 So, okay, I think it's time to continue. So, the next speaker is Cenk Sivu, and he will speak about the wrote and type coding for Clanian groups. Okay, so, yeah, firstly, thank you very much for inviting me here, and also thank you very much for giving me the chance to speak. So, the topic that I'll be talking about is not really related to either random matrix or point process, it's more closely related to egodic theory or random, or like random walks on discrete groups. So, the motivation is earlier paper by, oh yeah, so firstly, this is a collaboration with Sasha Bufenthoff and Alexei Klementko. So, the motivation is earlier paper by Bufenthoff, Klementko and series, Caroline series. So, it's about, it's some result about convergence of spherical averages on function groups. So, the main result of that paper is this theorem. If G is a function group, those are discrete subgroup of the isometric groups of a hyperbolic plan such that it has an even angle fundamental domain. So, in other words, it has a fundamental domain which is a polygon where the angles are of the form two pi over two n, and it has, its fundamental domain has either five sides or three or four sides and a non-compact or four sides but no two opposing angle in like pi over two. So, if you satisfy this property, then if there is a measure preserving action of G on a probability space X or any function on log L of X, the spherical average the function at two n steps and then let n go to infinity. So, this is the spherical average which is defined as of a function and it will just be the sum of like the average, I'm sorry, sum of all the group elements of a word length n and then G inverse of f. So, here S of two n is the subset of G consisting of a group elements of word length two n. Here the word length is defined using the generating set consisting of elements that sends a fundamental domain to an adjacent fundamental domain. So, earlier Alexander Buffenthal did this in the case of three groups but one of the key arguments in the case of three group is that there's a way to like code all the elements in three groups using a mark of coding which has a inverse and I'm going to define those concepts carefully later. And the idea is for this, the main idea of this one is that we can define similar inverse mark of coding not just for three groups but for a more general case for this kind of function groups. In other words, even though these function groups are generally not free, they are like good enough for this purpose. So, the main tool of one of the main construction of this of their paper is that they are using a coding which is a invertible coding of elements in some group. So, I'm going to define this concept. So, firstly, by a mark of coding of a group G with generating set S, we mean the following data. Firstly, we have a gamma, a finite directed graph and then a starting vertices and ending vertices. So, two subsets of, two non-anti subsets of the vertices of gamma. Secondly, we have a group element G on each vertex of gamma. And, thirdly, we have a map from... So, when we have the first one and the second one then we have always have the map. Map from all paths on gamma to group G sending a path, say I1, go to I2, go to In, 2G, In, G, In minus one, all the way to G I1. But for this graph together with these group elements to be a good coding, we will require this map to be a one-to-one from paths of length N to group elements of word length N. Well, because this is just like when we are using these codings we are doing averages, sometimes the one-to-one can be slightly weakened. In other words, sometimes just almost one-to-one would be enough. In other words, for some purposes, we just want, we want this map to be mostly one-to-one but like for some set with a really small number comparing to the size of the group elements of word length and they may be like two-to-one or multiple-to-one as long as those numbers are neglectable as N increases. This is also fine. So yeah, using the coding we can actually already get many interesting results about egoticity for the groups. But because this is like the goal is to get estimates on the spherical average is we require some additional information on the coding. Additional structure on the coding. So that's why we need to strengthen this concept into invertible coding. So invertible coding. The limit is in the, where? Oh yeah, yeah, yeah, yeah, yeah, yeah, yeah, yeah, yeah, yeah, yeah, yeah, yeah, yeah, yeah, yeah, yeah, yeah, yeah, yeah, yeah, equals just the limit that one would expect. So this is as square. So this is like the sigma algebra from a generated by subsets of x that are invariant on the s-square. So the limit that one should expect. So in particular, if there's no invariant subsets then the limit would just be a constant. And then this is almost everywhere and then in the sense of measure. And also L1. So, yeah, so in order to deal with something, this spherical average which is more subtle than a spherical average, for example, one need to add some additional conditions on the coding. So this is what we call invertible coding. So firstly we need an evolution on the graph gamma sending starting sets. Oh, by the way, there's a mistake in this statement. All paths on gamma, not all paths on gamma allowed. It has to be a path starting with some point in the starting set and ending at some point in the ending set. This is going to be useful later. So sending starting set to ending set and then ending set to starting set. And secondly, for every vertex, we need an additional group element such that G i equals h i if i is in ending set. And furthermore, we need that if there is an edge from i to j then h j g i equals g sigma j inverse h i. So this condition guarantees that under the, so if we have a sequence i one or the i two all the way to i sub n and then we apply the evolution sending them to sigma i sub n goes all the way to sigma i one. So this condition guarantees that, oh, mistake, still in the same direction. Guarantees that the group element corresponding to this string will equal, yeah, it's the opposite direction. The group element corresponding to this string will be the inverse of group elements corresponding to this string. So this is guaranteed by that statement. And also the fact that this is true. So this evolution will reverse the edges. The motivation of this coding in the case of the function group case, firstly, like one of the earlier motivations for this coding is the Bowen series coding for function groups. And then, but what's actually used in this argument because there's need for the coding being invertible is a kind of modification of the Bowen series coding by Roten. I think the original purpose for Roten to develop this coding is to put a central limit theorem for the intersections of geodesics in hyperbolic surfaces. But because it's invertible, it can also be applied to this case. So I will just first reveal the motivations like the basic ideas behind the coding that's used to pull this theorem. So firstly, there's the Bowen series coding. So if we have G function group with even angle fundamental domain, then we can say fix a fundamental domain like this P and then for any group element, G. We just, so firstly because it's even angle, we see that the union of all the sides of the fundamental domain will just form some, a collection of geodesics in the hyperbolic space. So we know, we can think of them as like the walls. So it's easy to also just buy some geometry, have a bulge of geometry to see that a path from a fundamental domain to another fundamental domain is a combinatorially geodesic. In other words, it's the shortest path even only if it passes through each wall only once. So yeah, the coding can be done as follows. So if we start with a P and then want to end up in another fundamental domain, we can just look at all the geodesics that passes through the size of P and then if there's only a single geodesic passing up between the block P and GP, then we just move this block, say if there's only a single geodesic between P and our goal, then we just move this block in this direction crossing this geodesic. So if there are multiple geodesics, we can just, we need to make a choice. So if in this case, we just apply the group element corresponding to sending this fundamental domain to this adjacent fundamental domain, while in the case where there may be multiple, we will need to make a choice. So in this case, we have to make a choice. So because of this process of making a choice, this coding is not exactly one-to-one. So it's one-to-one for most of the strings, but there will be, but firstly, it's a suggestion, but in a small set of group elements, it will not be one-to-one. And also, because of these choices, obviously not preserved if we reverse the direction of the words. In other words, if we have a combinatorial geodesics obtained using this procedure, the inverse of the combinatorial geodesic will almost impose, I think it's almost never possible to still keep the same choice. So this is definitely not invertible coding. But an observation by Rontan is that if we extend this construction a little bit, and not just consider the geodesic that respect the choice we made when we need to go through this kind of procedure, but consider just all the combinatorial geodesics. So if instead of considering one combinatorial geodesic, consider all the geodesics from a fundamental domain to its image on the G, then this construction is definitely going to be reversible because if something is a geodesic, then obviously of it will still be a geodesic. So like if we have this geodesic and this, then we just take the union of all the possible geodesics. And a very detailed analysis in the paper by Puffin-Touff-Klemenko and Siris will show that if we consider the union of all these geodesics, then actually they have, at them is a rather simple combinatorial structure. So this is what they call the Fickon paths. Some further analysis, those that, this set must be of the following structure. So we start and end at a fundamental domain and we are, we sometimes may just move from one fundamental domain to another. Sometimes it may branch out and then maybe stay branch out for a while and it may, this kind of phenomenon may happen. So in other words, if we consider the levels of vertices or if we consider all the fundamental domains in this set and divide them according to the distance to the starting point and the ending point into different levels, then between every two levels, there are only four possibilities. We divide the fundamental domains into levels according to the distance, start and ending point. What happens between two levels can only be one of these cases. Either both levels consist of a single fundamental domain or one level may have two and the next we have one or one have one, the next have two or both of them have two and then there's nothing or we can have this or this. So in terms of pictures, this is like if we have just going there, something like going here and for this one, something like this. So this should be for this one. So yeah, there's pictures for each of them and this is not sufficient and also because we want the coding to be mark of the other words, we want to make a guarantee that what happens in the next step only depends on what happens in the previous step. So in order to make that be true, they also analyze what happens to the boundary of these vacant paths. There is also explicitly geometric conditions. The boundary of vacant paths, which guarantees that like to check a sequence of geodesics connected this way is a vacant path. We only need to check whether or not when we connect to adjacent levels, the resulting thing is satisfied this geometric condition. We only need to adjacent levels. So this is their construction. So the motivation, yeah, so what we are trying to do for the generalization of this construction is that we are thinking about whether or not it's possible to generalize this to the setting of small cancellation groups. So there are many, of course, there are many small cancellation conditions. So the first small cancellation condition that we are considering is the C4T4P condition. One motivation for choosing C4T4P is because if we are in the case of even angle fundamental domain and we have at least four sites for each, so at least five sites for each fundamental domain, then it will satisfy C4T4P. So the meaning of this is that in the presentation complex, like universal cover of presentation complex, group G, any tool cell has at least four sites any two cells intersect at most one site. And in any disk diagram about which we just mean, yeah, this is the standard definition. So this diagram is just a cellular map, two-dimensional disk to the complex. In any disk diagram, any interior vertex has at least four edges connected to that. So yeah, so this four is this four and the four sides, this four is this four and P means any two cells intersect at most one edge. So in particular, if the function group itself satisfy the even angle condition together with the condition that it has at most, it has at least five sites for each fundamental domain and it satisfies the C4T4P condition. Actually at least four sites would be enough. So in other words, four sites in fundamental domain and also even angle. The way to build a presentation complex is that we can just consider the dual complex of the cell decomposition of H2 into the fundamental domains. And also if it's clientian group, then we just want it has a polygonal fundamental domain with each vertex having at least four edges and all dihedral angle less than or equal to pi over two. They also satisfy this condition. And yeah, the motivation. So in this setting, the idea is we are just mostly following the argument by Gersten and Schott. So they have two papers on the automatic property on small cancellation groups being automatic. So the construction we are doing is mostly follow their idea. So the idea is that for this C4T4P complex, we can just consider any two cell. So this is a two cell with five sites. We can decompose it by adding vertices in the middle, the two cell as well as adding vertices in the middle. And to the midpoint of edges. And we can see that this is, the result will be what we call non-positively curved Q-conplex. Then we get non-positively curved Q-conplex. And there's many tools to dealing with non-positively curved Q-conplexes. So the specific coding that we used, which is also inspired by the automatic structure constructed in Gersten Schott's paper is that we are considering the union of admissible geodesics. But of course, I don't think this is the only possible way to do coding in this case because there are many other tools in non-positively curved Q-conplex theory. So this is one possibility. So the idea is that we assign geometry to each Q and it into Euclidean Q. In other words, all the angles will just be right angle. Then a geodesic. Then a path is called admissible. If every, yeah, this is something that I don't remember. Every positive term must be followed by a negative term. So here the positive and negative is from the perspective of a disk diagram. So suppose we have a disk diagram. In other words, we have an embedded disk into the Q-conplex and then we have, so that the path lies on the sides of the disk diagram. Then this will be positive. So say this is the path, then this will just be the angle. So if the angle is less than pi, then it's positively turned. If it is greater than pi, then it's negatively turned. So if the angle is less than pi, then this is positively turned and greater than pi, then it's negatively turned. And this construction will give us a coding, but this is not an invertible coding. So we have to add some additional constructions. In other words, we have to also allow paths going in the other direction. But if we allow paths going in both directions, potentially the two paths can be really far from each other. So actually we do still need to, at the moment we do still need to assume that this is hyperbolic. Well, if it is further hyperbolic, then the paths in the positive and negative direction are going to stay not very far from each other. Then we can just do everything. Not exactly everything actually. So yeah, after building this invertible coding, we still need some further properties in order to make the argument for spherical average work. So the starting point will be firstly, we need to show that the coding, the graph is strongly connected and also a periodic. So in this case, if we just consider the original graph, it's not yet a periodic, but if we consider like the sub graph, then it will become a periodic. In other words, we consider some vertices and then allow, so yeah, in other words, basically we just consider the second power of the incidence matrix of the graph, then it will be a periodic. And also a strong connectivity argument is similar to the argument in the earlier paper. And next, we need an additional condition. Oh yeah, I think I don't have enough time to write down the additional condition, but the rough idea is that if we have a path of length two n, we want to break up this path into something that's of length n in the positive direction and something of roughly of length n in the negative direction and then they share a starting point, like a starting edge. So for this one, we are still not completely able to do this. So I think this is the main missing step for this argument to work. So yeah, since it's only three minutes, two minutes, I don't think I can continue with the next section, so maybe that's all.