 It's a pleasure to be here and to wish happy birthday to Tomá. In contrast with some other speakers, I am not a student or a direct collaborator of Tomá, but I just wanted to mention about 20 years ago when I came to France. It was very influential for me even to have some short discussion with Tomá in Strasbourg or in Paris. I will speak indeed about Poisson boundaries. There are several ways to define the boundary. We can consider a group and a random walk on a group, given a probability measurement. If I want to mention, don't hesitate to ask if you have any questions. Don't hesitate to interrupt me at any moment if something is not clear. And we consider random walk. So, at time n, we stay at position xn in our group, and we go to xn times something, small g, and small g is chosen with probability mu of g. And we define the boundary, for example, in the following way. When we have two trajectories, xy and yi, we say that they are equivalent if xy is equal to yi plus k. Let's say for any i greater than some constant n. We say trajectories are equivalent if up to a time shift they really coincide after some time instant. There are many other equivalent definitions. This is called exit boundary. This is one of the equivalent definitions of Poisson boundary. One can speak about tail boundary, not allowing time shift. We can just say really equal from some time instant. For general Markov chains, these definitions are a bit different, but for random walks on groups, this gives the same object. And there is also a way to define Poisson boundary in terms of mu harmonic functions. Maybe I mentioned it a little bit later. And our goal to understand what is the boundary of the group. So, something that may be more well known for this audience here. As you all know, if the group is hyperbolic, or at least if it has some hyperbolic feature, I don't say really hyperbolic. You have heard in the talk of Glenara this morning, many other definitions, which are hyperbolicly spirited. And in many situations, when the group is a bit hyperbolic, you know there are some geometric notions of the boundary. Here, one of the advantages of the Poisson boundary, this can be defined for any random walk, for any group. In some situation, when we have a geometric boundary, it is closely related, in fact, to some more geometrical or topological notions of the boundary. But in general, this is not a topological space. This is not a metric space. This is just some space with probability measure, which I haven't defined yet. So, we can see that the space of infinite trajectories, one-sided trajectories, is infinity. And we quotient over measurable hull of this relation, and we get Poisson boundary. One well-known question, which I would like to discuss today. First of all, given a group, given a measure on a group, we want to understand whether the boundary is trivial or not. This is the boundary trivial. And then, we want also to understand, given an element in a group, we want to understand whether the group acts trivial or not trivial on the boundary. I did not define the action yet, but given J in J, does it act trivial? Which elements can act trivial? And these are very well-known questions. We don't know, but in some situations, we know what happens. And I will try to recall some general facts. In many situations, we don't know. Even there are many examples of groups and random walks where it could be tricky even to determine whether the boundary is trivial or not. And I will speak about these questions. And today, I also want to speak about much more new questions concerning the action of the boundary. I will speak about, on one hand, free actions, and I will speak about totally non-free actions. And this is something, a new topic for random walks of life, totally non-free. Well, when I speak about free action, maybe I give already definition, precisely I will speak about effectively free, what is called a bit misleading term, but it's called effectively free actions. What do I mean by this? So it happens in some groups that some elements really have to act trivial. It happens even in some very well-known examples, hyperbolic groups, right? And there is no notion of hyper-FC-centre. Well, hyperbolic group is enough to define just FC-centre, hyper-FC-centre. And this is a generalization of notion of a center. So central elements which compute with everything, right? And FC-central elements which are elements that have and, finally, many conjugates. And so you can take FC-centre in principle, you can take ascending sequence of FC-central extension and what you get at the end is the hyper-FC-centre of your group. For example, in returning, as I said, to hyperbolic groups, taking limits is not necessary. It's well-known, right? There is this maximal normal finite subgroup. If just remark, if J is hyperbolic, hyperbolic, this is just one finite group. In general, it could be a larger group, obviously. And what is known that if you have all hyper-FC-central elements, it should act trivial on the boundary. So we have our group. We have really to quotient if you want something free. We have to quotient over, let's say, hyper-FC-centre of G, because this is something, the group that acts completely on the boundary. And you can ask. So if you have some group, if you quotient over some group that acts trivial in general, if you have G in G over H, Hx trivially on the boundary, then if the quotient over Hx freely, freely mod 0, so we always speak of the boundary as a probability space, right? So then we say that the action on the boundary is effectively free. It's not difficult to check that, for example, if G is hyperbolic, whatever measure you take, the action G acts effectively free on the pass on boundary. In fact, in many situations, not only in the hyperbolic one, it's quite a common situation, not so many things are known maybe, but it's very natural to expect in many situations to have effectively free action. So and maybe I mentioned a very general result. So first I recall, yes, I didn't mention some general things about the reality of the boundary. So first of all, which group admit measures with trivial boundary? Which groups admit measures with non-trivial boundary? So just some first theorem of, let's say, Furstenberg, and in the difficult direction due to Kamanovic-Wershik and no old theorem in the early 80s, Kamanovic-Wershik and Rosenblatt, independently proven by Rosenblatt at that time. As in code. By the way, even for hyperFC centrality, I will mention some things that in the last years, so one learned importance of notions related to hyperFC centrality, particularly in some work that I will mention now, but just the fact that FC central elements act trivially, so one of the reference also, so he didn't use this terminology. Basically, it's already in the old works of Azenkot also, but more explicitly in the work of Jaworski much later. And exactly this statement we explained, for example, in our paper with Vadim Kamanovic. Yes, I didn't say so among things that I will explain to you. So first I will speak about some joint result with Vadim Kamanovic. And then I will speak also about other results with Josh Frisch and maybe some other results as well. So this old theorem says that G is amenable if and only if there exists a non-degenerate, non-degenerate in the sense that the support of the measure generates the group, otherwise we will leave on a smaller group. Non-degenerate measure, such that the boundary of Poisson boundary of G-mu is trivial. So this is one of the equivalent definition of amenability. I guess you have seen some definitions mentioned in the Koldara talk, right, but one knows also very good that Poisson boundary also can be used to give an equivalent definition of amenability. So if you can't have a measure with trivial boundary, this means that the group is amenable. And if you have a non-omenal group, all random walks have non-trivial boundary. But I mentioned this is, as I said, theorem proven in the early 80s. And surprisingly, it took much more time for us to understand which measures admit measures with non-trivial boundary. So in particular, in the old conjecture of Comanowich Wierschek was asking, does every group of exponential growth has a non-trivial, has a measure with non-trivial boundary? It's important to say, has a measure. Maybe we'll discuss some, we'll know the examples a bit in detail. So for simple random walks, it can happen the boundary can be trivial, there is a whole world of what can happen. But for the moment, I discuss existence of measures with trivial and non-trivial boundary. And there is a remarkable result by Frisch Hartmann, Tommus and Vahidi Ferdowsi, proven, let's say, two years ago in the 2000s. That says that the g admits, countable group g admits mu with non-trivial boundary, if and only if g is not hyper fc-central, is not hyper fc-central. And without going in details, if you don't know this stuff, the most important for us, if you take a finitely generated group, the only hyper fc-central group, it's exactly virtually nilpotent groups. So the theorem says in particular, for finitely generated groups, the group admits a measure with non-trivial boundary, if and only if the group is not virtually nilpotent. It's well known for many years, for simple random walks, proven by Dinkin-Malchanov in the 60s, then in more general case, by Azenkott again and some other people. So we know very good for virtually nilpotent group, the boundaries, trivial, whatever measure you take. And the theorem of this for all of us say it's if and only if in all the other situation you have some measures with non-trivial boundary. And so this theorem proves non-triviality of the boundary using some quantitative observation on the convolution of measures. And in subsequent work of myself and Vadim Kamanovich, we study some class of measures closely related to the measures defined in this paper, a bit more general. And for a certain condition, we describe completely the boundary. So we give for some measures here, we give complete description of the boundary. So I don't have time to speak about this today because I want to speak about more recent results. I wanted to mention briefly some parallels. I really describe the boundary and this boundary resembles a little bit what happens in a tree. So in some sense what we prove with Vadim, maybe it's surprisingly for myself, that the convergence to the boundary for these particular measures resemble very much what happens not only in hyperbolic group, but really in a tree in some sense. But this is in some sense some tree with infinite valency and I don't want to describe it. Now I just wanted to mention some corollaries of this description, just some corollaries of this description. So we prove in particular, so one of the properties we prove here, so for any measure there exists this measure with no trivial boundary where the action is effectively free. So one of the things we prove. So for each G like this, for each G, which is not hyperepsicentral, what we prove with Vadim, in particular there exists measure, the action on the boundary is effectively free. Let us be quotient over hyperepsicentral, we prove the action is free. Now I want to mention some other results. So when it's free, first of all, as I said, for any group there exists some measure where the action is effectively free. Then for some groups, for free group for example, for any hyperbolic group, whatever measure you choose without any assumption on the measure, the action is effectively free. I would like to mention that some question about Poisson boundary remains difficult surprisingly even for free groups and hyperbolic groups. There is a well-known question unfortunately still quite open, even in the case of a free group or even in the case of a free semi-group. So what happens in hyperbolic group, what is well-known, you have your trajectories and they converge to the points of the geometric boundary. So they lie not so far from the geodesics and they converge to points of geometric boundary. And this is well-known, this happens for any measure. But a well-known open question that we take the exit measure, does it provide a complete description of the boundary? So in some situation under some moment condition it's known. So again for free group very old result, even not using this language, but description of the Poisson boundaries then for hypergroup due to Ankana and then for more general class of groups due to Kamanovic and again not only in hyperbolic, in many hyperbolic-like contexts you can have a complete description of the boundary in terms of the exit measure to the geometric boundary. But it's an open question even for free group, whether the convergence to the boundary, so in the case of free group it's just your tree, right? You converge just to the boundary of your regulatory. Does it provide a complete description of the boundary? No, this is not known. Do you mean by the description that there's full support or something like that? Just a complete description that, first of all, what does it mean that the boundary is trivial? Maybe I didn't say it explicitly. Everything is mode zero, right? So boundary is non-trivial, meaning there exists some set of probability strictly between zero and one, which maybe gives some definition. The definition of a new boundary, which is basically a quotient of the Poisson boundary, if there is some event that are well defined on this exit boundary, on this equivalence classes, this is a new boundary. So when you have convergence to something as it happens in the geometric situation, this means that you have a new boundary, so you have a quotient of your boundary. And as a space with measure, equal means really as a probability space, equal to this natural mapping is an isomorphism as a space of measures. There are many ways to define it. For example, if you prefer using a rather harmonic function, the reason Poisson boundary is called Poisson boundary is sometimes called Poisson first boundary, due to his great results about the boundary. But Poisson, because of the Poisson formula for harmonic functions in the circle, you integrate your boundary condition to get harmonic functions. Similar here for underbooks, a harmonic function you get from the exit boundary. You put a probability measure and you integrate, you get a harmonic function. Complete description here, for example, means all harmonic functions you can obtain integrating over this exit measure on the geometric boundary. Yes, but these were some side remarks. I simply wanted to mention that though we don't know completely what happens with the boundary for general measures in the Habapol case, what can be shown that whatever measure you take, you have something effectively free. There are some situations also that could be far away from hyperbolic. There are many reasons why the action could be effectively free. For example, it can be shown that if the group has, let's say, CMS property, CMS property, if it has countably many subgroups, one can show that if the group has CMS property, the action is effectively free, whatever measure you take really. And for example, maybe some monsters were mentioned, I believe in the previous talk, if you take, for example, some of Alshansky monster with only finite subgroups, some of these monsters have only countably many subgroups, or you can take a polycyclic group, for example, which have countably many subgroups. There are various examples of groups and there is reasons for the action to be effectively free. And in fact, until recently, so one didn't know any non-free actions and my result was Vadim. There exists group G and I mentioned some examples we studied. As G, you can take, for example, infinite, maybe I mentioned already, symmetric group over countable set. Or there exists G, which could be chosen like this. Maybe I mentioned already, alternatively, it can take some, G can be locally finite, for example. But there exists some G, such that the action on the boundary is not only not effectively free, but maybe something which is called totally non-free. I want also to explain you quickly some other results about solvable groups, so maybe I don't describe you in detail the definition, but totally non-free is a strong negation of being free. In particular, it implies, but it's even stronger than that. If you take two points, almost surely they have different stabilizers. Here, effectively free, you just have this subgroup, trivially all your stabilizers are just a subgroup. Here, on the contrary, you really distinguish the points of the boundary. That's what we prove for some groups, including infinite symmetric groups with Vadim. I'm tempted to ask the question, so I have to admit that this we cannot do so far. Does there exist a simple random book on a finally generated group with this property? Does there exist a simple random book with this property? So far, we don't know even if there exist groups with finite entropy, so I did not discuss entropy in detail, but for the moment, our examples are quite dispersed measures, and it seems interesting to understand, but I don't see the reason why it cannot happen for finally generated group on one hand. On the other hand, as I said, there are many obstacles for many classes of groups we know it cannot happen, right? But now I want to explain you something else. What should I leave? Yes, maybe before moving to another topic, I wanted to mention briefly that this notion of totally non-frequency was introduced by Wierschek around 2010 in his work on invariant measures in the symmetric group, and quickly after that around the same time, the same notion, but using another language which became popular now, the language of invariant random subgroups was introduced in the work of Abert, Wirak, Glasner and Poin, right? So this statement is a part of trying to understand what measures live on the subgroup of the group, right? So here in this story of this totally non-frequency, so we have the points of our boundary. We can consider the map from the points to the stabilizer, so each point we look, what elements stabilize these points, and that's what Wierschek studied, not for random books, but just for abstract options of a symmetric group. And so we have the image of the measure, right? So we have a measure on the boundary. In the case of the boundary, we have an image on the subgroups, and so this is among these questions. We want to understand the measures living on the subgroup. In the case of the Poisson boundary, these measures are not invariant. They are quasi-invariant, right? So you never will have a really invariant measure. And this is a much more general notion, which seems interesting to understand for general groups. But I want to move back to more old and very usual questions about the Poisson boundary. As I said here, for this a bit exotic so far, maybe not exotic at all, property of total non-freeness, with Vadim recently we construct even some locally important examples, right? And if we say not locally important, but important as I said, this is a very old result, if you have an important group, the Poisson boundary is always trivial. Well, the important is like this, but if you take a soluble group, the question is very much open. We don't understand good. What happens with the Poisson boundary even when it's trivial and when it's not? So I remind very old examples of Kemanovich and Wyszyk. Probably everybody knows reef product, right? You have a reef product, lamplighter group, and if you take D1 or 2, if you can see the simple random walk, the growth of the group of course is exponential, but the boundary is trivial, and if D is equal to greater than 3 or any group of not quadratic growth, then the boundary is non-trivial, but the group is amenable. So as I said, the existence of the measure is important, not enough to consider just simple random walks, right? Non-trivial. The reason why the boundary is non-trivial, probably everybody knows, right? We walk in that free, for example, which is the random walk is transient, and so we look what happens with the lamp. For the simple random walk, we can change the situation with the lamp only when we visit a point. So meaning we see since the random walk is transient, the lamps converge, right? So for each point of the space, at the limit there is a well-defined notion that this lamp is on and off. There is this limiting operation of the lamps, but it was an occasion for quite a while, whether it's a complete description of the boundary, for D starting with 5, it's my result, and in a general case, for simple random walks, it's D greater or equal to 3, it's the result of lines and pairs. Lines and pairs. Two, three, I don't know, four years ago. It's lamps, complete description of the boundary, for simple random walk, there are still open questions, really, to characterize when it happens lamps, complete description of the boundary. And yes, maybe I mentioned, yes, still speaking about effectively free action, so another result with Vadim. If you have a reef product with something, and we have some mild condition on the measure, which is true, for example, for any simple random walk for a group with non-trivial boundary, we assume that there is convergence, the measure mu, such that there exists convergence of the lamps. For example, for any simple random walk on Z3 with Z2Z, with Vadim we prove the action is free. The action on the boundary is free. Again, here under this such general assumption we don't know complete description of the boundary, and in some cases it's really not, even if it could be more complicated than the lamps, but at least we can analyze the action on the configuration to conclude that the action is free. And it's a natural question to ask whether it's always effectively free. Well, if I said, since I didn't say whether it's trivial or not, let's say effectively free is effectively free. It could be trivial, for example, but anyway it's effectively free. And as I said, but on some other examples, as I mentioned here, our main result on symmetric groups and on some local unipotent groups, it could be totally non-free. But now I wanted to mention something about explaining our work with Josh Frisch about solvable groups. So for some solvable examples like this, we know for quite a while what happens, but there are many solvable groups for which we do not know what happens. And we don't know, so just I wanted to some remark about reef products. So one way to see reef products, one can write like this, we have the D, let's take D variables. You can take independent variables or you can take algebraic independent complex numbers, whatever you prefer. And you can see the such matrices. So you can see the, maybe I'll write it like this, and the matrix is one. You can see the two times two matrices. So this one corresponds to the lamp and this one generates a billion groups of D. This is just one way to explain what is a reef product, right? And Poisson boundaries for linear groups were studied in many cases, but it was not known in an open and still, not known even for solvable groups among linear groups, whether the boundary, when it's trivial and not. So just a remark, so, of this remark from, take a linear group, right, over some field. But this alternative we know, either this group contains a free subgroup, so in particular it's non-nominable, right? Either G non-nominable or G is virtually, virtually virtually solvable. To be more accurate, at least infinitely generated case, it's virtually solvable, but otherwise it's also more or less virtually solvable, right? And on one hand we know this, in non-nominable case, as I mentioned, we know in any group the boundary is non-trivial, right? If you're only interested in the triviality of the boundary, we know what happens if the boundary is non-trivial, but what happens if the group is amenable? Then if the group is solvable by Milner and Wolf, we know very good, right? What happens with the growth of the solvable group, right? So either the growth is exponential, and in this case there exists actually a free sub-semi-group. Either growth, let's say, either growth is polynomial or G contains a free semi-group, right? So we see by these classical results of Milner and Wolf at this, right? There is a very simple reason for a linear group to have exponential growth. Either it really has a free semi-group and obviously any group, not only linear, right? Containing free semi-group has exponential growth, or the growth is polynomial, right? And one can ask, can we find some reason for triviality of the boundary? And so far, nothing was known in this direction, and I want to explain my recent results with Josh Frisch, which says the following. So we have some linear group, so we have some n times n matrices. And I will mention at the end some more general question that makes sense even for non-adminable case. We have even much more open questions, but for the moment, we discuss just triviality and triviality. As I said, we know by this alternative the group is virtually solvable and by Marx's theorem, there is a finite index subgroup, so the group is apatriental, right? So I have some 0, 0, 0, and something over there. And so our strategy is like this. So first of all, whatever the characteristic of the field we have, we prove a theorem reducing any linear group like this to a particular metabilian two times the group of a special form. So one of our main results is like this. So we have some measure mu on this group. Let's say a finite entropy. I guess many of you know what is the entropy. If you don't know, take a simple random box. This statement, first of all, is important even if you consider simple random box. And we say that the boundary of j mu is non-trivial if and only if. There exists some i and j, such as the following hold. So we take some i, some position, i and j, so we have here i and j. And we consider something we call i, j block of these measures is the following thing. So we consider only four positions over here. If we have any raised zeros, we consider these two elements on the diagonal and the elements here corresponding to this coordinates i and j. And we consider the following two times two matrices, such to this linear group, we associate the following group. So we consider two times two matrices and on the diagonal we just put what elements we have here on diagonal if we have here something j, i, i. Here we have j, j, j. We just consider the diagonal two times two matrices like this. And now we look what happens, what we like to put in this position. Then we look in this position and we have to consider the following order on the positions. We say when we have these positions we consider partial order. We say... I was confused what I like to choose, bigger or larger. So we consider the order like on the picture. So when we move here it becomes smaller. If we move here in the vertical position it becomes smaller. And what we need to consider, as we say, there is a valid block in this position i, j if the following holds. If for this i, j there is some element in our group which has zero in all elements in all upper elements over diagonal which are bigger in this position i, j and some non-zero element here. So we need here... So we need something here, something non-zero here. Then we want zero, zero, zero here. Well, here on diagonal something. We don't care here, we don't care, right? So if there is some element in the group we say there is a valid group for this indexes i, j. And if the group is valid for these two associated two times two matrices we consider just matrix with one. One, one, zero, one. Not so important what non-zero element we can have here and it may happen we can have several elements here. It really influence the structure of our group. So this procedure may make sense even if we studied these two times two matrices. There are interesting two times two groups for which one didn't know whether the boundary is trivial or not. So here something that makes this reduction we make even if the matrix was already two times two matrices because we choose only one single element and we put one in this position. Yes, so I didn't finish the statement so we say our boundary on our linear group is non-trivial and only if the boundary is non-trivial if there exists i, j and a valid block in position i, j such that the boundary the boundary of let's say associated I will explain what it means associated measure on this block let's call this block B i, j on this group B i, j is non-trivial what do I mean when I say we take an appropriate measure on this block what only matters us is the projection on our billion group, right? Our group has an important subgroup and a billion quotient correspond to the diagonal so here there is this a billion quotient so what is important when we reduce our group to these blocks we need to consider the measures that have the same projection on this a billion group and we can state there are several definition we can say there exists here we can... so actually the non-triviality on the block wouldn't depend on our choice of associated measure so we can formulate a theorem the boundary is non-trivial even only if there exists some i, j and a measure appropriate measure on this block such that the boundary is non-trivial or we can work closely related from mentioned if there exists i, j and for all associated measures the boundary would be non-trivial so this is our first main step that works over any field so my conclusion I have a few minutes right so do I have five minutes or something you can take more if you need so this is the general statement that works for any field so there are two cases the characteristic could be positive or it could be zero now maybe I will try to in the remaining minutes briefly that in positive characteristic case we give a complete description containing many new examples both with trivial and non-trivial boundary so maybe I am explaining characteristic zero characteristic zero characteristic p first what happens in characteristic p just for example this classical lamp light or two lamps there is a particular case of characteristic two right so even characteristic p we have many interesting examples and some are really more complicated just if product but what we prove in characteristic p what we do next right we made this reduction our goal then to understand the reality of the boundary of the whole group is always to understand the reality of the corresponding two times two groups so if you want to make a classification for two times two upper delinquent groups to understand whether the boundary is trivial or not and so so maybe I don't have time to explain the criterion but I mentioned some corollaries from our complete description maybe one is also equivalent description actually so in characteristic p what we prove finally that for our group G so the boundary the boundary is not trivial is not trivial if and only if there exists a block I.G. and in this block we have just three dimensional reef product inside so a priori with some energy with what we know to grow as I said when we discuss the growth one direction was obvious when we have a semi group for example free semi group obviously the growth is exponential here in the situation of Poisson boundary it's a well known open question we don't know first of all whether the reality of the boundary depends on the simple random walk on a group unfortunately for many situations we know it's the case but it's still an open question whether we can have account example and there is also a related open old open question we have some subgroup with non-trivial boundary for example this one a priori even in this direction it's not clear even if you have a group which contains this as a subgroup should the boundary be non-trivial maybe yes but we don't know in general but here what happens for the linear group after all so we have this classification so after all everything is reduced to check whether in the block we have some standard object like this and then the boundary is non-trivial otherwise it is trivial and some further remarks in particular this shows that for these groups for example for this linear group of a characteristic P, the reality of the boundary for simple random walk doesn't depend on the choice of the simple random walk rather of this characterization this also one can also using as a primary inequalities for some new inequalities but related to further functions of these groups moreover one can show that if among these groups under this assumption if we have two quasi-asometric groups if for one group the boundary is simple random walk is non-trivial for the other it also will be non-trivial again in general it's an open question for graphs there are these old count examples so if you have two quasi-asometric graphs it was first 30 lines many years ago who has shown you can have quasi-asometric graphs with trivial boundary one the other is non-trivial further examples were constructed later by Benyamin who constructs even some polynomial growing examples but for the groups it's an open question but at least in this class of groups everything is stable now it's just maybe a remark some caution so in the block so we reduce the question even only if in the block there is some standard object just attention this subgroup in the block is not a subgroup not even a section of our group right so it's important we say there exists this block but we associate this block and we check do we have this three-dimensional product or not and this is what happens in criticism p now in criticism zero I don't have time to explain we have some partial results in criticism zero and we have some conjecture in criticism zero and which shows in criticism would be very different so here in some sense everything so the only blocks when we reduce the blocks we look really on these three products and more or less we get this well-known picture with these three products we know which are trivial or not so one can speak about the rank I don't really define the detail but just intuitively rank for if product for example if you have zd the rank is d and here if rank is at least free the boundary is not trivial if rank is one or two the boundary is trivial what what happens in criticism zero although we do prove even in criticism zero we give a sufficient criterion if rank is free with new examples if rank is free then the boundary is not trivial if rank is one the boundary is trivial but in rank two is more subtle in criticism zero case because in rank two the boundary can be both trivial and not trivial and just as a formulating our conjecture that the boundary would be would be trivial only if when we look the block of rank two it will be really two dimensional product so here here in criticism zero all rank two case after all are the same for the behavior of the boundary but in criticism zero two-dimensionary product we know of course that to have trivial boundary and if we look to the blocks we conjecture there is nothing that can happen so I guess I have to stop here Thank you Are there questions? Yes, there are It's not that certain groups which are not important they are there are certain projects and certain bigger groups like this Maybe Yes Maybe you could repeat the question because I'm not sure that people online but if Anna you could repeat the question I would be fine I didn't say that question yet Questions If you have this measure which is the trigger and etc can you make a subject of certain groups which are not important I try to repeat the question I'm not sure if I understood correctly so maybe first an obvious remark which I forgot to mention if you have a group and you have a quotient probably Tamar has this in mind so if you have a quotient of the group with non-trivial boundary the boundary of the group is non-trivial so if you map to something with non-trivial boundary then then the boundary is non-trivial but here I hope I don't say something stupid but here as I said so you don't that's why I mentioned sections you don't necessarily have it as a section you don't necessarily map it to I mentioned leaf product but if I say it correctly you don't necessarily can map your group No here it's a good question you certainly cannot map to a leaf product in general even in the tabillion case in these blocks you can have a block of rank 3 not mapping to the corresponding leaf product but your question can you map to a metabillion group with non-trivial which is not just very important I need to assume group of finite index which maps onto a metabillion group Yes but we want preserve the boundary of the boundary so my question is if you assume that your server will go as non-trivial can you make your metabillion group also as non-trivial it's a good question right so these blocks are not quotients right but it's good to check in it whether you can map to a metabillion one as I said what I know you certainly cannot map to a leaf product but it's true you can yes this is a question I should with Josh we should know the answer but we did not check it actually so this is a very natural question we were thinking about but I'm not completely sure about the answer in this situation can you find a quotient to the middle you ask not only for linear this is a very interesting question I understand that all right the boundaries of the boundary in many solvable groups so it could be a conjecture but I don't have a count example I guess even in general solvable context your question if you have a solvable group maybe I do have I have to think a little bit so you ask you have a solvable group with non-trivial boundary let's say for a simple round of work can you map to a metabillion group such that the quotient also have non-trivial boundary or we are up to find it never mind I have to think I think maybe the answer is no but I have to think before saying something but thanks a lot this is a very relevant question maybe I made some comment related I forgot I wanted to make this comment at this conference but forgot during the talk so some comment about this theme of office so they provide a measure with non-trivial boundary but another question whether this boundary can have a nice decay and this is an open question I just wanted to mention very briefly the story I didn't discuss here but my very old more than 15 years result ago which constructed some measures with non-trivial boundary on some trigger to groups I just wanted to mention in that old work that I think Toma in my paper because he some lemma explaining this theory about decay of the measure that can be chosen with non-trivial boundary I think Toma at discussion the scheme improved the condition of this lemma and but some much more recent results for all Grigorych group after all we know a measure with some finite moment alpha moment certainly it couldn't be first moment because growth is sub-exponential but the question we asked recently on any group it makes sense to ask does there exist a measure not only with non-trivial boundary in this general theorem but can we have a nice measure nice measure with some alpha alpha moment which is strengthening of well-known gap conjecture for growth right but I don't have the time for that next time I guess other questions Alex on this side is there any question online I don't see any question online if there are people online who want to ask a question now well if there is no more question I think we can thank another guest