of infinite rank. And this is in a way the best that one can get for these lattices. So this is an example of groups like sort of familiar that are really in the weakest level of compatibility with median geometry while obviously being very strongly incompatible with either kubelable or strongly medianizable. And one should also mention, oh okay so again there is still one other question, natural question that one may ask in terms of median geometry of such lattices, can they act properly on an infinite dimensional casual kub complex? So just properly not co-compactly. As it happens, the result we have with Indira is also interesting for only one factor. So let me draw your attention to the fact that in our theorem we can also have only one factor and there are certain arithmetic uniform lattices that are not known to be kubelable and at least for some particular example of arithmetic uniform lattices in SO7-1 that are constructed using the field of octaves, at least some of the experts think that they may not be kubelable. In case we believe that kubelable is equivalent to strongly medianizable, maybe they're not even strongly medianizable, which means that the result we have with Indira shows that this is the best one can get in terms of compatibility, in terms of say kubelable, so non-discreet kubelable. That's the best one can get for such lattices, just the fact that they act co-compactly on a median space of infinite rank. And as I mentioned before, another interesting question related to these three degrees of compatibility with median geometry is, well, it's the equivalent of what was no, I mean, the same question for trees. So in the case of trees, the question was if we have an action on a real tree, can we extract an action on a simplicial tree? In this case, the question would be if we have an action on a median space, can we extract an action on a k0 kub complex? And under what condition can we do that? And as I mentioned before, this has a translation in terms of amenability, because proper action on median spaces, for instance, is equivalent to at amenability, while proper action on k0 kub complexes implies weak amenability. So if we manage to get a ribs type theorem, that means that we will manage to say under which condition at amenability implies weaker amenability. So, yes, so this is just an overview of sort of what I said. The initial ribs type theorem started from this question, when can one extract from the action on a real tree? Really not the line. And obviously with natural assumptions of minimal non-trivial, can one extract from such an action, an action on a simplicial tree? And the key assumption here was stable action, which essentially means that the family of stabilizers of non-trivial arcs satisfies the ascending chain condition. So, say the first example, the first theorem in this direction was the one of best enough pain, which says that if we have that the action is stabilizer is stable, meaning that I mean, that the ascending chain condition is satisfied for stabilizers of non-trivial arcs, and the group is finally presented. Then from the action on the real tree, we can extract the natural and simplicial tree with stabilizers of edges being stabilizers of the arcs in a real tree arcs by cyclic, so extended by a cyclic group. Actually, so most of the people who contributed to this are probably in the audience, so I should really apologize to them because they know this very well. So, CERLA later replaced finally presented with trivial stabilizers by tripods, and then actually the strongest rip-stab theorems are now due to Vansan, Vansan de Radel, but I didn't add those because there is a lot to be said in terms of improvements for rip-stab theorem. I just wanted to recall the basics because as far as median spaces are concerned, so first of all here I'm recalling the interest of a median version of a rip-stab theorem. First of all, it would relate the negation of property T with a geometric statement. So the negation of property T by our result with Indira and Frédéric is equivalent to the existence of an action on a median space with infinite orbits. So if we would have a rip-stab theorem, so that would mean under certain conditions I can extract from the action on a median space an action on a k0 cube complex that has pretty much the same features. So if that would be the case, then the negation of property T, an action on a infinite orbit on a median space under certain assumption will imply an action on a k0 cube complex. So it would be a geometric definition, like really purely geometric definition of the negation of property T and that we all agree would be nice. Obviously it has to be an assumption same as for the classical rip-stab, but still it would be very, very interesting. And then the second reason why a rip-stab theorem would be interesting for median spaces is that it would provide a connection between at amenability and weak amenability. So again, under some assumption. Now in this case, it is absolutely clear that an assumption is needed because the general statement is not true. So at amenability does not imply weak amenability in general. And an example showing this is the rip product of a finite group with F2. And this such a group is always at amenable. It is a result of Connolly Estadler Ballet. While a group like that cannot be weak amenable with Cowling-Hagerup constant one, this was proved by Ozawa and Popa. So one cannot hope for an implication like that without an extra assumption. But even with an extra assumption that would be interesting. And in this sense, actually, the result we have with Indira is bad news. Because it shows that a theorem like the best enough feign one is not going to work in general. So maybe an extra assumption is needed because uniform lattices of isometries of products of hyperbolic spaces are finally presented. So they are finally presented. They act on median spaces because they are amenable. But still, well, I mean, and we prove that they are they are medianizable. But so yeah, so they are properly minimally and co-compactly on median spaces. This is our result with Indira. But they cannot act non-trivially co-compactly and on casual cube complexes by the result of Chatterjee, Furnace, and Goethe. So for for these groups, so this shows that just being finitely presented is not enough, maybe. So maybe I mean, there is an extra assumption, but this type of lattices, for instance, are not going to satisfy it. It may still be possible to obtain ribs type theorems for actions on median spaces of finite rank. So as before, it looks like median spaces of finite rank have stronger connection with casual cube complexes. This makes sense because ribs type theorems work for real trees and real trees are a median spaces of rank one. So for those it works. And yeah, just just to mention, in the classical ribs type theorems, there is always one always needs to assume that the tree is not aligned. In the median setting, the natural condition is is the one that appears there. So it should be a median space with no global fixed point at infinity under the full isometry group and not within bounded house or distance from a space RN with the L1 norm. So that is very likely the assumption to make to rule out the real line case. But again, one needs to see what kind of stability is needed. And yeah, I mean, what what what would work? So where does so how do we how do we prove our theorem with Indira? So how comes that the uniform lattices of isomet is products of of hyperbolic spaces is active compactly on median spaces. So how is this obtained? Because the main ingredient actually in in our proof is this one is the this theorem here. So I should write actually in here that it's again, joint work with with India. And actually, maybe it's part of part of we were also using a lot of the results that we obtained jointly with with Frederick. So this is because the real the real hyperbolic space embeds isometrically and equivariantly with respect to the isometric group into a median space that is proper, that means that the balls are compact, all the balls are compact. And the embedding is so that the image of a 10 is a finite household distance. So we can embed a 10 into a median space that is proper. And that is just the thickening of a 10. And then that means that any group that acts by isometrically is on a 10 is going to act by isometrically is on that median space. And if my group was acting co compactly on a 10 or on products of a 10 is going to act co compactly on the median space or the product of median spaces and the product of median spaces has also a median geometry, it's also a median space. So this is this is the main ingredient that goes into into our proof. And again, actually, it shows that the real hyperbolic geometry. So the hyperbolic geometry of of the real spaces is is strongly compatible with the median geometry. And without going into further details, this embedding is embedding is constructed using another structure that is closely connected with the median geometry, which is the structure of measured walls introduced by Ayrshire X, Martha and Valet. So the walls in this case are the co dimensional one hyperbolic hyperplanes in a 10. And that is a way of introducing a measure on them. And this structure helps a lot with the whole thing. So with constructing the embedding and with many other things. But I will not say that. Right, so somehow I seem to have lost the complex hyperbolic case. So I should maybe then just mention that none of this works in the complex hyperbolic case. So in the complex hyperbolic case, actually, we proved with Indira that complex hyperbolic spaces can can so they do embed into median spaces. But they can it's never it can never be with their own metric. So it's always going to be with a snowflake metric. And the the image of the complex hyperbolic space will always be an infinite host of distance from the from the median space it embedding. So in the complex hyperbolic geometry, none of this will work. And the compatibility with the median geometry will become weaker. So this in a way is again, consistent with what we know. So if we look at the hyperbolic spaces of rank one, so we have the real hyperbolic space, the complex hyperbolic space and then the quaternion and the octaves two dimensional plane. So we know that we are basically moving from a team inability to property key. So the real hyperbolic space and the complex hyperbolic space isometry groups are a team in the ball, and therefore so are all their lattices, while for the quaternionic hyperbolic space and for the KLA octaves plane, we know that the group of isometries and all the lattices have property key. And in a way, the team inability gets weaker as we move from real to complex. So as you saw, the real hyperbolic geometry is has has a very strong flavor. I mean, it's very strongly compatible with the median geometry, while when we move to the complex hyperbolic space, it already becomes weaker. So again, in the complex hyperbolic space, the space itself cannot be as geometrically embedded into a median space. Instead, it can be if instead of the distance of the distance, the natural distance, one considers the square root of the distance. And in this, when the embedding is done, the image can never be at finite house of distance from the median space. So therefore a lot of the compatibility is lost. Now, this is just to mention what happens in the case of lattices and sort of in a way, these are the easiest examples or sort of the first examples of hyperbolic groups. And here is how their compatibility with median geometry looks like. Now, if we go to a much larger class that has a hyperbolic hyperbolic geometry, so the acylindrically hyperbolic groups. Now, I'm just recalling the definition to make sure we're all at the same page, but I'm sure that everybody knows what an acylindrically hyperbolic group is. It's acylindrically, a group is acylindrically hyperbolic if it just admits an acylindical non-elementary action on a gromo-hyperbolic space. And well, obviously, it's a class that includes non-elementary hyperbolic groups, also mapping class groups, except the virtually abelian and the groups of auto-automorphisms of flinolabelian groups. So it's a really large class and it's probably sort of the weakest kind of hyperbolic geometry. And therefore one cannot expect results as strong because those results have to be pretty general to cover all the groups that are here and many others. To begin with, there are a number of results that one cannot hope to get. So unlike the examples I mentioned before, acylindrically hyperbolic groups cannot act by affine isometries on LP spaces such that the orbits are infinite. So for instance, for lattices in the real and the complex case, they even act properly on LP spaces, but we cannot have a general result like that. Every acylindrically hyperbolic group acts by affine isometries on an LP space with infinite orbits. And the reason for that is an example of Minasian and Osin. So they proved that there exists a finitely genetic acylindrically hyperbolic group that is the quotient of all hyperbolic groups. And with John McKay, we prove that random groups, that means random hyperbolic groups, so within density between one third and one half, have the fixed point property for larger and larger classes of LP spaces. So if you look at the refined actions on LP spaces, they tend to have the fixed point properties as we increase the class of groups in the random model. They tend to have the fixed point property for more and more classes of groups. So that means that we have hyperbolic groups that have the fixed point property for LP spaces between one and anything and any P with P going to infinity. So that means that the Minasian and Osin example are acylindrically hyperbolic groups that have the fixed point property for any action by a finite geometry on any LP space. But there is even more than that, because later Tim Delat and Mikhail Delassal improved our result and proved that random groups satisfy the fixed point property for larger and larger classes of uniformly curved Banach spaces. So this class of uniformly curved Banach spaces was introduced by Gilles Pizier and he essentially, there is a definition. I personally don't find it particularly enlightening probably because I don't have the right background, but I think it suffices to say that it basically covers, no, not basically, it just covers all the known examples of super reflexive Banach spaces. So there is no known example of a super reflexive Banach space that is not uniformly curved. So besides the LP spaces and interpolation of LP spaces, subspaces of LP spaces, quotients of LP spaces, all those are inside this class. So it really covers all the examples that we know. So probably if there is an example, it's going to be very pathological and I'm not sure it's going to be very interesting. Now, because of the result, this result implies that even if we weaker, even weaken what we would like. So instead of looking at actions by isometrists, by a finite isometrists, we're just going to look at actions that are by a fine transformations that are uniformly bi-lipschitz. So what does that mean? So we're looking at actions. So this is known for isometrists. So isometrists on LP spaces and more generally on Banach spaces by an old theorem of Meyser and Ulam are always of this kind. That is, they look like the groups of isometrists of Rn. So they have a linear part and they have a translational part. So if it is an isometric action, that means that the linear part is a unitary transformation. Now, if instead of this kind of actions, we look at actions that are still a fine. So they have a linear part and a translational part. But the linear part, we're not asking for it to be unitary. We're just asking for it to be bounded. And we're asking that the linear part is uniformly bounded for the whole group. So that means uniformly bi-lipschitz or fine uniformly bi-lipschitz. Right. So this is a much weaker kind of action. So we may maybe then hope to construct for, to get a theorem of this kind, a cylindrically hyperbolic groups can act by a fine transformations with uniformly bounded linear part such that the orbits are infinite. Well, even that is not true. So it can't hold in full generality because of the same example. So the Minasian example cannot act by uniformly bi-lipschitz transformation on LP spaces such that the orbits are infinite because this example has the fixed point property for any uniformly curved Banas space. So if it would have an action like that by a fine uniformly bi-lipschitz transformation on an LP space, we would be able to to renorm the LP space. So in a certain way, such that the action becomes isometric. And then we would have an isometric action on a uniformly curved Banas space. And for this one, the group would have to have the fixed point property. So that means that for, if we look at the whole class of a cylindrically hyperbolic groups, there is really no hope if we want to create any kind of action on an LP space. But, okay, and then maybe before announcing what one can get, let me say another thing that one cannot hope to have. Again, some people in the room are super experts, so apologies for going a bit slower on this bit. So cylindrically hyperbolic groups cannot act properly by a fine uniformly bi-lipschitz transformations on L1 spaces. So we agreed that there is no hope if we look at LP spaces for P larger than one because of this uniformly curved business. Because they have this strong fixed point property on uniformly curved Banas spaces. But we can still look at L1. So again, let me maybe go back here. So this whole argument showed that if we look at actions on LP spaces for P larger than one, we are in the realm of uniformly curved Banas spaces. And therefore, we cannot hope to have an interesting action. All the actions have to have, there are cylindrically hyperbolic groups for which all the actions have to have bounded orbits. So the only sort of interesting space at which we can look is the L1 space. And that's nice, because as it happens, it's also a median space. So we're going back to actually a question of compatibility with a median geometry. So another thing that we can't hope to have for the whole class of a cylindrically hyperbolic groups, we cannot hope to have an action on L1 spaces that is proper. So we can't hope to have, obviously, we can't hope to have it for isometric actions on an L1 space, because proper action on an L1 space by our result with indirect feathery is equivalent to with at amenability. And the Minasian also an example shows that there are Icelandic hyperbolic groups that are highly non at amenable. Actually, they have all the fixed point properties you can think of. But maybe if instead of isometric actions on L1 spaces, we look at again, a fine actions that are uniformly by lip sheets. So again, rotational part, translational part. And for the translational part, for the rotational part, the norm has a uniform bound. Maybe then we can have a proper action. The answer is no, though. And in for this case, the example comes from graphical small cancellation groups. And this is because of the Gromov monsters. So this class of groups were introduced by Gromov with the view to construct groups in such that their Cayley graphs have uniform copies of graphs and, well, in fact, exponders into their Cayley graphs. Gornara gave a talk where she covered this in detail. I'm just mentioning this briefly for the sake of completeness. So Gromov used this technique of graphical small cancellation groups to construct the so-called Gromov monsters, that is groups that contain in their Cayley graphs uniform copies of families of exponders. And this was developed in the set of notes of Gornara and Thomas. And in particular, because the Gromov monsters contain weak copies of exponders, they can't embed uniformly into any LP space. That means that they can't act properly by a fine uniformly by Lipschitz transformation on an L1 space. So to sum up, if we want a result about a cylindrically hyperbolic groups, the whole class, we can't hope to have actions on LP spaces for P larger than one, not even a fine uniformly by Lipschitz. And we cannot have hope to have a fine uniformly by Lipschitz actions on L1 spaces that are proper. But luckily, at least one bit is true. Right. So I forgot to say here that Gromov monsters are a cylindrically hyperbolic. This is a result due to Grubber, Dominic Grubber and Alessandro Sisto. So with John McKay, we proved that all the cylindrically hyperbolic groups have a uniformly Lipschitz or by Lipschitz, it's equivalent a fine action on an L1 space with unbounded orbits, which is all that one can hope to say. Because as I said for LP, can't work and proper action can't work. So unbounded orbits, that is all. And in particular, that implies that the graphical small cancellation groups, in particular the Gromov monsters, have a uniformly Lipschitz action on an L1 space with unbounded orbits. Because such groups are, sorry, because such groups, so the graphical small cancellation groups are a cylindrically hyperbolic due to the result of Grubber and Sisto that I mentioned before, while the cubical small cancellation groups are a cylindrically hyperbolic due to the result of Archanzavan and Hega. So to sum up, in the case of a cylindrically hyperbolic groups, this is the only kind of compatibility with the median geometry that we can get, which is not surprising because really this is such a large class, so we could not hope for more. So there's no hope of isometric action on an L1 space with unbounded orbits, because it would have to have a fixed bounded orbits, I mean, by our result of, with indeed anthroporetic and by the construction of Minasian and Osim. So I can't replace uniformly Lipschitz affine by isometric affine. So yeah, so it won't have that kind of compatibility with the median geometry. It will only have this sort of much weaker compatibility with the median geometry described in this journal. And I will stop here. Thank you very much. Is there any question? Maybe online? I don't know. Alex, if you have seen something. I'm not seeing any question, but please unmute yourselves if you want to ask Cornelia something. Can you make more explicit the construction of this median space from the standard hyperbolic space? Okay, so I'll, I'll share the screen, I'll share the screen from my tablet, then that means that, so let me see. Yes, okay, so. Oh, right, so I have to stop sharing from here and share the screen for the other. So let me see, how can I do this? Yes, sorry, I'm more familiar with stop sharing and I'll start sharing from the tablet, should work? Yes, yes, perfect. All right, so let me explain a bit more. So in the case of, it is actually the, the construction that we had with, with Indira and for the very, it's simple. So HN, on HN, we have the structure of measured walls. So these are the hyperplanes. So co-dimension one hyperplanes. So this can be identified with SON1 mod out by SON-11. And therefore, this class has a measure, natural measure, coming from the hall measure, because the group is modular. And that gives a structure of measured walls. So let's call this, so this, this class of hyperplanes. So the class of hyperplanes H. And then we have this measure, new H. And now the way to define the median space is as follows. So we can see the, say, the set of, right. So, so then we have the following. Yeah, I'm sorry, I'll have to, I'll have to be a bit, to bore you with some definitions. So we have, we have two classes here. So H are the set of hyperplanes. And then we have the so-called set of walls. Sorry, so the set of walls, walls here are, so pairs of half spaces. So spaces, determined by the hyperplanes. So we're going to see them like that. Now clearly the set of walls is the same as the set of hyperplanes. So in fact, what I explained before, so if this is the set of walls, so I did mention that the set of hyperplanes has a natural measure. So, and so by this identification, we have that the set of walls will have a natural measure. And now we consider, we consider the two, so the two sheet cover that is defined by the half spaces. So consider also the set of half spaces. And these are, so well, I'm going to define them also by, I'm going to write H. And it has a measure that comes from the measure of walls, because it's just a two-sheeped cover. So it has a measure induced spread. And now we're going to do the following. So we define, we consider, say, the so-called admissible sections. Actually, I shouldn't put that's exactly the name, admissible sections. So these are maps from the set of walls to the set of half spaces. The sections, so it's clearly, so it's a section such that, so it has, so it always selects. So one of the two, and such that if, if Sigma selects a certain half space, a certain H, and we have another half space contained in this half space, then Sigma selects that one too. So it's a way of choosing whenever we have a wall that is a splitting into convex subsets, half spaces. Sigma is selecting one of the two with this condition of compatibility. So if it selected one half space, it's going to select all the half spaces containing it. Now a particular case of, so an example of such admissible section is when we have a point in the plane. So Sigma X is just the half space containing X. Now there is a bit of an ambiguity for the walls determined by hyperplanes containing that point. But if we, that's not really very important. So the half space containing X. So one can arrange that. Sorry, maybe I should write here. And now what we are going to consider is the, is say M of X is going to be the set of all admissible sections with the property that when we look at the set of, so we can look at admissible sections as a set of half spaces. Right. So let's maybe say that we fix, we now fix a base point. And here we consider all the admissible sections such that when we look at the symmetric difference between such a section and the section determined by a base point, this one has finite measure. And the choice, it doesn't depend on the choice of X naught, this set. And the distance on it is just going to be the measure of the symmetric difference. And this makes this space a median space. So if I can see the M of X with this distance, this one is a median space. And the embedding of, we can embed X into M X isometrically. Well, as you probably have guessed, I associate to every X the admissible section sigma X. And it's going to be an isometric embedding. And it's also going to be equivalent. So the group of isometries of X is going to act obviously on M of X. And the embedding is going to be a sum of X equivalent. Now, in most most cases, as I said, M of X is very far from X. But for X, any real hyperbolic space, we can say that M of X is proper. And that the household distance between X and M of X is finite. And that's what makes things work. I mean, there is a reason for that. But yeah, maybe I've tortured you enough with the definitions as it is. So this is the median space. The space of admissible sections that differ by a set of half spaces of finite measure from a section determined by a point. Yes, could I ask some sort of question? Okay, I'm interesting. If we have a hyperbolic lattice acting, for example, it is generated by reflections. Can we understand what will be the fundamental domain in this median space? For example, it is right angled polytop. Will we have some? Right. So another thing that we proved is that the half planes extend. So it's true. So the walls, yeah, there is more connection actually. If I can see that. Excuse me, if you can make your answer quick, because then we'll have to add something to the next talk. Right. So yeah, yeah. So okay, so then I won't write. I'll just say that the walls of M of X cut the space X into its own wall. So the traces are on X are the hyperplanes. So in a way, the hyperplanes in X extend to hyperplanes in M of X. And in fact, in the case of HN, I think they do extend very nicely. We did not quite prove this, but I think with a bit of extra effort, one could say that the, for instance, a group of reflections would act on M of X also by some reflections. It had to be proved. It has to be proved. So what I can say is that the hyperplanes that the reflection hyperplanes extend to hyperplanes in M of X. So I see it's very likely that then this group of reflections is going to be a group of reflections in M of X as well. Okay, thanks. Okay, I think that was a good time to stop. Thank you again. Very much.