 going to over life a little bit with what Francois said. So just to review and get us all on the same page. So I apologize if you know that well, but hopefully you'll enjoy that. So the boundary of a hyperbolic group. So in general, this is a topological object that's going to be canonically associated with a hyperbolic group. And it's going to tell us a lot of information. So I'm going to show that this is a canonical object or topological object. It actually also has a metric structure, which I won't go into today. And then I'm going to show some things that it's good for. What does it tell you? It tells you a lot. So the spoiler is a lot. And then I'm going to talk a little bit about relatively hyperbolic parameters. So this is going to be a lot. So I want to just quickly review this. And then I want to spend most of my time on what the boundary of a hyperbolic group actually tells you about that hyperbolic group. So x is going to be a proper. Proper is important. Geodesic isn't so important, but I'll usually be implicitly assuming it. Hyperbolic metrics ways. So let me just give three very common definitions. So definition one. Boundary x is the set of geodesic rays from some point. The topology won't depend on the point. And modulo equivalence. So what's that equivalence going to be? So we want the geodesic rays to be equivalent if they're in the same house or they have bounded house or distance. So let me just get an example to keep in mind that I think you've done is H2, delta hyperbolic space. I'm going to have rays that go out here. And if I think about it, maybe I have some slightly wiggly hyperbolic space and there's some other geodesic ray that stays the same. And I kind of think it ends at the same point. If they stayed in the same house or distance, they would end at the same point in H2. So this is bounded house or distance. So I'll say that gamma 1 is equivalent to gamma 2 if the image of gamma 1 is contained in the neighborhood of c of the image of gamma 2 and vice versa. There's some c that just works. So this is a good definition just for getting your intuition. So lots of times we're going to be dealing with quasi-isometry. So we're not going to want to have geodesic rays. So definition 2 is really similar to this. This is one that I'll use. And I'm thinking about quasi-geodesic rays. So just quasi-geodesic rays at some x not contained in x. And again, modulate the same equivalence. And this is bounded house or distance. So you can just think about wiggly rays. And you want them to stay in some distance. And remember that in a delta hyperbolic space, if I have a quasi-geodesic, it's within bounded distance of some geodesic. So you can take a geodesic in this class if you have a geodesic metric space and just think about that. So let's think about definition 3, which is a one I want to talk a little bit about. So now instead of thinking about the actual ray, I'm just going to think about points going off to infinity. So this turns out to be really useful. Often we think of a group as just the orbit of its point acting on some delta hyperbolic space. So we might just want to think about a bunch of points. It actually is very helpful. It's a little bit less intuitive, but it's very helpful. And this is sequences tending to infinity. So I'm going to look at, so let me just say what that means to tend to infinity. I'm going to look at, so the Gromov product, which I think you've seen, at least Endero is probably telling you about it. So I have x, y at some point p. It's just defined to be 1 half the distance, I'm in a metric space, xp plus the distance yp minus the distance xy. So let's draw a little picture of this. So if I'm in hyperbolic space, I've got some p. I'm going to look at my two points out here. And I'm going to have the distance from x to p, the distance of y to p, minus the distance between them. So I'm drawing a little picture of xp. So notice when they get way far away, this distance is actually going to be going to infinity. So these will be big. And if they're close together, if they're sort of going to the same point, so this is sort of an idea of a proof that what happens if these are points along rays that are staying within bounded house dwarf of each other, way out here, they're going to look exactly the same. And this is going to go to infinity. And another way to think about, a good way to think about the Gromov product, I think, is in a tree. So if I have like, here's my p, and this is x. So I'm drawing part of the Cayley graph for f2, x and y. OK, so let's look at what the Gromov product is here. OK, the distance from p to x. So this is just going to be 3, right? This is going to be 1, 2, 1. I wanted this to be 4. 1, 2, 3, 1, 2, 3. I'm going to put another one here, plus 4 minus 5, right? So this is going to be what? 1. OK, so what is this? So where's my geodesic between x and y? So it's fun to do this in a tree if you haven't done this for a while. So this distance in this space is going to measure the distance from p to the geodesic from x to y. So here's my geodesic between x and y. I've got to go through that point. And then here's this distance to 1 to that geodesic. So in a tree you can start to see if I have, I want to look at sequences going to infinity. So let me say what that means. I'm going to say that tends to infinity. Or sometimes I'll write actually right error infinity. But I'll have lots of different points on the boundary. If the limit as i goes to infinity and j goes to infinity of ai, aj equals infinity. So that means whenever i and j are bigger than some number, then this is as big as I want it. I can make that as big as I want. So it's a good idea to play with that in a tree. Tree is zero hyperbolic. Everything is kind of nice. Eraser is tricky. Oh yeah, there's more boards here. Oh, there's even three. All right, I'll put some stuff on the side. OK, so in the exercises this shows that there's, you can think of this as like thin rectangles to be thin. So definition, maybe I'll put this over here. x is delta hyperbolic if and only if for all x, y, z, w, xy of w is greater than or equal to the minimum of x, z, w, y, z, w minus delta. OK, so you can think of, you can do everything you could never know about geodesics and be happy in a hyperbolic metric space. OK, I send rectangles, see the exercises. OK, and also in a delta hyperbolic space, let me just write this down, I have nx delta hyperbolic of p is going to be less than or equal to the distance between p. And this notation here means the geodesic between x and y. And this is x, y plus delta. OK, so in a zero delta hyperbolic space, this is what you're starting to get thinking about the graph. OK, so I'll leave these up here. I'm going to use this in a minute. All right. Yeah? Oh, I haven't said I haven't finished. Thank you. So very good. Modulo equivalence, let me write that down. So I'll just write definition 3, and I'll say, sequence is tending to infinity. Now I've said what that means. Now let me say modulo equivalence to infinity. And then I want to say that these sequences are equivalent if the limit as i goes to infinity and j goes to infinity of ai vj equals infinity. OK, so this is an equivalence relation for a hyperbolic space. It's not necessarily equivalence relation if you don't have a hyperbolic space. OK, so it's not completely obvious that this is transitive. So you need delta hyperbolicity. OK, and then this is my set. Thank you. OK, so now what I want to do, so this is just the boundary of a hyperbolic metric space. I want to think about the boundary of a group. So the boundary of a group that acts geometrically on a delta hyperbolic metric space, that's the definition of a hyperbolic group. But maybe it acts on different hyperbolic metric spaces. It'll all be quasi-isometric. I want the boundary to be well-defined. Now do I want just the set to be well-defined? I want the topology. So let me put a topology on it. And then I want to show that this is or give you some idea why this is well-defined for the quasi-isometry type of the space. So now let's put a topology. And as Anna mentioned yesterday, just like in H2, the isometries at the hyperbolic space are going to induce homeomorphisms of the boundary. And so that helps you see the action of the group really helpfully. OK, so what I want to do is I'm going to extend this Gram-Wolff product to the boundary. OK, so I have mn of w. So m and n are points in the boundary. OK, this is going to be equal to the supremum over xi tending to m. OK, so what is xi tending to m? That means the sequence xi, its equivalence class under this equivalence as a point in the boundary, is m. I won't write that. But that's what I mean by xi tending to m. It means the equivalence class of the sequence xi, I'm calling that m, because that's what points are on the boundary. And yi tending to n of the limb, mf, xi, yj, w. OK, so what that's supposed to be indicating is sort of like roughly the distance to this kind of geodesic out here as I go out to it. That's a good way to think of this. OK, why did I put supremum and mf just to be a pain in the but there's actually examples that maybe I should have put in the exercises. But if you want, you can think about this space, which is quasi-isometric to z. So it has two points of the boundary. And you can find sequences going to positive and negative infinity so that this wouldn't be well-defined unless you did this. OK, so that's a fun thing to play with too. OK, so you need that supremum and mf. I've actually seen in the literature mf of mf, but you could have like 0 and 1 going in that example. OK, so now let's put a topology on this space using the boundary, the group union the boundary. Can you see this board under here? OK, which equals x union boundary x. OK, so I want to compactification. So if you think if you're h3 person like me, you can think of x using the boundary as a solid ball. So I've got, if I have x as some point in the big s, I'm going to just look at epsilon ball. So I'm going to look at a ball of radius r around x. So these are metric balls. I'm doing a basis for the topology. OK, and what about x contained in the boundary x? We have a neighborhood of r of x equals y, such that x, y, w is greater than r. OK, so some of these points, y, OK, so what does this mean? Some of these points, y could be in the boundary, right? OK, so I'm going to have, I've got eventually a raw example that's not h2, so I have some point x in the boundary. OK, so some of the points will be y in the boundary, such that this Gromov product from some distance is very far, is very large. OK, and some of these, also, I want to get a little neighborhood here. I want to glue this boundary to the space. OK, so some of these will be points. So if y is contained in the boundary of x, it's a definition above this definition of the Gromov product, OK, above. OK, if y is actually in x, then I use x, y, w. It's just going to equal the supremum as xi goes to x of the limit of xi and y, w. OK, so I just modified this definition. So one of them is in the space, and one of them is at the boundary. I don't need to take both sequences, OK, because y is actually in the space. OK, all right, so this is my topology. And my space is going to be a neighborhood basis. OK, so here's some facts. I'm not going to prove everything. Try to give you some idea why if I have a quasi-isometry of the spaces, we have a homeomorphism. OK, so it turns out that when x is proper, so x is proper, and I'm always assuming hyperbolic metric space, then we're going to have that x hat and boundary x are compact. OK, they're also house dwarf. So dx is also house dwarf. This is not so hard to prove. That is also not so bad. So I'm going to show that the map is continuous, that I have a continuous bijection when I have an isometry. Or show that, oh, now I need to use my hook. So we're going to show that if our x is quasi-isometric to x, there's a continuous map, and it's bijection. OK, so let me just outline that using a gromov product. OK, so let's let, so I'm going to kind of go between definitions so bear with me. So just to define the map, I'm going to use a definition of quasi-G at sx. So we're going to let f take x to x prime, be quasi-isometry, z inverse g. OK, so m, equivalence class in the boundary, is represented by quasi-G at s at gray gamma. OK, so f, because this is a quasi-isometry, f of gamma, is also quasi-G at s at gray. OK, so I'm going to set boundary f of the equivalence class of gamma just equal to the equivalence class of f gamma. OK, so this will be in boundary x. OK, so now if, so I want to show that, so if I have two things that are equivalent, right, so if they're in bounded distance, so if distance, let me just say this real quick, I could just exercise. If the Hausdorff distance between these two is less than k, OK, then the Hausdorff distance between gamma 1 and f of gamma 2 is going to be less than or equal to, say, lambda k plus c. OK, so then this is a well-defined map. OK, so if I look at the n, the inverse is also bounded g composed with f of gamma and gamma, OK, is bounded. OK, so this is going to give me a bi-adjection. So now I get to use my hook. Oh, that's pretty good. That's actually way better than the ones that have buttons. Because buttons, you can get mixed up. Now I'm just going to be messing with these all the time. So you know the definition now. Oh, you can't, which one do you want? All right. Oh, you know what I should do, I should do this. OK, so now let's look at, I want to show that the pre-emptive open set is open. I'm going to go back to my Gromov product definition, where I use this topology, all right. So let's look at some y contained in. I want to show that if I have this, you can sort of think of this as like a delta, but that goes the other way, I want it to be big, is going to imply that, where am I? Boundary F, this is my map of y, is contained in some other little neighborhood of boundary F of x. OK, so I want to be able to get my neighborhood small enough around here, so that I can get inside some open neighborhood in here, just like delta epsilon definition. So let's look at the Gromov product. So I have F of xi, F of yi. So here I'm thinking xi is going to be tending to x. This is equivalent class, and yi is going to be tending to y. It doesn't matter which sequences I choose. OK, so I'm going to see how, what this neighborhood looks like. I'm going to mess around with it to get something, thinking about how close I need to get the sequences x and y to be. This is F of x0 greater than or equal to the distance F of x0. This means the geodesic F of xi, F of yi minus delta. So that comes from this fact that I, this inequality right here. So this means the geodesic between F of xi and F of yi. And this just equals, so there's some point on this geodesic that realizes this. So this is just equal to the distance between F of x0 and t prime minus delta for some t prime on, because it's just realized. OK, so the image of this, of xi, yi, is going to just be a quasi-geodesic. So this is the geodesic, but there's some quasi-geodesic. So it's the image of the geodesic between xi, yi. So this is going to be greater than or equal to, which is within bounded distance of this geodesic. OK, this is going to be the distance between F of x0, F of t. OK, so this is some point on the geodesic, minus k minus delta, where the actual geodesic between F of xi and F of yi is within the bounded distance of the image of the geodesic between xi and yi. Let me write that out for some t, because this maps to a geodesic, this maps to a quasi-geodesic. No, this geodesic is in some, this is some, but this is not any particular k. This is just, it's within some bound. So it's just some k. So it's not the k, if I use that k before, it's just it's in a bounded distance. OK, and this k depends on that, what that quasi-geodesic constant is. I'm just not worrying about it right now. OK, so this is going to be greater than or equal to, now I'm going to use it, this is my quasi-geodesic, distance between x0 and t minus c. So there's a quasi-geodesic, minus k minus delta. OK, so I just carry those down. And this is going to be equal to 1 over lambda. OK, the distance between x0 and xi, yi, because this t was in xi, yi, minus c, minus k, minus delta. OK, OK, so what did we, what did that tell us? So look, that we related the grommet product of the image to this grommet product of the original thing. Maybe I missed a line. Wait, I did miss a line. Let me put it on this thing. OK, so remember this, the distance to the geodesic between xi and yi is bounded by the grommet product. So this is going to be, this is x0. This is again from that inequality over there, minus c plus k plus delta. OK, so this relates to the grommet product of the image to the grommet product of the original thing. So we have if that grommet product is bigger than that number, x0 is going to be greater than lambda times n plus c plus k plus delta. OK, then, OK, so we're going to start off in this small neighborhood. And I claim that we're going to get within a n neighborhood when we map in. So this is like the R from the beginning. OK, then this is going to be 1 over lambda, xi, yi at x0 is going to be greater than n plus c plus k plus delta. OK, so this implies that 1 over lambda, xi, yi at x0 minus c plus k plus delta is greater than n. OK, and the grommet product of the images is bigger than this thing. OK, so if I get in a small enough neighborhood, I can get my other thing in a small enough neighborhood, that's continuous. OK, so now we have an invariant of the group. OK, so now the boundary of g, gx, geometrically on x, proper hyperbolic metric space, we have a well-defined, the boundary of g is just going to be defined to be the boundary of any such x. OK, so now we can go to town. So when this was figured out, I guess by Gromov in his article where he uses grommet product, the people were wondering what kind of stuff can this tell us. It actually can tell you a lot of stuff. OK, this can tell you a lot. So let's look at some examples, some quick examples. OK, so examples. So the boundary of a free group, say f2, is an equal canter set. So you can think about this if you want. I put a couple of little exercises about this. And actually, so you can think about sequences going this way. You can start to have these disjoint neighborhoods and then you can get disjoint and disjoint. It's really fun. Think about this. And for a trivalent tree, it's a little easier if you like the middle thirds canter set, because everybody is prejudiced towards. And then actually, if g is hyperbolic, the boundary of g is equal to a canter set, then g is virtually free. So another nice example is the boundary of some group, closed surface group, like you saw yesterday. So a closed surface group acts geometrically eight tiles h2. OK, so the boundary of a closed surface group is going to equal s1, OK, the boundary of h2. And if boundary of g equals s1 and g is hyperbolic, then g is virtually a surface group. This is much harder. This is go by Tukea, Kass and Jungrys, a lot of work, a lot of non-trivial work. OK, so some open problems that you can immediately start formulating as soon as you know the definition. So you can take your favorite boundary and try to figure out all the things that have that boundary. Problem 3m, which you'll need a piano continuum, have to have a lot of symmetry, OK, and which hyperbolic g have the boundary of g homomorphic to m. OK, so one interesting question that you might want to work on that's actually been resolved. So this proof about quasi-isometry shows that if the groups are quasi-isometric, then they have homeomorphic boundaries. OK, there are examples of non-quasi-isometric groups that have homeomorphic examples, so be a little careful. OK, that will use the quasi-conformal structure at infinity that I'm not going into, but they exist. OK, and then you might even wonder, maybe you take your favorite space and you want to know is, so an easier version of this, maybe this is the empty set, is the boundary of anything of any hyperbolic group, so maybe, or characterize the spaces which occur. OK, there's been some work done on this. OK, those are both kind of hard, but there's things you can start thinking about. All right, so one thing that you can see is you can see the boundaries of subgroups. So the geometry of subgroups is really interesting, and certain subgroups you can see in the boundary very nicely, so let me go over that. The geometry of subgroups can really tell you a lot about the group, and I'm happy to go on about that later. I like thinking about geometry of subgroups on subgroups. So some of these subgroups you can actually see in the boundary. For example, if you have, you can kind of think, I've got a z in here. Let me just do this example. OK, I've got two points in the boundary that are invariant by that g. I can see the boundary of g. What's the boundary of z? Two points, good, and it's right there. OK, maybe it's not distinguished from any other two points, but it is there. It might not be easy to see, but it's there. OK, so this is an example of a quasi-convex subgroup, so let me just say what that is. So a subset A of a geodesic matrix phase is, I'll say, e-quasi-convex, and I won't care about the e. OK, if any geodesic, 1, a, 2, and x, so where a, 1, and a, 2 are in my subset A between points of a, is a bounded distance from a. So an e, so is distance e from a, neighborhood of a. OK, so the picture you might want to have in mind is some kind of blob. OK, it might not be convex, but it's quasi-convex. So, and then I'm going to say a, so all my quasi-convex subgroups are going to be infinite, so a, an infinite group, subgroup, is quasi-convex, if it is quasi-convex in some Cayley graph. So g is going to be finitely generated. If it is quasi-convex in the Cayley graph of g, s, or s is some finite generating set. So I'm being a little careful here. I won't have to be careful in a minute, but I haven't said anything about hyperbolic yet, so there are actually groups that can be quasi-convex in one generating set and not the other, so that's horrible, so let's go to hyperbolic groups. So for hyperbolic groups, we can just say, oh, it is quasi-convex in some group, some space on which the group acts geometrically, which is what we like. So g is hyperbolic, doesn't depend on s, and I can think a quasi-convex, because I'd rather think about just some space, my favorite space on which the group acts geometrically if and only if the orbit of a, a x naught, is quasi-convex in x, where g acts geometrically on x, so x itself will be hyperbolic that acts geometrically. Okay, so this is just, pick some point and then move it around by your subgroup. Okay, so let's talk about how we can get something. So now we have a hyperbolic group, we've got a wonderful boundary, we can think, how is this going to tell us about the boundary? So the limit set, so a contained in g hyperbolic, the limit set, you could talk about the limit set of some subset of x, but let's just think about an orbit here. I'm going to have the orbit of some point is going to be a subset of x. Okay, and then the limit set of the subgroup a is just going to equal all the equivalence classes of sequences that just come from that orbit. Okay, so let me just write that down. The equivalence class of sequences, where xi is somehow in this, say this orbit. Okay, so for example, yesterday you saw a quasi-fuchsin group here, where you had some group, a surface group that actually was acting on H3, but wasn't exactly like acting as PSL2Rs that was acting in PSL2C, and the limit set here was still topologically a circle, because it'll turn out that this, when the group is quasi-convex, this is going to be a quasi-convex subset, I'll just say what I mean by this, but this was the limit set of the orbit of that subgroup of PSL2C. Okay, so example quasi-fuchsin. Okay, and this has been used for cloning groups for a long time to understand a lot about the subgroups. Okay, so the limit set was really used in cloning groups. In fact, in my mind, the whole definition of the boundary is a generalization of the limit set of a cloning group. Okay, so let me just say one thing. So I'm on my own timeframe, but I'll stop in like two minutes, is that okay? Okay, so let's talk about the cloning convex subgroups. Okay, so now I have this X, proper hyperbolic, and I have some closed subset of the boundary. I can look at the convex hole of N, which is going to equal the union of AB. Now what do I mean by these ABs? Okay, so I want to look at lines between points on the boundary in this set. So we're AB, R, and N. Okay, so in a hyperbolic space, so if I have AB contained in the boundary of X, there exists, this is geodesic space, there exists a bi-infinite geodesic. You can get these by approximating the sequences tending to these two points. RI, so R that takes negative infinity to infinity to X, and I want this side of the sequence to converge to A and this side of the sequence to converge to B. So I mean the limit as I goes to infinity of R, negative I equals A. Okay, so this means that the sequence, R to the negative I, its equivalence class is A, and the limit as I goes to infinity of R of I equals B. So I can't stop drawing H2, but it looks like this. Okay, and I have this geodesic. Okay, so to form the convex hull, I'm going to take all of those lines between points of the boundary. Okay, and we're going to call this AB. So this is the convex hull. Okay, so if I have a, so let's give some definitions for quasi-convex. Did I say what a quasi-convex subgroup was? Or the orbit to be quasi-convex. Did I say that? Okay. Yeah? In this AB, do we pick choose a geodesic or do we take all that may exist? Take all of them if you want. So there, it's going to be fine. The set of lines between points of. Okay, so just proposition about what happens with quasi-convex groups. So if I have G acting geometrically on X. So the convex, so if I have quasi-convex subgroups of G, of G are hyperbolic. So I like to think of this. So it turns out that the convex hull of the limit set of a quasi-convex subgroup is going to be itself delta hyperbolic with maybe some other delta. It's quasi-convex. And you can think of the group acting on that or you can think of the group acting on its orbit if you like because that's quasi-convex. And that's going to be a hyperbolic space for this subgroup and it acts geometrically on that. So A contained in G is quasi-convex. Exactly when, so I can think of this convex hull, the convex hull of the limit set of A, remember that's the limit set of things coming from the orbit, all over A has finite diameter. Okay, so this is exactly analogous to what happens in H3 where you have a limit set, you take the convex hull of the limit set, this gives you a space that it acts on, your group acts on it, and downstairs you get a compact manifold. Okay, in general we're not going to get a manifold, but we'll get some object that's compact. So that's why the geometry of it is easier to deal with. And also we have that if A contains in G is quasi-convex, then the boundary of A is going to embed in the boundary of G, which just equals the limit set of A. Okay, so we have these hyperbolic subgroups and we can see their boundaries in the boundary of our whole group. Okay, so, oh yeah, yeah, yeah, sorry. Okay, so let me just introduce, talk one minute on the next topic, or I'll just stop, I can just stop right there.