 Thank you all for coming, and thanks to the organizers for inviting me to speak here. It's my first time in Uruguay for mathematics, and it's been a real pleasure so far. So yes, this work is joined with Sang Hyun Kim from Seoul National University who was here, but just left this afternoon. Right, so what we'd like to talk about is some structure of diffeomorphism groups of various regularities for some fixed manifolds in very low dimensions, so dimension one. So throughout, when I write M, this is either going to be either the interval or the circle, and we're going to consider the groups dif k of M, and maybe some further interpolations between these, where this is the group of ck orientation preserving diffeomorphisms. So I'd like to involve groups, geometry and dynamics in this talk as much as possible, so I mostly do group theory myself, so that's one check mark. So we're going to look at the interplay between algebra on one side and analysis. So what I mean by these things is the structure of groups of diffeomorphisms, in particular finitely generated or more generally countable groups of diffeomorphisms. And here I mean, as suggested here, regularity. And the interplay here is mediated by the dynamics in some way. Precisely, there's dynamics which is forced by algebra on the one hand, and on the other hand, we have dynamical interpretations of regularity. So that's the general framework of what I'd like to talk about, so now maybe I'll say some specifics. So the invariant to a group of homeomorphisms that I'd like to consider is its critical regularity. So let G be a subgroup of the group of orientation preserving homeomorphisms of M. Generally I consider this to be a countable subgroup, it need not be a priori, but for the most part you should think of it as a countable one. And then we'll write CR of G and maybe I'll decorate this with an M when it makes a difference, although generally it won't in this talk. This is going to be the critical regularity of G and it's defined to be the supremum of the values of alpha such that G is realized as a subgroup of diff alpha of M. So a few remarks here are in order. So first what is diff alpha? So here if we write alpha is equal to K plus epsilon where K is the floor function of alpha, then diff alpha of M means the CK diffeomorphisms of M with epsilon holder continuous Kth derivatives. So that's what I mean then by defining this group diff alpha of M. And now this is going to be defined for any subgroup of homeo plus of M and what we'd like to do is to study it a little bit. So let me just remark another thing about, so this is a supremum. So it may be extremely difficult to calculate unless it's sort of some known value that's somehow obvious. So what I mean by that is you have examples of groups with critical regularity equal to infinity which are, say, countable. So for example if you look at finitely generated torsion free abelian groups, then it's very easy to have them act by C infinity diffeomorphisms on any one of these manifolds. You can even make them act analytically or free groups, closed surface groups. For these the critical regularity is just infinity. In these cases the studying this invariant isn't that interesting. So I'm going to be mostly interested in groups which have finite critical regularity. And part of the difficulty with that is there are very, very few examples which are known where the critical regularity is both finite and something that we can write down. So let's give some background. So one of the categories of groups that we can think about where critical regularity is something we can begin to at least explore in some reasonable way are torsion free no-potent groups. So there's a theorem due to Farbe in France and independently by Horkera which implies that if n is finitely generated torsion free and no-potent then this implies that n is a subgroup of diff one of m. So it can act faithfully by C one diffeomorphisms of the interval end of the circle. And there's always an upper bound on the critical regularity of a non-Abelian torsion free no-potent group that comes from a more classical theorem of Plant and Thurston. And that says that so if n is as here torsion free, so let's say finitely generated torsion free no-potent and n is a subgroup of diff two of m then n is Abelian. In which case its critical regularity will then be infinite. So what the Plant-Thurston theorem together with the theorem of Farbe, Franks and Horkera says that if you look at torsion free non-Abelian no-potent groups then they populate the interval between one and two in some way. And there are only two examples for which the exact value of the critical regularity is known. So the first is a theorem of Kastro-Horkera and Navas and so what they consider is the Heisenberg group. So if you take H which has this presentation then they compute that the critical regularity of H is exactly equal to two. So what they prove in particular is that H embeds into diff two minus epsilon of m for all epsilon greater than zero. So in particular when I define critical regularity as a supremum, this supremum may or may not be realized. And this is an example of a case where it's not realized. So if it were actually realized then that would violate the Plant-Thurston theorem. And the other known example is due to Horkera Navas-Rivas and so they take N4 which is this group of nil-potent matrices of four by four nil-potent matrices like this. And they prove that the critical regularity of N4 is exactly equal to 1.5. And it is unknown whether or not this critical regularity is realized or not. And so as you might be convinced now if I hand you a group and I ask you, okay, this group is known to have finite critical regularity, what is the critical regularity? This is an extremely difficult problem. And one can oftentimes only give estimates. So let me tell you some general facts. I'm sorry. Yes. This I don't know. I would only put my, so I don't know the exact answer. Yeah, I would suspect that it works for all of them. So the first general fact is a result of Derwea, Klebsen and Navas, which says that if, so if G is inside of homeo plus of M and it's countable, then G embeds in the group of, right, this is diff zero lip of M. So this is going to be the group of bilipschitz homeomorphisms of M. And so there's some sort of reasonable way now in which we can think of any countable subgroup of homeo of M as having critical regularity at least one because it acts by bilipschitz homeomorphisms, which are from many dynamical perspectives, that's almost as good as differentiable, though they're not definitely cannot always be realized by differentiable, but by differentiable diffeomorphisms. Caligari gave the first example of a finitely generated group which acts faithfully by homeomorphisms on the circle, but does not act by diffeomorphisms. And then there's a subtle issue here passing between lipchitz regularities and sort of the next higher level of differential ability, which I'll address a bit later. But, so this will, I'll sort of take this as a given and a justification for saying that the critical regularity of any countable group of homeomorphisms has critical regularity at least one. And then as far as regularity and its interplay with the structure of the group itself, so this morning Yashlota told us about property T a little bit. So there's a theorem of Navas, which was then strengthened by Badr, Furman, Gelander, and Mano, which said that so if G has property T, then its critical regularity acting on a circle, is at most 1.5. It's strictly less than 1.5. So Navas proved that it was less than or equal to 1.5 and these guys proved that it can't be exactly 1.5. And as Yash had indicated earlier, that conjecturally property T is an obstruction for groups to act at all in any interesting way on the interval around the circle. But this is the state of the art here. And another interesting relationship between critical regularity and algebraic structure of groups comes to another theorem of Navas, which says that if the critical regularity of G is greater than 1, then G has either polynomial or exponential growth. Just to say that if you have critical regularity above 1, then you have no groups of intermediate growth there, which is a... And then groups of polynomial growth are always virtually no-potent by a theorem, a famous theorem of Gromov. So those are general facts. And before trying to talk about more about specific values of critical regularities, let me give some more estimates for various classically studied classes of groups. So the first that I'll tell you is if G is a right-angled art and group, so that is to say it's defined by some gamma, which is a finite graph, and the right-angled art and group A of gamma is generated by the vertices of gamma and then two vertices commute if and only if they're connected by an edge. So this is a very classically studied class of groups in geometric group theory. They form a nice sandbox where you can explore conjectures about mapping class groups and other sort of these cat zero groups and so on. And these groups are always what are called a residually torsion-free nil-potent, which means that every element in here that's non-trivial survives in a torsion-free nil-potent quotient of the ambient group. And so then a strengthening of this theorem that I mentioned earlier of Farb and Franks and Horker independently implies that A of gamma always embeds inside of diff one of m. And certainly sometimes A of gamma embeds in C infinity diffeomorphisms of m. If gamma for example were a complete graph then this group would just be a billion. If gamma had no edges then this group would just be free. But then there's this entire sort of spectrum of groups that's in between the abelian and free ones and it's not clear how to analyze the regularity of actions there. So combining some results of Beck, Kim and myself and then Kim and myself as well is that you either have either the critical regularity of A of gamma is equal to infinity or the critical regularity of A of gamma is less than or equal to two. And this first conclusion happens only if A of gamma is isomorphic to a possibly trivial direct product of other right-angled art and groups where each of these separately is a free product of free abelian groups. So this is a very limited collection of these right-angled art and groups. So most right-angled art and groups again populate the same stretch of the critical regularity spectrum as nilpotent groups do somewhere between one and two. And the people who study mapping class groups and their actions on the circle are sometimes interested in regularity of such actions too. And one can show basically the same conclusion. So if S is an orientable surface and G is inside of the mapping class group of S, which is again the orientation preserving homeomorphisms of S up to isotope. And if this is here a finite index subgroup, then either the critical regularity of G is infinite and this can only happen if S is homeomorphic to S11. So a torus with one boundary component and S0n where here n is less than or equal to four. So a sphere with four or fewer punctures. So these are sort of sporadic surfaces in which case the mapping class group is basically just a free group possibly crossing an abelian group. Or the critical regularity of G is again less than or equal to two. So just one is the lower bound. That's the best that we can do. So if you took any of these right-angled art and groups which have finite critical regularity and asked me to tell you the critical regularity, I have no idea. I don't even know how to guess. And so what I'll do instead is I'll reverse the question and I'll say give me a regularity and I'll give you a group which has that exact critical regularity. So to state the theorem precisely let me just introduce a bit of notation. So I'll write fancy G alpha which may or may not be decorated by M. I won't for now when it's not going to cause confusion. It's going to be the class of countable subgroups of diff alpha of M up to isomorphism. Just abstract isomorphism types of countable groups which occur inside of diff alpha of M. And so let me state some version of the main result and that is let alpha be in one infinity. Then the collection script G alpha without union beta greater than alpha of script G beta contains a finitely generated group and a simple group. So in particular this means that if I look at all countable groups with critical regularity exactly alpha where alpha is realized and there is a finitely generated group which satisfies which falls into that class and there is also a non finitely generated simple group which falls in that class. And here I'll just remark that here finitely generated in the case of the interval is 5 and in the case of the circle we had to bump up the number of generators but it's at most 9 and that's uniformly independent of alpha. And so there's one more conclusion so remember when I stated these results about the critical regularity of certain nilpotent groups there's an issue about whether the critical regularity is realized or not. So also the intersection beta less than alpha G beta without G alpha has the same conclusions. So this contains a finitely generated group and a countable simple group. So just to make sense this has to require some explanation when alpha is equal to 1. So when alpha is equal to 1 exactly what I mean here is that there exists a finitely generated group and a countable simple group which acts by by Lipschitz homomorphisms on the interval and on the circle but not by diffeomorphisms. So let me state some corollaries. So one corollary is that the critical regularity spectrum which is to say the set of alpha such that alpha is equal to the critical regularity of some group G is equal to 1 infinity. So everything is possible and another corollary which goes into I guess the original reasons why these issues of regularity of group actions on one manifold played any role in geometry or topology is about the smooth ability of certain foliations. So Tsuboi and Cantwell and Conlin had produced examples of foliations on closed three manifolds with various integer regularities which didn't admit any smoothings to higher integer regularities. So what we can do is we can give examples for non-integer regularities as well. So the particular the precise statement is as follows. So let alpha be in one infinity again. So some real number that lies in the critical regularity spectrum of groups. And let Y be a closed orientable three manifold. So we need some sort of condition on it to guarantee the existence of foliations and with which have services which are transverse to all the leaves with H2 of Y with coefficients in Z not equal to zero. Then there exists a co-dimension one foliation which is C alpha. So its charts are given by C alpha maps which is not homeomorphic to a union over beta greater than alpha of C beta foliations. So in particular what I mean by this is that it's not even homeomorphic as a pair manifold foliation to a foliated manifold with a higher degree of smoothness of its leaves. So to give an idea of why you might believe something like this you start with maybe some C infinity foliation co-dimension one foliation on Y and you use some general results about three manifolds to say well okay there's somewhere there's a surface a closed surface of some sufficiently large genus which is transverse to the foliation and then you cut out a tubular neighborhood of it and you replace the you can use the action of pi one of that surface on the interval to modify that foliation inside of that tubular neighborhood in such a way that it becomes exactly a C alpha foliation and then any homeomorphism from that new foliation to a C beta foliation for some beta larger than alpha would require a topological conjugacy between the surface group action on the interval and this higher regularity surface group action on the interval and that is precluded by the choosing of exactly C alpha critical regularity action. Right, so because these are all finitely generated groups of some uniformly bounded number of generators you can say okay they're quotients of a surface group. So I'll say a little bit more about the exact structure. One thing I'd like to say is about other moduli of continuity. So here we're looking at holder moduli of continuity for derivatives but there are many more that one can consider. So if say omega from say zero infinity to zero infinity is some concave function then one can talk about omega continuity which is defined by analogy to holder continuity. So f is omega continuous if, so this is considering this as maybe a real valued function on the interval, if f of x minus f of y in absolute value is bounded by a universal constant C times omega of x minus y for, this is taken over x different from y. So if omega were just say the function absolute value of x to the epsilon then this would just say that f is epsilon holder continuous. And so there's actually many many more examples of groups that are stated here in this theorem. It turns out that there's a, you can put a partial order on concave moduli of continuity and produce examples of groups which satisfy a weaker modulus of continuity but not a stronger one and that in particular says that not only is there one example in each of these classes that I've defined here but there are in fact uncountably many. So let me just state that a little bit more precisely. So if omega and mu are concave moduli of continuity then we can write, so there's a strict partial order. Omega is much less than mu if this following technical condition holds which is a little bit hard to parse at first. So this ratio of omega of x to mu of x tends to zero even when rescaled by this log to the k of 1 over x factor. So this is just a definition, it's not too important to parse its exact meaning but one can actually prove is a strengthening of this theorem which is as follows. So let k be a natural number and let mu be much larger than omega in this sense there exists a funnily generated group q which depends on k and the modulus mu and it's a subgroup of diff k mu of m. So these are ck diffeomorphisms whose k-th derivatives are mu continuous such that, hold on I'll write this just for the interval here for now it will become clear why in just a moment such that first of all the commutator subgroup of q is simple and every proper quotient of q is a billion and second is that if omega is as above then qq is not in script gk omega of m for m equals irs1. So you have this strengthening of this theorem to basically you produce this group of diff k with k-th derivatives mu continuous group of diffeomorphisms of the interval where the commutator subgroup is simple and retains the same critical regularity as q itself and whereas here q is only defined as diffeomorphisms of the interval you have this critical regularity property holding for both the interval and the circle and so in between any two real holder moduli of continuity you can find an entire universe of sort of intermediate moduli where you have examples of all these possible critical regularities and so in particular that shows that each of these classes has infinitely many different uncountably many different distinct examples. So in the remaining two hours I'll go into all the technical details. So let me just say a few words about some of the ideas that go into this. So what you need to do to have any hope of proving a theorem like this is to have what I said earlier is a dynamical interpretation of regularity. So the mantra that one can state at least in not so precise language and then maybe I'll state it in precise language is that there exists so let's see that their k is fixed and mu is much greater than omega as in this theorem here. Here secretly you can just consider alpha and beta where alpha is less than beta. So there exists an element f inside of diff k mu of let's just say i for now which is faster than every element of diff k omega of i. So what on earth does that mean? So what that means is that so if I want to write down examples of groups like this I'm going to have to write down pretty explicit diffeomorphisms and so how might I do that? Well I'm going to have a sequence of disjoint intervals inside of the unit interval which are getting smaller and smaller and I'm going to build by hand essentially a diffeomorphism which moves points inside of each one of those disjoint intervals to the right or maybe to the left but let's just say to the right some amount and it turns out that the rate at which the points in these intervals get moved to the right compared to their lengths is intimately related to the regularity of the diffeomorphism you get. So let me state the following sort of technical fact which there are really two technical facts which make precise sense out of this gobbledygook so let k be an n and let delta be something between 0 and 1 fixed and mu a concave modulus and let's let j sub i be disjoint intervals in the unit interval and n sub i inside of n a sequence such that if I look at n sub i times ji to the k minus 1 times mu of the length of ji and that this is greater than or equal to 1 then there exists an f in diffe k mu of i such that it satisfies the following conditions so one is that the support of f is exactly equal to the union of these ji's the second is that if I look at f and I raise it to the n sub i power then this is what I would call delta fast on ji so what that means is that there is some point inside of ji that when I apply f to the n i to it it moves it the fraction delta of the way across the interval ji so what's going to happen is that I have a sequence of intervals like this which are getting smaller and smaller and now after the n sub 1 moves a point here say 90% of the way across the first interval and then after the n2 takes a point inside of ji2 and moves it 90% of the way across and after the n3 moves a point inside of ji3 90% of the way across and so on and so to make sense of faster than every element of diffe k omega of i is the following which I'll just state in words because I'm getting low on time is that if instead you considered some diffeomorphism g there's strictly higher regularity so that is to say you increase the strength of the modulus of continuity say you replace mu by omega and then you consider say g with the same exponents n sub i then not only do you not move 90% of the way across each of these intervals when you apply g instead of f you start to move almost not at all that is to say outside of a subset of the natural numbers of density zero the limit of the percentage of the length of the interval you moved across divided by the length tends to zero and so that makes that mantra precise so that's what I said at the very beginning I had this arrow between dynamics and analysis and this is a dynamical interpretation of regularity so then how does one then use this to prove a theorem like this well what you can do is you start with some mystery group I'll call it g plus I'll tell you what g plus is in just a second g plus is now going to act in two different ways on the interval one is going to be a particular let's say diff k mu action so it's going to be the weaker modulus of continuity and the other one this is going to be an action that we build by hand and this is one here which is arbitrary but of a higher regularity so this is diff k omega this one is prescribed and this one is arbitrary and so what we start out with is some element u inside of g plus which is not the identity and which has this magical property that its support is compactly contained inside of the union of the supports of all elements of g plus acting on the interval so it has in some reasonable sense finite complexity and so that allows us now to take some word here u and then we start iterating some conjugates we take some conjugates inside of g plus and have them act here and here so here u is again acting but here is something that we can prescribe here is something arbitrary now we use this dynamical interpretation of regularity to say we're going to do some conjugates by elements of g plus here and stretch out the support of u and here we apply the same conjugates and stretch out the support of u and here we can measure how much the support gets stretched because we use sort of some pre-built diffeomorphism that we get from this theorem and here we use the fact that any element of dif k omega removes points more slowly than our pre-built diffeomorphism in dif k mu and so after some number of iterations we get this element here u say conjugated by some element g which is going to depend on this action here which whose support is much larger than the corresponding support down here and what this allows us to do then is to say look I can now conjugate by some further element to pull the support off of itself and take the commutator and I get the identity because the supports are disjoint and here the support is much bigger so if I move over the supports still intersect each other and the commutator is not the identity and so what that means then is I have some sequence of elements inside of g plus and in some particular action of g plus on i by ck diffeomorphism term mu continuous so that if I look at any action whatsoever at least one of those elements is going to be the identity under this action and so no homomorphism from the image of this group here into this group can possibly be injective and that's the basic idea and I don't think that I can talk about any more of the details right now in three minutes maybe I'll just say a little bit about what u is so g plus is ends up being the group free group on two generators free product with bomb slog solitar 2 1 or 1 2 cross z so I said before for the interval yet five generators so 1 2 3 4 5 there they are and one can now play with these with different actions of various regularities here using results of of gelman louise alius sorry and bonati montverde navas and rivas I think yes to say ok well you can have reasonably good control of how elements of this group act on the interval if the regularities is at least say C1 and then one can play all sorts of games with commuting diffe amorphisms to say ok well now I can build some word by hand so that if I look at any representation of this group into diffe amorphisms of the interval which are at least C1 then this particular non-trivial element gets sent to a diffe amorphism whose support is a compact subset of the support of the group itself ok so I'm going to stop right there yeah it's some it's some countable subset of the intervals between 1 and 2 and I don't know what it is exactly so it's a as far as this critical regularity spectrum for holder moduli it's a very interesting question also to ask which regularities occur as critical regularities are finitely presented I have very little idea of what the answer there might be it's some countable subset and I have no idea what it is I don't we don't know of any examples of finitely presented groups with critical regularities strictly greater than 2 which is also and finite ones G plus is it's critical regularity and finite ones G plus is fixed so when I have k and mu here I have some the group that's acting here is some phi which is phi of k mu applied to g G plus is acting here this is something that one builds directly and this one is an arbitrary action so here the G plus is some group that we begin with we build all the examples that we get are quotients of G plus yeah yeah yeah yeah they're certainly not faithful they're acting here by this representation which we build depending on k and mu and then we compare it to one which is arbitrary sorry what? right so you mean little u okay so alright so support of say wrote down phi or psi is going to be some arbitrary action to some diff of m here goes whatever you want so the support of psi is the this is by definition the set of points x such that psi of g of x is different from x for some g in in big g okay so this is an open subset now of the interval as I've defined it and the property that u has is that if I look at the support of psi of u and look at its closure then this is contained inside of the support of psi the support of psi of u is compactly contained in the support of psi independently of psi and sort of one third of the proof of the theorem is proving that such a u exists it generally doesn't exist for homeomorphism actions but one really needs some sort of a priori regularity to even begin to talk about it