Contracting self-similar groups and conformal dimension
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Nekrashevych, Volodymyr
One can associate with every finitely generated contracting self-similar group (for example, with the iterated monodromy group of a sub-hyperbolic rational function) and every positive p the associated \ell_{p}-contraction coefficient. The critical exponent of the group is the infimum of the set of values of p for which the \ell_{p}-contraction coefficient is less than 1. Another number associated with a contracting self-similar group is the Ahlfors-regular conformal dimension of its limit space. One can show that the critical exponent is not greater than the conformal dimension. However, the inequality may be strict. For example, the critical exponent is less than 1 for many groups of intermediate growth (while the corresponding conformal dimension is equal to 1 ). We will also discuss a related notion of the degree of complexity of an action of a group on a set.
 So, thank you very much for inviting me. I'm thrilled to be on a conference for the first time since almost two years, since actually end of March of 2020. So, I'll talk today about, basically it's a project more than, I'll formulate several theorems, but it's more like project, an attempt to study alphor's regular conformal dimension using group theory. So, how could that be achieved? So, let me start from the setting. So, the setting is a covering map, where, so this is a covering map, where M is a compact matrizable space, or metric space, though metric is not very important here. Yeah, it will be. And such that it's locally expanding, in the sense that there exists constants epsilon greater than zero and L greater than one, such that distance from f of x to f of y is greater than L times distance from x to y for all x, y in M of such that distance between them is less than epsilon. So, an example of this would be a hyperbolic complex rational map restricted to the Julia set. You will have then this expansion condition for the restriction of the concurrent metric. And many other examples say a class of examples where M is a manifold are precisely when M is a infranily manifold, so when it's a quotient of an important league group by a discrete affine group and f is induced by an expanding automorphism of the league group. So, now I want to associate an algebraic object with this situation. And for that I will assume that M is path connected, though it's not really needed. It just easier notationally, but everything what I say will be true for any expanding map, a covering map. So, if you have a point in M, two points in M, t1, t2, and the path L from t1 to t2, then you can lift this path using the map f. And so, t1 will have some pre-images, t2 will have some pre-images. So, if you take this path L, you can lift it and you will get a bijection between the pre-images and the f. Then these pre-images will have pre-images. You can lift the lifts and you'll get bijection between these pre-images and so on. If you take the union of all of this, you can think t, you will get a bijection between the backward iterations of f of t2. So, this is arranged naturally into a tree, which I will denote as t1. These also form a tree, t2, and we get an isomorphism of these trees, which I will denote as SL. And you can look at the boundary of these trees. So, boundaries are cantorsets of all paths, infinite paths going, infinity, and this isomorphism of trees will define a homomorphism of the boundaries of the trees. And we obviously have actually an action of the fundamental groupoid on these boundaries of the trees. So, if you take one path and then continue it, the corresponding, the isomorphism induced by concatenation of the paths is the same as composition of the isomorphism induced by each of the paths. In particular, if we fix a point, if we take one point t, then we have a semi-group of fractal isomorphisms of sub-trees of this tree defined by paths connecting different vertices of this tree. So, we take two vertices of this tree. So, we have the tree, let's take the case when degree is 2. So, we have a binary tree, a rooted tree, and then for any path inside m from one point to another, when we lift these paths, we'll get an isomorphism from this tree to this tree. So, we get on the boundary of the tree, we'll get a homomorphism between a clope and subset of that boundary and another clope and subset of that boundary. And the set of such homomorphisms will be what is called an inverse semi-group. So, it's not a group because homomorphism is not defined on the whole campus set, the whole boundary of the tree, but it's a set of homomorphisms between clope and sets and this set of homomorphisms is closed undertaking inverse and composition. The composition is a composition of fractal maps, so sometimes you can get empty map and so on, but it's something that algebra is called an inverse semi-group. And this inverse semi-group is finely generated. So, I didn't claim that m is locally simply connected, but it does matter. Well, let's pretend that it is for a minute, then the fundamental group will be finely generated. So, this semi-group will be generated by the generating set of the fundamental group and any collection of paths connecting the base point to the immediate parameters. If you take these paths, this inverse semi-group will be generated by these paths, by the corresponding of the morphisms of the tree. So, there will be some of the morphisms of the tree and then partial of the morphisms which map the whole tree to this sub-tree and whole tree to this sub-tree. The semi-group will be generated by these. Even if the space is not locally simply connected, the set of automorphisms that you get from the loops will be still finely generated because of the expansion. The fact is that any loop which is small, efficiently small, when lifted, will be definitely a loop even smaller and so on. So, all small loops act trivially on this tree. So, even if the space is not locally simply connected, the group is still finely generated. The group defined by loops, the sub-group defined by loops in this semi-group will be still finely generated. So, the group defined by the loops is called the iterated monodromy group of the map F. But in addition to iterated monodromy group, we have these maps which map the whole tree to sub-trees. Now, this group, the iterated monodromy group, so we have the group, the iterated monodromy group, the sub-group of the semi-group generated by the loops. And as I said, there are additional generators which in the case of degree 2, there are two additional generators which map the whole tree to the sub-tree growing from the first level. So, let's denote the set of those additional generators by x. And then for every element of the iterated monodromy group, so for every loop, and for every element of this additional generating set, so for every path connecting the base point to a pre-image, we'll have the following relation that if you apply Sx and then, I compose them this way and sorry, if you apply Sg and then Sx, you can rewrite it as applying, sorry, I confuse the sides again. If you apply Sx and then Sg, that will be the same as applying sum Sy and then Sh. So, for all g and x, there will exist h and g and y and x, so I get the following relation holds in our semi-group, namely what are these? So, in the semi-group that I just mentioned, the semi-group of all these homomorphs, so the question was if where this relation holds in the semi-group of those partial isomorphism between sub-trees defined by paths. Let me explain it. Since I see inverse semi-group, I can rewrite it in a different way. So, if you take, so this is our path g, this is our path x and so what are these h and y? So, this is a path in the base point, this is a pre-image of the base point, so we can lift this path g in a unique way and this by the map f, by the covering, so the map fx like this, it maps both these points to the base point. So, this is a lift and this is one of the pre-images, immediate pre-images, so there is an element of x connecting it to the base point and that element is our y and our h will be this triangle. So, you go along x, then along this lift, less than or the lift by gx, so the relation will be that, yes, so h will be the concatenation of the paths x, gx and then y. Why this is so? Because just from the definitions, if you take these three paths, the corresponding automorphism, the corresponding element of the semi-group, so if you take this path, the corresponding element of the semi-group will be sx, sgx and then sy inverse and by definition of gx, by definition of g, g is the sg, sg is the automorphism of the tree defined by g, so it's defined by lifts of g everywhere, so restriction of this sg to the subtree starting in this vertex is exactly, so if you restrict it to the subtree starting in this vertex, let's call it tx, then this is exactly sgx, because by definition, sg is defined by taking lifts of the path g everywhere, so it's the isomorphism between the trees defined by these lifts of sg, one of these lifts is gx, so if you start lifting gx, you get the same isomorphism, but you started only from this vertex, you didn't start from the root of the tree, so you have your binary tree describing the action of f and sg is the automorphism of this tree induced by lift, defined by lifting the path g, but now if you restrict it to this subtree, it will be, this map is defined by lifting this frame image of g to this subtree, and so restriction of sg to this subtree is exactly sgx, and you can check that this is exactly what it should be, yes, essentially this is what I wrote here, so you can hear what we do, we map the whole tree to the subtree and then apply sg, but applying sg is the same in this case would be as applying restriction of sg to this vertex, so it's applying gx, and then we come back using y, and so you can check, and you can check that you have this relation. Okay, so what we get here is, so let me give you an example of such computation, so for example if you take z square minus one, so the basilica, you will have two generators of the iterated mod dummy group A and B, I just want to show you how these relations here, and then there will be two elements of the set x, so if you have a base point it will have two pre-images, so let's take the base point, so these elements A and B will be defined by loops going around two post-critical points, zero and minus one, and if you take the base point it will have two pre-images, let's say take the base point, the fixed point between zero and one, so it will have two pre-images, and let's take one connecting path, the trivial one, denoted x0, and another one, the non-trivial one, denoted x1, and then one can show that, so if you take A to be the loop around critical value and B a loop around zero, in that semi-group you will get that SA, if you do Sx0 and then xA it will be Sx1 and identity, because if you lift A you will get two curves like this, and so if you do X0 then you take the lift of A will be this path and then you come back in the triangle using the connecting path you will get a trivial path, so it will give you this relation and then you can check that this relation and then for B you will get these relations. It's annoying to write these S's, so we'll just write it this way, Ax0 is x1 Ax1 is x0B, Bx0 is x0, Bx1 is x1A, and this is something that was known in group theory as a self-similar group for a biset, mainly if you take any product of these X's, X1, X2, XIN and multiply by A, you can use these formulas to rewrite it, so you have A times x0, x1, you can rewrite it now as xj1 times some group element, time another group element, well a generator A or B or identity, then again you can rewrite this using the same formulas, again you can switch letter and group element, this x and a generator of the group and rewrite it again as some other letter and some other element of the group. Could I ask a quick question, can you hear me? Yeah, yeah. Just to clarify, so the x0, so let's say your base point is the alpha fixed point, then your x0. In this case is the trivial path, yeah. No, no, just the x1 and the x0 to begin with, so you're connecting one, you're connecting your base point to the preimage of the alpha fixed point, one's on the top and one's on the bottom of something, that's your x0 and your x1. So the whole semi-group is generated by two sets, by union of two sets, one set is the generating set of the fundamental group for example, and the other set is the set of connecting paths from the base point to immediate pre-images of the base point. Yes, did I answer the question? Yes, thank you, sorry, I just realized my video was off, sorry. Yeah, we can take the fixed point between 0 and minus 1 of Basilica, which is what I forgot. But this square minus 1 will have two pre-images, alpha and beta, and that's the better one. This is alpha 1, okay, not the landing point of ray 0, but the other one of landing point. No. Yeah, so I take this, this is called alpha, the one which is just two rays, yeah, so you take alpha and minus alpha, sorry, I thought that you say that minus alpha is a fixed point, no, no, no, no, correct. Alpha and minus alpha, yeah. So X0 is the trivial path at alpha because alpha is the ray measure itself, so we can take the trivial path as the connector, connecting path, and then you take X1 is this path in the upper half plane from alpha to minus alpha. Did I? In these formulas? Okay, I'm sorry for that. Yeah, I'll change this. Thank you. Okay, so we get these, these can be seen as some kind of rewriting rules that allow us to write any product, an element of the fundamental group times a product of these Xs, two product of these Xs times an element of the fundamental group. So and this is exactly what is a similar group. Now, this inverse semi-group, which I mentioned at the beginning, it's a semi-group of partial homomorphisms of the boundary of the tree, so it's a semi-group of homomorphisms between clop and sets of the boundary. In fact, it's more natural, so this semi-group is nice, it's given by finite number of kind of defining relations which I wrote above there. But okay, sorry. So let me repeat once more, one more time what I wrote at the end there. We have self-similar group, which means that, so there is the iterated-mordemy group, the subgroup of the semi-group, which is defined by elements, consisting of elements defined by loops. Then there is this set of paths connecting the base point to the, to the pre-images. And in this semi-group, we have relations of the form that if you take any element G in the, in this subgroup, and you take any element V in the semi-group generated by this set of connecting paths, so all products of these elements xi, then there will exist a unique element g of, unique element h of the group and a unique element u of this semi-group generated by x such that g times V is u times h. So you can rewrite, and actually it's easy to write it explicitly what it is. So this V is a product of, of these isomorphisms of the whole tree with a sub-tree of the first level. And so when you compose two of such, you will get an isomorphism of the whole tree with a sub-tree of the second level, and so on, if you compose n such, you will get an isomorphism of the whole tree with a sub-tree of the nth level. So it will be defined by a path connecting the base point to that point of the nth level. So there will be some path l1 connecting the base point to that point of the nth level, and then g will be a loop. So you can lift this loop to that point of the nth level, so to nth preimage by nth iterative f. So that will be your lift of g. And then there will be one path corresponding to V, and there will be a unique path corresponding, unique path such that the corresponding isomorphism of trees will map the whole tree to the sub-tree of nth level. So that's your l2, and that l2 will correspond to u, and then h will correspond to concatenation of these paths. In particular, you see that because we assume that our map is expanding, this lift will become shorter and shorter. And this can be used to prove that eventually these elements h, these triangles, these concatenations, will belong to one fixed finite subset of our iterated modeming group. So there will exist a finite subset of the iterated modeming group of the subgroup of our semi-group generated by loops such that when you do this rewriting, so when you're lifting paths and completing these paths by the connecting paths, you will reach the set n. So in the group element, there will exist n such that for every V in x star of length n or more, let me denote this element h by g with index V. This g with index V, so this element which you get over here belongs to this finite set. And such groups are called contracting self-similar groups, such iterated modeming groups, contracting self-similar groups. And this is an algebraic counter part of expanding covering maps, that conversely every contracting self-similar group, so every group given by relations like above there, which satisfies this condition, appears in this construction. So it is iterated modeming group of some expanding map with one caveat, you have to consider orbit spaces, like you have to include subhypropolic cases here. And that's a little bit technical in this case. But modulo this technicality, you have a complete like a bijection between expanding covering maps, modulo topological conjugacy, and contracting self-similar groups, modulo some type of algebraic equivalence, which is also a little bit non-trivial, but very can be easily formulated. So we have an algebraic theory in this way, we get an algebraic theory of expanding covering maps. In particular, this implies that there are only countably many of them, up to topological conjugacy. That's one of the proofs that you can use. So how you can reconstruct from this semi-group this expanding map. So the idea is very similar to the idea of hyperbolic, gromov hyperbolic groups and boundaries of gromov hyperbolic groups. In a sense, this semi-group should be thought of as a gromov hyperbolic group, even though it's not hyperbolic, and not a group. I mean, it is not a group, but it is hyperbolic. So let me explain how it is hyperbolic. So if you have a group, you draw the kilograph, and it's negatively curved, and then the boundary is the space that you're interested in. Here, the kilograph of the semi-group is not what you should think about. What you should think about is the kilograph of the associated groupoid of germs. So you have this semi-group acts on the boundary of the tree. So it's a group of partial homomorphisms of the boundary of the tree, of the boundary of the pre-images. So you can look at germs of this semi-group, germs of its action on the boundary. So you take a point on the boundary. So you have your tree of pre-images, it has your boundary, it has the boundary. And then you pick some point of that boundary, just pick some backward orbit. So let's call it xi. And then take your generating set of this semi-group, and then for every element of the semi-group, you have the corresponding germ, if that element is defined on xi, you have your element as defined by some path L, it acts on some clop and set, and if that clop and set contains xi, you have the germ of the action. So you identify two semi-group elements if they act the same way around this point xi. And then you draw the calligraph. Among these germs, if you have one germ and another germ, so the germs will be vertices of your calligraph. And you connect one germ to another if you have a generator such that this germ is composition of this germ and the germ of the generator. So it's exactly the same as for calligraphs of groups. The only difference is that you have to fix a point and look at germs of this group at a point. If you have a group action, you will get every element will be defined on this point. So these germs will come from group elements and every element will have the germ at that point and it will be closely related to the calligraph of the group. In this case, there are many elements of the semi-group which are not defined on xi, you ignore them. So how will this graph look like in our case? So in our case, our generating set is split into two parts. There is the generating set coming from loops and the generating set coming from connecting base point to immediate frame edges. And so the edges of the calligraph will also split into two parts. There will be edges which correspond to loops and edges corresponding to the connecting paths and it's natural to draw edges corresponding to groups kind of horizontally, edges corresponding to connecting paths vertically. If you take the vertical part of the calligraph, you will get a regular tree. So for example, in degree two, you will get a tree which has one edge going down and two edges going up. So these are our x's. X0, X1, X0, X1, X0, X1 and so on. The edges corresponding to loops, so generators of the iterated modem group, they will connect these vertices horizontally. So in this tree, they will, because the loops will preserve the levels on the tree, you will get some connections horizontally. So in each level of this tree, there will be some horizontal edges. You'll get this infinite graph which is naturally graded by this subtree corresponding to the set x, these connecting paths. And edges either these connecting paths arranged into a tree or some horizontal paths connecting these vertices of the same level. And the fact is that this graph, so let me denote this graph by gamma, it depends on xi. This graph is grommar hyperbolic. So that's what I meant when I said that this is a hyperbolic semi-group in a way. But it's not just this, graph has a special point on the boundary corresponding to going down in this tree. So the boundary has a favorite point. Let me call it omega xi. And then we have the grommar boundary and it's natural to remove that point. Because then the boundary will correspond to going up in this tree. And it will be a non-compact space kind of completing these three up stairs. So it will be like a half plane model of the hyperbolic plane. So there is a special point and then there is the real kind of real line. And so what is that space? What is that boundary? So remember xi is an inverse or is a point of the boundary. So it's an inverse orbit of t. You choose each time one pre-image. So this boundary will be a leaf of the natural extension of the covering. So if you have our covering map f, we can look at the inverse image, the projective limit, the inverse limit of iterations of f. And then it is naturally a fiber bundle where fibers are the boundaries of these trees that I was talking about. And base is m. And so in that fiber bundle horizontally you get leaves. And the case when m is path connected, those leaves are exactly the path connected components of the inverse limit. And xi corresponds to one point of the inverse limit. And then this boundary is exactly the connected component, the component corresponding to that point. And more precisely, let's m hat will be the inverse limit of these spaces. Then f induces a homomorphism of this inverse limit. It's called the natural extension. And this natural extension is a hyperbolic dynamical system. So there are stable and unstable equivalence classes. And this boundary of this chromo-hyprobolic graph is the unstable equivalence class of the point xi. So it's, yeah. OK. So now, yeah. Yes. Yes. Yes. Yes. So the cantor sets, if you take the basilica, natural extension of basilica, the cantor sets are the stable equivalence classes. So if you take two points in that cantor set and iterate your natural extension, they will converge. The distances will go to zero. Now if you take the inverse map, then the set of sets, you have the unstable equivalence relation. So it's the pairs of points where distances go to zero if you take the inverse map. So these equivalence classes, the unstable equivalence classes, are exactly the boundaries of these graphs. And they are exactly the path-connected components of that bundle. The path-connected components with a natural inductive limit topology. So when you unwrap it. OK. So let me go, OK. Sorry. So this is a particular case of a more general setting of so-called hyperbolic group points. So that includes many more examples, not only coming from expanding maps. And in all these cases, the calligraphs have this, the calligraph of these hyperbolic group points, they have this special point on the boundary. And so they are also stratified like this by horocycles corresponding to that special point. So these levels of the trees are the horoscales corresponding to the, sorry, horospheres corresponding to this point on the boundary. And we have the levels. Levels of the levels can be described using the Buseman cos-cycle. The Buseman cos-cycle, in this case, it just measures the difference between the levels of this graph. So we have, if you take two germs, then you can close this diagram. So you get another germ closing the diagram. And the value of this cos-cycle on this germ, let this germ be gamma, the Buseman cos-cycle beta of gamma is just the difference of levels. The level of the end of the germ minus the level, well, it's positive if you go up and take the difference between these levels. So that's the classical Buseman cos-cycle for a hyperbolic graph associated with a point on the boundary. But that's not the only one Buseman cos-cycle, because if you forget where this graph came from, you look only at quasi-ozometry class of this graph, you just remember this graph up to quasi-ozometry, and you remember this special point. There will be many Buseman cos-cycles associated with each realization of this graph in the quasi-ozometry class. So let me give you a general definition. And then I will, in the remaining 10 minutes, talk about how it is related to the dimension. So let G be a topological groupoid of germs of some semi-group. So in our case, we have a semi-group. It consists of homomorphisms. You take germs of these homomorphisms, and the topology is natural. If you have some partial homomorphism, then germs of these partial homomorphisms, the set of germs of these partial homomorphisms, is homomorphic to the domain. So you have your homomorphism, it's a bunch of germs, and this bunch of germs has the same topology as by the domain of rain. So you get, if you have a semi-group of local homomorphisms, that defines a topological groupoid, so the set of these germs. And then a quasi-co-cycle on G is a map better from G to R, such that there exists epsilon or not epsilon, eta greater than 0, such that first, this is almost continuous in the sense that for every germ gamma, there exists a neighborhood U of gamma such that the values of beta vary at most by eta for all elements of this neighborhood. And the co-cycle property is that for all composable germs, gamma 1, gamma 2, you have the product gamma, gamma 2 of gamma 1, beta of gamma 2, gamma 1 minus beta of gamma 2 plus beta of gamma 1 is less than eta. So it's almost additive, and it's almost continuous. So an example is, well, where eta is 0, actually, is that co-cycle coming from those graphs. So for the semi-group, which I was talking about today, you look how much you change the level, from which level you go to which level, that's an integer. And it's continuous, it's locally constant, actually, and if you go up or down, you add the differences. So that's an honest continuous co-cycle. Here's another example, which is kind of dual. Yeah? Sorry? Yeah, yeah, I know. So here's another example, you take F to be the groupoid of germs defined, so generated by F. F is a local homomorphism. So its germs are invertible. So you can generate a semi-group of local homomorphisms by F. That will give you a groupoid of germs, and there are two natural choices. So every element of this group point looks like that, like this. You take a point, you apply F some number of times, and then you apply branches of F inverse some number of times. So if you compose F and then F inverse, that will be a germ in this groupoid, and all elements of this groupoid are like that. So you go N times here, and say M times here, so you can define the value of the germ of the Bussmann co-cycle as say M minus N, for example. And this will be a co-cycle, if I did it right order. If you draw the kilograms of the kilograms, same way as we did over there, they will be exactly trees without any horizontal edges. And this Bussmann co-cycle is just the difference of the levels. But suppose that M, F is an expanding map on, say, for example, it is a rational function. Then another choice would be to take logarithm, negative logarithm of the absolute value of the derivative at your point, at your base point. So this is point Z, at point Z. And this is also a co-cycle. And it is also a Bussmann co-cycle for the same graph, but for a different realization, for different lengths of edges. So in the same quasi-zometry class, you have these two co-cycles. Okay, unfortunately, I don't have time to tell what I wanted to say, all what I wanted to say. I'm going to finish now by formulating some statements, and that will be it. So the point is that these two pictures are dual to each other. This groupoid and this groupoid, in the sense that, okay, let me, so if you have a Bussmann quasi-co-cycle, so we have two groupoys. One groupoid is what I was talking, what I defined at the beginning of the lecture, the groupoid defined by path. Another groupoid is groupoid of germs generated by the expanding maps. And if you have better Bussmann co-cycle on G, then it can be in a unique way up to bounded additive constant associated with a Bussmann co-cycle on the natural extension. So it induces a Bussmann co-cycle, sorry, eco-cycle on the groupoid generated by the natural extension. And then this co-cycle induces a Bussmann co-cycle on the other groupoid. So if you start from a Bussmann co-cycle on a Kili graph or one groupoid, through the natural extension you get a dual Bussmann co-cycle on the other groupoid. So if you start from the derivative, for example, rational function, you get a co-cycle on the groupoid for the self-similar group. So the project which I mentioned is to study Bussmann co-cycles on self-similar groups. And each Bussmann co-cycle on this self-similar group will define a measure on the Julia set, so on the gram of a hyperbolic boundary, and a class of metrics. So if you have a Bussmann co-cycle beta, then it associates the visual metric, so this visual metric exists starting from some exponent, and you have the measure, basically the Patterson-Salivan measure associated with this Bussmann co-cycle. So this Bussmann co-cycle will be, for the measure, the Radon-Nikodem derivative will be exponent of this Bussmann co-cycle with some coefficient, and the visual metric will be the map, the expanding map will be a similarity with some coefficient, with some derivative, and the logarithm of the derivative will be this Bussmann co-cycle with some coefficient. So you can associate with each Bussmann co-cycle its Hausdorff dimension, so that's the minimum of the Hausdorff dimensions of the visual metric, and one can associate then the smallest such Hausdorff dimension, which will be the isomorphous regular conformal dimension. And one last quick theorem is the following. So I mentioned that these self-similar iterative modem groups are contracting, and contracting means that, as I probably raised that definition already, that these elements belong to a finite set, and that can be in a different way formulated as contraction of lengths. So if you have a group element, and you take its length in the generating set, if you look at how this group element acts on the subtree, so you look at these gv for some fixed level n, you take all of them, you take all these elements, if you remember they were defined by triangles or paths, and you take the vector in of lengths of this, of all these elements, so you take lengths of all these restrictions on that level, x to the n, this is a vector in r to x to the n, actually in z, and you look how much the LP length, so you take the norm, the LP norm of this vector, and you look how much the LP norm is contracted, and you take the exponential contraction rate of this LP norm. So being contracting is equivalent for L infinity norm to be contracted, and then there will be a critical value of p for which you still have contraction, and the theorem is that this critical value is very interesting, by the way I include here also p less than 1, so theorem is that the alphos regular conformal dimension of m is greater or equal to the critical value of p for which g is LP contracting. Okay, so the project is to study these Busseman co-cycles and LP contractions also similar groups which are purely algebraic object and apply it to the corresponding expanding method. Thank you very much, let's thank the speaker. So we have time for some quick questions. I think your measures by this graph are somehow related to geometry coding, your graphs are related to geometry coding trees, I developed a theory and transport of measure from symbolic space to… Yes, so this… There's another graph, but maybe it's very similar to yours by pilgrim and… It is the same graph, I mean, the kind of… my graph is a covering graph of the graph. Another visual measure… Yeah, yeah, it is related to the work, yes. I should have mentioned that, of course. So in the case of the rational map, so we have this three-dimensional hyperbolic spaces attached to the leaves of the natural extension and I guess your heligraph or the group point is some discrete object whether it's a metric… Yes. …to this hyperbolic space. Yes, that's right, yeah. And so if you consider general situation to expanding map, do you have… well, you can… You mentioned that you can also consider one-dimensional extensions of the unstable leaves, you have a natural hyperbolic structure, the two-pitches… Yeah, but it's only discrete. Only… I mean, I have only graphs, I don't have any three-dimensional… So there is no sort of intrinsic good structure… Well, it is up to quasi-zometry, it's a unique structure, but it's like a general… case of general hyperbolic groups that you don't have any manifolds or things like that. So it's the only way to get it through the calligraph? Yeah. Yeah. What about the upper bound for the alpha-cellular guillotine, for what I mentioned? Do you have anything to say about this? So it's not… So this can be less than one. So in the cases when M is topologically one-dimensional. So the inequality can be strict, but the only case is when it's strict is sub-hyperbolic cases with really weird orbit spaces. Like for the Grigorychou group, this number is exactly the number which appears in the growth of the Grigorychou group. So it's really interesting when do we have a stricter inequality here? And so this is like maybe in the… actually in the expanding case, not sub-hyperbolic, but when you really have an expansion the way I defined it, maybe we have equality here, but I don't know. I had a question. Yeah. So you defined, I think you defined what is a quasi-co-cycle. Yeah. But then you are also speaking about a Boozeman co-cycle. Yes. So in these four hyperbolic group-oids, if you have a hyperbolic group-oid like these two examples, G and F, for each of these hyperbolic group-oids there in the calligraph, there is a special point on the boundary. And then you define one has to consider a special class of quasi-co-cycles which I call Boozeman quasi-co-cycles related to that point on the boundary. For example, in the case of the gromov hyperbolic groups, Boozeman co-cycle is a quasi-co-cycle always. There is also always this error. And so inside this class of quasi-co-cycles, yeah. So I give definition what is a quasi-co-cycle, but I didn't give a definition which is a Boozeman quasi-co-cycle. So it's some special type of co-cycles associated with hyperbolic group-oids. Okay, thank you. Any more questions in the room or remote? Okay, so then let's thank the speaker again.
 So, thank you very much for inviting me. I'm thrilled to be on a conference for the first  time since almost two years, since actually end of March of 2020. So, I'll talk today  about, basically it's a project more than, I'll formulate several theorems, but it's  more like project, an attempt to study alphor's regular conformal dimension using group theory.  So, how could that be achieved? So, let me start from the setting. So, the setting is  a covering map, where, so this is a covering map, where M is a compact matrizable space,  or metric space, though metric is not very important here. Yeah, it will be. And such  that it's locally expanding, in the sense that there exists constants epsilon greater  than zero and L greater than one, such that distance from f of x to f of y is greater  than L times distance from x to y for all x, y in M of such that distance between them  is less than epsilon. So, an example of this would be a hyperbolic complex rational map  restricted to the Julia set. You will have then this expansion condition for the restriction  of the concurrent metric. And many other examples say a class of examples where M is a manifold  are precisely when M is a infranily manifold, so when it's a quotient of an important  league group by a discrete affine group and f is induced by an expanding automorphism  of the league group. So, now I want to associate an algebraic object with this situation. And  for that I will assume that M is path connected, though it's not really needed. It just easier  notationally, but everything what I say will be true for any expanding map, a covering  map. So, if you have a point in M, two points in M, t1, t2, and the path L from t1 to t2,  then you can lift this path using the map f. And so, t1 will have some pre-images, t2  will have some pre-images. So, if you take this path L, you can lift it and you will  get a bijection between the pre-images and the f. Then these pre-images will have pre-images.  You can lift the lifts and you'll get bijection between these pre-images and so on. If you  take the union of all of this, you can think t, you will get a bijection between the backward  iterations of f of t2. So, this is arranged naturally into a tree, which I will denote  as t1. These also form a tree, t2, and we get an isomorphism of these trees, which I  will denote as SL. And you can look at the boundary of these trees. So, boundaries are  cantorsets of all paths, infinite paths going, infinity, and this isomorphism of trees will  define a homomorphism of the boundaries of the trees. And we obviously have actually  an action of the fundamental groupoid on these boundaries of the trees. So, if you take one  path and then continue it, the corresponding, the isomorphism induced by concatenation of  the paths is the same as composition of the isomorphism induced by each of the paths.  In particular, if we fix a point, if we take one point t, then we have a semi-group of  fractal isomorphisms of sub-trees of this tree defined by paths connecting different  vertices of this tree. So, we take two vertices of this tree. So, we have the tree, let's  take the case when degree is 2. So, we have a binary tree, a rooted tree, and then for  any path inside m from one point to another, when we lift these paths, we'll get an isomorphism  from this tree to this tree. So, we get on the boundary of the tree, we'll get a homomorphism  between a clope and subset of that boundary and another clope and subset of that boundary.  And the set of such homomorphisms will be what is called an inverse semi-group. So,  it's not a group because homomorphism is not defined on the whole campus set, the whole  boundary of the tree, but it's a set of homomorphisms between clope and sets and this set of homomorphisms  is closed undertaking inverse and composition. The composition is a composition of fractal  maps, so sometimes you can get empty map and so on, but it's something that algebra is  called an inverse semi-group. And this inverse semi-group is finely generated. So, I didn't  claim that m is locally simply connected, but it does matter. Well, let's pretend that  it is for a minute, then the fundamental group will be finely generated. So, this semi-group  will be generated by the generating set of the fundamental group and any collection of  paths connecting the base point to the immediate parameters. If you take these paths, this  inverse semi-group will be generated by these paths, by the corresponding of the morphisms  of the tree. So, there will be some of the morphisms of the tree and then partial of  the morphisms which map the whole tree to this sub-tree and whole tree to this sub-tree.  The semi-group will be generated by these. Even if the space is not locally simply connected,  the set of automorphisms that you get from the loops will be still finely generated because  of the expansion. The fact is that any loop which is small, efficiently small, when lifted,  will be definitely a loop even smaller and so on. So, all small loops act trivially on  this tree. So, even if the space is not locally simply connected, the group is still finely  generated. The group defined by loops, the sub-group defined by loops in this semi-group  will be still finely generated. So, the group defined by the loops is called the iterated  monodromy group of the map F. But in addition to iterated monodromy group, we have these  maps which map the whole tree to sub-trees. Now, this group, the iterated monodromy group,  so we have the group, the iterated monodromy group, the sub-group of the semi-group generated  by the loops. And as I said, there are additional generators which in the case of degree 2, there  are two additional generators which map the whole tree to the sub-tree growing from the  first level. So, let's denote the set of those additional generators by x. And then for every  element of the iterated monodromy group, so for every loop, and for every element of this  additional generating set, so for every path connecting the base point to a pre-image,  we'll have the following relation that if you apply Sx and then, I compose them this  way and sorry, if you apply Sg and then Sx, you can rewrite it as applying, sorry, I confuse  the sides again. If you apply Sx and then Sg, that will be the same as applying sum  Sy and then Sh. So, for all g and x, there will exist h and g and y and x, so I get the  following relation holds in our semi-group, namely what are these? So, in the semi-group  that I just mentioned, the semi-group of all these homomorphs, so the question was if where  this relation holds in the semi-group of those partial isomorphism between sub-trees  defined by paths. Let me explain it. Since I see inverse semi-group, I can rewrite it  in a different way. So, if you take, so this is our path g, this is our path x and so what  are these h and y? So, this is a path in the base point, this is a pre-image of the base  point, so we can lift this path g in a unique way and this by the map f, by the covering,  so the map fx like this, it maps both these points to the base point. So, this is a lift  and this is one of the pre-images, immediate pre-images, so there is an element of x connecting  it to the base point and that element is our y and our h will be this triangle. So, you  go along x, then along this lift, less than or the lift by gx, so the relation will be  that, yes, so h will be the concatenation of the paths x, gx and then y. Why this is  so? Because just from the definitions, if you take these three paths, the corresponding  automorphism, the corresponding element of the semi-group, so if you take this path,  the corresponding element of the semi-group will be sx, sgx and then sy inverse and by  definition of gx, by definition of g, g is the sg, sg is the automorphism of the tree  defined by g, so it's defined by lifts of g everywhere, so restriction of this sg to  the subtree starting in this vertex is exactly, so if you restrict it to the subtree starting  in this vertex, let's call it tx, then this is exactly sgx, because by definition, sg  is defined by taking lifts of the path g everywhere, so it's the isomorphism between  the trees defined by these lifts of sg, one of these lifts is gx, so if you start lifting  gx, you get the same isomorphism, but you started only from this vertex, you didn't start from  the root of the tree, so you have your binary tree describing the action of f and sg is  the automorphism of this tree induced by lift, defined by lifting the path g, but now if  you restrict it to this subtree, it will be, this map is defined by lifting this frame  image of g to this subtree, and so restriction of sg to this subtree is exactly sgx, and  you can check that this is exactly what it should be, yes, essentially this is what I  wrote here, so you can hear what we do, we map the whole tree to the subtree and then  apply sg, but applying sg is the same in this case would be as applying restriction of sg  to this vertex, so it's applying gx, and then we come back using y, and so you can check,  and you can check that you have this relation.  Okay, so what we get here is, so let me give you an example of such computation, so for  example if you take z square minus one, so the basilica, you will have two generators  of the iterated mod dummy group A and B, I just want to show you how these relations  here, and then there will be two elements of the set x, so if you have a base point  it will have two pre-images, so let's take the base point, so these elements A and B  will be defined by loops going around two post-critical points, zero and minus one,  and if you take the base point it will have two pre-images, let's say take the base point,  the fixed point between zero and one, so it will have two pre-images, and let's take  one connecting path, the trivial one, denoted x0, and another one, the non-trivial one,  denoted x1, and then one can show that, so if you take A to be the loop around critical  value and B a loop around zero, in that semi-group you will get that SA, if you do Sx0 and then  xA it will be Sx1 and identity, because if you lift A you will get two curves like this,  and so if you do X0 then you take the lift of A will be this path and then you come back  in the triangle using the connecting path you will get a trivial path, so it will give  you this relation and then you can check that this relation and then for B you will get these  relations. It's annoying to write these S's, so we'll just write it this way, Ax0 is x1  Ax1 is x0B, Bx0 is x0, Bx1 is x1A, and this is something that was known in group theory  as a self-similar group for a biset, mainly if you take any product of these X's, X1,  X2, XIN and multiply by A, you can use these formulas to rewrite it, so you have A times  x0, x1, you can rewrite it now as xj1 times some group element, time another group element,  well a generator A or B or identity, then again you can rewrite this using the same formulas,  again you can switch letter and group element, this x and a generator of the group and rewrite  it again as some other letter and some other element of the group.  Could I ask a quick question, can you hear me?  Yeah, yeah.  Just to clarify, so the x0, so let's say your base point is the alpha fixed point, then  your x0.  In this case is the trivial path, yeah.  No, no, just the x1 and the x0 to begin with, so you're connecting one, you're connecting  your base point to the preimage of the alpha fixed point, one's on the top and one's on  the bottom of something, that's your x0 and your x1.  So the whole semi-group is generated by two sets, by union of two sets, one set is the  generating set of the fundamental group for example, and the other set is the set of connecting  paths from the base point to immediate pre-images of the base point.  Yes, did I answer the question?  Yes, thank you, sorry, I just realized my video was off, sorry.  Yeah, we can take the fixed point between 0 and minus 1 of Basilica, which is what I  forgot.  But this square minus 1 will have two pre-images, alpha and beta, and that's the better one.  This is alpha 1, okay, not the landing point of ray 0, but the other one of landing point.  No.  Yeah, so I take this, this is called alpha, the one which is just two rays, yeah, so  you take alpha and minus alpha, sorry, I thought that you say that minus alpha is a fixed point,  no, no, no, no, correct.  Alpha and minus alpha, yeah.  So X0 is the trivial path at alpha because alpha is the ray measure itself, so we can  take the trivial path as the connector, connecting path, and then you take X1 is this path in  the upper half plane from alpha to minus alpha.  Did I?  In these formulas?  Okay, I'm sorry for that.  Yeah, I'll change this.  Thank you.  Okay, so we get these, these can be seen as some kind of rewriting rules that allow us  to write any product, an element of the fundamental group times a product of these Xs, two product  of these Xs times an element of the fundamental group.  So and this is exactly what is a similar group.  Now, this inverse semi-group, which I mentioned at the beginning, it's a semi-group of partial  homomorphisms of the boundary of the tree, so it's a semi-group of homomorphisms between  clop and sets of the boundary.  In fact, it's more natural, so this semi-group is nice, it's given by finite number of kind  of defining relations which I wrote above there.  But okay, sorry.  So let me repeat once more, one more time what I wrote at the end there.  We have self-similar group, which means that, so there is the iterated-mordemy group, the  subgroup of the semi-group, which is defined by elements, consisting of elements defined  by loops.  Then there is this set of paths connecting the base point to the, to the pre-images.  And in this semi-group, we have relations of the form that if you take any element G  in the, in this subgroup, and you take any element V in the semi-group generated by this  set of connecting paths, so all products of these elements xi, then there will exist  a unique element g of, unique element h of the group and a unique element u of this semi-group  generated by x such that g times V is u times h.  So you can rewrite, and actually it's easy to write it explicitly what it is.  So this V is a product of, of these isomorphisms of the whole tree with a sub-tree of the first  level.  And so when you compose two of such, you will get an isomorphism of the whole tree with  a sub-tree of the second level, and so on, if you compose n such, you will get an isomorphism  of the whole tree with a sub-tree of the nth level.  So it will be defined by a path connecting the base point to that point of the nth level.  So there will be some path l1 connecting the base point to that point of the nth level,  and then g will be a loop.  So you can lift this loop to that point of the nth level, so to nth preimage by nth iterative  f.  So that will be your lift of g.  And then there will be one path corresponding to V, and there will be a unique path corresponding,  unique path such that the corresponding isomorphism of trees will map the whole tree to the sub-tree  of nth level.  So that's your l2, and that l2 will correspond to u, and then h will correspond to concatenation  of these paths.  In particular, you see that because we assume that our map is expanding, this lift will  become shorter and shorter.  And this can be used to prove that eventually these elements h, these triangles, these concatenations,  will belong to one fixed finite subset of our iterated modeming group.  So there will exist a finite subset of the iterated modeming group of the subgroup of  our semi-group generated by loops such that when you do this rewriting, so when you're  lifting paths and completing these paths by the connecting paths, you will reach the set  n.  So in the group element, there will exist n such that for every V in x star of length  n or more, let me denote this element h by g with index V. This g with index V, so this  element which you get over here belongs to this finite set.  And such groups are called contracting self-similar groups, such iterated modeming groups, contracting  self-similar groups.  And this is an algebraic counter part of expanding covering maps, that conversely every  contracting self-similar group, so every group given by relations like above there, which  satisfies this condition, appears in this construction.  So it is iterated modeming group of some expanding map with one caveat, you have to consider  orbit spaces, like you have to include subhypropolic cases here.  And that's a little bit technical in this case.  But modulo this technicality, you have a complete like a bijection between expanding  covering maps, modulo topological conjugacy, and contracting self-similar groups, modulo  some type of algebraic equivalence, which is also a little bit non-trivial, but very  can be easily formulated.  So we have an algebraic theory in this way, we get an algebraic theory of expanding covering  maps.  In particular, this implies that there are only countably many of them, up to topological  conjugacy.  That's one of the proofs that you can use.  So how you can reconstruct from this semi-group this expanding map.  So the idea is very similar to the idea of hyperbolic, gromov hyperbolic groups and boundaries  of gromov hyperbolic groups.  In a sense, this semi-group should be thought of as a gromov hyperbolic group, even though  it's not hyperbolic, and not a group.  I mean, it is not a group, but it is hyperbolic.  So let me explain how it is hyperbolic.  So if you have a group, you draw the kilograph, and it's negatively curved, and then the boundary  is the space that you're interested in.  Here, the kilograph of the semi-group is not what you should think about.  What you should think about is the kilograph of the associated groupoid of germs.  So you have this semi-group acts on the boundary of the tree.  So it's a group of partial homomorphisms of the boundary of the tree, of the boundary  of the pre-images.  So you can look at germs of this semi-group, germs of its action on the boundary.  So you take a point on the boundary.  So you have your tree of pre-images, it has your boundary, it has the boundary.  And then you pick some point of that boundary, just pick some backward orbit.  So let's call it xi.  And then take your generating set of this semi-group, and then for every element of the  semi-group, you have the corresponding germ, if that element is defined on xi, you have  your element as defined by some path L, it acts on some clop and set, and if that clop  and set contains xi, you have the germ of the action.  So you identify two semi-group elements if they act the same way around this point xi.  And then you draw the calligraph.  Among these germs, if you have one germ and another germ, so the germs will be vertices  of your calligraph.  And you connect one germ to another if you have a generator such that this germ is composition  of this germ and the germ of the generator.  So it's exactly the same as for calligraphs of groups.  The only difference is that you have to fix a point and look at germs of this group at  a point.  If you have a group action, you will get every element will be defined on this point.  So these germs will come from group elements and every element will have the germ at that  point and it will be closely related to the calligraph of the group.  In this case, there are many elements of the semi-group which are not defined on xi, you  ignore them.  So how will this graph look like in our case?  So in our case, our generating set is split into two parts.  There is the generating set coming from loops and the generating set coming from connecting  base point to immediate frame edges.  And so the edges of the calligraph will also split into two parts.  There will be edges which correspond to loops and edges corresponding to the connecting  paths and it's natural to draw edges corresponding to groups kind of horizontally, edges corresponding  to connecting paths vertically.  If you take the vertical part of the calligraph, you will get a regular tree.  So for example, in degree two, you will get a tree which has one edge going down and two  edges going up.  So these are our x's.  X0, X1, X0, X1, X0, X1 and so on.  The edges corresponding to loops, so generators of the iterated modem group, they will connect  these vertices horizontally.  So in this tree, they will, because the loops will preserve the levels on the tree, you  will get some connections horizontally.  So in each level of this tree, there will be some horizontal edges.  You'll get this infinite graph which is naturally graded by this subtree corresponding to the  set x, these connecting paths.  And edges either these connecting paths arranged into a tree or some horizontal paths connecting  these vertices of the same level.  And the fact is that this graph, so let me denote this graph by gamma, it depends on  xi.  This graph is grommar hyperbolic.  So that's what I meant when I said that this is a hyperbolic semi-group in a way.  But it's not just this, graph has a special point on the boundary corresponding to going  down in this tree.  So the boundary has a favorite point.  Let me call it omega xi.  And then we have the grommar boundary and it's natural to remove that point.  Because then the boundary will correspond to going up in this tree.  And it will be a non-compact space kind of completing these three up stairs.  So it will be like a half plane model of the hyperbolic plane.  So there is a special point and then there is the real kind of real line.  And so what is that space?  What is that boundary?  So remember xi is an inverse or is a point of the boundary.  So it's an inverse orbit of t.  You choose each time one pre-image.  So this boundary will be a leaf of the natural extension of the covering.  So if you have our covering map f, we can look at the inverse image, the projective limit,  the inverse limit of iterations of f.  And then it is naturally a fiber bundle where fibers are the boundaries of these trees that  I was talking about.  And base is m.  And so in that fiber bundle horizontally you get leaves.  And the case when m is path connected, those leaves are exactly the path connected components  of the inverse limit.  And xi corresponds to one point of the inverse limit.  And then this boundary is exactly the connected component, the component corresponding to  that point.  And more precisely, let's m hat will be the inverse limit of these spaces.  Then f induces a homomorphism of this inverse limit.  It's called the natural extension.  And this natural extension is a hyperbolic dynamical system.  So there are stable and unstable equivalence classes.  And this boundary of this chromo-hyprobolic graph is the unstable equivalence class of  the point xi.  So it's, yeah.  OK.  So now, yeah.  Yes.  Yes.  Yes.  Yes.  So the cantor sets, if you take the basilica, natural extension of basilica, the cantor  sets are the stable equivalence classes.  So if you take two points in that cantor set and iterate your natural extension, they will  converge.  The distances will go to zero.  Now if you take the inverse map, then the set of sets, you have the unstable equivalence  relation.  So it's the pairs of points where distances go to zero if you take the inverse map.  So these equivalence classes, the unstable equivalence classes, are exactly the boundaries  of these graphs.  And they are exactly the path-connected components of that bundle.  The path-connected components with a natural inductive limit topology.  So when you unwrap it.  OK.  So let me go, OK.  Sorry.  So this is a particular case of a more general setting of so-called hyperbolic group points.  So that includes many more examples, not only coming from expanding maps.  And in all these cases, the calligraphs have this, the calligraph of these hyperbolic group  points, they have this special point on the boundary.  And so they are also stratified like this by horocycles corresponding to that special  point.  So these levels of the trees are the horoscales corresponding to the, sorry, horospheres corresponding  to this point on the boundary.  And we have the levels.  Levels of the levels can be described using the Buseman cos-cycle.  The Buseman cos-cycle, in this case, it just measures the difference between the levels  of this graph.  So we have, if you take two germs, then you can close this diagram.  So you get another germ closing the diagram.  And the value of this cos-cycle on this germ, let this germ be gamma, the Buseman cos-cycle  beta of gamma is just the difference of levels.  The level of the end of the germ minus the level, well, it's positive if you go up and  take the difference between these levels.  So that's the classical Buseman cos-cycle for a hyperbolic graph associated with a point  on the boundary.  But that's not the only one Buseman cos-cycle, because if you forget where this graph came  from, you look only at quasi-ozometry class of this graph, you just remember this graph  up to quasi-ozometry, and you remember this special point.  There will be many Buseman cos-cycles associated with each realization of this graph in the  quasi-ozometry class.  So let me give you a general definition.  And then I will, in the remaining 10 minutes, talk about how it is related to the dimension.  So let G be a topological groupoid of germs of some semi-group.  So in our case, we have a semi-group.  It consists of homomorphisms.  You take germs of these homomorphisms, and the topology is natural.  If you have some partial homomorphism, then germs of these partial homomorphisms, the  set of germs of these partial homomorphisms, is homomorphic to the domain.  So you have your homomorphism, it's a bunch of germs, and this bunch of germs has the  same topology as by the domain of rain.  So you get, if you have a semi-group of local homomorphisms, that defines a topological  groupoid, so the set of these germs.  And then a quasi-co-cycle on G is a map better from G to R, such that there exists epsilon  or not epsilon, eta greater than 0, such that first, this is almost continuous in the sense  that for every germ gamma, there exists a neighborhood U of gamma such that the values  of beta vary at most by eta for all elements of this neighborhood.  And the co-cycle property is that for all composable germs, gamma 1, gamma 2, you have  the product gamma, gamma 2 of gamma 1, beta of gamma 2, gamma 1 minus beta of gamma 2  plus beta of gamma 1 is less than eta.  So it's almost additive, and it's almost continuous.  So an example is, well, where eta is 0, actually, is that co-cycle coming from those graphs.  So for the semi-group, which I was talking about today, you look how much you change  the level, from which level you go to which level, that's an integer.  And it's continuous, it's locally constant, actually, and if you go up or down, you add  the differences.  So that's an honest continuous co-cycle.  Here's another example, which is kind of dual.  Yeah?  Sorry?  Yeah, yeah, I know.  So here's another example, you take F to be the groupoid of germs  defined, so generated by F.  F is a local homomorphism.  So its germs are invertible.  So you can generate a semi-group of local homomorphisms by F.  That will give you a groupoid of germs, and there are two natural choices.  So every element of this group point looks like that, like this.  You take a point, you apply F some number of times, and then you apply branches of F inverse  some number of times.  So if you compose F and then F inverse, that will be a germ in this groupoid, and all elements  of this groupoid are like that.  So you go N times here, and say M times here, so you can define the value of the germ of  the Bussmann co-cycle as say M minus N, for example.  And this will be a co-cycle, if I did it right order.  If you draw the kilograms of the kilograms, same way as we did over there, they will be  exactly trees without any horizontal edges.  And this Bussmann co-cycle is just the difference of the levels.  But suppose that M, F is an expanding map on, say, for example, it is a rational function.  Then another choice would be to take logarithm, negative logarithm of the absolute value of  the derivative at your point, at your base point.  So this is point Z, at point Z.  And this is also a co-cycle.  And it is also a Bussmann co-cycle for the same graph, but for a different realization,  for different lengths of edges.  So in the same quasi-zometry class, you have these two co-cycles.  Okay, unfortunately, I don't have time to tell what I wanted to say, all what I wanted  to say.  I'm going to finish now by formulating some statements, and that will be it.  So the point is that these two pictures are dual to each other.  This groupoid and this groupoid, in the sense that, okay, let me, so if you have a Bussmann  quasi-co-cycle, so we have two groupoys.  One groupoid is what I was talking, what I defined at the beginning of the lecture,  the groupoid defined by path.  Another groupoid is groupoid of germs generated by the expanding maps.  And if you have better Bussmann co-cycle on G, then it can be in a unique way up to bounded  additive constant associated with a Bussmann co-cycle on the natural extension.  So it induces a Bussmann co-cycle, sorry, eco-cycle on the groupoid generated by the  natural extension.  And then this co-cycle induces a Bussmann co-cycle on the other groupoid.  So if you start from a Bussmann co-cycle on a Kili graph or one groupoid, through the  natural extension you get a dual Bussmann co-cycle on the other groupoid.  So if you start from the derivative, for example, rational function, you get a co-cycle  on the groupoid for the self-similar group.  So the project which I mentioned is to study Bussmann co-cycles on self-similar groups.  And each Bussmann co-cycle on this self-similar group will define a measure on the Julia set,  so on the gram of a hyperbolic boundary, and a class of metrics.  So if you have a Bussmann co-cycle beta, then it associates the visual metric, so this  visual metric exists starting from some exponent, and you have the measure, basically the Patterson-Salivan  measure associated with this Bussmann co-cycle.  So this Bussmann co-cycle will be, for the measure, the Radon-Nikodem derivative will  be exponent of this Bussmann co-cycle with some coefficient, and the visual metric will  be the map, the expanding map will be a similarity with some coefficient, with some derivative,  and the logarithm of the derivative will be this Bussmann co-cycle with some coefficient.  So you can associate with each Bussmann co-cycle its Hausdorff dimension, so that's the minimum  of the Hausdorff dimensions of the visual metric, and one can associate then the smallest  such Hausdorff dimension, which will be the isomorphous regular conformal dimension.  And one last quick theorem is the following.  So I mentioned that these self-similar iterative modem groups are contracting, and contracting  means that, as I probably raised that definition already, that these elements belong to a finite  set, and that can be in a different way formulated as contraction of lengths.  So if you have a group element, and you take its length in the generating set, if you look  at how this group element acts on the subtree, so you look at these gv for some fixed level  n, you take all of them, you take all these elements, if you remember they were defined  by triangles or paths, and you take the vector in of lengths of this, of all these elements,  so you take lengths of all these restrictions on that level, x to the n, this is a vector  in r to x to the n, actually in z, and you look how much the LP length, so you take the  norm, the LP norm of this vector, and you look how much the LP norm is contracted, and  you take the exponential contraction rate of this LP norm.  So being contracting is equivalent for L infinity norm to be contracted, and then there  will be a critical value of p for which you still have contraction, and the theorem is  that this critical value is very interesting, by the way I include here also p less than  1, so theorem is that the alphos regular conformal dimension of m is greater or equal to the  critical value of p for which g is LP contracting.  Okay, so the project is to study these Busseman co-cycles and LP contractions also similar  groups which are purely algebraic object and apply it to the corresponding expanding  method.  Thank you very much, let's thank the speaker.  So we have time for some quick questions.  I think your measures by this graph are somehow related to geometry coding, your graphs are  related to geometry coding trees, I developed a theory and transport of measure from symbolic  space to…  Yes, so this…  There's another graph, but maybe it's very similar to yours by pilgrim and…  It is the same graph, I mean, the kind of… my graph is a covering graph of the graph.  Another visual measure…  Yeah, yeah, it is related to the work, yes.  I should have mentioned that, of course.  So in the case of the rational map, so we have this three-dimensional hyperbolic spaces  attached to the leaves of the natural extension and I guess your heligraph or the group point  is some discrete object whether it's a metric…  Yes.  …to this hyperbolic space.  Yes, that's right, yeah.  And so if you consider general situation to expanding map, do you have… well, you can…  You mentioned that you can also consider one-dimensional extensions of the unstable leaves, you have  a natural hyperbolic structure, the two-pitches…  Yeah, but it's only discrete.  Only…  I mean, I have only graphs, I don't have any three-dimensional…  So there is no sort of intrinsic good structure…  Well, it is up to quasi-zometry, it's a unique structure, but it's like a general… case  of general hyperbolic groups that you don't have any manifolds or things like that.  So it's the only way to get it through the calligraph?  Yeah.  Yeah.  What about the upper bound for the alpha-cellular guillotine, for what I mentioned?  Do you have anything to say about this?  So it's not…  So this can be less than one.  So in the cases when M is topologically one-dimensional.  So the inequality can be strict, but the only case is when it's strict is sub-hyperbolic  cases with really weird orbit spaces.  Like for the Grigorychou group, this number is exactly the number which appears in the  growth of the Grigorychou group.  So it's really interesting when do we have a stricter inequality here?  And so this is like maybe in the… actually in the expanding case, not sub-hyperbolic,  but when you really have an expansion the way I defined it, maybe we have equality here,  but I don't know.  I had a question.  Yeah.  So you defined, I think you defined what is a quasi-co-cycle.  Yeah.  But then you are also speaking about a Boozeman co-cycle.  Yes.  So in these four hyperbolic group-oids, if you have a hyperbolic group-oid like these  two examples, G and F, for each of these hyperbolic group-oids there in the calligraph, there  is a special point on the boundary.  And then you define one has to consider a special class of quasi-co-cycles which I call  Boozeman quasi-co-cycles related to that point on the boundary.  For example, in the case of the gromov hyperbolic groups, Boozeman co-cycle is a quasi-co-cycle  always.  There is also always this error.  And so inside this class of quasi-co-cycles, yeah.  So I give definition what is a quasi-co-cycle, but I didn't give a definition which is a  Boozeman quasi-co-cycle.  So it's some special type of co-cycles associated with hyperbolic group-oids.  Okay, thank you.  Any more questions in the room or remote?  Okay, so then let's thank the speaker again.
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