 Okay, first thank you very much for the invitation to speak and I'm really very happy to speak at the conference in honour of the MA. We started about the same time, I mean, it just started a bit earlier and I'm really happy actually that he's part of the geometric group theory community, if he's still part of it. And yeah, I think one of the great things that are still true for academia is to have people like him around, unfortunately now it's quite a big distance, but much less conferences in person, but still it's a great advantage. Okay, what I decided to talk about actually I myself now in recent years work less on groups, but what they choose, I choose this topic which actually emerged from a question that Nathalia Rips asked me exactly a year ago and somehow completely took me out of what I'm doing in this question and this was an attempt to answer what he asked, I'll get to that. So actually I worked in the JSJ starting at the ITS more or less 30 years ago in 1991. I will say something about how it started and for me I looked mainly for applications or for appearances of the JSJ in other algebraic categories, but apparently there are new things that are related to the JSJ or anyway some extensions of the JSJ in groups. I was rather surprised by that. And I should say also that when I was 30 years ago, when I started my career basically, I was very fascinated by the fact that theories from three-dimensional topology not only that they can be transferred to groups, but they are crucial to groups in order to prove all sorts of things. Now I think the JSJ is also fundamentally logic. And by moving to other categories, I thought that it's about time, first of all I found out the JSJ is probably fundamentally not algebraic categories such as people didn't know about this and one of the reasons to go to other categories is to try eventually to find some sort of categorical formulation of the JSJ because I think it's the general. But on the other hand what happened is that whenever I go to another algebraic category there are really lots of new phenomenas and now also in groups. So it's very difficult to make things as a theory categorical if whenever you go to any direction you find some new phenomenas that were not familiar, didn't appear before. I should say also just the closest thing, I plan to write this paper before the conference but I didn't manage to and so I really consider this still as a work in progress. It's not complete and it's not easy so take it with this, I even mentioned at some point what part I didn't write. But I think the work in general is conceptual so I decided to speak about it. It also as you'll see well hopefully if I'm still remembered I mean we had a conversation maybe 25 years ago on things that are related to what I'm going to talk about related at least in terms of the subject if I'll mention that and it's in the appropriate but it's still not completely written so I cannot really regard it as a complete work but anyway it suggests some new directions. So I'll start with the Josh J the composition, it will be a review I think many of you probably know it but it's important also for the continuation so I'll also to put it in context so I'll mention that so there is a now I think well-known theorem of Frédéric Poulenc for I think from 89 maybe 88, I'm not sure about the years that I've written here so it doesn't matter if it's plus minus few years. So what pull up proof is that if the outer automorphism group of hyperbody group is finite is infinite sorry then the group acts in a really free with virtually a billion stabilizers of segments or in fact okay virtually a billion is fine which in this case means either virtually cyclic or finite and then there's a fundamental work by Riebs and by Stina and Fein who prove that okay if you wanted to it's a core well it's a core of their work and the theorem of Poulin that if the outer automorphism group is infinite then a gamma acts on a cyclician tree with virtually a billion edge stabilizers okay so this is a fundamental fact and at the same time I was working on the isomorphism problem for hyperbody groups and during my thesis I managed to prove there's a morphism problem in the torsion free case when the group is rigid so when the outer automorphism group of the hyperbody group is finite or if you want according to these theorems if it doesn't have now it's okay the group is torsion free so it doesn't have cyclic or cyclic splitting or and it's really in the composable so for rigid groups there was a solution and then I tried to generalize it actually while being in the ITS it was my first positive position and I tried all kind of tricks to to get around the the the fact that the outer automorphism group is infinite see it's pretty clear that that when when when the outer automorphism group is infinite the group has many symmetries and this this may be an obstacle for solving the isomorphism problem indeed it is a problem and and I should say also that part of the solution to the to the isomorphism problems in the rigid case was that it's possible to decide if the if the if the group has a splitting or not this this is an output this is in a corollary of this of this of the algorithm for solving the isomorphism problem in there in the rigid case and as I said I tried all kind of tricks to get around the fact that the to get around when the automorphism group is infinite and I couldn't and then it was clear to me that if I want to to to generalize it I must find a theory to that will explain what is the structure of the outer automorphism group and what what are all the splitings of of how to find all the splitings of the group actually it's turkey I'm sorry whatever Bernie musket stated the ITS at the same time and it happened that at the same time he found a proof of the of the jacochelin and Johansen work for for a geometrically finite glanian groups actually a rather difficult proof and I managed to to go to his talk actually together with with l'unique putigailo and that that actually the proof itself has nothing to do with with the JSJ but it's a special case of what the jacochelin and Johansen did actually a rather complicated proof but but but this actually dragged me to the to the direction that I better find some sort of splitting like this for everybody groups and this will explain the structure of the automorphism group of the outer automorphism group and all the splitings of the group so really the lecture of musket somehow without him knowing and never discuss it with him but but this was somehow the trigger okay so so again these are these are some basics about the JSJ but but this will be important for the continuation so I am still going over that so gamma will be a one-ended group and okay I'll discuss it in the torsion free case actually for this method there's no much difference between the torsion free and and case we torsion if you consider automorphisms at least it's readings that come from automorphisms okay so so just think about the torsion free for that matter cycle splitting that divided into three categories okay I'll explain it in a second there are elliptic elliptic splitings in which case one can find a common refinement for the two splitings there's elliptic hyperbolic it's a dual position between two to the compositions but then in the torsion free case there will be a free splitting of the group so if you assume the what the group is one-ended and this doesn't appear and to sort of extend the most important case is the hyperbolic hyperbolic case so this is when splitings are somehow not compatible and it turns out that that in this case okay like in the in the three-dimensional picture there is there is a quadratically hanging subgroup okay in the case of torsion free here we are in the hyperbolic case there will be some some fundamental group of the punctured surface inside unless the group is a surface group itself and and then the two splitings come from intersecting simple closed curves on the on the on this surface and actually these two simple closed curve must fill the surface okay so this is the these are these are the three sort of trichotomy of the compositions that will play a role also in the future but but it's also it will be important for me afterwards to explain why the machine that constructed jfj will fail in the in the place that in the in the situation that I'm going to discuss okay so this is just the picture of what I what I just wrote in words so we have two splitings and we say that they are elliptic elliptic if the extra okay I'm just discussing in the torsion free case later on I'm not going to assume that anymore but but it doesn't really it's not it's not much different so so we say that it's elliptic elliptic if the edge group say z1 here can be conjugated into one of the vertices in the other and okay it's a dual position so z2 is also can be conjugated to the vertus group of the of the first one and in this case it's possible to find common refinement for both so we look at what splitings this vertus group inherit from the from the from the other splitting in the case of hyperbolic hyperbolic okay so a priori we just know that these two splitings are splitings of an abstract group or at the body group later finally presented you but in the beginning it was just for the body group and in this case the way I approached it is by building a machine so each each with each splitting we can associate the cycle group of automorphisms then twist so conjugating one of the vertices this is amalgamated product the hnn has a similar similar group of automorphism so we conjugate one of the vertus group and leave the other one fixed and we look at at the sequence of automorphism that is obtained by each time finite but it's growing and growing to the left we take an automorphism from the first side from the first side group of automorphism then automorphism from the second etc with powers which are growing fast so n1 will be much bigger than m1 m2 will be much bigger than n1 and so on to satisfy certain combinatorial properties that I'm not going to get into and okay using this this machine and and the refinement it's possible to find the composition which will be canonical actually in the case of hyperbolic group it will be invariant under automorphisms there's no problem with that and what happened is that I wrote maybe that's not a common notation but a finite subgroup of the outer automorphism group this is assuming the group is one ended now we'll have a map onto the direct sum okay each each automorphism because it preserve this splitting is up to conjugacy it acts on each of the surfaces okay so it acts in some automorphism and therefore it's mapped into the automorphism in the into the outer automorphism group and we get a map from each automorphism from each automorphism therefore from the whole from this whole group into the mapping class groups of these surfaces and the kernel will be some some free or someone asked something okay I should say actually in my written paper I assumed that this this the short attack sequence please later Gilbert Levit pointed out to me that it's not always the case okay I don't hear if someone asked something but all right I will continue because it's okay okay but this is this is the picture and this will be important also for the for the continuation okay this this short exact sequence sequence yeah is there a question I'm sorry I'm not from the audience okay all right so let's continue okay there was a whole list of continuation actually now it's it's interesting to say somehow sociologically that in the first couple of years practically no one believed that the JFJ actually exists somehow it wasn't accepted at all until what changed the situation was my work with Rips that generalized this work to finally presented groups actually with essentially with the same machine but we just just looking at finally presented groups instead of just hyperbole groups and here here there's no torsion free assumption but we looked only in cyclic splitting and shortly after bodice constructed the JFJ for hyperbole groups again with torsion and not not using automorphism but using the topology of the boundary okay later there were there were other I wrote some I mean there are many papers now the JFJ probably goes in dozens or even hundreds done with the instagive constructed the JFJ for finally presented groups but but over more general groups so say over polycyclic groups and then Koji Fujiwara and Panas Papasoglu constructed actually it's a slightly more general result but over the same class of groups using complexes of groups and the last two are two asterisk groups more recent one by Scotland's were group somehow at least were group was rather from the onto Peter Scott they were rather bothered by the fact that the JFJ that that we constructed is not exactly like the three-dimensional ones there are some some distinctions so they they managed to do it and they also constructed the composition which is environment and automorphism and the most I don't know more than anyway the most recent I think common reference for that is the work of you know then and the beat um that that that deals a lot with the with the question also of the canonicality of the gesture in case of a world group there's no problem really about canonicality but but the general finally presented groups there is and this is sometimes a problem also for in the applications okay but but what I want to talk about today will not be about general finitely presented groups but um but one should note you know when when I worked on on the JFJ for for at least for ocean free acrobatic groups the JFJ encoded both the automorphisms and the cycling split okay at least for one in the groups but okay but it can be generalized when you go to general finitely presented groups the JFJ in more complicated way encodes still encodes the splitings that's how it's constructed but it doesn't it cannot anymore encode all automorphisms I mean even subgroups of acrobatic groups right like like the example of of no already and you know all kind of example of finitely presented groups say where acrobatic groups fiber over them you know where the fiber group is this finitely presented group um it's impossible that the JFJ will encode this the automorphisms of these groups because if he did then uh mapping tower mapping towers over these groups will not be hyperbolic but um but he doesn't call the splitings and the question is what happened with with automorphisms in uh in other cases so so there are classes like like the body groups like limit groups where sometimes the relatively acrobatic groups where um where the JFJ encode all the automorphisms but in general one cannot expect that and apparently there are some natural questions when um a generalization of the JFJ is supposed to is supposed to do this work okay so so in some sense this work as I said it really started with this question of ribs but but before that okay this is a question that I answered at the time but but okay it didn't at least for me it didn't ring any bell um so Ruchani asked me I think around 2007 it was the time that she worked with Karin Vogman on automorphisms of private and clouding groups I'm not sure about the year but anyway it was around that and she asked if the JFJ encode all the automorphisms of of right-handed outing groups my answer was rather simple that okay it will give some of the automorphism but not all of them this is this is a correct answer but apparently okay if one if one generalize the JFJ then um then it will encode them all but anyway it it left like this there's a very okay I don't know if I managed to speak about right-handed outing groups in this talk but but there's a very nice work by Ruchani and Vogman and part of it with Chris and also there was another group Duncan, Kazachkov and Ramesnikov on automorphisms of of frogs but somehow not from this point of view I mean so one has to go a little further to to find really okay a generous form of JFJ that will encode this these automorphisms and the question that really somehow dragged me into this project was let me left my project and read so for now for a year actually and it was a question of rips he just called me one day during this pandemic and he asked what can one say about the automorphism of cubulated groups I mean he asked that because because he was familiar with with the application of the JFJ for the automorphisms of hyperbole groups and he asked whether whether one can do something like that for for cut zero cube complexes and you know because we know that that lattices in higher-ranked semi-centilit groups don't really have automorphism because of because of super rigidity so and and automorphism say three manifolds all come from the JFJ so so the question is whether all automorphism are somehow low-dimensional as in the case of in the sense of the JFJ so as in the case of hyperbole group and it was clear to me that in order to answer this question please partially it's better to look at cubulated groups although not all of them satisfy that but still okay it's better to look to them as as hierarchically hyperbole spaces over the fundamental groups I look at hierarchically hyperbole groups so HHGs so I don't know if anyone is familiar with it I'm not going to get into definition here because it's technical and and in some sense from my work although I meant to answer this question but but it's not the how to say it's not the main point it's a generalization but but a generalization if you make some additional assumption is not that difficult to to achieve after some preliminary so anyway these spaces were were introduced by Burstock, Hagen and Sisto I wrote the the this is really the year when the paper appeared they produced it you know three years before and what they did is they they took the very important work of major honesty that found hierarchical structure of the of the mapping class group okay starting with the curve complex and using other curve complexes of some subsurfaces again I'm not going to talk about this either and they they axiomatized their work and and talked about spaces that that satisfy these properties and it turns out that many examples satisfy that and therefore it's it's very it's very useful to study certain even certain example using this using the the axiomatization and using some results that they proved for this for these spaces so okay so it holds for the mapping class group but this is the place where they started it holds for example for right angle art in groups it holds for special or visually special cut zero cube cubicle groups and they have many more examples okay this is not anyway it's not the main thing at my top and and really the main thing to for me the main thing to to actually study and to try to understand is what's what is the automorphism group of a group that act on a product of hyperbolic spaces okay the jshc already failed sorry okay sometimes there's noise and I don't know if someone asked something okay so so really is what can one say about the group that acts on on on product like this okay so what is it first so products of hyperbolic spaces we take finitely many so m hyperbolic spaces we can assume that they're all not bounded otherwise the divided ones are not of course not interesting because otherwise we will assume that the the action is is proper properly discontinuous so okay so xj are actually hyperbolic spaces there's a group g that act oh I see that I didn't write here but but I meant cocompactly it doesn't have here it doesn't have to be isometric action because isometric I really mean in the sense you look at if you look at the axiomization of these hhgs but in this very special case of a product somehow this is the the most important case at least under the assumption that I'm going to make and okay what happens in this case yeah that see there are the projection spaces the assumption is that first of all okay the group g it may permute the the projection spaces so we can we can pass to a finite index subgroup if we want the characteristic subgroup well it will be important later to move the characteristic subgroup anyway it's a finite index okay that this finite index subgroup doesn't permute these spaces so it acts as it moves each projection space to itself and these ones I assume that they are isometric this is the assumption in the in the hhg case again well one can relax slightly these assumptions but that's on the point and and the question is what can be said about the structure of the outer automorphism of such a group and what can one say about the dynamics of individual automorphisms as I said okay I thought that if I want to have time I mean I still regard this case as the most important for for hhgs and if I have time I'll say something about how it's how it's generalized but this is anyway this is the in a way this is the main this is the heart of the problem now you may think that I don't okay someone who in in some sense if you look at I don't know pathologies or contrary to group theory then you know product spaces are are rather problematic actually not only in group theory but but but certainly in group theory but it's interesting to know maybe if I if I talk about this just conceptually in logic somehow product spaces are not a problem okay if you if you ask questions first of the question only about elements if you understand the factors you understand the product but but if you ask questions about subgroups about automorphism okay questions that are not first order then products are much more complicated than than than the factors so okay here is here are some problems that they're just good to keep in mind so actually there are two talks in this conference by PRP yesterday and by Martin Breitzen about sub-direct products of of sub-direct products of some direct products of free groups surface groups etc or other groups well also other groups in PRP talk yesterday but anyway Michailo was at least one of the first ones she what she did see this group is a direct sum of free groups is reasonably finite so so the word problem is solvable but what she did is she took a finite represented group that has unsolvable word problem and showed how to translate it using fiber product so that there is a finitely generated subgroup of this direct sum which which doesn't with where the membership problem is unsolvable and since then there are I don't know lots and lots of pathologies that were constructed using fiber products using other things sometimes sometimes problem to to even improve properties of pathologies or counter examples so this is this is one one thing that one should keep in mind subgroups it's impossible to control in in direct products in general even if you know everything about the subgroups of the factors okay another thing that is just good to keep in mind what what I'm going to say will not shed any light on this and these things but still you know if you consider three-dimensional topology in general in some sense this was this was one of the direction in three-dimensional topology before Thurston so any three-dimensional any compact three manifolds let's say close without boundary has a he got splitting so it can be made out of two handled bodies when we identified the boundary and this would be the van Kampen diagram for this for this he got splitting okay this is the fundamental group of the of the he got surface the free group is the fundamental group of the handle body and this is the van Kampen diagram I'm sorry I made a mistake in this notation but anyway it's a fundamental group of this so this handle bodies and the Poincare conjecture for instance ask in this in this formulation this is Stolling's formulation which is equivalent to the Poincare conjecture ask well if we know that this is an epimorphism is it true that there is a simple closed curve in the kernel okay this is equivalent to the to the Poincare conjecture and it's it's important to know that until now there's there's no algebraic approach that gets anywhere near the Poincare conjecture although the Poincare conjecture has okay different proof geometric proof by permanent but that's the only proof that we have to this algebraic statement okay and and one of the reasons that you know at least from my point of view that we that this is hard to approach is that we work with two factors and we want to make any changes in one factor it completely it immediately affects the other factor and controlling both of them somehow coordinating between the factors is a very it's a very major problem and if you think that you know just if you try to apply all sorts of techniques it's better to remember that if you don't look at epimorphism but just sub direct product then all three-dimensional topology are just in this in this in these types of diagrams so you know it's it's it's rather again it's rather pathological situation algebraically okay another another example to to remember is the following so we take a surface and we take we take a pseudonosoph of the surface and we look at the backward and forward iteration of this pseudonosoph and taking the limit we get actions on two trees which are dual to the stable and unstable filiations on the surface coming from the forward and backward iterations of this of this pseudonosoph so we get an action of the surface on the product of two trees and if you look if you look at the action on the product the action will be properly discontinuous i think it's a reason to throw some but i'm not sure um so so you see what it means that if you look at the two-dimensional object in a sense you don't see the dynamics but you see the dynamics locally but you see the dynamics if you look at the projection if you look at the projection then you'll see the dynamics of the pseudonosoph on each of the of the real trees separately but if you look at the two-dimensional object somehow okay locally we lose the dynamics we have to go further to see what's going on okay so so another example that i want to mention uh which i didn't write here which i'm going to exclude uh by by assuming by by adding an assumption in the minute uh is the is the example of borda and mozes okay mark borda and schacher mozes of a group that act on product of two locally finite trees and which is simple okay and i'm going to make an assumption that in particular we will exclude this example the last one okay so the assumption that i'm going to make is that um you know when you go to when you go to a factor there may be a kernel to the action um so okay in the action of course we assume it's free on the space but um free and properly discontinuous but when we go to the factors okay that it may not be free and it's not properly discontinuous anymore so we assume that the the action is weakly a cylindrical now and this is probably overly overly used terms even tomah has a notion of quickly a cylindrical but somehow tomah notion of quickly a cylindrical implies us okay this is this is a weaker notion than a cylindrical and it's supposed to to to make a classification of the fact that there is a kernel and model of the kernel the action is a cylindrical so what it means okay that if two points that that's how to think about it but if two points are far or far enough okay we fix an epsilon first and if two points are far enough uh there will be boundedly many elements so that if there is an element h that move both of them sorry here i made a mistake uh move both of them more than epsilon that's a mistake in the inequality it's supposed to be bigger than epsilon then okay it's not true that there will be only boundedly many elements but but this element h is belong to boundedly many cosets and cosets are thoughts of elements that acts quasi-trivial in the whole space okay that they move any point not more than that doesn't have to be there i mean uniform constant doesn't have to be the delta of a publicity but but if the space is not aligned it's not easy to see that it's i don't know two deltas of their publicity but but that's what okay that it belongs to to find boundedly many cosets that's what's important and i view that as a classification of of um of the slina bromide and fujibara condition of wwpd but in the h-cylindrical case i don't look only on axis but in general okay so or if you want it's a classification of of just a cylindrical of a cylindrical model okay um so that's that's our assumption on the factors and by the way again i don't know if i managed to get to that but if i go if i talk about it it is then then we also assume somehow there that the stabilizer of each projection space there also acts weakly a cylindrical this is an assumption that okay i have to assume in order for these techniques to work okay so um so okay so we ask again uh what can be said about the automorphism group of a group that act on um um on the product okay we don't care what the group is but only what the automorphism is so we start with the sequence of automorphisms and we look at we look at the action of this of this group eight that stabilize set wise each of the projection space okay and we twist it by the automorphism it's a standard thing and we look at the set sequences that converge should should have written after rescaling for all the for the projection space it at once okay it converges for all of them it's a if the stretching factor goes to infinity so it's a standard now compactness argument that there will always be a subsequence that we that we converge okay it will convert to each to each space so by by passing from subsequence to another n times it will converge to all of them from them okay and the limit gives an action on a real tree and we said um um we said l to be l okay this is also pretty standard now model okay infinity you just we call l a limit group in this setting it just imported me to know for me to know that uh you know there's a various definition for limit groups um you can define it using Gromov-Hausdorf convergence and you can always also define it using more spaces for limit groups over free groups or like a body groups it doesn't matter but here the definition varies so okay so I really could consider here the k infinity in terms of the of the action of the Gromov-Hausdorf convergence okay these are elements that are trivial but we consider automorphism because we consider automorphism there's no there are no kernels for these maps group algebraically but there will be element that act trivially on the on the tree at infinity and therefore this group in general is a is a quotient of H and usually it will be a proper quotient okay so the definition here varies it's important all right okay sorry I'm going much lower than I thought but okay um so I'll explain the principles maybe okay so um so so again I'm just repeating here the the the setup uh X is a is a product space um H okay we pass this we start with G but but we pass to this finite index characteristic subgroup so it's characteristic so the automorphism of the big group act on this as H as well because it's characteristic um so so the group H act on this space and the advantage of H that it preserves set wise each of the factor spaces and we have we have the sequence of automorphisms that uh with which the action using this using this this automorphism and we get that we get a limit action on a real tree okay and for this for this sequence we get a corresponding corresponding limit group which is H modern the kernel now if if the yes okay um now oh I should say maybe I thought it even appeared before but but anyway um this this was very convenient for me uh there is a paper by Daniel Brooks and Michael hard on on um okay they they deal with um with equation in Ethereum families of of a cylindrical everybody groups anyway that they um they do all the technical setup is needed here and uh I can always I can I can refer to them I don't need the analysis of equation Ethereum and so on but but but they are and they all all that is needed in order to to analyze uh this this limit action was done by them um okay so so from the action of L on it's action of H but but this k infinity actually so so it's really a faithful action of L it's possible to write to to read off splitting of the splitting of the group okay it doesn't have to be like this but could be surfaces etc could be splitting it over finite groups free groups etc free splitting etc and each time we do that we get a map to uh from H to a limit group which has a splitting the big problem here okay that doesn't appear in the original construction of the JSJ is that these limit groups are different okay the groups are different the kernels are different you have them uh also correspond to each of these factors and um you know the whole machine that was used to to construct the the JSJ and it doesn't really matter what what machine you use um I mean all all the construction of the JSJ the machine will fail because because it has a kernel there's another um there's another conceptual big problem here and this is you see we look at quotients here and splitting off quotients now the big advantage of the JSJ is that that the automorphism of the group can be read from at one level from the splitting of the group okay it's almost an ideal situation here when you when you look at uh when you look at um at spittings that are obtained from quotients in general there's no there's no chance that something like this will be true even if you consider only free splitting okay all not these spittings but only free splitting it there is no something similar to the JSJ that will that will uh somehow describe all the free spittings of quotients of a given finite represented group actually I have a work on that with Eric Zaligo and reconstructed the McCann-Rosberg diagram that encode all the free spittings of quotients of a finite represented group this was needed afterwards for logic purposes but but there is no chance that it will be in one that okay there could be there is a McCann-Rosberg diagram but it cannot appear in one level and it took me quite a long time to decide whether if one tries to develop a theory here whether it should be in one level or it should be something like McCann-Rosberg diagram and you know McCann-Rosberg diagram in this setup is I'll explain that is much easier to construct but but it's much less powerful okay so what we want is something that will remind will be you know somehow connected to the JSJ in that case of hyperbolic group all right so uh so okay so how to match the different projections and uh you know I knew how to construct some sort of McCann-Rosberg diagram because these are tools that existed from before as I said in Groves and Hall they also constructed they don't construct actually a McCann-Rosberg diagram I think but anyway they they they prepare all the tools to construct maybe I'm wrong maybe they do but but I think not um but they prepare all the tools that are needed for that um as I said it it's it's it's much in a sense it's much weaker but uh but this is something that one could start with and I said that I was really puzzled whether to to look for some sort of JSJ or to look for McCann-Rosberg diagram and at the end I decided you know if I if I cannot beat this this this this McCann-Rosberg diagram but they join it so uh it means that we first construct a McCann-Rosberg diagram and then we use this to construct a JSJ um so so let me yeah I don't know how much I managed to say somehow this is the first time that I talk about it and I somehow assume that I'll be able to say much more um okay so um so how to construct this this McCann-Rosberg diagram um so we we said we start with the with automorphism p I I work now with each projection separately by the way I have an hour or I have less than that um about 55 minutes so you still have 10 minutes or so it's all right all right I'll get whatever I get um so um so so there's a sequence of automorphisms in the beginning we we construct one limit group now this limit group will have will as I said we will have splitting if it diverges it will have a splitting so we can we can associate the modular groups with that now from now on we don't study automorphisms anymore it comes from automorphism what we study quasi-morphisms and um or if you want if it was a real kernel we study homomorphisms this limit group which are in general quotients or proper quotients of the original one and we can construct we can construct that the diagram here but but we are interested only in splitting that are obtained originally from automorphism because this is what we want to study we're not interested in the algebraic structure of these limit groups we are just interested in automorphisms so we cannot use the JSJ of these limit groups although the JSJ exists but we use only splitting that come from these automorphisms and because of that some of this epimorphism will be actually isomorphisms but it's guaranteed that after finitely many steps in fact after bounding the many steps we will move to to a proper quotient and this is this this descending chain conditions was proved by Groves and Huy that after after such such a sequence terminate after finite time and when it terminates we get we get a quasi sequence of quasi-morphism obtained from this automorphism which is non-divergent anymore so there is a bound on that this is important and okay the whole things here are only finitely generated it's not finitely presented in general and finite generate is too big for us in terms of cardinality so we can replace each just resolution by finitely presented object just just for you know to to to think about it just to think that we replace each of these limit group with a finitely presented object this is not how it's done but never mind and therefore there are only countably many of them if you replace the whole this resolution you call it if you replace it by finitely presented object it is the only finitely there are only countably many of them and this is as I said there is at the end there is a bound on the on the sweating factor we can assume this bound as a positive integer so we can we can we can count all this all this finitely presented approximations of such things and using okay I think by now pretty standard compactness argument it's possible to to be left with only finitely many of them so so it's enough to to use only finitely many of this resolution and what we get is what they call a higher-ranked mechanical spore diagram for products so what does it mean we have m factors okay m factor spaces so each each resolution each element in our diagram will will have finitely many resolution for each factors because we start with sequence of automorphism and we look what happened in each of the factors but you see when we analyze them that's the disadvantage here when we analyze them we analyze them separately we move to quasimorphisms and these quasimorphisms are just appropriate just for the just for the appropriate factor there's no when we continue in this diagram from the first level we lose there's no coordination between the factors anymore so so we have we have this m collection of covers and finitely many of them are enough that means that every automorphism of the group will factor through one of these one of these finitely many m collections okay this is the this is the high-ranked mechanical spore diagram it was obtained from by compactness argument so there's no chance that any like anything like this will be canonical okay here is here is what I call the completion of resolution I'm not going to get into that actually I should say I don't know how much I'll get to that but but one thing that that is appealing for me and actually it appeared before in a in a work with Cody Fujiwara that that tools and techniques that I developed to study the first order the first order theory of free group actually appear here the tools not not the not not anything from logic and this is I think this should happen more and more I mean there are techniques there the solution of taskly problem in generalization that that should have application in group theory and in in in other places and this is an example you see studying automorphisms of groups in the beginning doesn't seem to be related but apparently the tools are related are useful not just really okay so um so the first maybe I'll just describe that I'm assuming that um you know at first that that each each of these resolution is only one level see if it has one level meaning just one map if there were no maps then there are bounds of the stretching factors okay and if there are bounds of the stretching factors and the group acts properly and co-compactly co-compactly it's not matter just act properly then it means that only uh finitely many automorphism will factors or something like this so this is not very interesting so I assume that there there's at most one so every resolution has um has at most two floors of two levels okay so this is this is the first the first thing and in this case okay there's um the group of automorphisms form okay what what I call in the work of doctor problem is a test sequence it's not really a test sequence it's a weak test sequence and what it really means is that this sequence of automorphism just converges to to the splitting that we see here one one is to explain what converge mean because we are going to see it in different scales like in our our entry or something but but anyway from this from the sequence of automorphism we can read off we can read off these splitings all the m-splitings and all factors and now one can one can define a complexity of these resolutions okay complexity of resolution is a is a is a major issue in in studying the first order theory of of of a free group but in there it's far far more complicated here it is it's rather simple we just define say naive uh complexity of of this a billion splitings okay first the the course rank the number of free factors and free elements then the topology these are orbit faults in general they're not not just surfaces um so so the the these orbit faults say the order curve receive the genus order will see graphically in in decreasing order and then then things to do with the with the a billion groups and the edges and we consider only only the resolution this is maximum complexity our order the complexity sorry is well ordered so we can we can look only at maximum complexity and we we look at these these weak decisions these test sequences that converge into maximum complexity resolution now if we take any automorphism any automorphism in in the group and we compose it pre-compose the the sequence with this automorphism we get another test sequence just okay maybe it doesn't factor through this resolution but we have universality of the of this of the diagram that we had before so it will if it doesn't factor through this one at least a subsequence of that will factor through another one okay again because because of universality and using that it's possible to prove okay this is i think this will be my i have more slides but i will i will stop here because of time that that for single for single level level resolution there is exist just one m collection of cover resolution that is preserved by finite index subgroup of the automorphism okay so so there is this you see m m splitings but it's important to know these are not splitings of the original group these are splitings of quotients and usually it will be splitings of proper quotients of the groups the the quotient itself is not canonical but the form of the the composition is is is canonical i didn't study much the the canonicality of this decomposition but but anyway all surfaces that should appear etc and of course it all depends on the finite index subgroup that we chose because if we choose if we choose another finite index then you know we get the splitting which will be in some sense commensurable with this one but it will be different um and and this this the composition will encode okay not all automorphism but the finite set a finite index subgroup of the automorphisms i won't i will not talk about resolutions with higher level now but let me just let me just customize last slide no this is right angle oh not a lot less than okay in general from this uh from this the composition okay i jumped several slides but from this the composition it now it's possible to say all sorts of things on the on the outer automorphism group so for example okay if the group if all splitings are of one ended let's say one ended splitting it's not important if the group is one ended but it's important whether these splitings don't contain um free the compositions or splitings over finite over finite groups then we get also a map from the from the from this finite index subgroup of the outer automorphism group uh onto mapping class group sorry not onto that's important into the mapping class group of these surfaces and outer automorphism group of some virtually virtually a billion um vertex groups as well so it's not the general linear group but outer automorphism group of finitely generated virtually a billion groups and the kernel the kernel has to be has to be finitely generated virtually a billion okay it's a it's a similar picture in the sense to what happened in in in hyperbolic groups or actually it's also similar to what the car what charny and botman did for for right angle awesome group just that there there are no surfaces so um okay now if the group is not one ended then um then you have to work then the outcome cannot be stated uh so so nicely in this well so in such a short way because we will also get the map into into these objects but then there will be a kernel to the map and the kernel to the map you know after we look in the kernel we can we can run off the construction of this hiring jj the composition just for the kernel because you know we we didn't really need to tell that this this machine that it is this we're all automorphism it's okay to take it just from a subgroup and it may be that new surfaces and new a billion groups will emerge from from the free the composition it may be that new of them will emerge so we get another map with another kernel this can happen boundedly many times and at the end what we will get let me see if i have a slide here at the end what we will get is uh you know we'll get factors with only virtually okay i didn't consider with only virtually cycling virtually cycling the composition and maybe some free elements i didn't draw here also edges with free with sorry with finite edge groups and if you like if you consider only stabilizer of the factors if there are no free the composition only stabilizes yes then of all the factors in in all the factors also in the in the in the space okay because the space was a product of certain factors it will be the stabilizer will be finally generated virtually a billion okay i will not talk about the generation to to uh hierarchy at the body group but this is anyway this is not the main the main thing i really think that the main thing is about products i didn't manage also to talk about them about the resolutions with more floors but okay i mean there's a limit what one can do or i didn't i didn't plan it properly okay well stop thank you very much are there questions either online or at the hos yes so you you mentioned the quasi-morphisms occurring somewhere in your in your in your resolutions what could you say more about those quasi-morphisms oh yes okay so let me see when yes because okay you see the okay let me let me just say that the limit group here okay the limit group here is is the group H model model of this kernel now this kernel in general it doesn't act trivially on the it doesn't act trivially on the on the space but but it will but what what it will happen it's this kernel okay not sorry not not this kernel contains more element but but anyway there will be element that move any point just move any point in the space or in the factor just a bounded amount so okay think think about for example think about the limit group here that that which is finally presented so it has finally finally many relations so these relations are not mapped to the identity but they are meant to element that move any point in the space by by this bounded amount and this is a normal support okay that's what quasi-morphisms here means it's actually not very far from from homomorphism just one needs to to get used to work with this thing but but the quasi-morphism we're not interested in all quasi-morphism we are interested only in quasi-morphism that are somehow obtained from automorphisms of the big group through these resolutions okay but again it's not that far from dealing with them using this this mechanical score of diagram machine is not very different in dealing with homomorphisms but again it's important that we are dealing only from those that come from automorphism of the ambient group and not not not just general quasi-morphism at the beginning you mentioned the Mikaerova groups and you have example of interesting Mikaerova groups with a lot of automorphisms okay I didn't I didn't start yeah yeah I didn't study subgroups I didn't study subgroups okay for this it's important Mikaerova group acts on the product of free yeah yeah no I assume that the action is co-compiled oh okay yeah also in the other group sorry on a group also in the antibody group right the JSJ will not give all automorphisms of all subgroups even the JSJ of the subgroup will not we'll often call all automorphisms are there other questions well if not let's thank the speaker again