 Hi, thanks. It's great to be here. Several of you here, I realize, have heard a version of this talk before, despite the result being like seven months old only. I guess we all go to the same conferences. But Maxime and I, so I'm going to talk about new joint work with Maxime Wolf. And we decided we would take the opportunity here the fact that we're giving two talks to an audience who we hope will be very interested to use both talks to give more details of the proof, explain something that maybe you can take home and use to do other stuff with. So I have the pleasure today of giving more surrounding motivation and ideas and tools than I usually would in such a talk. And Maxime tomorrow will try and get into a piece of more detail, I think. So that said, also having more time and flexibility, your questions are extra welcome today. So ask away. OK, so what are we interested in? I want to understand just rigidity of group actions. So let's start from a very broad setting and narrow down. I want to understand representations of a discrete group, so gamma is some discrete group. I should write bigger, excuse me. So here's our setup. I have gamma, some discrete group, think finitely generated. And I have G, some topological group. And I want to understand representations of gamma into G and how flexible and rigid they are. So the context that you're maybe used to is the case where G is a lead group or a linear group, and this is classical representation theory. Or for dynamicists, G is maybe a group of homeomorphisms of some manifold or diffeomorphisms of manifold if you're doing smooth dynamics. And this is giving you possible group actions on the manifold. And there's lots of ways that you can say how flexible or rigid these are. I'm going to take a strong definition of rigidity. So provisionary definition. So we might have to come back and adjust. A representation of this sort is rigid if, well, so I want to say that all deformations of this, even on a global scale, all deformations are just trivial. So what do I mean by trivial? How about attained by conjugation in G? That's always something you can do. So to say that this is trivial means that if I take the space of all representations, gamma G, I want to quotient by the conjugation action of G. So to say that the only deformations in here just came from this, which I have now killed, should be saying if rho inside of this space, when I pass to the quotient, is an isolated point. It's a very strong notion, but I will give you some examples in a minute of where this actually occurs. Bit of a problem. I just sort of very brazenly took a quotient to be like, I don't care about conjugacy. And what I've come up with is probably a very terrible space. So issue, even in the nicest cases, this quotient is typically very highly disgusting, like non-Hausdorff say. Let's do an easy example. How about gamma? It's just a cyclic group. And G is classical example SL2C. It's a familiar space to many people in this room. So now this space really is just homomorphisms of Z into SL2C. So that's just SL2C up to conjugation in SL2C. And so I'm just talking about conjugacy classes here. That's not very sophisticated, but already there's an issue. Here's a conjugacy class. Things that are upper triangular, where T is not 0. And here's a different conjugacy class. These are not conjugate. But I cannot separate them by open sets. Up in SL2C, before I quotient, I can take T as close to 0 as I want. And you can imagine that if I put in some more complicated group, this only gets worse. So I don't like non-Hausdorff spaces, because I don't know how to put coordinates on them and things like that, because we put coordinates in R or something. So let's fix this and take a bigger quotient. So does that line mean something to some people? But I won't put it there. That's meaningless. So here's the fix. Is define a character space x gamma g to be just abstractly the largest Hausdorff quotient, gamma g mod g by conjugation. This is a thing you can do using undergraduate topology. Any topological space has largest Hausdorff quotient with the property that a map to another Hausdorff space factors through this one. Great. And so now this is the thing that you could possibly hope to work with. And we'll go back to our definition and fix it up a little bit and just take this out and put this in. And now I'm saying that there is no, all the deformations are just by conjugacy as far as you could ever hope to measure in any kind of reasonable way. So that may be sounded like a bunch of setup and abstract nonsense, but this is a thing that people work with all the time. So for an example in SL2C, in the case where your group really is this, this is just, or I should say four, this is the character variety. So it's better than being Hausdorff. It actually has the structure of an affine variety. So there's all kinds of algebraic structure. And this actually is true for SLNC representations. And for a larger class of league groups, it's the quotient you get from geometric and variant theory. And the example that we're interested in, so I'll edit this to SLNC so you think it's very good. So it's just captured as parameterized by characters and representations. It's as nice as you could hope for. What I like is dynamics of group actions. And there's one other context in which this is beautiful and it makes sense. And this was perhaps sort of known or understood, but Maxime and I had to write down the details because it hadn't been done. So proposition. If your group is the group of homeomorphisms, I can't do any manifold. That would be an interesting question. But let's take a nice one manifold. Let's take the circle. Orientation preserving homeomorphisms of the circle. Gamma, anything. So any discrete group. This thing that I presented as some weird abstract quotient of things up to conjugation in so far as you can understand them. This has a nice interpretation. This analog of the character variety is representations to semi-conjugacy. So if you haven't seen thought of semi-conjugacy in this context before, probably this goes without saying, but just in case we're doing quickly. If I have, so conjugacy means there's some homeomorphism of the circle so that H composed with row of gamma. Maybe I have row one and row two and they're conjugate. Is the same thing as row two of gamma composed with H for all gamma. Gamma, that's conjugate. H is a homeomorphism. Semicondugate means that I don't require H to be a homomorphism. Just take H from the circle to the circle to be, should not use my fingers. Continuous, so I'm gonna let you collapse intervals down to points. So continuous and degree one and I want you to weekly preserve cyclic order. Okay, so you can collapse intervals to points but you can't go back and forth. Screw things up too badly. Okay, so this is an example of two representations being semi-conjugate. So it's like conjugate but slightly weaker and I'll say that semi-conjugacy is the equivalence relation generated by this. So I'll say that in this case, so if this happens for some H, say row one is equivalent to row two and semi-conjugacy is the equivalence relation this generates. So as I've stated it, it wasn't symmetric but really the symmetric version of this is you have two actions and if you can kind of collapse a wandering interval in one and maybe blow up some points also to get the other, blow up an orbit or collapse a wandering interval down that's all you need to do to pass within a semi-conjugacy class. So it's pretty easy to check that semi-conjugacy classes are connected and they're the closure of conjugacy classes and so that's how you start to recover this. Okay, but to say that this is exactly the largest house-door quotient on the nose is a little bit more work. But this is a natural notion and it's been studied for a long time. It turns out nicely though that it just fits into this context. Okay, great. Okay, so this is what I'm interested in rigidity and flexibility of actions. Here's a familiar context. Here's a kind of a nice way to talk about this in the group action setting. How about some examples of things that are actually rigid? So probably the most famous example of rigidity. I wanna keep up that definition there. So let me start again. Yes. Where does this come from? Yeah, yeah. This is a quotient you have example of this matrix today. There's the P of the E1. Yeah. The E1 non-separated is not a power space. How does it begin when you have this extension? What happened? How does? No, whatever, you have this quotient. You have the trouble that they ended up in your group separation. So it's not quotient, so I should identify these two points. You have the, well, okay, so when you have this extension, what happened to those two guys? When I have this further quotient or? Yeah, so you have the largest count or? Yeah, yeah, yeah. So if I'm really taking just the SL2C, so ham, Z, SL2C, mod, this and I take the largest house torque quotient. That is all I identify. Now I'm just, yeah, traces, it's now the space of traces of matrices. So X, yeah, Z, SL2C is in natural correspondence by a trace with, ah, so this is exactly what, and so more generally, if I add some other group, I should now recover traces of all the elements. If I have SLNC, I should look at characters, and I mean this is, you can prove that is exactly what happens, this is the largest house torque quotient. Yeah, cool. So one of the points of this talk is I'm kind of cooking this to try and draw parallels between the linear representations case and the groups acting on the circle case, okay, which is a theme that I've been motivated by for a long time is that dynamics of group actions with no regularity should, you know, a priori not be anything like linear representation theory, but if you look very hard, and you do a lot of work there, these surprising parallels that come up, and that's the whole point of this talk today. Okay, so let's draw another parallel. Let me do an example of rigidity in this lead groups case, how about Z, my favorite theorem since a long time, many of you will say this is master rigidity, but the form I'm gonna state it, I think is older and just do the kalabi, okay. It says the following, if my gamma is the fundamental group hyperbolic manifold of dimension at least three. So in other words, I can think of gamma inside of SON1, the group of isometries of hyperbolic space as something that where the quotient is compact and that will recover, I guess the unit tangent model of my manifold. So this is co-compact lattice, so IE. Then I just told you that gamma sits inside this lead group, this natural inclusion of gamma into SON1, this geometric one that comes here is rigid in the sense that it's an isolated point in this character space. So that's kind of the geometric version or the linear groups version and kind of the paradigm for all of this. IE groups acting on manifolds version V. So if I wanna recover this, I wanna substitute candidate for a theorem, I wanna substitute for being a co-compact lattice in a lead group. Well, here's the best substitute definition. Let's call representation of gamma into a group of homeomorphisms of M, geometric. This is not a lead group, it's not finite dimensional in any kind of way, it's not even locally compact. This is a disaster. How am I gonna try and recover something like this? I'll just artificially insist that there's a lead group in there. So it's geometric if it factors through an inclusion of gamma into some lead group as a co-compact lattice. Okay, so like this example here, and then I want G to act on M in a natural way. So G, I'm gonna require to be a connected lead group to act on the manifold transitive lead. So that I don't, I see all of M. Okay, in other words, M should be a homogeneous space for G. Okay, that's the definition. Gamma sits in the axon M in a way that sees a lot of geometry. Okay, it's a co-compact lattice in a group that makes this a homogeneous space. Maybe the example zero of this is you can put the fundamental group of surface of genus G, at least two. Okay, it's one dimension down from the clubby's theorem over there into PSL2R or SO21, if you like that notation better, which acts on the circle by homeomorphisms where this action is by mobius transformations, if you like, or if you think of this as the isometries of hyperbolic space and you take the Poincaré disc model, this is the action on the boundary. So this composition, if I call this row, that's an action of a surface group on the circle, which is geometric because I cooked it up that way. I put this inside of here, put a hyperbolic structure on the surface, if you like. It's now, its fundamental group is acting on the universal cover. The hyperbolic plane by isometries in a way that this is a lattice in this group and this acts nicely on the circle. I like this example particularly much because you might have run across it without this space in the middle. This is the sense, this is a hyperbolic group in the sense of Gromov and this is really the action on the Gromov boundary of the group, which is by homeomorphisms. There's no extra structure there. You're doing geometric group theory. This is all you've got and oh, nice. What a nice example of something that satisfies the definition of geometric. See, secretly there's a legal side there. Okay, great. So we have a definition, what's the theorem that goes along with it? An older theorem, due to Matsumoto from, I guess it's about 30 years old, that was from 1990. It's a remarkable statement that example zero is rigid. I.e. If you take this representation, it is an isolated point in the character space for actions of the fundamental group on a surface on the circle. In other words, any deformation even globally is semi-conjugate, dynamically essentially the same to the original action you started with. At the time, I don't think anyone thought that one would ever have sort of a master rigidity-ish theorem for things with no regularity and no group structure or anything in the target. This is the part in the talk where I have to tell everyone who studies surface groups in PSL2R and likes Teichmuller space, which is the space of all representations here up to conjugacy and it's large and it's interesting and it's rich, that I had just killed it entirely because all of these examples where you go into PSL2R first are conjugate in the group of homeomorphisms of the circle. Okay. So, although when you quotient by PSL2R you get something big and interesting. If I take the quotient by dynamism, topological dynamics if I care, but things up to topological conjugacy, that's now done. Okay, so this is not a contradiction in mathematics, this theorem makes sense. This example is one example and it is an isolated single point. Great. Okay, so here's my whole setup. What's the rest of the talk? It's a rigidity theorem for all the other geometric examples on the circle. So, rigidity of all other geometric actions on the circle. Let me tell you what all the other geometric examples are. So here's a fact. Exercise. Okay. If G, so I need, what do I need in this definition? I need to tell you I'm gamma, I want to put it in the homeomorphisms of the circle, but it's supposed to factor through some lead group. What are the options for the lead group? Let's be systematic. Okay. If G acts on the circle transitively, there's not very many options. G is either, I already gave you PSL2R as an example. There's the obvious example of just rotations. And then there's one more silly thing you can do, which is that the circle is a cover of itself. So I could lift the action of PSL2R to that cover and I'll get all the other lifts will just be some central extension of PSL2R by the finite deck group. Okay. So, or there's extension of PSL2R by some finite cyclic group, central extension, okay, in which G sits, okay. And if this is acting by Mobius transformations on this little circle, this is acting on the K-fold cover. Okay, so it's just the group of all the lifts. Okay. This is a fun thing to try and prove. What's the key here is that this, the circle is not very big. A lead group inside of its group of homeomorphisms can be at most three-dimensional, okay. So these are all the geometric examples and lattices in these. Well, this one's already compact. Here you have surface groups and here you can have either surface groups or things that are virtually surface groups. So the fundamental groups of surfaces, of G at least two, are the only candidates here. And the theorem, okay. One direction was worked from my thesis in 2014. And the other direction is with Maxime Wolf from last year, okay. It's the statement that actions of surface groups on the circle, an action of a surface group on the circle is rigid, namely an isolated point in this character space. If and only if it's one of the examples I just gave you, it's geometric. So secretly there's a lead group there. Could look like Matsumoto's example. It could be one of these central extensions. And those are exactly the ones that have no non-trivial deformations. Oh, and of course these arrows now make no sense. Okay, so the easy direction was the earlier one. Geometric implies rigid. Here you have specific constructions. I give you a surface group sitting in here. I know what its action is. I can write down matrices if you like, whatever. You wanna prove that there's no interesting deformations of this. So this is something you start attacking by hand and saying, well, if I systematically tried to do these things, nothing can happen. There's a bunch of machinery to do this. And this is the truly remarkable example that that's an example. At that point I conjectured out of optimism that I had found all the rigid examples because I tried to find more and I couldn't, you know, brute force could conjecture by lack of creativity. This direction is, how does this go? Someone who hands you a mystery representation about which you know nothing. You just have some group action. Okay, oh, but you do know one thing. If you tried to change it, you would fail. From that, you're supposed to recover a lead group in which your surface group happens to sit as a lattice. So this is the direction that we're gonna be talking about that required an invention of a bunch of techniques for doing these things. Questions so far? Well, remark, it's really, really easy to produce lots of non-rigid, non-geometric examples. Okay? So from Gamma, homey OS, I'll even say actions of, I'll say surface groups for concreteness. This space is big. In fact, even this character space, it's big and it's kind of wild. It's easy to make lots of non-geometric examples. For instance, maybe you like working with representations of surface groups into PSL2R. That's a thing people like a lot. If you take one of those that doesn't come from a hyperbolic structure on your surface, then it's not rigid even in the PSL2R character variety or deformation space. So it follows from work of Bill Goldman that the non-tychmuller examples have a large deformation space. And in particular, you'll find elements that act by rotations. And as you perturb this, you can change the amount they rotate. And that's a conjugacy invariant, even a semi-conjugacy invariant in the group of homeomorphisms of the circle. So even if you just work within representations to PSL2R, you don't find any other candidates for things that might be rigid by topological conjugacy. So even with image in PSL2R, lots of this, lots of these. And if you wanna go outside of, I know how to write down matrices and do these things, here's another good way to make lots of crazy examples. Okay, so I guess one, you could take representations in the PSL2R. Or two, here's my favorite strategy. Take your surface, think of it geometrically, cut it in half somewhere. And this lets me think of my surface by what, Van Kampen? I can think of the fundamental group as an amalgamated pre-product of the fundamental group of this part and that part. Okay, but these are just both free groups. So this lets me write my fundamental group as a free group. That's a free group on, here I've done it as, let's write it properly, that's a free group on two generators. This one on four, I guess, and it's amalgamated over the cyclic subgroup generated by this curve. So what is this saying? Free groups, that's easy, just take any two homomorphisms of the circle, there's no relations to satisfy to any two homomorphisms of the circle can be the generators of this group. Similarly, pick whatever you want here and you're subject to a single constraint that the action of this group and the action of that group agree on this common element. But that gives you lots of room for variation and make this one do whatever you want and now just cook up the other side so that product of commutators that gives this curve agrees. So free groups are easy to make act and deform. I think that's interesting here that so I wanna start sort of warming up towards some ideas of the proof. And one thing that I sort of, I didn't realize until afterwards and when I went back and read Matsumoto's proof again is that there's ideas in here sort of well under the surface that parallel some global strategy of what we did in here. So let me tell you one of these very important tools. So a key tool, which is also another kind of beautiful analogy between the linear case and the surface group case. And in fact is part of how you see that representations up to semi-conjugacy is a house dwarf space and a reasonable corrosion. It's a following theorem, which is do tijis and rephrased in the way I'm gonna state it by Matsumoto. And it says that coordinates, there are natural coordinates on the space of actions of any group on the circle up to semi-conjugacy. Analogous to characters or trace coordinates of representations into SLNC. So taking SL2C to be concrete, if you have a representation of a group into there and you know the trace of everyone, theorem you know the, well I mean you know the representation up to conjugacy unless it was upper triangular basically. Okay, this space has same kind of thing. So what's the analog of character? I should have some conjugacy invariant function. This is given by the only dynamical invariant really of circle homeomorphisms, the Poincaré rotation number. Okay, so if you haven't seen this definition before, this is to any homeomorphism, you assign a number in R mod Z or in S1 if you like that captures the average amount it rotates points. And the theorem here is stated very imprecisely, slightly more precisely, it is that a representation from any group you like into the group of homeomorphisms of the circle is determined essentially. So there's some finite ambiguity there but nothing that we need to worry about for the proof by the rotation numbers of each curve gamma in gamma. And maybe it's worthwhile to say what I, when I meet by essentially, there's slightly more information that you need. Okay, so here's really what's going on. Let's take my favorite group, my favorite representation, the one that gives you a surface group in PSL2R, the Fuxian action, the nice boundary one. There we know the dynamics of things. Every element of my group acts as hyperbolic transformation has two fixed points and has sourcing dynamics. Having a fixed point is equivalent to having rotation number zero. So this representation gets assigned zero for everybody. Okay. It is highly non-trivial but unfortunately the trivial representation also gets assigned zero for everybody. So that's what essentially is saying, it said I was not being precise enough. Okay. What's the difference? It's determined actually by, so it's determined up to semi-conjugacy by the rotation numbers of not single elements. Okay, but comparing pairs. So what you need to know is the function that takes two elements and counts how their product wraps around the circle. So in PSL2R, if I have two things, maybe someone with an axis like this and then someone else with an axis like this in very strong dynamics and I take their composition. The composition will quite likely, if I start with some point here, this guy, the blue guy will move it all the way over here and then the black one will move it back to where it started. It will have a fixed point but one where if you track how it winds around the circle, it will have made a full rotation. Okay, that's somehow different than the trivial representation where everyone's the identity and the point doesn't move ever. Okay, so this is given by lifting these, so I wanna track how much something winds around the circle. This function is given by taking an element, lifting it to a homeomorphism of the line that commutes with, I'm thinking of the line here as a universal cover of the circle, so I'll lift it up to, if my circle is r mod z, this will be something that commutes with the deck group acting by translations. So it is z-equivariant homeomorphisms of r. I can take a lift of the other one. I can look at the average amount these translate points. So if I took their lifts with fixed points, I would get zero and zero and I wanna compare this to how the product does. So in the cartoon I just pointed at, this is exactly capturing the fact that this product winds around the circle once. I've lifted through the universal cover so I could keep track of my movement back and forth. So this is what you need to keep track of really that distinguishes the trivial representation from this folksy and guide, but it's framed completely in terms of essentially a rotation or translation numbers of elements. Great. Another question I wanna point out now before I go on is that I would very much like to know an analogy of this theorem if you increase the regularity here. So instead of homeomorphisms up to semi-conjugacy, let's study actions of a group by diffeomorphism, C1 even on the circle. And I can think of those up to C1 conjugacy. And that space I'm sure is not house dwarf and is nasty and terrible, although I haven't quantified how bad it could be. But we can say, okay, pass to the largest house dwarf quotient. What is that? And if you like smooth, replace C1 with smooth. I know no nice analog of this in the higher regularity, but it's mostly because I haven't tried. Or who knows, maybe it's reasonable to say or maybe it's extremely difficult. I think it's a fascinating question. Okay, so what does this all boil down to? It means that both us and Matsumoto are trying to understand the deformations of representations by understanding how much curves rotate points around the circle. We're looking, we know things are non-conjugate. Basically, as soon as this number changes for some pair of gamma one, gamma two. So I wanted to give the technical correct one, but you should think of this as being sufficient to just know rotation numbers of individual elements. If you're working locally, that's good enough to recover this. That point is the bad example I gave of the trivial thing and the spruxian thing are extremely far apart from each other. So working locally, you just need to understand rotation numbers of elements. Let me give you the two-step outline of what we do and maybe a cartoon of one of the steps and then leave it to Maxime tomorrow to go into more detail. So here's an outline, a very broad outline of the proof rigid implies geometric for surface group actions. Get some more space here. The first thing we noticed is that this theorem would be a lot easier to prove. If instead of just knowing something was rigid, we knew we can make arguments like this. We knew that all the elements looked like they had sourcing dynamics or we're secretly living in PSL2R or something like this. I was surprised very recently that going back to reading a Matsumoto's original paper of Fuxing's example is rigid. He actually spends two pages being like, let's pretend the image was just in PSL2R because that'll make my argument easier. And then he's using sources and things of hyperbolic elements and gets a nice conclusion and then goes back and says like, well, you can't expect this in general at all, but we'll try and do some deformations and say that this argument kind of sort of can be pushed through and five pages later of like technical misery nails it. In the penalty, we had the same idea. Well, if this was kind of looking like everything was just sort of nice hyperbolic dynamics in this topological sense, we could almost do that. So then came the hard work of reducing to this case. So part one is to say that suppose that row is rigid, we can conclude that after semi-conjugacy, I can, well, after semi-conjugacy, I can always pass to a minimal action, one where all orbits of dense. So I'll add rigid and minimal to kind of pick out my preferred semi-conjugacy class. I won't blow up anything I don't need to. Then, and this is the hard part, this is sort of the bulk of the work. Individual elements look like they sit inside of one of these lead groups. So then very locally, we look kind of geometric. There's a local geometric-ish picture. Precisely, if you take two curves on your surface, two elements of the fundamental group represented by simple closed curves, that have intersection number one. So if I have A and B are curves, here's maybe A, here's B. So it's up to some action of a mapping class group I can draw my curves like this, okay. Here's the dynamics of row A and row B. So then there exists some K, which is actually independent of row A, sorry, independent of A and B. So the same K works for all the curves on your surface. So that A and B each have two K fixed points, not necessarily fixed periodic points, okay. Half of which are attracting, half of which are repelling. Maybe this is the attracting one and the other repelling. So alternating, attracting and repelling periodic points, and they also alternate in the surface. So this is exactly the dynamical picture of how periodic points look in one of these lifts from an action of PSO2R, right. If you look at the geometric picture of this, that's K equals one. This is, these are two hyperbolic elements with crossed axes in the way we usually draw this picture. Okay, NFI. So there's an attractor, a repeller of this one, an attractor, a repeller, an attractor. If I lift this to a K-fold cover, I take any of the lifts, these will become periodic points alternating attracting and repelling. So here's the picture where K equals two, okay. So there's a local geometric picture. I say geometric-ish, because all we know is the kind of combinatorics of attracting and repelling periodic points. So this really is the the headache to go from nothing to, ah, a nice combinatorial picture. Step two is to just do a combinatorial argument and recover your surface. So step two is local to global argument, okay. So this is a statement that if row is rigid and satisfies the local picture on pairs of simple closed curves given by one, okay. Row is in fact geometric. And the idea of this is just to use very specific deformations, is we use very specific deformations, analogous to bending or doing an earthquake from Teichmiller theory, but really we're just, instead of elimination, we use a one single simple closed curve, okay. So it's almost as if you're putting a dain twist around the curve in a one parameter family. So bending from, say, the theory of clining in groups, you like. To say that if you're not in exactly the right configuration all your curves, so here's something only on pairs, but I really want to recover my entire surface, okay. To reconstruct the combinatorial configuration of all curves, all simple closed curves on the surface. So a hypothesis says that some pairs look good and now I want to say, well like now, but now I have a curve over here that's disjoint, maybe this one. And the geometric picture, it's supposed to have an axis that's disjoint, so in the lift I'm supposed to see, what if I see green attractor, then before I see the red repeller, I should see two points here, plus and minus. And then, oh, I have to go around a whole time and then I should see the other two lifts of this, plus and minus there. That's something that this hypothesis doesn't give me, and we have to recover this kind of information, okay. And we do this by producing, should this not be true, if this plus was actually somewhere over here, goodness forbid, we write down a specific deformation that moves it somewhere else, giving us the wrong combinatorics or the wrong rotation numbers for pairs of elements and say, oh wait, but it was rigid, that could not have happened. So the whole idea of here is to use specific deformations to say you had to be in the right combinatorial configuration. And once you know that the attractors and repellers of all your simple closed curves are in the right place, it's not so bad after that to say, your surface group had to be acting as it was in the geometric picture. But I have run out of time, and so hopefully that has given us kind of a starting point or an advertisement for Maxime's talk tomorrow. So I'll conclude with that. Like non-surface groups. So I, there's two ways I know to produce examples. Okay, so one is if you finite order elements, that will help you because, well, if you find an order, you're conjugate to a rotation. So you can play with this and do some things. So things like even SL2Z, free product of Z2 and ZM3Z is known to be rigid for an argument like that. I forget, I should cite who, I think Michele Tuestino and friends have written this down recently. In some work that they've been doing. But I'm gonna get the wrong people attached to the wrong theorems. And I think you can do is you could take a hyperbolic three manifold group from a three manifold that fibers over the circle and take the example, this is sort of, there's a nice geometric action of this group on the circle where the surface fiber subgroup is in PSL2R. And because the rest of your fundamental group normalizes this, then that forces that action to be rigid. So you can do sort of tricks like this. I don't know, so a question I would like to answer is that's an example of Thurston's universal circle construction. To what extent are the, but that's a very general construction that applies to a wider class of examples. Are these all rigid? What do they look like, if not? So it's trickier, because you have some choices in the construction more generally. So I think that's good, that's a good, it's a good long term project. In which part of the book do you use these coordinates? Yeah, so this is going, so first, well it's here everywhere. So first, the conclusion of this is that everything has a periodic orbit of period 2K. So first thing we have to show is that the rotation number of a simple closed curve is rational. And then that they're all the same denominator. And that's done by again producing an explicit deformation. We're saying, oh, if it was irrational, then I can do a particular thing and change it's rotation number. Oops, not rigid. So it must have been rational. So that's actually like a lemma zero is that all simple closed curves have rational rotation number. And then we want to put their, their put orbits in a good configuration. And that's Maxime's talk, definitely. So. Any more questions? So otherwise, let's get it again.