 Thanks to you all for being here. Okay, so I'm going to be talking about limit sets of groups, discrete groups. But I'm going to start from the bottom. I'm going to, today I'm going to focus on the classical setting, which is groups of isometrists, groups of isometrists of H, hyperbolic N space for N equals 2 and 3. So a lot of what I'm going to say in the first part today has already been said today in the various lectures. So that's very good for me and also for you. So, so let me be in a few words about this guy. So tomorrow, sorry, yesterday we saw several ways of thinking of the real hyperbolic space. Let me, let me redo it and you will see that many of yesterday things will come about. Let me start with H2. Let me consider H2R. And remember, we have the disk model, Poincaré's disk model. We have a two-dimensional disk, D2 in R. We consider, we can think of it as being, there are many ways of thinking of it. Let me think of it as being the unit disk in R2, which we equip with a special metric, the hyperbolic metric. And in order to do that, I'm going to, okay, let me, before giving the metric, I'm going to give you the group of isometries for the metric. So John described it beautifully yesterday using linear algebra in a way which works in all dimensions and for real complex and quaternionic spaces. Let me do it in a geometric way, also using some of the material introduced yesterday in Francois's lecture. Remember, we spoke about the inversions. So what is inversion in R2? Suppose you have, I take, suppose now this is a sphere of some radius R in R2 with some center at some point P. Okay, so yesterday we defined inversion in Francois's lecture as follows. If you want to know who is the inverse of this point, X, what you do is you take this ray that joins X to the center and you map this point to the unique point in this line, this ray, such that this distance times this distance is this radius to the square. There's a unique point like that. Okay, so that is the inversion in the circle in R2 with center of P. And that gives you a map from R2 minus P into R2 minus P and it can be extended to R2 union infinity by mapping P to infinity and infinity to zero. Okay, let me recall that R2 also union infinity is nothing but the two-dimensional sphere. And in general, Rn union infinity is Sn. Okay, this we all know. I guess. And if we don't know it, we can prove it with one picture. Here. Yes, yes, yes, yes, yes, yes. I knew you were the symmetric. Yes. Okay. Okay. Okay, very good. Okay, now that way of doing it and also this way, also something important, also applying all dimensions. See? Yeah, yeah, yeah, yeah, yeah, yeah. Okay, so the same notion applies for whenever you're in Rn and you take the co-dimension of one sphere, you define inversion in exactly the same way. Okay, that's very important. Now, one very important property of inversions that comes out from yesterday's talk, essentially, is the following. Suppose, suppose you have two circles that intersect. Okay, when those, the inversion in one leaves the other one invariant. Okay, that's an easy exercise coming from basic properties of inversions, which is the inversion in one of these leaves the other one invariant if and only if they are orthogonal. Okay? So now, let G be a group generated by inversions, circles, infinity, which is S2. And when I write it like this, I'm including the lines in R2. Okay, those become circles here. Okay, and then the inversion in one line is just a reflection. So let us look at the group generated by all these inversions. Okay, then it's an exercise to show that G transitively, the disc, the unit disc, that means if now this is the unit disc in R2, if we take any point and we take the only, if we take any point, there's one transformation mapping the origin to that point or taking one point to any other point. The action is transitive. And it's also an exercise to show that the isotropic group is O2. Yes, you're right. Yeah, you're right, you're right. Yeah, I'm taking, yes, you're right. But yes, that's very important. Yeah, yeah. Yeah, very, very important. Thank you. This is the eraser. It's horrible. This is wet. Every time you use it, you make a mess. Okay. Okay. Okay. So you are right. Let me be not like G with a hat like that. And then let me be not by G. It's a group of G. Generated by, sorry, I messed it. Generated by inversions, circles and lines, orthogonal, the boundary of the unit disc. Okay. Then, transitively on the disc, isotropy. Okay. And now we can do exactly what Bertrand did, I think, yesterday. We have the disc. We put the usual metric at the tangent space at the origin. And we use the action to spray it around. To define the metric in this tangent space, what do you do? You take a transformation taking this point to this one. Being the few morphisms takes the tangent space into the tangent space. And you use a metric here. Now, if you go there by a different element, if you go there by two ways, then if you take one followed by the inverse of the other, you are in the stabilizer of this point. And the elements preserve the metric. So there's no ambiguity here. It doesn't matter if you go by this transformation or by any other transformation, the induced metric is the same. So you get a metric. And obviously the group acts by isometrics with respect to that metric. Okay. That is a hyperbolic metric. And so this is very nice. Because linear algebra gives us a very tangible way to work with the isometrics of hyperbolic space. This gives us a very natural, very tangible way to see what isometrics are. Every isometry of the hyperbolic disk is generated as a composition of inversions on circles or lines or togono to the boundaries. Beautiful. Okay. And that goes through exactly the same proof to all dimensions. If you want to construct hyperbolic three-space, what do you do? You do the same, but now you consider the spheres of dimension two in R3 union infinity. This includes planes. For each such sphere, you have inversion. You look at the group of inversions. You look at those which are inversions with respect to spheres which are orthogonal to the boundary of the unit sphere. Those generate a group. The group acts transitively on the inside, on the ball. And you give the ball with a metric and you get the hyperbolic three-space and so on. Okay. So those are hyperbolic spaces. Very nice. Now, something which comes out beautifully from this description, I presume is going to be somehow related to something that John is going to talk tomorrow. You will do it in the complex case. This isn't in the real case. You said you were going to speak about this, I think. Let's see. No. No, no, no. Now it's something very nice. No. Look at the following thing. Look at hyperbolic three-space, which is the inside of a unit ball in R3. Okay. Now look at the isometrics, the group of isometrics of this guy. And let me just look at the generators. So generator is one inversion in one sphere orthogonal to the boundary. Okay. So let me take this here. No. So the inversion on this is here takes the inside of the ball, which is outside this one here. And there is little part here. It leaves the ball invariant will take the outside inside and the inside outside. That's the isometry we have of hyperbolic space. But notice that this map naturally extends to a map of the boundary. Okay. When you do that, if you look just at the boundary, the boundary is a two-sphere. What you have on this sphere is intersection with this sphere, which is a circle. And that circle defines an inversion in S2. And you have a map of S in S2. You have an inversion. Okay. And that's a conformal map that he preserves angles. Okay. So every isometry of hyperbolic three-space defines a conformal automorphism of the boundary of the two-sphere, which is beautiful. Vice versa. All conformal isometry, all conformal maps of the two-sphere are inversions on circles. Take one of the generators. Say you have this circle on the sphere and you take the inversion in S2 on that circle. Now, how many spheres you have in three-space, which meet the unit sphere, or totally exactly at the circle? You agree? Okay. Very nice. Thank you. Okay. So if you are given a circle in the sphere, there's a unique two-sphere in the ambient space, which meets the unit sphere or totally at the given circle. Now you make an inversion in that sphere and that gives you an isometry of hyperbolic space, which extends to the interior of the map you have on the boundary. Okay. So we have proved theorem. Isometries of hyperbolic three-space are the same thing as conformal automorphisms. Here conformal means preserving angles. I'm not talking about orientations. Often conformal means preserving the orientations. If you want to preserve orientation, then we put a plus here and we put a plus here. Okay. That's very nice. So we already have this. And if you see, we haven't really used that we are in dimension three. We used it just for making drawings. But exactly same argument tells you that isometries of hyperbolic n plus one space are the conformal automorphisms of the boundary sphere. Okay. That's very, very nice. So we have hyperbolic geometry inside the ball, conformal geometry in the boundary sphere. Now let me focus. Yesterday, John mentioned this low dimensional accidents, which are fantastic. And also we're trying made a dictionary, beautiful dictionary in dimension complex dimension one. So let me now focus on the case n equals two. So isometries of hyperbolic three-space. So in that case, we are looking at conformal automorphisms of the two dimensional sphere. And let me put a plus here. Okay. Then we know it's not trivial, but we, it's elementary from elementary courses and analysis that every conformal automorphism orientation preserving of the two sphere satisfies the Cauchy Riemann equations. And it's holomorphic. And actually it's not only holomorphic, but this group is the same as the group of transformations that we saw yesterday, complex coefficients. And we can normalize them to have a determinant one. The Mobius group or the group of tomographies is exactly the group of orientation preserving isometries of hyperbolic three-space. And that's very nice because, you know, hyperbolic three-space cannot be isometrically embedded in this world. But we can construct the isometries. It's a sum just by looking at things of this form, which is a two dimensional sphere. And this sphere is just the plane plus one point. So we can make drawings in the plane and we are defining isometries of hyperbolic three-space, which is very nice. Okay. And let's see something else. Yesterday we saw that this is the same thing as the group PSO to C. This means two by two complex matrices with a terminal one. And then we identify those with opposite sign. V, C, D is mapped to this. This is easy to see that this is a group homomorphism. And if you go from here to here, it's a group homomorphism with kernel set, this group generated by minus identity. So that's why when you divide by this center, you get an isomorphism. And in fact, in fact, this can be seen also very, very nicely in the following way. We know it was mentioned yesterday and it's very important that Cp1, Cp1 is by definition the space of lines in C2. Every line through the origin in C2 minus the origin is identified to one point. Okay. This by definition. Minus the origin and you identify points in the same complex line. And it's clear that we know that this is same thing as S2. Okay. How you prove that is we will know it, but let's prove it. We are throwing. We are drawing. I'm going to do it because tomorrow I'm going to use it in one dimension more. I'm drawing over the quaternions for the same drawing. So let's see. Put C, you're in C2. A point here means a line here. I want to see that this is the same thing as S2, which is C union one point. Okay. So we have C here. There are many ways to prove this. Let's do the following. Put here a copy of the complex numbers parallel to this axis. Now take one line through the origin, which is not this axis, any line. That line meets this online complex line in exactly one point. So you have a map from CP1 minus one point corresponding to this line to C vice versa. If you're given any point in this line, there's a unique complex line joining this point to this one. So you are giving an inverse map. So there's a bijection between lines here minus one with the points here minus one. Now, when you go to infinity, you have one point and you're here. Okay. So that gives you the field morphism. Actually, I actually have a whole low morphism between CP1 and the Riemann sphere. And now, if you have a two by two complex matrix, that gives you a linear automorphism of C2. Okay. There is way. If you have ACV and you have a complex number, C1, C2, that goes to AC1 plus BC2 comma CC1 plus DC2. You have a linear transformation of C2. Now, a linear transformation takes lines into lines. Okay. So it induces a map from CP1, the space of lines. And which map is that? That is exactly this one, if you think on S2 has been C union infinity. So everything fits together in a very, very nice way. Okay. So, for example, an exercise, beautiful exercise. This is telling us that this group of matrices is exactly the conformal group of S2. In particular, rotations of the two sphere are conformal maps. Okay. Who is this subgroup here that corresponds to the rotations? Okay. Exercise. It's a beautiful thing. So, well, then we have this beautiful picture. And we can also go one dimension less. We can look at the isometries of a real hyperbolic two-space. And this is going to be the group generated by inversions S1. Okay. In other words, if you have the boundary of hyperbolic, you have D2. Now you have the boundary, which is S1. You take circles, which are orthogonal to the boundary. Okay. Each such circle gives you an isometry of hyperbolic two-space. And if you restrict that to the boundary, what you have is on the boundary, you have this interval. And you can define the inversion with respect to this interval that takes this point, the center to the antipodal point, then this segment to all this and so on. Okay. So, exactly the same picture. So, we have this, I mean, all these fantastic accidents in low dimension. Now, I next want to look at discrete isometries of hyperbolic space for n equals 2, 3, particularly. Just a comment is, well, let's see something. Suppose you have an isometry of hyperbolic two-space. Okay. So, this is hyperbolic three-space. And here you have hyperbolic two-space. Okay. It's a, there's a natural way to extend every isometry in dimension two to an isometry in dimension three, which is this famous Poincare extension. How do you do it? You have a map from, think of H2 as being this equator plane. Okay. So, you have a map that, an isometry of this plane. You want to extend it to the ball. How do you do it? Well, suppose this point goes to this point and so on. Now, you have a point here and you want to say, who is the image of that point? How do you do it? In the hyperbolic three-space, there's a unique geodesic that passes through this point and is orthogonal to this plane, to the hyperbolic two-space. Okay. Now, you look at this point. Now, you know that this point goes, say, to this one. Through this point, there's a unique geodesic which is orthogonal to the plane. So, you go the geodesic. And then you know that this geodesic is going to go to this geodesic. Now, who is the image of this point? Not the geodesic, but this point. Well, you parameterize this by arc length and there's a unique point here whose distance to here is exactly the same one you have. Okay. Then you have an extension of the isometry in two dimensions to the isometry in three dimensions and you can keep going. Okay. Now, I'm telling you that the isometries of two-space H2 are invading those of H3, H4, blah, blah, blah. You know, in a very natural way. But there's another way to see this. Yesterday we spoke, we saw essentially the same arguments tell you that isometries, this guy is just exactly the same as the group PSL2R. Okay. These are two-by-two real matrices. Okay. You have to think it's a group of index two of an even number of invasions. Yeah. Okay. Now, okay. Okay. But now the beautiful thing is that it's important. Thank you. Yeah. Every such matrix without the P is also a complex matrix. So there's a natural embedding of the group into PSL2C. Okay. That gives you a way of embedding. And we also saw in John's lecture that this is the same as PU11. So this corresponds to different ways of thinking of hyperbolic two-space. This is for one model, the upper half space model, for the disk model, and so on. Now I want to construct subgroups. Now I want to construct subgroups. First, in H2R, an example is here. That means the groups of PSL2R. How do we do it? Examples. We saw in the trans lectures yesterday. One example is this Fuxian representations of the fundamental group of a S compact Riemann surface of genus more than one. Okay. So we have a Riemann surface. You can see that as a quotient of the upper half plane by a script group of isometrists. And that is giving you automatically a representation of the group inside here. Okay. And then you have a naturally a disk itself group here. And if you want to go to PSL2C, now you just embed this one in PSL2C, and you have natural groups of isometrists, discrete groups of isometrists in PSL2C. Okay. Other examples. Let me give one beautiful example, which was also somehow just mentioned by John yesterday. Triangle groups take the disk model and now take a triangle inside bounded by geodesics. Geodesics are segments of circles which are orthogonal to a boundary. Okay. So suppose you have one, another, and another. It's not very good. No, it's horrible. You make it. Suppose you do like this. Okay. This is a triangle with one vertex at infinity. If you want to make it a little bigger, now we have an actual triangle. Okay. This is supposed to be segments of circles, which are orthogonal to the boundary. Okay. Now, each one gives you an inversion. Each one inversion on each side of triangle gives you, gives you an isometry. Now, look, if you make the reflection of this one, the triangle becomes something like this. If you make an inversion in this one becomes something like this. And then if you keep reflecting the images of these triangles, fill out the space and if the angles are good, what you get is a discrete group. What means that the angles be good? Well, if you see, when you make the reflection on this geodesic, all this geodesic is invariant. When you make the reflection on this one, all this geodesic is invariant. So if you make the geodesic reflection in one and then in the other, the intersection point is a fixed point. And if you look carefully, similar exercise, if you do that, what you end up is by having infinitesimally a rotation around this point by an angle twice this angle. So if this angle is irrational after you trade, you come back with a new triangle, you overlap with the other one, and you will keep going there forever, and you are not going to have a discrete group. But if the angle is of the form pi over r, for example, after you iterate, inversion in one and then the other, r times, you are going to come exactly in times that are going to fit nicely. And if all angles are like that, you get a tessellation of this and you get a discrete group. These are, if the angles are, if you choose the triangle to be of the form pi over r, pi over p, pi over p, pi over q, these are the famous triangle groups, p, pi, q. OK? So you have this for every angle such that p, q, r are greater or equal to 2. And 1 over p plus 1 over q plus 1 over r is more than 1. Whenever you have triples like this, you have a triangle group. You can do the same for quadrangles with some kind of conditions and so on. So these are, you have many more fuxian groups. Fuxian groups by discrete definition are discrete subgroups of PSL2r. OK? Let's go one dimension higher. We could keep here for much longer time. But let's have some fun in one more dimension. Now I want isometry of hyperbolic tree space. But since we cannot draw hyperbolic tree space here, let's draw the boundary and let's work on the boundary. OK? So suppose you have one circle, inversion here. We know this is a conformal automorphism of the two sphere and that gives you an isometry of hyperbolic tree space. So look, for example, at the group generated by three elements or by four if you want. Or by r circles, which are all of them pairwise disjoint. What you get is examples of what are called Schottky groups. OK? Now suppose, for instance, you choose your circles in a very appropriate way. For example, perhaps you want to choose it in this way. Start with a unit circle, if you want. And now choose a family of circles which are orthogonal to this, which emit each other tangentially like this, forming a color in such a way that each of these yellow circles is orthogonal to the unit circle. When you make inversion in each of those yellow circles, that carries the pink circle into itself because they are orthogonal. OK? So this group, the group generated by inversions in these circles, they all leave this one invariant. OK? That's a very nice color, beautiful. But now you put it in your bag, you take it for a trip, and when you arrive, it has moved. OK? It has the form. So now, yeah. So maybe one of these circles is like this. So now there is not a common circle which is invariant to all of them. Now you make your construction again and you again have a group of isometrists. But something has changed. What has changed? Well, that there is not an invariant circle. OK? And can we say that in some intrinsic way? OK? This brings us to the concept I want to speak about, which is the limit set. Let G in isometrists of hyperbolic N space be a discrete subgroup. By the way, let me just say that discrete subgroup means that it is a subgroup, and that as a topological space, it is discrete. OK? Because it's of isolated points. Suppose you have a discrete subgroup of isometrists of hyperbolic N space. Then the limit set is the subset lambda in bar, which means H and R union, the sphere at infinity, the boundary sphere of points, relation points of orbits. So you have the ball. If you look at one point and for each element here you have a transformation of the hyperbolic space union debondering to itself. So you apply the element and this point goes to some other point. Then to some other point. Then to some other point. Let me assume this has infinite cardinality. If it is a final group, then the orbits have far far far far many points, and they don't accumulate anywhere. Suppose the group is infinite. OK? Then for each point you have the orbit. Now, when you look at this, this is a compact space. Every sequence in a compact space has convergence of sequences. So you have accumulation points of the orbits. You look at those for all orbits and what you get is, by definition, the limit set. So let's go back to see what happens here. When we had our original color and we made the inversions, the pink-yellow is invariant. The pink circle is invariant. OK? So if we have one point here and we make the inversion, we will come to another point here. So orbits in the pink circle will always remain in the pink circle. It's an invariant set. OK? Now it's not trivial, but I think we will be convinced easily that if you pick your preferred point in the circle, any one you want, this one, now take any point outside this one. Then we can find a sequence of inversions in these circles that takes this one as close as we want to this one. We first make an inversion in this one, in the circle. So the outside comes inside. This will come to some point here. And then you keep iterating and you will get closer and closer and closer to every point. So what I'm saying is that the pink circle is the set of accumulation points of every orbit in this sphere. OK? Now what happens if you perturb this slightly? So if you have the notes, there's an example in page, let me see. Well, in page 15, you will see something like this with the circles in the right place, orthogonal to the boundary. And then in page 16, you will see examples of what is happening. So in those examples, if you have something like this, the circles are touching each other tangentially, but they have no longer a common orthogonal circle. What is happening is that the limit set is going to be something more perfect to S1, but with infinite length. What is called a quasi circle, which is quasi homomorphic to a circle. But now it's no longer a round circle. OK? So these are beautiful examples. We could, for example, go the other way around and spread the circles, separating them slightly. So they don't touch each other. Then the limit set becomes a counter set. And then, well, you have a fascinating geometry and dynamics involved here. And in this case, no. If you separate the circles slightly so that they don't touch each other, then yes. Yeah. You want me to make one now? No? In the whole circle? Yeah. Yeah. OK. OK. So let me give a couple of definitions which are useful. Definition, a discrete subgroup isometric fuchsian. It's limit set contained in a circle, in a geometric circle, in a round circle of maximum length in the two sphere. OK? Let me say G. G is quasi fuchsian if its limit set is contained quasi. That means a closed curve which is quasi homomorphic to a circle. If you don't know what this means, just keep it in memory and you will know. I will not enter into that because I'm not going to use it. OK? And somehow bounded distortion from being a real circle. So these are two interesting concepts. And then nowadays, a discrete subgroup isometrist of age 3R is called a clinium. So clinium groups are by definition discrete subgroups of isometrist of hyperbolic space. But formerly, the term clinium was reserved to a special type, very general about the special type of clinium groups, of discrete groups. Those whose limit set was not the whole two sphere. OK. Let me say some other examples. Consider G and H. So we know that we can represent by a mobius. Yes, I saw, yeah, I saw me pre-preserving orientation. Yeah. Sorry, thank you. OK. We know we can represent this by a mobius transformation. A C plus D over C C plus D. OK. With complex numbers and determinant and one. And, you know, we know that every such map is conjugate to one of the four set goes to either set plus one or to lambda set with lambda and on zero complex number. It's an elementary exercise. Every mobius transformation is conjugate to one of this type. OK. In the first case, a translation and G is parabolic. Otherwise, G is elliptic. Lambda has norm one or loxodromic if lambda has norm different to one. This classification was mentioned yesterday. We were just looking at the cases where the coefficients were real numbers. So the term loxodromic was not mentioned because they were called hyperbolic. So hyperbolic is a special case of loxodromic. So what is the geometry of those maps? Did we look at this here? Look, this is just a translation. OK. So if we look at this here, we have one fixed point, which is infinite. And then all orbits move things like this. OK. In other words, we have the plane and we are translating in that direction. All these lines are invariant. And when we compactify the plane to be this here, these lines become circles like this. OK. So that is the first case. The other two cases are, let's see. In the other two cases, where we have this, we have two fixed points, infinite and here, the origin. Those are two fixed points. OK. Now, if the norm is one, what we are having is a rotation. If the rotation is by a rational angle, after finitely many times, we come back to the identity. So the orbits are finite and there's no dynamics, there's no limits set. In the other case, each orbit is infinite, but the whole sphere is a set of accumulation points. And actually, when we look inside, I mean, in the other case, we don't have a discrete group. OK. So if we have a rational rotation, the corresponding group obtained by all iterations is not a discrete group. It's not a discrete representation of the integers, so to say. OK. What about limit sets? If we look at what is happening in hyperbolic space, so this is the boundary. We have S2 and we have hyperbolic three-space. So in this case, in the parabolic case, we have one fixed point on the boundary. And then we have all the horoscales and the points move around this horoscales. OK. In the other case, it's very interesting. We have two fixed points. We have one geodesic, which is one in this case. And then the orbits, I mean, along this geodesic, we have something like a translation. And then all the other orbits move from one to the other, spiraling away. When we come to the boundary, what you have is, depending if the norm is greater than one or smaller than one, then the origin is repellent and if it's attractive or vice versa. OK. And then in that case, the limit set consists of just these two points. Orbits accumulate in the future to one and in the past to the other. Sorry. I have no idea. The etymology. Some basic properties. So for these examples I gave now of cyclic groups, the limit set for parabolic elements, the limit set was consistent of one point. For loxodronic elements, it consisted of two points. OK. So basic properties. One, if lambda, the limit set, has finite cardinality, it consists one or two. If it has more than two points, it has infinitely many points. I'm assuming the group is infinite. Well, OK. Or the group is finite. The definition, the group is elementary, lambda has finite. If it is not elementary, it's called non-elementary. OK. Non-elementary is the set accumulation points every orbit. So we don't need to look at all orbits. It's just any orbit accumulates at the same set. And the action, lambda. So lambda is an invariant set. The action is minimal. That means there is not closed invariant subset contained inside it, which is a proper subset. Or equivalently, i.e. every orbit in lambda is dense. Let me give you normal properties, but let me give you a quasi-proof of this. OK. Let's see. Suppose it's the same dimension to make it moving. Suppose this point x is in the limit set. OK. That means i.e. there exists some y in the disk. Let me say here. And a sequence gn of elements in the group such that gn of x, sorry, gn of y, the sequence of points accumulates to x. OK. By definition. If this point is in the limit set, it means that there is some point here and a sequence of points in the elements in the group that make this one converge here. OK. So my claim is that if you take some other point, any other point, there is a sequence taking this one, making this one accumulate here. OK. That is the claim. Let's see. Let me call this one y. Sorry. Yes. Yes. Yes. Exactly. No. Look at the distance between these two points. Now look at the first iterate. You're moving by an isometry. So the distance is the same. Next iterate. The distance is the same. Next iterate. The distance is the same. You're always moving, keeping the hyperbolic distance fixed. OK. But now this point, something I didn't mention and one more property. So property zero. The limit set is necessarily contained in the boundary. OK. The idea is that this happens because inside you're moving by isometrics. So the only way if the group is discreet, the only way how orbits can accumulate is when they go to infinity. OK. You have to do a little more. That's basically the idea. So the limit set is contained in the boundary. So this point is in the boundary. And then these two are moving by a sequence of points and the hyperbolic distance is fixed as you're probably in the boundary. That can only happen if the Euclidean distance tends to zero. OK. So these two sequences converge to the same point. That's almost a proof because, for example, what if your group is generated by a loxatronic element and then you have this is one of the limit points. And you have another one here. And this one never moves. So the argument fails. You agree? So you have to prove that if the group is non-elementary, there are elements in the group that move all points in the boundary. Every point in the boundary gets moved. OK. Well, then that's essentially the idea for proving this. Three. Look at the complement. Set omega by definition equals the complement in the boundary sphere. S2 minus the limit set. OK. So inside the ball, we are acting by asymmetries and the group acts properly discontinuously. We heard the definition yesterday. The usual, I mean, the definition. Todd gave us yesterday a characterization. The group acts properly discontinuously. Suppose the action is free. The action is properly discontinuous if the quotient is a manifold. The definition in general is that the action is properly discontinuously. If whenever you have a compact set and you look at the orbit of that compact set, that intersects itself at most finitely many times. OK. It's equivalent. Now, inside the ball, the action is properly discontinuous. But when you go to the boundary, you don't know. For example, going one dimension less. No, but yes. But if the action is free, then it's a manifold. Yeah. In general, it's a manifold. Yeah. No. Take a circle and take my goodness. OK. Take one of these examples I gave before the triangle group. Then when you look at the iteration of this group, the limit set is the whole circle. So the action. So the limit set is the whole circle. However, if you take, well, if you take a virtual triangle, like the case of the mobile group is not like this. Sorry. If you take a virtual circle again, the limit set is the whole as one. But what did you remember? This corresponds to circles in the space, which meet the boundary orthogonally. But what if you split your circles a little more? So you have a unit circle and now you're trying to become something like this. Then when you look at the action in the boundary, all of these will be outside the limit set. OK. And all the orbits of these intervals will be in the limit set within the complement of the limit set. In that case, the limit set will be a counter set contained in the circle. OK. So in the in that set, so this is the set I'm looking at the complement of the limit set in the in the boundary. Here's in one dimension. Same picture. So this is what is called is by definition. Well, this is the region of discontinuity. And it has several remarkable properties. This is largest set properly discontinuous. OK. So if you have any set in the sphere where the action is properly discontinuous, that is contained inside here. And this is also the region of continuity of the group. What does that mean? It's a concept which is very useful. It's very important in analysis. It's very, very useful. Basically, it means that, I mean, if you have one map, continuous means that it continues at one point, means that whenever you have epsilon greater than zero, there is delta greater than zero such that blah, blah, blah. OK. When you have a family of maps, continuity means that whenever you have epsilon greater than zero, you have delta greater than zero that works for all elements in the family. OK. And this is an important concept. OK. And these are some, there are many, many, many, many more very important properties. These are some properties I want to mention. I will stop here. Yes. It can be empty. The limit set, if the group is not finite, then the limit set cannot be empty. But this one can be empty. OK. Yeah. Probably. Omega can be empty. So for example, if you take any compact hyperbolic three-manifolder and you take the fundamental group, then the limit set, this region is empty. And there are many more examples. Or you can have one connected component or else it has a field remaining. So if this region of this continuity has more than two components, it has infinitely many components. Sorry? I don't know. Contably or in contrary, I better don't say because I will be inventing. The limit set is a perfect set. The limit set is a perfect set. I would believe it is so, but I would believe. I should think so. The components of lambda. The components of the limit set. Well, if it's a counter set, yeah. No, no, no, still there's a beautiful, very, very, very, very deep theorem. Suppose that the group is finitely presented. OK. Look at the region of this continuity and look at the quotient. So you identify points in the same orbit. Then you get the Riemann surface of finite type. That means finitely many components and each component either is compact or compact minus finitely many points. OK. That's a deep theorem of Denis Sullivan. Sullivan's finalist theorem. Sorry. Sorry. Sorry. Sorry. Sorry. Sorry. Sorry. Sorry. He even made the interpretation for rational maps and that's his final, his non-guandering domain theorem. Sorry. OK. Well, I will stop here today and then tomorrow. Tomorrow I will start speaking about higher dimensions.