 So it's a great pleasure to give a talk here and a great honor for me to celebrate Anna's birthday. So I will talk about this large girth graph, so it's bounded diameter by girth ratio. And this is our plan. So the goal is to construct new explicit sequences of such large girth-digi bounded kelly graphs. So we would like to have explicit and kelly. And the plan is this. I will explain a little bit exactly, actually the definition, what we would like. My motivation, because there are probably many different motivations for these graphs and the construction. And a little bit about applications and interesting, in my opinion, open questions. So let us start. The girth of the graph, you know that this is the length of h-length of the shortest non-trivial path. Diameter is the greatest h-length distance between any pair of vertices. So quite clear definition. And what we would like? We would like to have an infinite sequence of finite kelly graphs. Sequence of graphs is infinite, but graphs are finite. Such that the girth, so for every finite graph you can compute girth and diameter. So we would like to have girth in this sequence going to infinity. This is one condition, this is large girth. And the second condition is that the diameter of the girth, you would like to have a uniform bound, not depending on where you are for the sequence. So this diameter of the girth, the uniform bound, I will call it dg, diameter girth, dg-bounded graph. So we would like to have large girth, dg-bounded graph. Okay, this is a goal. Great. So this is a goal, so I ask you immediately, okay, give me an example. Give me an example, the first example. Circle. Circle. So you have, yeah, circle bigger and bigger and bigger. So girth, don't run in front of the train, yeah. So, yes, so for the moment I say just I'd like to have these two conditions. And of course this is a basic example, yeah, degree two. But of course, as it was mentioned, degree two is too simple, yeah, too easy. So we would like to have a degree bigger than two and a little bit, you will see, and many of you know probably that why we would like to have a degree bigger than two, yeah. This is, so the goal is to have such a sequence with a degree of graphs bigger than two, okay. This, as I said, calligraphs, so they will be regular graphs, so not growing degree, okay, fixed but bigger than two. Okay, great. So what about, so you remember, we have two conditions, large girth and dg-bounded. And we would like to have explicit and cally. So, okay, what about the first condition, large girth? So, a lot of people did this in combinatorics and the first probably think people cite in this area, combinatorics cite in this area, this is a result of Erdos and Saks probabilistic construction. So, you take the model, Erdos and a model random graph and there is a little computation you can try to do. It's not very complicated to prove that actually there exists a graph of the given girth you want, yeah. Okay, there is another reference, same year but only by Saks. And this is an interesting reference actually, sometimes it is, so many people cite this Erdos Saks, but not many people cite this another paper by Saks, where there is some recursive construction and recursive on the degree actually, so it's not really, we are not interested in this kind of stuff. But why I cite it also, because it seems to me, this is a side remark, it's the first appearance of kind of zigzag construction. Probably some of you know zigzag construction for expanders. And in this paper it's really zigzag construction he's explaining, but the goal is to produce graphs of large girths. Okay, so another way is to do a mod to cover of a given graph, so you take a graph, you all know that the fundamental group of a graph is a free group, yes, and you take a rank of this free group and take a group of two elements, direct product, how many times exactly this rank, and take the finite cover of this graph with this deck transformation group. This is a very quick definition of mod to cover of a graph. And the graph you will get here, which is a finite cover of the given graph, will be a bigger girth, and you can, by construction, just to see that the girth is growing. So if you do enough, you will have a big girth, and if you start it with some regular thing, you will get something nice. Okay, so this is another way to do it, and probably easiest way now, but you see the result is from 98 by bigs. The easiest way for us is just to remember that a free group is rigidly-finite, and remember that rigidly-finite or fully rigidly-finite is the same thing. So if you take a definition of rigidly-finiteness, like you take a non-trivial element and it will survive in the finite quotient, or you take the whole finite ball in the free group, and it will survive in the finite quotient, these two definitions are equivalent, of course, because the direct product or finitely many finite group is finite. So it means what? It means that if you start with a free group and you start with a ball, in the ball of course there is no any cycle, and you will get some finite group where you will have exactly this ball, so no any cycle, so you will have a low bound on the girth, and by taking a bigger ball you will get a regular graph with a bigger girth. This is the easiest way to do this construction. But this was about large girth graphs, and we would like two conditions, right? We would like large girth and also diameter of the girth bounded. So what about diameters in these constructions? And actually it doesn't look okay at all. In all these constructions you can estimate what is happening, but it will not give what we want. So, okay, now how we can still have what we want, large girth, gg bounded, so there is an interesting result of bigs. This is about so-called cages, this is a terminology from combinatorics, there is a whole subarea in combinatorics where they study cages, so graphs of smallest size of a given girth. So you fix girth, and among all graphs of this given girth you take with smallest number of vertices, so cages. So there is this nice result, a very easy result of bigs, which says that if you take our regular graph with minimal size among all graphs of a girth at least g, then the diameter is smaller or equal to girth of the graph. So in particular our bound will be 1, right, diameter of the girth. So let us try to prove this result, I think it's very nice and very easy to prove. So suppose you have this graph, gamma, and then you take two vertices in this graph. So you take this graph at, yeah, the distance g plus 1, and so we are in gamma and we have some edges here, right, some edges here, and we would like to produce a new graph, gamma, I call it gamma 0. So what we do, we remove these vertices, which are at distance 2 plus 1, we remove these vertices, we remove all these edges, but we add new edges. So for all vertices incident to these edges to V, these edges we make a new one, for example, this is just an example, right? So in such way we do, and we get a new graph, gamma 0. So what we can say about this graph, so the claim is that the girth is at least g, again the same g, little g here, so how you see this? Okay, so you would like to estimate the girth of this graph, so you would like to take, okay, estimate the length of the non-trivial cycle, right? And suppose you take a cycle and you have a new, this new edge just once, and something happens here. So what is happening here, this is coming from the graph gamma, right, and this is a new edge which is in our graph gamma 0. So because this is in the previous one, and by construction you remove these things here, right? So the length of this is at least g minus 1, and then plus 1, so g. And then of course you have to consider, this is just the situation in the first case, you have to consider when you have also a situation when you have a cycle, when you have suppose two new edges and something happening here, and by the same just easy picture you see that the length of these things is at least g minus 2, and then plus 2, and okay. So it's very easy argument to show this thing. So it means what we show, this new graph gamma 0, it has a smaller number of vertices because you removed 2, right? But the condition on the girth is the same, at least g. So by minimality, so it means that the graph we started with was not minimal, okay? Clear argument, quite easy. Yes, so we can say we are happy with this result, and why not to use it to produce what we want, yeah? And actually it is really, many people work in combinatorics on these cages, in all kind of algorithmic sense also, but it is very, there are no actually many explicit examples, so mostly I listed them here. So some descriptions of cages are known, but for small values of degree or girth. So they are not explicit, neither explicit, no calligraphs. So we would like to have explicit large girth-dg bounded calligraphs, and actually there is a classical, the first classical example, and this is a construction of Margulis, and very nice construction. So you start with two matrices given there, right? Two matrices we all know, we like them. And the result of Margulis was that the calligraph of SL2FP with respect to these generators, so P is running actually, P is running, so these matrices, they are matrices in SL2Z, but then you take a modulo P, prime P, and P is running, okay? So each time here there are two, actually two results. First, that this matrix is when you take the modulo of P, they will really generate this group. This is the first result. And the second result is that the girth will go to infinity. Okay, there are two parts. And in addition, by the proof actually, he saw that the girth is going to infinity and how it goes logarithmically. And this gives this terminology logarithmic girth, when the girth is at least logarithmic in size of the graph. Okay? So when I say logarithmic girth, this will be this condition. Great, there is another example, explicit example known, and this is Ramanujan graphs which were mentioned. So this is a famous construction of Lubbocki-Philips-Sarlak, also matrices of size 2 by 2, specific matrices, and specific generating sets. Again, the same thing that there are two ingredients that each time this matrices really generate the group. And in addition, the girth is going to infinity, and actually in addition, it goes logarithmically. Okay? So in both examples, we have 2 by 2 matrices, and they go to infinity, the girth is going to infinity logarithmically. Okay? And of course, you all know that in addition, this is not our goal, but in addition, in these two examples, these graphs are expanders. Right? Our goal was like a large girth, a dg-bound, with no expansion in the goal. Yeah? But here, 2 explicit examples. Okay? I remind very quickly, it was already said during the conference, but the definition of expansion. So we define a suprimetric constant for the graph. Right? Any finite graph we can define as a suprimetric constant, like a further ratio. And then an infinite family of finite graphs is an expander. If there are these three conditions, the degree is uniformly bounded for every vertex, and the sequence, it is a sequence of growing graphs, growing in size, and the suprimetric constant is uniformly bounded, uniformly isolated from zero. And there is a general result, and actually it's a nice exercise to try to prove it using this definition. Really a nice exercise. It's not like it is done usually, but it's easy to see using this definition, that all expanders, they have logarithmic diameter. And the constant here, you all know this notation, big O notation. There is some constant involved. And the constant here depends, of course, on the expansion, so on this epsilon. Okay? So, okay. So, as I said, these were explicit examples, and they were in size 2, meaning the size of matrices. We'll say dimension 2, 2 by 2 matrices. And this, mostly up to now, there are some kind of versions, but on the same constructions. So, these were what we knew up to now about explicit large graphs, DG bounded calligraphs. Okay? And as I noticed, if you use the expansion property, and you know already logarithmic graphs, then automatically you get DG boundedness. Right? So, this is, if you allow yourself to use expansion, you have logarithmic diameter automatically. This was not our goal at the beginning. The problem we posed, just large graphs, DG bounded without any expansion involved. Okay, so I tell you my motivation. Why I posed this question like this? Motivation, there are two reasons. So, this geometric group theory reason, because we would like to use these graphs to construct monsters. And these are some examples of monsters. And actually these kind of graphs, so not necessarily expanders, large graphs and DG bounded, they come as input to these constructions. And in one of these constructions, I write it later, it's also important to have calligraphs. So, we define, we start with these kind of sequences. We label them how we can, so randomly or not. And then define groups generated by, generators will be labels, so finally many labels, finally many generators. And the related will be all words we read on cycles of these graphs. So, these are infinitely presented groups where related words which you read on cycles of these graphs. And to make the whole construction work, we really need these two conditions, and DG bounded. For the existing construction, of course if somebody invents some other construction, but for the existing construction, really these two properties are unnecessary. This is why I was motivated to make explicit constructions like this, because in which the construction in a side result, right, it's exactly the assumption, the first, right, your assumption on diameter. For general construction, you mentioned Gromov's construction there in a weekly sense, you don't necessarily need this condition. You need, you need. It is a condition formulated by Gromov at the very beginning. Both you need DG bounded and large graphs. It is written and it's not outside this invention at all. So, it isn't necessary. For isometric embedding, it's outside this result, right? I will mention probably this result, but this was not... I mean, it is originally formulated like this in Gromov's construction. But in Gromov's it's not isometric embedding. We are not talking about isometric, or not isometric. In the question for isometric embedding, your condition is a sufficient assumption, right, as a result to embed the sequence isometric. So, let us take, again, we would like to have infinite sequence of graphs with two conditions, large graphs and DG bounded. Then, depending on the construction you would like to make, depending on the embedding you would like to have, you will put different labelings on certain graphs. Okay? So, for some labelings, you will get isometric embedding. For some other labelings, you will get coarse embedding. Of the graph, of the sequence of graphs, into the resulting group. For originally in Gromov's construction, you also have some labeling. You get so-called weak embedding, or almost quasi-isometric embedding. But you still need in all these constructions, you still need these two conditions, large graphs and DG bounded. Okay? So, this is what is written here. And this is a little summary. This is not what I would like to discuss. These constructions is just a motivation here. So, in particular, the constructions of most, as I mentioned, they answer kind of, they clarify the picture between these four properties, like for amenability, high group property, and cos amenability of a group. Of course, amenability of a group into a Hilbert space. And so, mostly, okay, Gromov's most shows a group which is not costly embeddable into a Hilbert space. High group monster shows that this application is not true. So, it is a group with high group property, but not costly amenable. And here, in particular, this is very important, that the degree of the graph is at least three. It's really crucial. And, by the way, here also. And then, there is also what we call the true course monster. And here, a group which is not costly embeddable into a Hilbert space, but at the same time doesn't contain in any weak sense, even expand inside. Right. So, great. So, this is a motivation here. And there is another motivation which is probably less known. It is a motivation from metric geometry that actually you can use these sequences of graphs also to construct interesting metric spaces and with wild distortion properties when you would like to embed them into some good spaces like Adomar spaces or Banach spaces like L1. And in this case, you really use graphs themselves so you don't need to put some labeling so no any labeling involved. Okay. So, it's more metric combinatorial. So, there is a motivation to produce such explicit examples also for this metric geometry, metric embeddings people. Okay. So, the result what I'd like to present actually I don't remember till when I have time. Then. Okay. Good. So, new large-grid-dg-bound calligraphs and so this will be calligraphs of SLNFP. So, we give a particular generator so the group is clear, right? But we give a particular, we are talking about graphs so we need to precise what are generators so we will give particular generators so degree I say at least four so it will be two generators or more. And the dimension is any dimension starting from two. Okay. So, up to now explicit things where as I said only in dimension two. Okay. So, these are our matrices and this is a joint work with Arindam Biswas, a fantastic post-doc I have in Vienna so he is somewhere there. So, these are matrices just they are so nice that we call them magic. So, any dimension and you have one on the diagonal and say A, upper diagonal and B, lower diagonal and okay, to simplify a little bit calculation we said that A and B at least two but you can take absolute value. Yeah. Not necessary. Okay. And then as before I will call A, P, B, P when you take these matrices and take modulo prime P. Okay. So, great. And again an example of course if you consider this in size two you recognize Margulis matrices and this is the famous result of Sanoff that you all teach probably in your group theory course that they generate a free subgroup. Yeah. And I already mentioned the result of Margulis that this is the girth of these graphs is logarithmic. Okay, these are classical results. And now our theorem this is a schematic picture of the theorem the theorem comes on the next slide. So, our theorem we proved the following so there are three parts mostly. So, the first we say that these matrices in any dimension they we can take some powers of these matrices and then they will generate a free subgroup and we precise actually which kind of powers we are allowed to take and there are many you will see. The second part is that when we take a modular prime then the images in these finite groups will generate these finite groups so not a subgroup but a whole group which is kind of a surprising thing I would say because this is you started with a free group. Yeah. And the third property is what we really need that the girth and diameter are logarithmic but of course for girth we have low bound and logarithmic and for diameter upper bound logarithmic. Okay. So, these are DG bounded large girth and even logarithmic graphs and in particular this is just a chance they are expanders. Okay. This is a schematic picture of our result but this is really the theorem. Yes. So, I really succeeded to put the whole theorem on one slide but you see I can guide you again you know you have these parts like freeness, generation, girth and diameter and then expansion but I'd like to guide you a little bit I don't suppose that you read all things but again these are our matrices. Okay. Then in each part we have what we have we have a part in dimension two which is a known part actually. Yeah. But okay we precise a little bit and we should say yeah in original Margulis paper it is written for two but we have A and B. Yeah. But okay. So, then there is a dimension three and then there is dimension at least four so all dimensions all higher dimensions here. So, it seems and just our proof it gives there is some difference between dimension three and higher. Just we have some yeah and then we precise what kind of powers we can take to have our statements. So, for example here in dimension three we can take the power at least four and then everything will go. So, this is what happens here but again it is in every dimension for every n. Then the statement is large girls and dg bounded and logarithmic actually logarithmic thing. Again it happens that these graphs are expanders and also an important part that all absolute constants are effective. So, there are some constant appearing like you have to go the prime you have to go far enough and how far you go everything is effective. Is it hard to see that they are expanders? It is hard to see we don't do anything I will explain it is really I am not saying we proved it I am saying it happens that they are expanded so we use very deep results of other people. Yeah. So, this is in particular if you take an example of some power and some dimension and some a and b then these are concrete examples of logarithmic girth for regular expanders for example. You choose your favorite l your favorite a and b your favorite matrices and you do. Okay. Again, so our generators, our matrices they are not generators of slnz. Yeah. It was mentioned that you of course you can start with generators of slnz and then take modulo and then you will get expand of course because for n at least 3 this is a property t group but it doesn't answer our question this kind of expander because it will never be large girth. Okay. So, to have large girth you really need to start with a free group but if you start with a free group and then you will take modulo then you are not sure that actually what you will get will be the whole slnfp and this is our result in particular. So, another thing because we posed the question in such a way that we were not really interested in expanders we were interested in large girth and gg bounded so from the very beginning we decided we are not using expansion to prove logarithmic girth or to prove actually logarithmic diameter so because I explained to you that if you know expansion then automatically it will be logarithmic diameter but actually you don't need it for this and this is how this is our way, how we choose this way and okay, this is already said that this explicit large girth gg bounded expanders, logarithmic girth expanders for higher dimensions okay, great probably a few words about about proof so first we show freeness and of course when you have two matrices like this and if you are, yeah you can give it to your students and you ask, okay please show freeness what they will do the first reaction ping pong, yes great, you will be in trouble with this matrix complete trouble actually it is interesting to see that of course there is a lot of literature about ping pong actually probably you don't know but all kind of conditions ping pong conditions if you compare them they are not always the same yeah so and we tried a couple of conditions here just directly it's for high dimensional things it's not easy it's not easy so because you really you need to choose this areas where it will act and ping pong and for yeah so we finally didn't do it in this way so there is some induction involved some lexicographic order on coefficients of matrices etc so we did it in a more explicit way without going to some conditions because some of the conditions they are really not you can really not apply I mean ping pong things which you cannot apply for these matrices so it will not work but we succeeded to show fairness in a different way great the second is to prove generation so when you take modulo prime that it will generate the finite group this SLN Fp first we proved that it generates for a suitable prime P ok so this is some work to do and then we used strong approximation to conclude that it will generate actually for almost all primes so if you prove for one then by this argument it will generate for almost all ok so and there are several formulations but we used this formulation of Lubbocki but mostly it's strong approximation and we know that starting from some prime it will generate ok but again starting from some prime because we would like to have things explicit we should then quantify how far we have to go and it is possible mostly due to norris result about quantified strong approximation ok so we used strong approximation here then we estimate girth actually this is not very complicated and the argument yeah this is in all dimensions so our matrices so we need a kind of general argument but conceptually it is not very far from the argument of Margulis from our original argument of Margulis ok and then as I said we would like to estimate the diameter without using expansion yet and here we use a combination of also very strong results of other people so in dimension 2 it is a result of health got and in dimension higher and I will mention a little bit more explicitly how it goes so in dimension 2 this is the result of health got we are using so this is quite famous now result of health got about the growth of small subsets in SL2 FP and here how we use it so what it says it says if you take a subset which is proper subgroup which is not generating everything and it is big enough then there is this K such that it is expressed so this K is an estimate on the diameter right so now what we take here so sorry, not contained in any proper subgroup so generating generating the whole group so we use our steps 1 and 3 so 1 was freeness and 3 was logarithmic girth so because it is free and the girth is logarithmic this is what I explained about residual fineness when you take a ball in the free group and the girth is logarithmic if you take a ball of radius which is smaller than half of the girth minus 1 etc so small enough but not big enough and small enough at the same time so then when you have it the size of this ball will be exactly at least like in the free group so this is what is written here and what is important that the girth is logarithmic so we can write the radius of this ball we can express as a constant times logarithm of prime then we just take this set this set as prime and we use half god's result to get this k which is an estimate on the diameter so there are two important ingredients to be able to use this half god's result freeness and logarithmic girth and of course generation also otherwise we cannot use it so these are exactly the steps which we proved to be able to estimate the diameter great so in dimension 2 it works like this so it is an alternative way in particular to see logarithmic girth for Margulis sorry, logarithmic diameter diameter for these graphs without using expansion great so in higher dimensions it's a little bit more complicated but the argument like conceptually is similar but we need stronger results of course so but it works in a analogous way so either the subset is small or it is big enough to cover a big part of the group up to some constant and then if you if the product of three sets is big enough then you will cover actually the whole group this is the result of Gauss but the strategy is the same I write it here so we use freeness we proved and logarithmic girth then we get a similar statement because we know that it is free and yes and then we use the Bernoulli-Grins-Tau result so second if our subset we take this subset if it is small we use the second part of the result of Bernoulli-Grins-Tau constant number of times this is important and what is this constant number of times it depends on the size of the group where we are so we use this to go to the first step to cover almost the whole group okay and then we proceed with Gauss result to conclude the the diameter can you go back one slide please yes oh yeah no I was wondering about the degree of the representation oh yeah it appears in the denominator yes thank you so I hope the strategy this is just a strategy then you need some computation okay but all the ingredients we got in our proof they are really all necessary freeness, logarithmic girth and generation yes just proceed the logarithmic girth in chi-streamers doesn't it so it's other way around I would say right because if the group is free and you are talking about matrices then you will get logarithmic girth if you would like the quotient which is which is bigger yes not quite you could have three products of two element group yeah but here the generators have the generators downstairs yes so there is no depends if you consider group the generated group yeah you can you can end up with finite index free subgroup something like this we use it really this direction you would like to have other direction but we don't need it of course if there is a loop of length two you cannot have a small loop no it's good to think about everything yeah so as I said we did nothing about expansion we just used results of others and actually we were able to use results of others because of our results so we really need to use freeness of operation to get this for example in case dimension two so it is okay in dimension two we know that this equivalence between the risky dance and non-elementary but of course we proved that it is free so it's not elementary and then we use this strong result of Burge and Gamble that actually the risky dance is enough is equivalent to have this expand okay so in dimension higher it will not be anymore like automatic like this just freeness is not is not enough but the generation we proved so the fact that we proved that there is some prime P such that it will generate will be enough to conclude that actually our subgroup is again the risky dance and then we use we can use this result of Burgen value you see this kind of journals so we found magic matrices this is what I told you right so in addition they give us expanses in all dimensions so now we reduced remember we had two matrices and then it means that it gives four regular graph right generators and inverses and now if we would like to have we would like to have other degree it's possible also by our construction we just take in this free group we proved we take a finite index subgroup and then everything works again so on the same group meaning on the same SLNFP we can have other generators and it will give again so other degree but it will give again the same kind of graphs okay two K regular logarithmic graphs and it's also possible to have similar results for congruence coefficients like model or not a prime but integer it will not always be a result about generation but you will take some subgroups of these groups so congruence also we have some result another corollary that actually the group we have this generated by matrices A and B or finite index subgroups there also called thin groups the terminology was introduced by Sarnak and it is a very nice class of groups so I think two days ago not 10 already 10 10 days ago appeared for example a paper what is a thin group on archive by Kantarowicz long Lubocki read you can I invite you to read it it's not because Sarnak said use that it's not a bad terminology it's a bad terminology if you definitely recommend in any case our example our example is a thin group and this is an interesting so here is an argument how to see it so we have in any dimension and you can vary this L and you can have explicitly this matrices which generate a thin group for you and then you can do whatever you want with this thin group okay this is another okay so this corollary is that so you see we have several expanders and you vary powers you vary dimensions and this corollary says just by a trick which he was I think made by David Hume you can take some subsequences there and to produce expanders which are causally geometrically very very different this is what it says mostly not causally equivalent okay we can discuss this okay I'd like to mention a few open questions so one is is it true that this expanders we have now are super expanders in the sense that they also don't embed causally into uniformly convex banach spaces for example so why actually it is even open for Margulius expander but why kind of it's not a completely speculative question because the difference between dimension 2 and dimension higher is really on the construction you really feel it because of this free group which is indexed in higher dimensions so probably going to this is a hope that going to higher dimensions you can have probably some other tools to prove about these new graphs super expanders and why we would like to have them because this will be large girth gg bound and so we can this will be the only the first example and we can again this technology for monster groups to produce a new monster okay so this is very interesting question in my opinion and okay this is is it possible to make such graphs which are not not necessarily expanders as I said we just we found some graphs and they happen to be expanders but who knows if we can make also things which are not not necessarily expanders but have the same properties and calligraphs and explicit and then there are more precise question about Hadamard spaces but probably because of time I I don't really mention them but this question but we can discuss later if somebody is interested and this is really about distinguishing random graphs and random expanders coming from random graphs and other expanders from geometric very geometric point of view and this is around works of Manor Mandel and Asaf Nawal okay this is a summary so these are first explicit large graphs in higher dimensions made on calligraphs of SLNFP PU is a prime everything is effective so we can really have an algorithm to give all constants necessary and it seems that it's really it completes this list because previous examples were in dimension 2 and there logarithmic expanders and this is my happy birthday can you please repeat you made some comments that your sequence of graphs is not logarithmically equivalent of something to already known results you have mentioned if I understood correctly that the sequence of graphs you construct is not in some sense no no no I think you talk about this slide right so the point is that the previous explicit sequences of infinite graphs with these properties were known only in dimension 2 and now we have all dimensions this is comparison with the previous explicitly calligraphs known here I said something different in the sense that starting from this sequence we have any sequence I can take a sequence in a given dimension like say in dimension 4 size size of matrices suppose I take a sequence in dimension 4 my matrices are in dimension 4 I take my sequence then I can take sub sequences appropriately to produce different sequences which will be non-cost equivalent for example this is one way to produce more and more and more examples like this now I can even mix dimensions I can take some part from dimension 3 some from dimension 4 you know and also to produce appropriate sequences such that my different sequences will be not costly equivalent even not regular equivalent it's just playing playing with choosing sub sequences and choosing the fact that we proved for all dimensions this is now I think I understand thank you but I wanted to ask do you have maybe some geometrical properties of your sequence to claim that maybe you have some geometric property that in the previously known as an example you don't have it I would say this is the opposite in the sense that these are more examples with the same properties at least in the same properties but the hope is that this is my question about for example about Super Expandos or about this Adamark spaces the hope is that the fact that you are now free to choose your dimension even to play with different dimensions probably it can help to use to get results you could not get up to now for dimension 2 because you are fixed with this particular things but these are open questions I mentioned I didn't claim anything about this now these are more examples for the moment I don't know maybe privately it's a tea time so that you have more time to discuss and to present ok so thank you