 Okay. Thanks so much. I'd like to thank the organizers for putting together such a nice workshop and inviting me to participate. I'm going to talk today about affine manifolds and some of the group actions used to construct such manifolds. Let me start by introducing the affine group. So start with a real vector space, rn. I'll denote by aff of rn the group of invertible affine transformations. So those maps of the form x maps to ax plus b, where a is an invertible linear map, an element of gln, and this is called the linear part of this affine transformation. And b is a translation vector called the translational part. Okay. So I think we've all thought about affine transformations before in some elementary course. Just to remind you, they include things like pure translations of rn, Euclidean motions, rotations, reflections, things like that. And also transformations that to our Euclidean point of view seem to distort rn, like shears and stretching dilation, Lorentzian transformations, things like that. And of course any compositions of such things are also affine transformations. As a group, it's a semi-direct product, aff of rn is the semi-direct product of gln with rn, which is to say that when you compose two affine transformations, their linear parts compose as usual, but their translational parts don't quite add. They add with a twisting by the linear part. Alright. So what I want to talk about today are what are called proper affine actions. So let gamma be a discrete group. Okay. So I want to study proper affine actions. And these are of course actions on some rn by affine transformations. So you have some representation into the affine group which defines this action, which are properly discontinuous. Okay. And of course, properly discontinuous as we all know is precisely the condition so that we can take a nice quotient by the action that the orbit space of the action is a nice Hausdorff manifold or more generally an orbifold. So this orbifold or manifold has an affine structure naturally coming from the quotient and that affine structure is complete in the sense that the geodesic flow coming from parallel translation on rn is complete. So complete affine manifolds are in one-to-one correspondence with proper affine actions. So I'm going to study these things more from the group action point of view rather than the orbifold point of view. Okay. What are some examples of these things? Let me give you some examples coming from sub geometries of affine geometry that I think many of you have thought about. Okay. So I'll make a little table. Sub geometry and possible gamma. Okay. So first of all, I could work with the translation subgroup. Okay. So if I want to build proper actions by translations, it's clear what to do. I take k translations and provided that they're linearly independent, I'll get a nice copy of z to the k acting properly by translations on rn. Okay. If k equals n, I'll make in the quotient a complete affine end torus. Okay. Another well-known sub geometry of affine geometry is Euclidean. So if I studied proper Euclidean actions, this is a sort of well-studied subject that was described. Yeah. There's powerful theorems of Beiberbach from the early 1900s, I think 1911, which say that while the groups that are able to act properly by Euclidean transformations are all virtually abelian. In particular, if the action is co-compact, then gamma is virtually z to the n acting by translations, like in the first case. So we don't get much more interesting discrete groups acting using Euclidean transformations. However, if we allow ourselves to use some slightly more complicated affine transformations, which distort rn a little bit, we can get some more interesting groups. So there are examples coming from the three-dimensional Thurston geometries nil and sol. These are both sub geometries of affine geometry as well. In fact, these are the only three for which that's true. And so any complete three-dimensional nil manifold or sol manifold is actually a complete affine manifold as well. And the fundamental groups of these things are nilpotent but not abelian, in the case of nil and solvable, but not nilpotent in the case of sol. So even using some slightly complicated looking affine transformations, we're still only able to get solvable group actions here. So the theme of this talk is to explore the question of what other discrete groups gamma are able to act properly by affine transformations. So I really just want to know, there's of course a lot of interesting questions about what those proper actions might look like, but let's just try to figure out which gammas admit proper actions, proper affine actions. There's a famous conjecture having to do with this. It's now known as the Auslander conjecture. It goes back to work of Auslander in the 60s, I think 64. And it says that if gamma admits an action on Rn, which is proper affine and co-compact, gamma is virtually solvable. Okay, so this conjecture is known, let me just say, in a way, this conjecture is known up to dimension n equals six now by some recent work of Abel's Margulis and Soyfer, but it remains wide open in general, and I think it's, well, it seems like it's pretty hard. So what's the thing about the conjecture a little bit? So first of all, in this setting of linear groups, as Moon had mentioned, virtually solvable is the same thing as saying that gamma does not contain a free group of rank R greater than or equal to two, in other words, a non-Abelian free group. Okay, so the intuition, the naive intuition might be, well, affine geometry is clearly a little more flexible than Euclidean geometry, but it's still flat, and so it shouldn't have enough room inside of it for big complicated groups to act properly. Okay, I think that's the intuition that Aslander sort of had in the 60s. Now, the problem is that things are actually a bit more subtle than you might think at first. So in fact, liberally applied that intuition is wrong because Margulis, in 1983, found examples of the conjecture of proper affine actions by free groups in dimension three. Okay, so free groups are not solvable, they're in a sense big and complicated, but they can still act properly by affine transformations. They still emit proper affine actions. The quotients by these free group actions in R3 are now popularly known as Margulis space times, space times because there's a Laurentian metric in the background that I don't want to discuss right now, and a lot of people, including myself and my collaborators, have been studying these things and getting a better understanding of them over the years. Okay, I also want to say Abel's Margulis and Soyfer have also studied certain free group actions in higher dimensions and found some situations where they again occur in higher dimensions. More recently, Ilya Smilga has been studying proper affine actions by free groups where the linear part lies in the adjoint representation of some semi-simple Lie group, and that's the context that we'll work in coming up today. Okay, two remarks at this point. So in case it's not clear, these examples of Margulis are not counter examples to the Outlander conjecture. The quotient of R3 by these proper free group actions are not compact. Okay, and why is that? Well, it's for very simple reason. The free group doesn't have high enough dimension. The virtual co-emogical dimension of the free group is one, and one is less than three. So if the action was co-compact, you'd see, at least in some finite cover, you'd see some co-emology in dimension three. The free group doesn't have any, so that's not possible. Okay, so if you want to disprove the Outlander conjecture, if you want to find counter-examples, you should find some discrete subgroups gamma, or some discrete groups gamma with higher dimension that are able to act properly by affine transformations. Okay, so the next remark is that there's essentially no other examples known at this point. So no other examples of non-virtually solvable discrete groups that emit proper affine actions except for dumb things you can do with the free group. You can take a product of two free groups and let it act on R6. You can make a proper action that way. You can take some finite extensions and do various things. But essentially there's no other source of any non-solvable proper affine actions. Okay. Yeah? Okay, so the main theorem is joint with Francois Gerito and Fanny Kassel gives a new source of lots of discrete groups that emit proper affine actions. So a whole new class of groups are able to do this. The theorem is any right-angled coxiderm group, okay, I'm going to abbreviate that RACG from now on, admits a proper affine action, Rn, and some affine space, Rn, where here n is K choose 2 and K is the number of generators of the coxiderm group. Okay, so this gives many new examples of discrete groups that emit proper affine actions. And as was alluded to in Ian's talk this morning, these groups are very interesting groups. They have a rich subgroup structure and they're very, even though they're relatively uncomplicated to define, they're actually quite interesting groups. Okay, so as a corollary, we get proper affine actions by surface groups. Why is that? Because the right-angled pentagon group contains a surface group. Okay, these have co-ological dimension 2. We get proper actions by hyperbolic 3-manifold groups. These have a co-ological dimension 3. This of course uses Hegel's virtual specialness theorem. Okay, and in general we get proper affine actions by hyperbolic groups, word hyperbolic groups, of arbitrarily large co-ological dimension. That's using the work of Janoskiewicz and Sviakewski, which says that there's actually right-angled coxiderm groups which are hyperbolic and have arbitrarily large VCD. That's still something that kind of amazes me, but it's true. Okay, any questions? Great, so let me now show you how to do this. We can construct these very directly. Right, so these are definitely not co-compact. So none of these are counter examples to this conjecture. They're pretty far from being co-compact. It turns out it's easy, not hard to see that the virtual co-mological dimension of a coxiderm group is definitely bounded by the number of generators. And of course the dimension of the affine space is quadratic in the number of generators. So these are, in a sense, quadratically far away from being co-compact. But nonetheless, it shows that there's actually a lot of groups that emit proper affine actions. I don't claim that this is the smallest dimensional affine space that these could act on. We just have found some examples. Figuring out whether this conjecture is true might come down to a race between co-mological dimension of the groups and the dimension of the affine space they're able to fit inside of. Okay, so let me describe how to make these proper affine actions. Start by reminding you what a right-angled coxiderm group is. Instead of gamma, we'll call it W now, because that's what's commonly done in the literature. I think W is for vile. It's a group with k generators, let's say, gamma 1 up to gamma k, which are all order 2. And then there may be some additional relations in the group of the form for each i not equal to j. There might be a relation of the form gamma i, gamma j equals gamma j, gamma i. In other words, the i and j generators commute. Or there might be no relation. Okay, so it's just a group generated by k involutions, some of which commute and others of which don't. Okay, so here's a simple example which we'll use to illustrate the sort of inspiration for this technique coming from the setting of Margolis space times. Okay, it's a right-angled coxiderm group with 3 generators. Of course, they're all order 2 as required. And then I'm going to add just one more relation that the first and third generators commute. Okay, but I'm not going to impose any relation on, say, the first and second or the second and third generators. Okay, so this is a very uncomplicated example of a right-angled coxiderm group. Now, how do we think of these geometrically? Well, a right-angled coxiderm group is supposed to be a group generated by reflections in some right-angled polytope. It's an abstraction of that geometric idea. And in this case, this right-angled coxiderm group can be realized as a reflection group in the hyperbolic plane. So let me draw a picture over here. Okay, so here's the hyperbolic plane. And I want to think of it in the projective model. I'm thinking of it as lying inside of RP2. It's a round disk in RP2. So now I'm going to draw a reflection triangle. So here are two walls meeting a right angle. And then a third wall here, which doesn't meet either of these two walls inside of the hyperbolic plane. Of course, I've extended the walls to meet outside of the hyperbolic plane past infinity. It's just convenient for drawing this picture in the projective space. Okay. So how do I see this group in this picture? Well, I'm going to think of gamma 1 as reflection in this wall. I'm going to think of gamma 2 as reflection in this wall. And I'm going to think of gamma 3 as reflection in this wall. And then, of course, when I reflect in the first wall and then I reflect in the third wall, I get the same thing as when I reflect in the third wall and then reflect in the first wall. And that's precisely the relation that I wanted to have in the group. Okay, so the right angle gives me the commutativity relation that I wanted. And then no other relations appear by accident. Okay, I can reflect in, say, this copy of the second wall. And it sort of, you know, goes in its own direction from there. Okay. Great. So this is a geometric sort of realization of this group as an actual reflection group. So it gives me a representation of the Coxeter group into Po21, which is the isometry group of the hyperbolic plane, which is a discrete embedding, in fact. Okay, so realizing this group as a reflection group allows me to think of this group as a discrete group inside of Po21. But in fact, there's actually a whole module I have choices in the way that I constructed this fundamental domain. In fact, since wall number two doesn't meet wall number one, I get to choose the distance d12 between these two walls. And I get to choose the distance d23 between wall number two and wall number three. And if I adjust those distances, say by moving the second wall closer in or tilting it a little bit or moving it out, I get a different picture. So the tiling generated by all of these reflections will look a little bit deformed. So I actually get a two-parameter family of essentially different representations here, parametrized by these two distances. Okay. So that's very good because there's actually a very straightforward way to produce affine actions from deformations of discrete embeddings in lead groups. So let me explain how that works. I'm sorry, affine action. Okay, so for this, this is more general than just this situation here. So let's let G be a semi-simple lead group. Let's say it has trivial center. I'll denote by frack G, the lead algebra. Okay, so definition, an infinitesimal deformation of a discrete embedding row from some discrete group gamma in the same way as into G is a lift. Okay, so I've got gamma. It's embedded in G. It's a lift up to the tangent bundle of G. I'll call that phi. Okay, so it's essentially, it's just an assignment of tangent vectors. If I think of gamma as being a bunch of points inside of G, I'm now going to assign tangent vectors to each point in G so that the group laws and relations all hold to first order. Okay, so it's just a formal assignment of tangent vectors. Okay, great. So the point is now that, well the tangent bundle of G is just the semi-direct product as a group. It's just the semi-direct product of the lead group G with its lead algebra, frack G, twisted by the adjoint representation. Okay, so infinitesimal deformation of a discrete embedding row has two pieces. It's got the original discrete embedding row and it's got a function U, a map U that assigns elements of the lead algebra to each element of the group gamma. It's not a homomorphism. It's twisted. So it's called a row co-cycle. Okay, and this gives you an affine action simply because this semi-direct product naturally goes inside of the general linear group of the lead algebra. So now we just think of the lead algebra as a vector space with the usual twisting now, with the standard twisting. And this is just the affine group of the lead algebra thought of as a copy of Rn. Okay, so essentially to be really explicit, the affine action on the line of the line of the lead algebra is just gamma acts on V by the adjoint action of row of gamma on V plus U of gamma. So linear part is the adjoint action of row. Translational part is this co-cycle of U. Okay, so purely formally, any infinitesimal deformation is just a random deformation of a discrete embedding gives me an affine action. So particularly here we get many, by taking tangent vectors to this two parameter family, we get many affine actions. Of course, there's no reason for any one of them to be proper. So what we'd really like to have is some way to understand the dynamics of the affine action in terms of the geometry of the infinitesimal deformation. Something like as long as the infinitesimal deformation does x, y, and z, then the affine action will be proper. Okay, so there's no such, well, at least not a complete dictionary developed yet for such questions except in the case where G is Po21, then work of Goldman-Labry and Margulis and also myself with Frantzell Geritot and Fanny Kassel have completely determined exactly when the corresponding affine action is proper and it has to do with whether or not this co-cycle, this infinitesimal deformation is contracting, is shrinking the geometry. So let me show you in this example how to make that work and then we'll generalize that to the general case. Okay, so also people tend to think this formal translation from infinitesimal deformation to affine action is kind of, you know, not so intuitive to think about. I'll tell you a very intuitive way to think about this affine action in a minute and I hope that it will seem intuitive after that. Okay, so, yeah, no, I'm not saying that. In this case, I showed you how to make a two-parameter family of deformations. I can take a tangent vector to that family and that will give me such a co-cycle. But sometimes there might only be the zero co-cycle. Yeah, yeah, yeah, these are really, yeah, these are elements in the first co-emology with twisted coefficients, you know, in this, yeah. So if that co-emology is trivial, then there's nothing to do here. Yeah, yeah, definitely, yeah, it does. Okay, so I told you very vaguely that getting a proper affine action over here has to do with some sort of contraction of geometry over here. So what I'm gonna do in this picture is I'm gonna take a path where these distances are shrinking. Okay, so let's choose D12 equals some constant capital D minus T and let's let D23 be the same thing. These are the distances in the hyperbolic metric between these two geodesics. Okay, so it doesn't, because I'm drawing this in the projective model, it looks like these geodesics touch each other, but that's at infinity, so there's actually some shortest distance between them. So I've got these distances, these are orthogonal distances. I think the projective model is sort of unfairly unpopular, I think it's really nice. Anyway, but the angles don't look nice. Okay, so I'm choosing these distances to be equal and shrinking at rate one. Okay, so this gives me now a path, row T, of representations, discrete embeddings into PO21. I take the derivative of that path. Okay, that gives me an infinitesimal deformation, phi, which is the discrete bedding row zero with the tangent information U, where U is some row co-cycle describing the first order, how the group is deforming as I push these walls together. So yeah, so in this, what do I want to say? Yeah, so here U, it's a FracG valued co-cycle, here FracG is just the Lie algebra of O21, which we'll think of now as the killing vector fields on the hyperbolic plane. They're infinitesimal isometries, they're vector fields whose flow is isometric to first order. Okay, so what is this co-cycle telling us? It's telling us something very concrete in this case. If I apply it to one of my generators, one of my reflections, it's telling me the infinitesimal motion of the corresponding reflection wall. Okay, so of course there's some choice about how to normalize here. Let's keep wall number one and three fixed and just move wall two so that these distances shrink. My co-cycle when I apply it to gamma two will tell me how this wall is moving. So it literally gives me a vector field on the entire hyperbolic plane, but if I restrict it to this wall, I see some picture like this. I see a vector field just telling me how the wall is moving. Okay, and of course this co-cycle, it's not just random, it doesn't have random values, it has to obey a certain equivariance. So you see that if this wall is moving like this, then of course this copy of the same wall has to move similarly, same over here. Okay, so you see these four tiles are shrinking towards the center. What about this tile? Well note that the co-cycle is not actually invariant, it's not just the image of this vector field over here because it has to take into account the deformation of the group. So really since this wall is moving in this way, these two walls need to move even faster towards this wall. So this tile is both shrinking and moving towards the center of the picture. Okay, and similarly if you were to draw the rest of the tiling, tiles very far away from the center of this picture would be shrinking just like this tile is and moving very rapidly towards the center of the picture. So you see this sort of contraction focused on the origin here. Okay, so what I'll do now, I won't tell you how to do this, it's straightforward in this situation, is I'm just going to extend this vector field to the entire tiling. It's only defined on the walls right now, but let's just extend it inside in some reasonable way. So I wanted to, it's a vector field now defined on all of the tiling, so I guess I should name, let's name this first tile delta. So the orbit of the tile actually covers the entire hyperbolic plane. I actually want to think of it as a bigger convex set inside of RP2, but it contains H2, so it's fine to just think about the hyperbolic plane right now. Okay, and of course, similar to the co-cycle, to these vector fields on the walls that we drew, it satisfies an equivariance property, which is also called automorphic sometimes. Okay, so I've got this vector field and it's not really important what it does on small scale, but at large scale, it's really pushing points in far away tiles together very quickly. So I won't say precisely what this means, but it's uniformly contracting. So as a heuristic picture, here's a point in one tile, here's a point in a far away tile, those points should be moving such that the distance between them is shrinking to first order at a rate, roughly proportional to the distance apart. That's satisfied in this picture. Okay, so I promised you I would tell you how to think about the corresponding affine action and how to see immediately that it's properly discontinuous. So let me do that. Okay, so remember in this correspondence, the affine space that we're acting on in the end is the Lie algebra. I'd rather think, now instead of thinking of the Lie algebra, I want to think of an affine subspace of the space of deformation vector fields. So I'm going to consider my affine space to be x, my deformation vector field describing this particular deformation. And then I want to subtract, I'm going to, don't worry about the minus sign, you can just think add, it just makes things work out a little nicer. I'm going to add or subtract from x any infinitesimal isometry, any killing vector field. Okay, so you can, as we had already said, there was some choice about what walls number one and three did. Of course I could sort of add a drift here, I could just add an isometric vector field that makes this whole picture drift. That's essentially, that's the infinitesimal version of conjugating, of conjugating my discrete embedding. And if I do that, the vector field looks totally different, but it still has the same contraction properties because I'm adding an isometric vector field there. So I should say x minus v is a different deformation vector field but it's still uniformly contracting with the same constants. Okay, so, you know, essentially by the Brouwer fixed point theorem, or the vector field version of that, if I add a killing field, this picture, you know, it doesn't look like this anymore, it's not centered nicely in the origin, but it focuses somewhere else. It has a zero, it has a tile that doesn't move very much and the rest of the tiles move very quickly towards that tile. Okay, so there exists a tile for a zero gamma delta, so the gamma translate of the base tile which moves at least under x minus v and all other tiles that move toward it. So essentially when I add a killing field, the picture refocuses somewhere else on some other tile, perhaps far away, but it's still essentially the same picture, you know, one or maybe four tiles not moving very much and all the others contracting towards that tile. Okay, so this is going to be, this is going to tell me immediately that the action is proper, so let me say what the action is now, so in this setting. So we think of this as an affine subspace of the deformation vector fields and what's the action? The action is, I take, let's see, I have an element of the group, and I act on this affirmation vector field. Well, what can I possibly do? I have a vector field. I'm just going to use deck transformations by the original discreet embedding and just lift up the picture and move it and put it back down. Okay, so that in symbols is, okay, and this if you work it out is precisely X minus the adjoint action of rho zero of gamma on V plus U of gamma. Okay, so you see, I think it's more convenient to think of the affine space as a subspace of the space of vector fields, but really we're just doing that same affine action that I wrote down before. Okay, so why is it proper? Well, if you just think about it for a minute, it looks pretty proper, right? I've got a vector field with a zero here, I pick it up and no other zeros, it's very big everywhere else. I pick it up and I move it far away and put it back down. Now it looks totally different. It has a zero somewhere else, very far away, and in this original compact portion of the picture that I was looking at, it's now enormous. Okay, so essentially that's the proof. You can formalize that. So the action X minus G is proper. You can formalize that idea by saying, okay, I'm just going to define a projection X minus G down to the group. It's going to be equivariant with respect to the affine action up here and just left multiplication down here, and all it does is it takes X minus V and it maps it to the tile for which X minus V is focusing on. So essentially it maps it to the zero of the vector field. So the action down here is proper, so the action up here is also proper. Okay, any questions about that? So that's the basic idea behind everything that happens in Margulis space times actually. Margulis space times don't involve reflection groups usually, but essentially this type of contraction is what's producing all proper affine actions by free groups on R3. Of course, note that the Lie algebra of Po21 is a three dimensional affine space. Okay, so we want to do this now. We want to do this in general for any right angle coxeter group. Okay, so it would be nice if we could just do the same thing and find a reflection, a reflection polytope in hyperbolic end space for any right angle coxeter group. As far as I know, that's, I don't know if that's possible actually. It may not be known, but I think it's unlikely that that would be true. So we need to use some other source of discrete embeddings of right angle coxeter groups to try to do this. So, okay, general right angle coxeter group, W. Okay, so it's got K generators. Okay, so luckily there is a very nice source of representations of a right angle coxeter group and any coxeter group really. Coming from a construction that goes back to TIPS really, I think things like this construction were studied by Wienberg as well. And more recently, these very representations that I'll tell you about were studied by someone named Kramer in 94 and also very recently by Dyer, Ripple and Holweg in the United States. So I think that's a good point. Well, a series of papers sort of around 2013 and I think still continuing. Their work is in the setting of Katz-Mudi Algebras. But really they're describing some of the very geometry that we'll need for this construction. Okay, so it's very simple. What's the construction? Okay, so I'm going to choose, yeah, so if I have two generators which don't commute as I did over there, I'm going to choose a distance, Dij greater than zero. Okay, and I'm going to define bilinear form, okay, that takes into account those distances. And from my bilinear form, yeah, I guess maybe I'll write it, the bilinear form B of EI, EJ will be zero if the corresponding generators commute. So the faces dual to EI and EJ meet at a right angle. Otherwise, let's say minus cosh of Dij. If not, remember we picked a Dij specifically for this purpose. Okay, now I get a representation, rho, which depends on my choices, simply sends x, let's see, gamma sends x to x minus 2, sorry, I guess gamma i, the ife generator does the usual, just a reflection, standard reflection in the dual to the basis vector EI using this bilinear form. Okay, great, so of course, this is secretly exactly what we did in this setting. It just so happens that the representation landed in a nice copy of Po21. Here, it may be that the signature of this bilinear form, well, we can't control it. So it may not be N1, it's going to be some PQ. Okay, so the signature of B is PQ, some PQ. And so, and that'll remain constant at least for an opening set of choices of the distances. So I can think, let's see, yeah. Okay, before I deform this, let me just say that, okay, so what is the analog of the hyperbolic plane here? So we don't have a nice picture in hyperbolic geometry. The orthogonal group of our form is a copy of OPQ, or I guess I should say P, PoPQ. But there is a nice replacement for hyperbolic geometry here. This acts on what's called HPQ-1. It's the semi-remonian hyperbolic space of signature PQ-1. Okay, so this is a really nice geometry. If you like hyperbolic geometry, it's a lot like hyperbolic geometry except that its metric is not definite. It's semi-remonian, so some distances are negative, some are zero. Here's a picture of one of my favorite examples, H21, which is also known by the name anti-decider 3 space. Okay, so it's projective 3 space. It's the inside of a quadric. Okay, and I'll just say, at any given point, in the tangent space, there's a cone of directions that have length zero. Inside that cone, those directions have negative length. So for example, this geodesic has negative length, but outside the cone the directions have positive length. So for example, this geodesic has positive length. Okay, so working with a semi-remonian metric is quite annoying as you can imagine, but it's the best we've got here. So what do we do? Okay, yeah. So let me say these representations have a nice convex geometry associated to them. So in fact, the dual planes to the standard basis with respect to B form a nice simplex. So this is really a realization of W as the group generated by reflections in a K simplex, delta contained in RPK minus one. Okay, so let me draw that here. This is a lie, but let's pretend it looks like this. Okay, so here's delta. Okay, and it's got a reflection phase for each generator. And when you take the orbit under this discrete embedding, it fills out a convex domain. Omega is the... Okay, so like in this picture, we had a nice tiling of the two-dimensional tiling of a convex set in RP2. Similarly, we get a tiling of a convex set in RPK minus one. I'm drawing it as if it's contained inside of this HPQ minus one. It might not be, but you can work with it anyway. Okay, and now we're going to deform... Now we're going to deform this picture and try to make it contract like we did here. Let's see if we can get a proper action. So let me say this very quickly. Again, I'm going to choose the Dij to be some constant D minus t for all ij, for all relevant ij. Okay, now it's a little harder to see what it means that these spaces are coming closer together in this picture, but let me not say more about that at the moment. Okay, so I get from this a path of representations, really discreet embeddings, into POPQ. Okay, and I take the derivative along this path, by the way. I see this delta, this reflection simplex start deforming and the corresponding convex set also deforms. I take the derivative of my path at t equals zero. I get a deformation co-cycle u, and again, the values of u describe how the walls of my simplices are deforming. Okay, so again, I'm going to produce a deformation vector field, okay, which describes how this whole tiling is deforming. And now I need to check if it's contracting. So in fact, in this setting, we define x very explicitly. It's piecewise projective, and it's very natural. Lemma, x is uniformly contracting, but only in positive directions. Okay, so the deformation is shrinking the geometry in these positive directions, but we can't control what it's doing in negative directions. Side note, what would you even want it to do in negative directions? Do you want the negative distances to become more negative or less negative? I don't know. It's just annoying to think about that. And luckily, it doesn't matter, because essentially, this is okay, because the limits that is seen by any point p in omega, they're compact family of positive directions. So although the vector field might be doing weird things in these negative directions, if I start to orbit my tile with a very large element of the group, it's going to be very far away, very close to the limit set. There's some limit set here. You're meant to be imagining a picture somewhat like the limit set of a quasi-Fuxian group in H3. So this tile sees one of its orbits in a positive direction. So indeed, we can conclude that any too far away, like tiles, are moving towards each other. And by projecting onto that compactly many positive directions, we can conclude that we have the uniform contraction that we need and we can apply the same argument. So the action of W on the Lie algebra of OPQ