 I would like to thank organizers for inviting me here. Very pleasant, nice place. Also, I would like to remind that there was announcement about the conference dedicated to UNOSO. The poster is outside with details. I would like to emphasize that two organizers, namely Andrea Agarachio and Alexander Bufiato, worked in the department that was led by UNOSO. And I am affiliated with this department. So it was a big pleasure for us to work and to communicate with this great mathematician whose name is very often pronounced during this meeting. And also, Yakov Pesin was one of students of UNOSO. So everybody is welcome to attend this meeting. Today, I will speak about group actions of special sort, so-called totally non-free actions. I have a problem with the voice, but hope you are able to hear me, particularly because of this equipment. And invariant random subgroups. So I will begin with listing some classes of groups. Before, mostly, I understand some lead groups, like GL and R, where we are figurative and lettuces in these groups or subgroups in lead groups. But there is a wealth of discrete groups. There is abstract group theory. There is a combinatorial group theory. There is geometric group theory. And recently, measured group theory started to develop. And what is measured group theory is at the end of 60s, 70s, when Yakov Pesin, when me were students in Moscow State University of Lomonoso, attending seminars of UNOSO, of Sinai, and other great mathematicians. So sometimes, talks were considered triples of the forum group acting on probability measure space X mu. And at that time, it was called dynamical system with non-community time. And group G, again, can be topological group, can be lead group, but also abstract, can be abelian group, non-abelian group, but abstract group time to time were appearing at that time. Now, the study of such triples is a very popular subject. And in contrast with 70s, 80s, where mostly attention was paid to the measure space, group didn't play so much role. Now, more attention is paid to the group. What can be said about group if you know some properties of the action? Now, I will list you several, just sporadically taken from my mind classes of groups, just to refresh your memory, the varieties of the world of discrete groups. In my talk, I will consider only commutative groups, abelian, sorry, only countable groups. So, among the simplest is class of abelian groups, which is subclass of the class of nilpotent groups, subclass of the class of solvable groups. These are classical classes that people study in undergraduate courses of algebra. Then there is splitting of groups on amenable and non-amenable, very important in the notion, due to for Neyman discrete case and Bogolubu in topological case and Bogolubu definition for general topological group was a group with invariant mean. And the main result was that such a group acting on compact space has invariant probability measure, generalization of Kralov-Bogolubu theorem, one of the most important and often we use theorem in dynamical system. There is subclass of amenable groups called elementary amenable, which is subclass of groups without free the subgroup on two generators. This is just, it is not very, almost not important at the moment, like really like to understand, to don't ask me for definition. What are elementary amenable group? Also at the end I can define. Now, okay, some other list, three groups which will play some role. In my talk, the realization of them, grommu hyperbole group, there is so-called automatic group, even more general class. This is a subset of this set of SOFI groups, very popular subject of recent investigation and many interesting results are obtained for the SOFI groups. The problem is that nobody is able to produce example of non-SOFI group. Now, okay, such classical groups as group of auto-automorphism of free group, which is closely related to mapping class groups. There are so-called Tarski monsters. Higman-Thomson, finally presented infinite groups and so on. More reasoned are groups associated with minimal homeomorphism of a counter set. If you have such minimal homeomorphism, counter set T acting on X, a counter set, then you can associate set in a countable group, which is called topological full group. This is standard notation, which perhaps at the very end, if time permit will be mentioned again. As in the class of rigidly finite groups, these are infinite groups that can be approximated by a certain sense by finite groups, group acting on trees, buildings. Now, this slide shows something completely different and basically introduced and studied by speaker and his students and collaborators, classes of so-called so-called similar groups, fractal groups, iterated monodromy groups, branch groups, which will appear later and play some role now. Groups of intermediate growth in the intermediate growth in the sense of Milner-Schwarz. You count number of elements, so group has to be finitely generated. You count number of elements of lengths not greater than N. N is the natural number going to infinity and growth function, the number of such elements, grow faster than any polynomial, but slower than exponential. So there was question of Milner if such group exists and I constructed such a group and now these groups of intermediate growth are closely related to all these listed examples and appear in various situations as well. Now, what is totally non-free action? Again, I consider in my talk only countable groups and given a topological dynamical system, so group acting on say compact, not necessarily compact topological space by homeomorphism, so this is called topological dynamical system or metric dynamical system when you have group acting on measure space, in my talk measure will be probability and invariant. One can consider other situation when measure is quasi-invariant, but let it be probability invariant measure. Okay, so basically I will deal with this situation but also sometimes with this. There is well-known notion of a freeness of action, action of such dynamical system on commutative time or measured group GX mu is free if there is a subset of full measure with the properties the stabilizer of each point is trivial. So this is notation for stabilizer. Stabilizer of point in group G is a sub-group of elements which fix this point, it's a sub-group. It's clearly so freeness of action and means the triviality of stabilizer almost sure. And this can be formulated, here assumption that the group is countable is important that for every non-identity element, the measure of the set of fixed points of element G is zero. So fixed G is a set of fixed points for element G. So there are two approaches for the definition of freeness. In fact, it's essentially free. So because we assume that the first condition holds mu almost sure. Yeah, so here I remind what it means, this notation. Okay, and a few years ago, Anatoly Varshik, he came with very interesting definitions. He called triples action extremely non-free if again there is a subset of full measure aside that for any pair of different points from this subset, stabilizers are different. So if your space X, okay, is uncountable space, space like the back space isomorphic to interval zero one or whatever, so there are uncountable many points in X. So in particular, this means that the set of sub-groups of a group must be uncountable. There must be many, many sub-groups in your group. Okay, but here, okay, it requires that the different points almost surely have different stabilizers. So this is complete opposite to the freeness. In a free case, we assume that stabilizer is trivial group and here not only non-trivial but different points have different stabilizers. And he called triple totally non-free if the set of family of sets fixed points of elements generate sigma algebra of the measure space. So there is sigma algebra of measurable subsets of the X mu, so if it is generated as a set. So it means that this family of sets which are sets of fixed points of elements is rich family, it's base of the sigma algebra of measurable subsets. This young Ukrainian mathematician Artem.co we recently came to stronger condition. We call situation action absolutely non-free if every measurable, for every measurable subset and every positive epsilon, there is element in a group such that the measure of a different symmetric difference between E and the set of fixed points of G is less than epsilon. It's clear that this C is stronger than previous ZB and Wershik when he gave his definition he observed that A in fact is equivalent to B. So total non-freeness is equivalent to extreme non-freeness. And natural question arise what group have a faithful totally non-free action. So I would like to emphasize that before people were mostly interested in free actions if you look at classical books and so usually assumption of freeness play a role. And okay, Wershik came from completely different perspective now we are interested in totally non-free actions. Okay, and how this affects dynamics and how this affects group theory. Let us see. So the main point of my talk is that such actions lead to the so-called invariant random subgroups which will be defined soon and to the so-called factor representations which will be just mentioned and didn't discuss it because of absence of time. And both things are pretty important. So now I'm going to explain what is invariant random subgroup. Random subgroup, okay, so given a group again countable group in my talk but this works for locally compact groups and so on. By sub G I denote set of subgroups but I call it space because I consider is this topology. This topology is topology induced from the product topology of set of two elements zero one to the power G. So space of configurations on a group G just given subgroup H in G we identify its characteristic function which takes value zero or one depending G belongs to H or not. And this characteristic function is the element of this space. So here we have product topology which in case group is infinite makes this homomorphic to a counter set. And here we have close subset. Close subset in this space. So we induce topology. So now we have topology on the space of subgroups. And this is pretty interesting object to study of topological and combinatorial properties of this space of subgroups with this topology in the case of locally compact groups there is also definition more complicated and called shabati topology. So study of properties of this group. In our situation this space is a totally disconnected compact metrizable space. And converges in this space means the following sequence of subgroups HN converged to H. Even only if element G belongs to H then it must belong to all members of this sequence starting from some moment. If it doesn't belong to H it doesn't belong to all members of sequence starting from some moment. Now there is in this script to set theory notion of counter bandics on rank. So given a space that can be set of isolated points you can delete isolated points. This is called taking counter bandics on derivative new isolated points may appear and you continue this process. So in our situation when space is totally disconnected metrizable compact space this process of taking derivative will stabilize on certain countable ordinal and the first moment when stabilization whole is called counter bandics on rank. So and then when we finish this procedure of deleting isolated points eventually we come to perfect kernel that set without isolated points. And in our situation this set can be either empty or a counter set. And it will be empty if and only if the cardinality of the set of subgroups is countable. And otherwise it will be a counter set. So again for interesting group even if a group has only countable many subgroups and for instance nilpotent countable groups have only countable many subgroups and many solvable groups and Tarski monsters and so on. Still it is interesting method just to understand the topology of this discrete topological space but if there are uncountable many subgroups it's even more interesting you should like realize what group subgroups should be deleted just to reach this perfect kernel which is homomorphic to a counter set and this perfect kernel will be invariant. Yeah, I forget to say and this will appear immediately that we have action of a group on this space by conjugation. Group acts on the set of subgroups by conjugation. And this is action by homomorphism in our situation and this perfect kernel is invariant subset for this action. So restriction of action on this perfect kernel gives you action of your group on a counter set. Pretty interesting situation which can be used in many situation. Yeah, so one of the first question I will formulate few problems. Interesting group that the mind counter bendixon rank. By the way, in space of subgroups there is a subset of normal subgroups which is close subset which also is interesting for study. It's a counter bendixon rank. For instance for free group it is not known even for free group is two generators. It is not known what is counter bendixon rank of the space of normal subgroup. Of the space of subgroup is one, it's easy. Okay, so now what I already said that group G acts on the space of subgroup by conjugation. Given a subgroup H in G and given element G it maps H to H conjugated by G which is this expression. This action is by homomorphisms. What is a random subgroup? We have compact space of subgroups with random subgroup is random subgroup. It has a low or distribution which is probability measure. So we can identify random subgroups with probability measures on the space of subgroups. What is invariant random subgroup is random subgroup whose distribution, whose low, whose probability measure is invariant with respect to action by conjugation. So we are looking for those invariant, for those probability measures on the space of subgroups which are invariant with respect to this action. Action, it's called a joint action, action by conjugation. And again it's not difficult to understand that you can in certain sense it's important to restrict attention only on a perfect kernel on a counter set because it is invariant and on counter sets the space, there are many probabilities measure while surrounding things is a countable set of isolated points and so invariant measures are just data mass supported on finite orbits. But on the space of subgroups acts not only a group G itself by conjugation and this can be viewed as action of the group of in-automorphism. What I was speaking before is action of the group of in-automorphism on the space of subgroups. But now we can consider action of a large group, a group of all-automorphisms. Just given an automorphism phi and the subgroup H, you take image and this is action and again this is action by homomorphism. Because now you may look for random subgroup which is invariant with respect to action of this larger group. The larger group is, the set of invariant measures is less, less. So probability measure on the space of subgroups which are invariant with respect to action of group of all-automorphism are called characteristic random subgroup. And the characteristic comes from terminology in group theory. Subgroup of a group is called characteristic if it is invariant with respect to all-automorphism. Yeah, instead of subgroup to be invariant we consider probability measure which is invariant with respect to the action of automorphism group. And by the way, in previous definition in my opinion it was better to call not invariant random subgroup but normal random subgroup. But people started to call invariant random subgroups so I don't want to change terminology. Okay, so now we have like two situation. This action of a group or group or the same group of in-automorphism basically. And we are looking for invariant measure and the triples of this sort. Also one can consider certain subgroup which is intermediate. So group of in-automorphism is a subgroup of the group of all-automorphisms and quotient of it is normal subgroup and quotient is called group of auto-automorphism and one can take arbitrary intermediate group between these two and consider for random subgroup which are invariant with respect to action of this intermediate group. But we will not go into details of this. Examples, examples. If you have normal subgroup in G and put just delta mass, so H is a point in the space of subgroup, you put delta mass, it's invariant random subgroup just because of normality. It's fixed point for action by conjugation or if a group has finite conjugacy class and conjugation of H, H1, H2, HN, you put delta masses in HI and take some of them and average. These are examples of invariant random subgroup but these are kind of not interesting examples. And we are interested in continuous, continuous invariant random subgroups. Continuous means this measure should be continuous, no points of positive measure. And additionally, we are interested in ergodic measures. Everybody's, I hope, is familiar in this auditory. What means ergodic, so action is ergodic if invariant subsets have measure zero or one. Yeah, the set of invariant random subgroups is simple, showcase simple, as well as set of characteristic random subgroups is some sub-simplex of that previous simple. In any case, also showcase simple, so ergodic measures correspond to extreme points. So we are interested in finding extreme points in cardinality of the set of extremes points and so on. And general questions are what groups have ergodic continuous invariant random subgroups. In the future, I will abbreviate this like E, C, I, R, S. And given interesting group, how many such I, R, S it has. So as I told, we have simple of invariant random subgroups, simple of characteristic random subgroups. And we are interested in understanding these simplexes, counting the set of extreme points and so on. Two important cases are the so-called Paulson simplex. A simplex when extreme points are dense in the simplex. Take closer to the set of extreme points, you get your simplex back. And such a simplex by result of Lyndon Strauss, Olson, and some name, I'm sorry, I forgot, shows that such simplex is unique up to a fine isomorphism. And the example of such simplex can be produced very easy. You consider two sided shift over say binary alphabet 01. And consider the simplex of all invariant probability measures. And this is Paulson simplex. And another situation is Bauer simplex when the set of extreme points is closed. And the basic example is you take say compact metric space, consider simplex of probability measures. And it is Bauer simplex set of extreme points is just X itself. Now how these two notions are related. If you, let's go back to our dynamical system, group acting on space X mu, probability invariant. And then we have a map, which I denote phi from X to space of subgroups, namely to each point of the space we associate stabilizer of this point. So here point has subgroup. Now total non-freeness means that phi is almost sure injective. Different points have different images. And this leads to the invariant random subgroup. You take measure mu, which is image of mu and this map phi, mu is phi star mu. So now started from GX mu, you get G acting on space of subgroup by preserving measure probability measure mu. Mu is image of mu. And if mu was ergodic, mu will be ergodic. If mu was continuous without atoms, mu will be. Yeah, so it means that totally non-free actions on the back space produce invariant random subgroups. Ergodic actions produce ergodic. And so on. So this is a link between totally non-free actions and the invariant random subgroups. Totally non-free actions produce IRS. Now some results about IRS, very short account of results. So at the end I am going to list a certain result obtained by my collaborators and me in the last three, four years. But just short excerpts in the history. So first of all, there are groups without any continuous ergodic IRS. And as I told, I will abbreviate this ergodic a continuous IRS. Lettices in simple higher-rankly groups. Tak Tsimer 1994. Higman Thomson groups. So Thomson, he produced the first example and Higman then generalized that the first examples of finitely presented given by finite set of generators and finite set of relators. Infinite simple groups. Very famous in group theory, very studied. There is opinions that they are not so big. Nobody is able to prove this. In any case, these groups are very poor with respect to continuous ergodic IRS. They don't have this result due to... To us, Artem Dotkov was mentioned, and another young mathematician from Kharkov, Konstantin Medenets. Now, there are groups with uncountably... When I say uncountably, I mean cardinality of continuum. Tutti zhpave alef-0, alef-0. Rufs with uncountably many continuous ergodic IRS. For instance, non-Iberian free groups of FR, are number of generators from two till infinity, but countable infinity. This result is five years old, the result of Louis Bowen. So the name of Bowen was pronounced many times during this conference, but it was Rufus Bowen who had shot, but very bright mathematical life in the seventies. This is a reason for this, Louis Bowen, my reason colleague, but now in Austin, Texas. So he used various sophisticated ideas. Five years ago, so he proved that non-Iberian free groups have many, many invariant continuous ergodic invariant measures. Now, such group as a group of finite permutation of infinite countable set, set of natural numbers. For instance, this is due to worship. And some other, so this group is locally finite. Every finite generated subgroup is finite, so it's union of finite subgroups. Such group are called locally finite groups and there are results, again, due to co-inventedness. For instance, some special inductive limits of a finite group also have this property. Okay, now some recent results of speaking collaborators. It's not result, just observation, because nilpotent groups have only countable many subgroups, so we may not consider them. They are not interesting, not very interesting. But solvable groups usually have many ergodic continuous IRS. And as a first example that we considered carefully is the so-called lemplight group which play important role in a group theory, in theory of random walks, in theory of L2 co-homology and so on, L2 beta numbers. So this is just a risk product of group of order two and infinite cyclic group. What is that? You take a direct sum of infinitely many copies of group of order two indexed by integers and in a group of integers X on this direct sum by shift. And this action is action by automorphism, so you can consider corresponding semi-direct product of Z with this I billion group infinity to group of elementary to group of infinite rank and this is called lemplight group. In fact, it's too generated, can be generated by only two elements. It's a meta-billion, two-step solvable extension of I billion by I billion, has exponential growth, is not finally presented, it's quite non-trivial group, quite interesting group, has uncountably many subgroups. We computed perfect kernel and this Bendixon rank, but let me focus on the, just on this statement, it's due to Louis Bowen, Rostislav Krautchenko and me, that the simplest of invariant random subgroups on the lemplight group contain Paulson sub-symplex. So inside there is sub-symplex which is Paulson-symplex. So it shows that already a meta-billion group may have many, many invariant random subgroups and in fact it has uncountably many ergodic, ergodic continuous invariant random subgroup. So in fact, our statement is more precise but just to save time, I formulate it in this more simple form. Any questions so far, by the way, I know that I speak too fast. I will indicate and tell so that everything is clear. Now, you know that free group is a universal object in category of groups, it means that every group can be, is homomorphic image of a free group, in particular two group, group generated by two elements is homomorphic image of a free group with two generators. And now the following observations that if you have surjection from a group G onto H, so phi is epimorphism, then you get two maps from subgroups of G to subgroups of H, just you take image of subgroup and this is Borrell map and in opposite direction, subgroup of H to subgroup of G but containing kernel of phi. You take just pre-image, pre-image. And surprisingly this like opposite map is even continuous map. So consequence of this is that if factor group has many invariant random subgroups, then you can leave them to G. The group G itself will be. So now result of Lewis-Borlin that free group onto generators has many IRS just follows from the analysis result for the lamplighted group. Lamplighted group has many so in particular F2 but Lewis-Borlin was using percolation to show his result and so on. Okay, now kind of more. Another result due to Boevin crouching on me stays the following. Every non-Abelian free group, FR, group with R generators, number generators two or larger of infinite or countably infinite rank has uncountably many continuous. Weekly mixing characteristic random subgroups. So this is much more stronger than before. So first of all, instead invariant random subgroup. Here we may claim about characteristic random subgroup. And remember I discussed there are much less characteristic random subgroups than just invariant random subgroup. Secondly, instead ergodicity we claim that the action by in the automorphism action of free group by so here we mean action by automorphism group of free group. But we claim that mixing already hold if you restrict action of full group of automorphism on the subgroup of in the automorphism. We already have weekly mixing property which is much stronger than ergodicity. So previous results that free group has many continuous ergodic invariant random subgroups is made stronger in two directions. And using the fact that subgroup of characteristic group is again characteristic in contrast with normal subgroup. Normal subgroup of normal subgroup is not necessarily normal. So it's called subnormal. So, but the subgroup of characteristic subgroup of characteristic is characteristic. And now we can conclude from this result that if a group G has normal subgroup and this normal subgroup is isomorphic to free group non-community free group then G itself has uncountably many continuous weekly mixing IRS. And this query is applicable in the following situation, for instance, in the situation of non-elementary gromo hyperbolic groups. In the situation of the mapping class group of possibly punctured by finitely many points oriented surface of negative oil characteristic. And the auto-automorphism group of a free group, very important group started by many mathematicians. So here we use results of the mother's hand, the manager of Dell, Dennis Austin, who proved that the groups listed groups have normal subgroups which are isomorphic to non-community free groups. Among to prove that theorem we use some result of ADIAN in few days I will give a talk in Moscow at conference dedicated to 85th best day of ADIAN who has a few great results in group theory. One of them is related to so-called bounded Bersight problem and another here managed to construct infinite family of identities which such independent family none of identities like, identity I mean group identities like identity of commutativity commutator X, Y equals one or identity of nilpotence like chain of commutators is identically equal to one. So here built infinite family of independent identities and as one of corollies of this is that free group, non-community free group has uncountably many characteristics subgroups. It was known not known before. And another we take I billion group, just elementary P group of infinite rank, ZP is a group of odd P, P prime and on I billion groups in very random subgroups is just a space of all probability measures on the space of subgroups because action by conjugation is trivial action. But one can study characteristic random subgroups on I billion groups. And this is quite non-trivial thing and even for this particular case, it's not trivial. In fact, we proved some more general result but in this particular situation, the set of extreme points of the simplest of characteristic random subgroup is countable set. So simple itself is power. Simplex is a sequence of methods mu m is a probability measure in certain sense uniformly distributed on all subgroups of G of index P to the power m. P to the power m, sorry, not n but m. Now result by, I already mentioned that the nilpotent group is not interesting. Now, if you consider group from the point of view of growth, groups of polynomial growth are virtually nilpotent by famous theorem of Gromov. So this is not interesting case in our situation. Now, a group of exponential growth like free group, lamplight group, they may have uncountably many in very random subgroups, this was already mentioned. And what about groups of intermediate growth? And with Bindrin and Agnabida, we showed that there is group of intermediate between polynomial and exponential growth with uncountably many ergodic continuous invariant random subgroups. And perhaps this is the last statement that my time is going to the end that there is class of branch groups that was mentioned at the beginning. Next few slides we will tell few words, what is that? But right now assume you know what is branch group. Statement tells that each branch group has at least countably many ergodic continuous invariant random subgroups. And the first example of branch group is again group, sorry there was some notation, a group generated by four involution, group acting by automorphism of binary root three. This was the first example of group of intermediate growth constructed by me. And then later I understood, studying the properties of this group, I understood that it has branch structures, so introduce this class of branch group. So this statement, this theorem is applicable to this group which is studied by many people. So we know that at least countably many ergodic continuous IRS exists on this group, but we don't know if it is uncountable or countable. And this is one of open problems. Just I will finish in two minutes. Dr. Agracion, is it okay? Yeah, just a few words about branch groups. So people study actions of groups on trees, in particular on root trees. Here is binary root tree. Root is fixed point. And the group acts by automorphisms of the tree. There is boundary of this tree, the tree is infinite. The boundary consists of infinite geodesic rays and it can be identified with the space of binary one-sided infinity sequences. And group acts also on the boundary in natural way. And uniform Bernoulli measure is invariant for such actions. So we have this triple group acts on the boundary which is counter set by Bernoulli measure preserving transformation. And the fact is now group is branch, if it acts, if it has a faithful action on spherically homogenous root tree with the properties that action is level transitive, transitive on each level. And the structure of subnormal subgroup in certain sets imitates structure of the tree. And it was observed already in 2000 by Bartoldi and me and the branch group acts on the boundary completely non-free. So just different points have different stabilizers. So in particular total non-free with respect to uniform Bernoulli measure and the recent reviews that we understood that action is also absolutely non-free. Definition was told, say given at the beginning. And to prove this theorem, to prove the results that the branch group have at least countably many, we consider action diagonal action of a group on the ends power of boundary of the tree. But factorized by the action of symmetric group by permutation. We show that for each end such action is totally non-free. Has only countably many ergodic components and restriction on different connected components leads to different totally non-free actions. And therefore to different invariant random subgroups. Thank you for attention.