 Thank you very much Stefano for the invitation and to take me here this time it's a beautiful place and a great conference so thank you very much Stefano and the other organizers also of course. So okay so let me start with the the subject of my talk so what before the subject of my talk let me just tell you that this is joint work with Aaron Brown who is from Chicago University and with she didn't want which who is in in Penn State okay so once this is settled so we are trying to study actions of groups larger than the integers of the real line did much much larger than these groups and the way we started on thinking on this particular project was trying to understand when such an action has an invariant measure okay so groups for this matter could be separated in two giant classes one of the amenable groups where the the examples are finite groups compact groups abelian groups nil potent groups all the groups and not much more but something more and the other class which are the other the non amenable ones which is a much larger class so when you have a group acting on an amenable group acting on a compact metric space essentially by definition you have an invariant measure always for other groups is really a mystery so let me start with an example of an action not having an invariant measure which is very much to the point in our case so sample it's quite simple so let's say take SL to Z so it's integer 2 by 2 integer matrices or any other subgroup of SL to R will do essentially well not any but recently and then make it act on say the the projected line so rp1 case which is is the circle but is not the circle so it's is the set of directions okay so then the action is you take a direction here L and you pick a matrix a here and then you send this guy into a times L so you have a direction is a linear map so it will send the direction to the direction so then it takes a point here to a point here then you can see that if you take for instance your favorite matrix the two on one map then it has a stable space and unstable space okay so this implies that for this particular guy maybe let's do Victor here you will have two guys which I don't know where to draw them but let's say here and here you and ES which are the stable unstable lines for this matrix and this will be fixed point for this action of the matrix a and you will have that the dynamics goes from the south to the north okay so every line you take if it is not the stable line you start iterating it by a and it will go all the way to the unstable line okay so just from this picture you see that if such an action have an invariant measure it has to be essentially a combination of these two atoms because it has to be by and by this matrix and this is a more smooth system which just north south pole map or south north pole map and these are the environment measure now you pick another matrix now it's tough whatever other matrix and which has also in the short entries and assume so this is for a and essentially you the best thing will be just to conjugate by something which doesn't leave this but again anyway so you will have some the unstable for BC are the stable for B probably will be here maybe other configurations but let's say you have something like this and then you will have x in the same phenomenon for the for the B matrix so if the measures invariant by a and it's also invariant by B then it's impossible because on one hand it should be supported on these two points and on the other hand it should be supported on these two points and that's impossible okay so there you have an example of a non-ammonable group okay and here you have an action without an invariant measure okay and these are called the projective actions just obviously because this is projective okay same thing you can do instead of the RP1 you can take the circle and then you will have four points for these maps and similar example you can do if you take for instance SS3C you can make it act either on S2 or RP2 same way and you have exactly the same phenomenon okay so then since it is not obvious at all why there should be an invariant measure there we just start asking whether we can find invariant measures okay and if we were worried at the time we started with not this type of low dimensional system we were worried about an also of actions and we eventually did something about an also of actions but in the meantime we discovered something about this this more low dimensional cases so let me state the theorem we we found let me state it in this simplest way theorem so everything is which make also so it's I will not put names here so it says you take gamma lattice in SLMR gamma lattice this means discrete and SLMR mod gamma co-compact no co-compact finite volume it could be co-compact of course then we'll have finite volume and we can do for more general groups but let's let me stick to SLMR is everything simpler to state assume gamma acts let's just put this way in different systems of M infinity less differentiability is good enough but then I will need to discuss differentiability so I don't want so let's go to infinity C1 would not be enough for several stuff so let's say dimension of M equals to D then one if D is less than or equal to N minus 2 N minus 2 then I should call it alpha alpha has an embedded measure and the second part says that in the case D is equals to N minus 1 then of course I cannot say one because I have this example here I have SL3 acting on a two-dimensional manifold without an embedded measure so certainly I cannot say that you will have always an embedded measure but we can say that and alpha has no embedded measure then alpha is mostly conjugated to the projected action on SN minus 1 or RPN minus 1 and of course to say that I need to say something here which is M connected because if it is not connected you will have whatever combinations of this okay so that's that's what we found so the only abstraction for in those low dimensional settings to have the embedded measure is exactly this example yes compact compact compact compact indeed if it is not compact you can put some integrability conditions on the action and you will still get one for two we didn't check so we you cover the manifold with charts and we see how we use compactness in two essentially yes maybe something can be done but okay so this is essentially the main result I want to discuss I guess I will stick to this result okay let me say if I have to say something more about this the corollary so corollary when you go back to the equals to two and indeed whatever I'm saying for the equals to two should be exactly the same true for larger D and this will explain a lot the first part but we were not able to do anything in this case so assume gamma is SL and Z so now is you want this this particular lattice then either one gamma factors through a finite action so it means that you have that the kernel of this alpha is a finite index so it's really you are not acting by the by this SL and Z you are acting by something finite and then you are just having this this map or or two is the same as the two there so gamma or n equals to three and m is either s2 or rp2 and alpha is smoothly conjugate m never yes thanks yeah also here probably here it follows from the way it's written but let me just maybe not yeah I won't enlarge it so I was probably here yeah here for n equals to two is certainly not sms even since the circle so I wouldn't try to prove anything like that yet and for of course one for n equals to two is fine so and the corollary indeed let me just comment the proof so the main thing is that Franks and Handel and Handel prove that if there exists an invariant measure then the action is trivial so well trivial trivial in this sense action is like one so once you have the variant measure the action has to factor through a finite group okay so what we did is we prove that there is always the invariant measure and then you can use Franks Handel okay and the second is completely followed from this first so and in that's why so that's why we need here the the lattice so you need to apply Franks and Handel you need this unipotent elements of this this type of matrices something like that so which are distortion elements for the for the lattice and then they play a lot with these rotation numbers and all this stuff entropies but you need these elements so if the lattice were compact then what they do doesn't work maybe there are some modification we didn't try that but certainly will not be quite simple so but anyway anyone who is willing can try okay so this is the main theorem and the corollary so let me start discussing a little bit about the proof of the of the theorem and the construction behind it okay there is will be a lot of ergodic theory here there variant measure so we should expect ergodic theory and there will be smooth ergodic theory peasant theory and all this stuff okay so so the whole goal here is you look hard to get your invariant measure okay so you all the time to find your invariant measure and at some point you will fail and at the time you fail you will get a lot of information from the failure and then then you will have to work to win but still you win so so for that we need some starting point so we need to start with some invariant measure invariant by someone okay and then you promote this invariant by someone to more invariants and more invariants and more invariants but you need to start with something so let's let me tell you what is this something so for the ones that came to Jens Markloff course you will know if not you can get probably from linear algebra that you can this component is as k a n where this is compact is the diagonal and these are the important and these you call P so here k is essentially I never remember if it is O or O but a is diagonal I guess I should put the term so the sum minus 1 somewhere so let me put this there and a is e to the t 1 e to the t n where the sum of the t i is 0 and n is once in the diagonal 0 is here and whatever you want up here so the P is just whatever you want the owner whatever you want after they are zeros below and the first news is that P is solvable hence amenable which means that whenever P acts on a space on a nice space if it's compact metric then you are happy but if it is not compact metric still a nice space will do with a nice action then you will have an invariant measure so we have to make P act somewhere and get a P invariant measure from there okay so how to do that you do the same thing when you have one diffeomorphism so you do a suspension construction okay so what is the suspension so you have your manifold M you have your group G let me just write G for SL and R is shorter you have your gamma acting on your M so if gamma were the integers then G would be the real numbers and you have an iteration of F you can take the suspension and you can build a suspension flow so here we are doing the same and instead of having the integer and the real numbers we are doing the lattice gamma and the group G we have exactly the same picture so you do G times M and you quotient by some relation okay so what is the relation the relation says that you take a point G X and you relate here with G gamma alpha gamma inverse so that's what you do okay so you make this identification so how this business look like so this space that we call it M alpha so this M alpha will fiber over the G mod gamma which is this space you will have the G acting here naturally by left multiplication and you will have here also the G acting here also by left multiplication on the on the first factor so if you take a star the class of G X you just send it into a G X so you are not doing anything the second factor at least not upstairs but then the quotient so the identification is telling you that you are doing something so the same thing as suspension flow so and then if you draw something like that so this is the G mod maybe I will not have space here let's do it here so so here the G mod gamma then over each point of the G mod gamma you will have say one of your manifolds M and that's how this M alpha look like so for every point here you have that so you flow here like the G mod gamma G acting on G mod gamma is telling you you flow here down and then upstairs you essentially doing nothing but when you make a loop essentially you start doing something so you can take fundamental domains and see it more more detailed okay then we have a nice space so if gamma is co-compact then this is a compact metric space is gamma is not co-compact then it's not a compact metric space like in the picture you have here a gas but still the fiber is compact and uniformly compact and you can use some inability of P since G is acting then P is acting and then you just pick at P invariant measure so P invariant measure U P invariant measure for alpha projecting to hard measure on G mod gamma okay so we have this this projection P here so we have the alpha action acting here you have the natural action here which is essentially a joint action acting here and you have this too many piece let's call it pi so the pi projection here and since if you take the measure here and it's invariant by P then it will project to some measure which is invariant by P here and the only P invariant measure here is hard measure so it will be automatically hard but it just have this guy okay this is the invariant measure whenever you have the invariant measure okay so we are going to try to prove that this measure is invariant capital P is this group yes yes yes yes it is all these things so good so how to prove that this measure is invariant well you first define a group Q sub mu which is all the elements G in G such that alpha of G star mu equals to mu so all the elements of the group that leaves this measure invariant and I'm denoting here also by alpha the action on this suspension so we don't want to put till this for this time so so you take this group this is a close group because being leaving measure invariant is something continuous so it's a close subgroup of G let's not speak about connectedness so and it certainly contains the P okay so these type of groups are called essentially parabolic groups so okay so now to just to be able to to draw pictures let me assume that n equals to 3 dimension is 2 okay so let me just show you what's going on with there so here is P and I know that my measure is invariant by all these flow so when you put a number in any of these position and then a zero or one according what it must be then you have a one parameter flow there and you have invariance along all these guys so if you want to gain full invariance then you need to gain invariance along this this and this so you want to prove that the measure is invariant by the flow generated by this element by this element I basically have it so essentially you have these three flows this flow and then the similar flow when you put all all there okay and you want to prove that the the the measure is invariant by these guys okay so this is because you have n equals to 3 now comes the the first non-trivial thing so I need to introduce some terminology so she scores the upon of exponents okay so I have my pin variant measure so since the diagonal is inside the P in particular this measure is diagonally invariant and this diagonal group is so let's take it still maintain here so here a is essentially the same as R2 and I can take any element a and I can take a in this a I look at this action on this so in this case is what SL3 so it's 8 plus 2 is in this ten-dimensional manifold acting on the M alpha and I can look at the Lyapunov exponents and the oscillator splitting of this action okay so let me again draw this picture here so I will have that the tangent to M alpha I will not write down yet the selected splitting I will first write this splitting which is I have a tangent to the orbit foliation by the shifoliation times the tangent to the fibers here so I have this a dimensional space and I have this two dimensional space and this is an invariant splitting for the action just by construction then Ocelerates will tell me that this piece the compose and indeed for this I don't need Ocelerates so when I look at this part I already know what does G acting on G himself so I really know what's going on there and these are the so the derivative of this or the corresponding associated here spaces so this are the invariant splitting for the Ocelerates theorem so you put a non-zero here non-zero here here here so you have these six spaces and you have these six invariant spaces plus the two dimensional coming from this from this R2 so you have here some of some guys which are called psi ij which correspond to putting the ij's here plus some of the let's put the TA so the tangent to the to the diagram so this do this eight guys so these are beautiful and then I have two other guys which come from here which is some e eta one plus e eta two okay I have all these the composition so for the for the g part everything is known understood for the end part is Ocelerates theorem that tells me that you have two levels one they may coincide so I have no idea and this you have this direction okay but then let me do some more things so there is an a here so I pick this a and I applied Ocelerates theorem to this a so everything depends on a of course but since a is the an a billion group so guys they are commute so Ocelerates splitting once it works for one element as soon as this element is generic enough then it will work for all the other elements it's just the same reason why two commuting matrices are jointly diagonalizable so it's the very same reason is uniqueness of bundles in Ocelerates splitting so this part I already described is very well understood and here you will have these bundles that do not depend on a even though the exponent will depend on a okay it's not the same to have the identity or having a or two times a so you it has to be an even worse so let me now draw the picture of a so here comes a a will have first of all these three guys which is chi one two chi one three chi three one and the lines and the lines means so what I'm drawing is the kernel of these guys so it turns out again it just follow easily from Ocelerates theorem and it's linear algebra to see that when you bother your a these leapon of x one is become linear functionals so they have kernel of course the kernel could be everything but from this side you know that the kernel are only real lines so you have these corners of these functionals here okay so depending whether you pick the a here they will have positive or negative you pick them here then the corresponding will be zero and the other will be positive negative depending on the picture and by some simplicity property on these on these matrices so on how the a acts here you will have that the the kernels will be the same so the kernel of chi two three is here and you have that chi two three is minus chi I guess three two yeah and similarly so chi one three is minus chi three one and so on so one is the opposite of the other so the lines will be the same even though you have different leapon of exponents okay so this take charge of this part now comes this other part so I have data one data two these are two new functionals and it may happen essentially several things so but most importantly where they have color junk I hope you see it so let's say here is I see it white but maybe you see it the color it has to be so here you have data one and you will have some data two and I'm drawing it in the in in a sense that is good for me so data one is not coinciding with one of these guys here okay but deep they may coincide so it may be that the data two and and this will be also one possibility that the kernel of data two is the same as one below so let me make a definition now maybe two definitions one we call a leapon of exponent exponent for the G direction G direction a root this correspond to the the theory of Lea group so these guys are roots of the of the Lea group and these are the root space of the of the Lea group and second is we say a root is non-resonant with respect to alpha if let's just say call it the root R so R is one of these guy I J if R is different let's just write it with letter is not a positive multiple multiple of eta one so if it is if this is not going on so so here this root is a resonant root so this guy is resonant but this guy is non-resonant well this guy I'm sorry it's for the root so this guy is non-resonant this guy is non-resonant well the point is and since roots coming pair so once you have positive either one or the other will be bad but it still is important so you will see next theorem why some roots even if they are resonant you don't care and some other roots you really care whether they are resonant or not so here comes the next the first and theorem so but let me just attach here so attached to each of these roots you have your your group which is this group attached to this root space and then we have the tier which is completely general so it's just any dimension any group any whatever so even run one but run one is trivial statement so if a root R is non-resonant then mu the P by the measure mu is gr invariant so as soon as I have a non-resonant root I win in balance along this direction so if this the opponent corresponding to this direction is non-resonant with whatever happened in the M in this torus picture there up then you get in balance by this group here which means I complete this square here and I have to go to the next square so in this picture of capital exponents I draw there the bad guy is guy one three so guy one three if I understand it correctly is here so this is guy one three so I will win in balance by this direction and this direction okay then I know that Q mu will contain all matrices of this form but Q mu is a group so it will have to contain this direction also because any group containing this all these matrices will contain everything and then I won in balance at once and that's essentially the three when you go to higher dimensions so when not hiding so when then the N grows even if the D is more so I guess I have something like five minutes so let me instead of so you can think of this theorem let me not go into proofs of this theorem so it is very much related with this Le Drapier in variance principle and indeed when we write the proof we in the meantime prove some some conditional Abramov-Rochlin formula so it's a version of this standard Abramov-Rochlin formula and we also prove some kind of splitting of entropy along this type of directions here okay so but let me not enter into this so I prefer to move forward and show you how to get say the one and the two properties not here I guess so in the theorem from what I wrote proof of theorem so I didn't number so you should guess which theorem I'm proving but this is the this first one or the second one okay in this situation so n equals to three d equals to two so you have this group q mu and as I told you I have to add these guys so then I look at chi one three chi one two and chi two three so I have these three roots to check whether they are proportional or not now the news is that up to some reasonable isomorphism and have to what I really care so the groups containing P are either the P himself or this guy or this group or there is a symmetric which you put the start here instead of the start there but these are essentially the same thing up to some arrangement or everything okay so let's assume that the measure q mu is not everything so from everything so it's not I got them my invariant measure by G and once it is you have the better measure by G you can make build your gamma invariant measure on the manifold just some conditional measure construction so I have these three spaces and I have only two the upon of exponents on M so one of the three should not be should not be resonant so I win one extra invariant so if this extra invariant is the guy down here then you will get again the whole group so then the the guy that is not that is not restaurant should be one of these two guys and by some symmetric consideration I just have this so I can assume that Q mu is exactly that okay so that is what is Q mu okay so once I know that is Q mu the next news is that when I do G mod Q mu and while the gamma was questionable to the right the Q mu you have to caution it to the left to make it in the in the right way you get that this is whether you allow some positive for negative sign or not but this is essentially our P2 okay so if you allow only positive entry up here then it will be the S2 if not is the R2 so there is the sphere you are looking for the R2 you are looking for so here is the model so you want to prove that your M is equivalent to this portion and that your dynamics is the same as the gamma action here so that's what comes next I have two more minutes so that's let's say let me just comment on what's going on here so this type of suspension construction and this type of argument I was drawing here and the most important is the proof that would start here is essentially similar to the proof of Margolis normal subgroup theorem so in Margolis normal subgroup theorem he proved something similar to this type of statement so he proved that if you have a phi from G mod P into M where you have your gamma your natural gamma action here and your whatever gamma action here some question but a measure blah blah blah then this guy is measureally isomorphic to G mod this type of cumio okay and a very natural way so what we do is we just so we build the proof differently so our choice of cumio is not the same as him and all these subtleties technical subtleties but at the end of the day we just try some measureally the type of statement and you need to prove that when you take conditional measures along some passing manifolds then these conditional measures which will be a graph will have to be smooth graph and once you have these smooth graphs these smooth graphs are the consulgancy so you you win the consulgancy from the conditional measure so that's how you build the the the consulgancy so you you you think that an invariant measure as a joining and you think your mu as so you have this fiber bundle you think of mu as a joining from the or g mod gamma action here and your action on the M here but then you have to twist your understanding so your g mod gamma will become your g mod q and then your g mod q will be the same as this guy here so I think I don't have too much time to discuss it later. Thank you very much.