 Well, thank you very much, and I'd also like to Thank the organizers for having me. It's been a so far. It's been a very very wonderful visit and great conference Didn't see the summer school that much, but I've only heard good things so Let's get started So as you'll recall Fanny gave a very comprehensive Survey this morning, and so I can just highlight Recall that if you take a subgroup G of 0 to 1 semi-direct product with R3 So a discrete subgroup of affine isometries of Minkowski spacetime, and I'm going to Specialize to the case where the linear part is shot key By results including results of drum, but I've I Haven't mentioned everybody involved, but But Tadram Introduced crooked fundamental domains for such actions and was able to show concretely how to get Properly discontinuous actions, and then Danziger Kassel and Geritou Generalized to show in fact that Any such G that acts properly discontinuously admits a crooked fundamental domain, so that's my very quick Recall and what I want to do today The goal or what I want to describe today Is how one can Construct examples of what we could call Lorentz in shocky groups acting on the Einstein universe so three-dimensional Einstein universe and how we're going to do that is we're going to use Extensions of crooked planes actually Factifications of crooked planes called crooked surfaces, which were introduced by Shelf on says and I will Try to explain to you a little bit how you can get these things. Well, I'll describe what they are I'll describe try to describe how to get them apart from each other and we'll we'll recover some of this These ping-pong dynamics That we've that we've seen in the hyperbolic plane and in offline context so maybe what I'd like to say is Like maybe many people I Get a lot of Inspiration from the hyperbolic plane I mean we there's so much more to discover but on the other hand we know a lot about the hyperbolic plane and It's geometry and it seems like when you're trying to discover new worlds Such as the Einstein universe one thing that you might try to do is can I use things that I already know and look at them in Different context, so this is exactly what I'm proposing to do today take Lorentz and or take the shocky groups And see how we can extend them, but by no means am I trying to Describe a very general picture. I mean it's the basically I'm going to Give you pictures of examples, but I think it's we're far from a situation where we understand everything that's going on So just a few pictures for today Okay, so basically the Einstein universe is the projectivization of the light cone in our n2 So today we're going to focus on n equal to 3 and I think Todd discussed a little bit of Talked about this in his course by the way if you want I know that his slides are online So you can go check them if you need but So basically I'm going to let our 3 2 be five-dimensional real vector space, but I'm going to endow it with this This dot product of signature 3 2 so I've got 2 minus signs at the end and And and 3 2 is going to be the light cone of this space So I remove I remove the zero vector and I take all of the vectors whose dot product with themselves are equal to zero and The 3d Einstein universe is just the image of that cone under scaling So I mod out by all multiples so two vectors that are multiples of each other are Equivalent and I take the the quotient of that light and that's the Einstein universe and Maybe just a little bit of notation so I will use pi for the scaling map, but I'll also use these this notation for homogeneous coordinates when I find that it's useful and And Maybe is this the yeah, so actually so it's called the Einstein universe Einstein himself Studied this space will probably the higher-dimensional one, but Because it is it does admit what we call a conformal class of Lorentz in metrics So it's a space time One explicit way to do this is you take so I'm going to draw a picture which is a complete line But so you've got a light cone and I just take pick your favorite sphere and so the restriction so so that gives you actually a two-fold cover of The Einstein universe, but when you restrict the dot product to that to that set to that subset you actually get a Lorentz in metric so the indefinite dot product restricts to Lorentz in metric and Since you're just modding out by plus minus one after you can if you take something like this for instance, you just Have you have something very simple where you can get a new Lorentz in metric on the quotient Okay So it's a it's a conformal class of Lorentz in metrics because if you pick a different sphere You're going to get a different metric So if you like to work in a metric you just pick one But you have to keep in mind that the isometries that you're going to have might not preserve that metric But they're going to preserve angles so that's why we talk about conformal class But I just want to mention that but I don't want to insist too much on that today And if this is something that's a little mysterious to you Don't worry about it. You don't really need to to understand that Here Maybe I want to mention here that for some authors this two full cover is actually what they call The Einstein universe so the problem with our Einstein universe is it's not orientable because when you mod out by this Antibital map in this context you get something that's not orientable and some people like to work in orientable universes So they take the two full cover. They call that the Einstein universe So you just have to make sure that you read the introduction to the paper before you start going into chapter three Oh, you can just take for instance, you know, you take the induced apology and then take what is oh, it's it's a circle So there's a it's the pi one is the femal group is generated by By one By one line, so it's got it's got to apology. It's Yeah, it's it's it's s2. Yeah, it's s2 cross was s1. So it's a And you mod out by an antibital map and and you get another you know something that's but that antibital map reverses orientation Okay, I'll try to draw some pictures later. Yes Yes, so So, yeah, so Maybe so for me the Einstein universe is really the non-orientable one so One might say that it's the difference between taking projective space and Taking the sphere of directed directions, but you know, this is this is filmed right? So I think I coined the term thought Sorry, don't worry about that. You don't have to learn. You don't have to learn about the sphere of directed directions. Okay. I'm done Okay, sorry about that So I'm going to draw a few pictures of the Einstein universe. So as I said it's s2 cross s1 modded out by The product of that tip of lumps actually so maybe I was unclear on that but when you see that When you when you look at that equation then you could see that it's there's an s2 cross s1 going on right and I can't draw s2 cross s1 But what I can do is try to imagine how things would look if I if I remove a few things so So the first thing that I do when I'm drawing this So I imagine that s2 cross s1 is it's like a bracelet of pearls the pearls being the s2's and if I open the bracelet then I get a string of pearls Okay, and already that I can kind of imagine But now what I'm going to do is on each of these pearls. I'm going to remove a single point And so now I get like a shell bracelet. So I'm going to get a bunch of of discs so My picture of the Einstein universe after removal of These objects I'm going to get something that looks like this So all my pictures are going to be drawn like this. So you see you've got the you've got an s1 like this but I've removed a point so I get a an interval if you like and each of these discs is a sphere with I don't say you know remove remove the the South Pole off of all of it off of all of the spheres Okay, and and if if you're really paying attention You'll you might want to know that I have two points here. That's play very special rules There's one point here Which is that P not and And P not by the way It's here and it's also here because remember that this is glued to this via the antipodal map Okay, so North Pole South Pole and another important point is Called P infinity sometimes and so it's here, but on this copy, which I've actually removed but It's here Once again North Pole South Pole Okay, so I'm going to be drawing a lot of pictures and sometimes I'll write P not P infinity and those are actually the points that I mean Modulo calculation mistakes Okay, so And then I'll explain later if you're really trying to pay attention How I got those coordinates, but it'll come up Okay, so that's the picture of the Einstein universe that I'll be drawing in and It's already extremely crooked. So it's gonna it's only going to degenerate So I said this is a this is a it's a Lorenzian manifold It admits a Lorenzian class of metrics So you would you might be happy to know that there are certain Lorenzian objects that Translate well into this context It just has to be have to be careful of the conformal class business, but other than that you get a lot of Objects that you're you're relieved to to recognize So the first thing is what we call a photon, which is you know You might want to call it a light light geodesic or a light like line or something like that but We call it a photon and a photon is really There are several I think there are at least two Definitions here so the first thing is if you if you take two null vectors, so two vectors in the light cone of our three two Such that they themselves are orthogonal to each other. So their dot product is equal to zero so you get a you get a you get a plane and That's that's degenerate with respect to the to the dot product, but that projects To a line in the Einstein universe and so it's going to look something a photon is going to look something like this Of course, that depends on how you project face. This is a photon. Yes, I Didn't say that did I? Yeah, otherwise it would be born, but yeah, it's linearly independent So photons look like this you can also Okay, that's all That way And if you take a set of photons coming out of a point or passing through a point We're going to call that a light cone. So I've drawn two light cones here But if I continue this picture here if I take the set of all photons passing through p zero this is a light cone and This is another object that you find that arises quite naturally in the Einstein universe you can also realize a light cone as As the you take the so you take a vector here But I don't say what v is. Oh, yeah so if p is the projection of v so v is a null vector and And so I take the orthogonal plane to v which happens to contain itself and I intersect that with the light cone and then I projectivize I'm going to get a like Yes Okay, I'm sorry, but I'm not I shall not sorry It's yeah, it and a lot of this is dependent on how you so actually you can Yeah Yeah, yeah, so, okay Let me underscore that is what a what a light cone looks like is this if you took a torus and You took one of these marines and you pinched it to a point The picture and Ravi's talk picture and Fanny's talk everybody's been drawing this Yeah, exactly. I find this a little dramatic though. So I like this one. I'll have more Yeah, so those are light cones, I'll show you some more dramatic looking More like what did I say? Yeah, so you see now I've got I've got three light cones here So you've got this one here. So yeah, so I didn't I didn't say it But you probably read it is that these two light cones here Intersect in a circle, which is actually a space like circle, but Let's not let's not worry too much about that here and through that point This is this is the light cone going through that point Okay, and actually Okay, I'm not going to get ahead of myself, but you you kind of see the straight line and it kind of suggests What does it suggest? It suggests something like a tangent plane to a light cone Dot dot dot okay, so these are these are more like There's also something called what we call Einstein torus well Einstein torus is a two-dimensional Einstein universe and but they they arise in this context and How we get an Einstein Taurus here is so I'm going to draw the picture and you can read that here And I'm again going to try At some point I get tired of drawing cylinders, but Not quite yet What you want to do and I'm going to take my P not in my P infinity again So I've got P not I've got P infinity here. So these are two points This is a pair of points in the Einstein universe that do not lie on a common photon That's the first ingredient when you're building an Einstein torus That means that their light cones intersect in a circle That is space like and what we're going to do is we're going to pick any two points on this intersection So Maybe I'll pick these two points And I'll call this one F1 I'll call this F2. So maybe this is P1 if I'm using the notation here And I and this is P2 if I use the same notation. Okay So what that what happens is now once I've picked these four points this actually these four points They they come from four vectors Who span and the four vectors so they span a four-dimensional subspace of R32 which actually has a metric of signature 2 to so what we're going to do is we're going to get a the the project of the projectivization of the light cone of a copy of R22 and that's going to be an Einstein torus and any Any such surface that arises from such a choice of four points we're going to call an Einstein torus and so here for instance the The Einstein torus determined by these four points is really just you can think of it as I take this arc of great circle Which actually corresponds to this one two times? as one Okay, so remember all of these points here at one point so I get so I get something that closes up into a torus Okay, and Yes, there are a lot of words here, but that's pretty much it and So fun facts about the Einstein torus, so it is a torus It is a torus, but it's it's its complement is connected Because the Einstein universe three-dimensional Einstein universe is is non-orientable. So this by the way, it's a fun It's a fun Fun trick question to put on an exam if you if you want the fact that you've got this You've got a torus Which has only one side because it's in a non-orientable space No, this is really a torus it's really a torus, but it's only got one side so Exercise or I could say for instance So if you're on this side of the torus on this picture here and you come out here But this gets attached via the antipodal map to the other side. So it's So it's tricky, but it's something to contemplate while you're on the bus But not while you're driving Get you in trouble Also, and maybe this is a simpler way to think of it but it's it's also just the compactification of the the Orthogonal plane to a space like vector in R32, but somehow that's It's totally equivalent It's the it's just that vertical plane Yeah, so the edges are just getting glued Regular regularly and then you've got this antipodal map. It's a little funky, but But I promise it's a torus. We should we should teach a topology class. We should teach a topology class together drive the students nuts This gets mapped this gets mapped to here took me a while to There was a 50 50 chance that I wasn't gonna do that properly by the way I'm not even sure if I did somebody's going she's wrong, but we're gonna let her go. So I mentioned Something about conformal compactification and Todd talked about this last week, but I Just wanted to be just a little bit explicit here and just to show you exactly the map that I'm using to embed Minkowski spacetime or actually the affine space Modeled on Minkowski spacetime Into the Einstein universe. So there's my map so a vector v gets sent to this and It's this actually it's convenient. There are several different ways of mapping I'm willing to bet money that Todd didn't use the same map, but it's I don't know I'm I Like this one and and so you can you can map the you can map the vector space in that way and it maps R2 1 or what I call here v it maps it to a neighborhood of p0 But even much more explicitly than that and I think I say this on the next slide It actually the image of the map is everything except the light cone of P-infinity Okay, so if I if I remove this light cone What I'm going to have left is something that looks like Minkowski spacetime Okay, so this is what we mean. This is a this is a conformal compactification, but I don't want to get into the The nitty-gritty of that, but that's pretty much the the picture and in that sense I Don't so what I alluded to before if you take Remember I had these light cones like I had a light cone like this and that's the in that is actually the image of a Tangent plane to the light cone in Minkowski spacetime, so if you take the light cone based at zero so zero gets mapped to p0 or zero gets mapped to p0 and the Tangent space to the light cone a Tangent bit the light cone at P not which is a plane gets mapped to what something that looks like a light cone in this picture I find that so it looks like I'm going to talk about crooked surfaces now and I Think I'm going to finish early too. So yay for me right unless you ask me tons of questions So it crooked surfaces as I mentioned before were introduced by Charles Francis to study these conformal compactifications of marvelous space times which have been discussed and One way to define very quickly a crooked surface is you just take a crooked plane and You use you place it into the Einstein universe Versus the map Iota that I described before and You compactify so you take the conformal compactification of that Okay, we're going to discuss a way of defining a crooked surface that perhaps is more Synthetic or it looks more like how we define crooked planes In the Minkowski space but it's But the the key point that's going to come up is that it's it's also in a way Well, I say it is determined by a set of four points as for an Einstein tourist There'll be we'll have to make some choices But it's almost determined Okay, so how to make a crooked surface step one it's going to have a stamp and The stamp is going to lie in an Einstein tourists Okay, so what I want you to think of at the end is going to look like I drew the same picture over and over again Except this one is a little bigger It's in the same conformal class And then I forgot what I was going to say, okay, so I have So I want you to think of what a stem looks like for an actual crooked plane so it Right in in in R32 in R21 a stem looks something like this, right? Take a plane You take a plane that intersects the light cone Into lines and we take the part that's consists of time like factors time like directions so What you have to imagine is you if you if you start from this point and you look up what you're gonna see you're gonna see a wedge Okay, so Same thing here Actually, what you're going to do is this plane It's going to be Promoted to an Einstein tourists, so you're gonna take an Einstein tourists and What the stem is going to look like so I have to And there's some there's some little bit and that's in bolts here But you basically want to reproduce what you did in Minkowski space, so you're going to take All of these points here But you have these extra ones here This actually corresponds to the the bottom part okay, and You have to you have to think that here If I thought I was if I was thinking of this as being in Minkowski space this light cone Here's that infinity so as I go up like this. I'm going up like this Okay, so I have a stem like so in this picture. How clear is it? It's not really clear at all But it's it's this yellow part and this yellow part and a yellow part here And I underscored the fact that this part is glued to this so you really just get to two losses By the way, the pictures look here a little awful because I actually drew these in Mathematica using Using parametric so actually if you're if you're at all interested so I use Parameterizations in Minkowski space time and then I projected them into the Einstein universe and then pass them through My grinder of removing a line and removing a sphere so all these these were these were perfectly straight lines three Three maps ago and this is what happens and so Mathematica kind of freaks out here So but I think it's kind of interesting for archaeological purposes to see just how bad it can get and so somewhere in here You see you kind of see you know you've got the scrunching lines are all Getting packed together as they go to infinity Okay, so as they get close to infinity things start to bunch up it Am I too far away? You know what it might actually just be upset at the size of the next picture How way to get them? Well when you apparently when you speak French Canadian to the projector projector it it listens it knows I mean business So now you add two wings so what are wings what were wings in Minkowski space? They were you took tangent planes to the light cone and you put you pick the half plane right so those were wings You have one like this and one like that Well, what's a half? Tangent plane to a light cone. It's a it's a half light cone in the Einstein universe So here what you're going to do is you're just going to take a half light cone here and a half one there and You pick them so that they don't They only intersect In these two okay, so it's really just it really is just a conformal compactification of of a regular Crooked plane, but you get much more because you can you can get these crooked surfaces Base not only you can put you can base them anywhere including at points on the light cone of P infinity You've got a lot more you've got a lot more flexibility and you're moving them around with a lot more More maps, so that's just what it looks like so you've got So you've got this Lawson's here this one here, and then you're going to get I'm not going to draw it So I'll get too busy, but you've got you've got a half cone Coming out here, and it's going in the blackboard on the other side and on this side It's going in here Fold me out here. Yes Yeah, you would see yeah you see like these two Bulgy things. Yeah, really nice wallpaper. I Think so what's it going to be John? What's it going to look like? What do you think when we glue where we're going to get? That's okay. Why why am I picking on you? Yeah, I've got the memory of a goldfish so yeah welcome to my world Yeah, and I oh well I just showed the answer but So the light cones are going through f1 and f2 so that data that initial data that I told you about p1 p2 f1 f2 so p1 and p2 basically You take out you know the stem comes out from p1 and and that p2 if you want to think about it that way and The the wings that you're taking the light cones that you're taking are based at f1 and f2 Oh, it's so so it's homeomorphic to a Klein bottle So it's an unorientable surface But Einstein universe is still Not oriented so this time actually and this is nice. This is one of the nice things about Crooked surfaces is that they they're now there if you if you remove them The complement has to connect the components so there's sides There's two sides to a Cricket surface Which is the real reason why I don't work in the double cover because you in the double cover of course It's two-sided. There's nothing to it, but here is actually kind of a neat fact So so now you've got these things that look like Now this is really a cartoon image But if I were to draw the most A very general picture of photon would look something like this and A crooked surface in a cartoon way Would look something like this Okay, and I haven't I haven't that all explained why but you can you can use crooked surfaces as As bounding tubular neighborhoods of photons Okay, so there you can use them to build fundamental domains for a shocky type groups And actually this is exactly what chef on says did he He he he used this but he you know he was actually using compact fish Can fake compactifications of crooked planes, so they all intersected in one point and The game we're going to play in the next 20 minutes Is we're going to see how we can really pull them apart and really get them destroying from each other I've got to learn my slides by heart instead of just always saying them before I Wrote them and so what we're going to do here is just like I say here is we're going to remove the last point of intersection so in in Minkowski space we can move these away from each other as was Illustrated in several talks so far and we just need to do it One more time at the point at infinity So I'll I'll try to illustrate this with pictures and I'm going to do it in three steps So we're going to go through a little bit of a reminder Okay, so like I said I'm starting in my mind. I'm always starting with the hyperbolic plane Okay, so in the hyperbolic plane and I'm going to use I'm going to use a model of the hyperbolic plane Where geodesics look like straight lines? Okay, so Maybe something like this for instance Okay, this looks like a shockey group and So maybe I have something like this Okay, so I'm I'm illustrating a shockey group on two generators acting on H2 Okay, they're disjoint so you've got ping-pong dynamics. It's going to be everything you want it to be Okay, and then So then what do we do? When we go to when we promote to Minkowski space, so we in Minkowski space we're thinking of this Let me draw it this way. We're you know, maybe we're thinking of it this way, right? So we've got these things like this and now I'm going to take a bird's-eye view I'm going to look at a Z equal to something and I'm going to still see things like this But to get crooked planes. I'm going to add wings Right. This is what Todd explained to you and Fenny Reviewed this morning But we somehow need to get these apart from each other, right? Because what we want is we want them to be disjoint in order to debound a crooked plane. I'll domain and Perhaps this this this the simplest way we found to do this is to just Pull them apart in a parallel way. So let me let me draw it this way. So I've got my I've got my stem. So this is my crooked plane Okay, and I've got one wing coming out that way and I've got the other one going out that way and I have this neat little Quadrant here Just call it the stem quadrant and what you you you probably learned last week is if I if I just Slide the crooked plane So that the vertex lies in the stem quadrant. So literally I'm just using a translation That's parallel to the plane containing containing the stem But I place the vertex in the stem quadrant and now I'm going to get I know it's getting a little busy But you know, maybe I get something like this and I get a new crooked plane I just you know moved everything along and maybe what that looks like here It maybe it looks something like this Okay, I just I just slid it and I do this for every single one of them and One thing that we know is that as long as I stay within that stem quadrant In the interior of the stem corner everything is going to be disjoint from each other So I do that here or do that here and I do with the other two well Not that I'm obsessive compulsive or anything I think I have to do the plural for them now if you're just waking up. I'm sorry I know that looks terrible, but it's you have to follow the whole thing So you get them away from each other and that's how you build a crooked fundamental domain In and in Kowski space and now you just have to you know Remember what were the translations that you use to move one vertex to the other and that's how you assign Translational parts right so this is what we did here and we just moved the vertex in the stem quadrant. That was it, right and And one thing that we show is that as long as you move in a stem crop quadrant We actually have these crooked half spaces So the initial crooked half space Contains the new crooked half space Okay, so When you're sliding like this Maybe I want to make this next picture because it's kind of important, but You start with these two crooked planes that intersect in a point and Once you once you slide them away from each other you slide them away from that common vertex And the two crooked half spaces only intersected in that point so by sliding them away from each other you actually get your new Crooked half spaces are destroyed So that's how that's how you get this kind of stuff now as I know as I mentioned in The Einstein universe the thing is is you've got this vertex here p1, but crooked surfaces have a have a second vertex P2 and well it turns out Crooked surfaces are very Symmetric versus Relative to these two points and so yeah, that's what I just said So step three is we're just going to do the same thing at infinity Okay, now on the slides If you read the slides it looks a little complicated and When I first started thinking about these things I had a little bit of a complicated way of thinking about it but I'm starting to Starting to refine it, but it's it's still it's still up here. By the way, this is starting here, I think is It's a joint work with some former students of mine the Nick Francaire and Was my lab would so they're not here, but I Mentioned them anyways So you what you want to do is you you need to pull away not only at these vertices This p0 you have to move them away from the other but also at p infinity or p2 you have to move them away from as well and one way to describe this is To choose an evolution which exchanges Basically the two points and then you just whatever translation you did on the bottom Or whatever translation you could do on the bottom you just do to the top But let me let me draw a picture for you that I think is a little bit well Maybe let you see What we're really doing here It's the same picture again. I could just draw the the Einstein torus, but somehow I think it's helpful to see it in the context So yeah, so I have p1 here. I Have p2 here p1 So I can move p1 anywhere in its stem quadrant so I'm gonna put it here. So this is my new p1. I guess I'll call it p1 front and Well, okay, so now I have I have two photons coming out of it So it's the two there's a whole like home coming out of it, but I'm taking the pair of photons actually live in this Einstein torus Okay, it's a little crooked, but oh, I'm not done here. Okay, but you should think that So this point is identified to this one and then I have this It's glued to this What? No, it does not meet at that point. Okay. Let me let me just can I'm just gonna I Have to think about it. I mean this does this is fine This goes like this And this Yeah, but I still don't see how I'm supposed to Let me just erase and I'll start again Hey, something had to go wrong Okay, I'm gonna vouch for this one. So this actually gets mapped You know, this might actually just be too Okay, yeah, I'm I'm not gonna vouch for this one This is why we should always let machines make decisions for us Said the character in the prequel determinator Okay, I'm gonna draw this a little bit less Want to go so P1 was P1 prime was here. Yeah, that's and I'm gonna make it go a little bit So less likely to crash Okay, and then the other one I'm going to do like this and it's going to come up like this and So this gets mapped Okay, so, okay, so I've moved P1 now if I were doing this like a shelf contest was doing By the way, he didn't much more than this, okay, I'm not suggesting this is the work that he did was extremely Extremely central to to what we did later, but um So if I were just going to do this Then my new stem and I'm not going to draw it, but I'm just going to point to it. My new stem Would be I would have one loss in here because I've got the two lines coming out of P2 This is my new F1 and my new F2 Right, and then I've got another loss Right, but the thing is I'm going to do the same thing to P2 now. I'm gonna move P2 and And this is where I'm going to get in real trouble. So so you have to pay attention. So P2 prime is going to go I Don't want to get in too much trouble. So I'm okay. I'm going to be a little bit of a wuss and I'm going to put P2 prime here Okay, because now I'm a little afraid and And I'm going to do the same thing but I'm a wuss it should be a Little bit easier not to get into trouble and it's going here and then there's another one Going here, and it's going to look something like this So what's my new stem now for my new crooked surface? Well, it's this there's here's one When the cleaning ladies go through my room in the guest room, they're just going to find pictures of this all over the place It's they might get a little Concerned and there's another rhombus Kind of trying to hide out here and now to create a crooked surface You have to you have to fit in one half light cone here another light half light cone here and Yes Yeah, there's there's yeah, there's And and so but the end of but the way that I've constructed this thing it remained in this crooked Well this half space bounded by the crooked surface that I started with so I'm going to get everything disjoint from each other okay, and If if I were to take the last five minutes What I would try to do is to try to tie this in with what? Fanny and Jeff and Francois have done and and he decided Three space and tied in with stuff that some observations that Bill Goldman made about how crooked surfaces in the Einstein universe relate to anti-visitor crooked planes because Einstein universe Contains a double cop a double cover of any visitor space and if you do this really carefully you can build examples of Groups that act on that act properly on Anti-dissiduous space, but I'm not going to do this but what I will do is I will show you my very last picture. This is an awful picture. This is Mathematica telling me forget it I'm not gonna I'm not gonna work for you anymore and so but these are And these aren't even they're not even crooked surfaces. I think I only drew the stems But can you tell? But this is this is basically for crooked surfaces and you can imagine and of course look they're disjoint and You just map one like map the green to the yellow and map the magenta to the red and If you if you repeat this process You're gonna get these ping-pong dynamics and you're gonna get a a limit set Well, yeah, it's a carney limit set of of photons because you're gonna get a bunch of fixed points For all of the elements of the group, but you're also going to get the photons for each element the group You need to take the attracting the pair of attracting fixed points You take the photon that relates them and this is the limit set that you get but you get this cancer set worth of Photons, so this was actually first observed by Shelf on says and so these yield what he called shocky groups and I managed to use up all my time I am shockingly slow, but I will thank you very much for your patience