 And there's one more property which is very important that I want to point out, which is we see that we know that every such transformation and such transformations are classified into three types, elliptic, parabolic and loxodromic. Particular loxodromic elements have each two fixed points. One is attractive. One is repelling. Okay. And one very important property of this limit set is that it is the, when the group is non-elementary, you have infinitely many loxodromic elements. And in that case, each has two fixed points, each loxodromic element has two fixed points. They are part of the limit set. And you have that lambda is the closure of the set of fixed points of loxodromic elements. Yeah. Yeah. I don't know. Okay. Okay. So this is very, very beautiful. This means that whenever we have a discrete group of isometrists of hyperbolic tree space, we have the solid ball, three-dimensional ball, disc of dimension three, which the inside is the hyperbolic space when we have the hyperbolic metric, then when we have a discrete group, we have this group of transformations acting on the compact ball. Inside you have an action by isometrists. Outside on the boundary, you have the limit set. So you have this limit set, which as we saw, can be quite wild. I mean, it's a closed invariant set with a very, very rich geometry, can be a fractal set, can be very complicated, can have infinitely many components, and so on. But even so, you have this family of transformations on the two sphere, on the boundary sphere, for which this closed set is an invariant set. The complement is also an invariant set. Here is where the dynamics concentrate, the closed set, every orbit is dense, and the complement, the action is mild, it's properly discontinuous, you have a continuity. So if you look at the complement, that is the paradigm for complex geometry. When you look at the complement and you take the quotient by the action, you have a Riemann surface, very nice geometry with a projective structure, very nice geometry. And then when you look at the limit set, you have very, very, very rich dynamics, okay? So this has been studied for more than a century, many fantastic theorems, and still many things are being done. We've heard very, very interesting talks now in this by Bertrand, by Todd, by many people about still this very rich subject, which is studying geometry and dynamics associated to discrete groups of this space. It's a fantastic subject. There's also something which is very, very relevant, which is this addiction, Solivans, Solivans and Magnolins dictionary, which you have. So a client group is at the end, you have a collection of Russian, of Mobius transformations, and they generate a group. Each one has a very simple dynamics, but when you have several of them and you combine them, you have very rich dynamics, very rich geometry. So you have client groups in one side, and you can also consider, instead of this, you can consider, as in Roman's lecture, one rational map of degree, at least two, and all its iterates. That forms a semi-group, and you have a rational map, and so here you have the limit set, and the complement, which is the region of a continuity. Here you have the Julia set, and the complement, which is the Fatou set. And this dictionary tells you that, roughly speaking, ideas and results in one side correspond to ideas and results in the other side. It's not a perfect dictionary, not every sentence has a translation, but in general you do have a translation, and many, many important results have come like that. So for example, yesterday we heard about this non-wanderings domain theorem by Denis Sullivan. That is the translation, so to say, to this side of the dictionary of the Alpha sign of this theorem here. And so many things come like this. And so we have all this rich, fascinating theory. And in the, well, in the 1996, Alberto Berkhoffsky was a professor in Lee, France, and we brought him for one year to Mexico to start a new center by that time, and we started with the Geo-Dynamical Systems, and he was lecturing about iteration theory in several complex variables. Okay? So this is in one complex variable, and then he started lecturing about the work for Knys and Siboni in several complex variables. And then the natural question that came to us was, well, and what should there be in this side of the dictionary in several variables? Is there something like Kleinian groups in several complex variables? Okay? We didn't know much of the time. Then for Knys came to Mexico, Alberto Amon, anybody time. We talked to him, and he said he didn't know much of the time, but it should be an interesting topic. So we started working precisely on something like Kleinian groups in several complex variables that we call complex Kleinian. So by this we mean a discrete subgroup ESL M plus 1 C. This is the group of automorphisms of complex projective space of dimension N, which acts CPN with non-empty discontinuity. So when you have such an object, somehow you have a similar splitting to what I said. You have complex projective space, and if you have something like a limit set where the action concentrates, you have the complement where you can do geometry. So we started working on this object, and many questions, natural questions arose. Is or should be a notion of limit set, interesting families, etc. I can say that now we have several answers. Tomorrow I will focus on complex dimension 2. In complex dimension 2 now we are starting to have a good basic setup, but so far I can say that almost any question you make is an open question. Of course, this group is an ESL M plus 1 C. It's a semi-simply group, and we are talking about basically lattices or discrete subgroups of semi-simply groups. That's a classical theory. There's a lot known in that setting. So it's a very deep theorem. So you have all that general machinery. But then when you come down to studying something like limit sets where the action is properly discontinuous, portions of the region of discontinuity, properties of the limit sets, it's a subject which is very much in its childhood, and that's what I'll try to focus for today and tomorrow in studying that. So first about this thing of the notion of the limit set. Well, as Alberto explained to me 20 years ago or something like almost, there's really not in general one single notion that works in all contexts of limit set. At the end, definition is just a definition. You can define what you want provided you can prove something interesting with it. So this same question of what should be the limit set of a discrete group action in some space is a question that Ravi Kulkarni addressed in the 1973 thing in a paper in Invenzione. And he defined a concept which is pretty good for us in this setting, which we now call the Kulkarni limit set. I will define it in a moment. But before let me give a brief discussion of why should we bother? Why don't we take just the same definition? I mean in the classical setting it's just the set of accumulation points of the orbits. Why don't we do that? Of course we can do it. Let me give you an example. I'll give first an example that Ravi gives in his paper in R2 and then I'll give an equivalent example in the projective space. In R2, just take the map from R2 into itself, which carries x, y into 2x, one-half of y. Very simple map on automorphism. You take all the iterates and the inverse and the iterates of the inverse. You have a cycle group, a representation of the cycle group as a group of transformations on the plane. Now what are the orbits? Obviously the axes are invariant sets. And all orbits in future time they move like this. So you are expanding in this direction, contracting in this direction. So all orbits you have like this. Orbits are on the y-axis converge to the origin. In this axis they converge to infinity in the past, but they converge to the origin going backwards. So which is the set of accumulation points of the orbits? Just the origin. Or infinity if we really think of it in S2. So for the classical definition this would be our limit set. Why not? Well, there's something which is not so nice of this definition. Take for example one circle, any circle around the origin. And look at the orbit of that circle and all the backwards here. So the orbit of this circle under this group action will accumulate at both axes. The action is not properly discontinuous on the complement of the origin. So of course you can call this the limit set if you want. But you will not have the nice property that the action outside the limit set is mild in some sense. It's not such a... So it's not a really good notion. If we want to do it in our setting, the projective setting, you can do it very nicely. For example in PSL3C, consider the element represented by a matrix of the form lambda1, lambda2, lambda3, 0, 0. Where the norm of lambda1 is more than the norm of lambda2. It's greater than the norm of lambda3. It's greater than 0. Each of these gives you an eigen space in R3, in C3. When you come down to CP2, each of these gives you a fixed point in CP2. So you have three fixed points. One is attractive. Another is repelling. One is a saddle. Just because of the norm of eigenvalues. Now each two points, which are in projective space, give you a line. So you have three invariant lines. So this one is repelling. This one is a saddle. This one is attractive. And then it's an exercise to show that, well, these are the fixed points. Now these two lines, one is attracting, the other is repelling. And if you take the inverse matrix, you replace one is repelling, the other one is attractive. This one doesn't play such an important role. It's not obvious. And you can see several things. One, the action, if you just look at CP2 minus the three points, the action is not properly discontinuous here. Neither it is a key continuous. Okay. Now if you remove the two lines, then the action is properly continuous and a key continuous and the complement. Okay. But if you remove one line and the opposite points, what to say, you also have an action, the action is also properly discontinuous on the complement. Or if you remove this line and the opposite point, the action is properly discontinuous. So in these cases, we see that there is not one largest region where the action is properly discontinuous. There are two possible and none contains the other. None of them coincides with the region of the key continuity. None of them is a set of accumulation points of the orbits. So all the properties I listed of the limit set somehow break down here. And this is a very simple example. Okay. So what is the limit set? Then rather you have one definition which has several interesting properties. The definition which is canonically defined always. Two, you always have that the action on the complement is properly discontinuous. It's not always the largest region where it is properly discontinuous as in this example, but the action in the complement is properly discontinuous. You have always fundamental domains and you have a nice candidate up there. Let me give you the definition. So you have a group. Let me restrict the setting of groups of complex automorphisms, but this is a much more general definition. So you have a limit set that within all the lambda pool is lambda not union L1, union L2, union of three sets where L0 is the closure of set points with infinite isotropy. So for example, in this case, this is the origin. In this example, L0 is the origin. So in the example, this is just the origin, which is a fixed point for the whole group because it's infinite isotropy. Then L1 is the closure of set of accumulation points, of orbits of points in Cpn minus L0. So for example, in this case, if you remove the origin and you take, for example, orbits here, they will accumulate the origin. And this is the only one. So in this example, this is again just the origin. But this is not always like this. There are examples in which this one is contained in here, examples in which this one is contained in here, examples where they are different, all kind of things can happen. And then finally, L2 is the closure, accumulation points, orbits of compact sets in Cpn minus L0 union L1. So when you take this, you are forcing the action on the complement to be properly discontinuous. You are removing the set where you can have problems. So if you come to this set, then lambda 2, sorry, in this example, is the union of the axis. So in this example, the Kulkarni limit set is just the union of the two axes. In the example I gave before, in this projective example, the Kulkarni limit set is the union of these two lines. So now here's a definition. So what? So today I'll try to convince you that one, this subject is, I think, interesting. This is an interesting concept. But at the same time, there are many complications and things which are strange and which we still don't understand, especially in higher dimensions. And then tomorrow I will focus on dimension two in which we will see that this is the good notion in complex dimension two. So for the rest of today's talk, I will consider various families of groups in PSL and NC and see what happens. So the first case is cyclic groups. You have one transformation and you iterate it. Backwards and frontwards. Okay, and see what happens. So just a reminder, let us remind that in PSL to C, the elements are classified elliptic, parabolic, loxodromic. Many ways to do this classification. I did, I said it before using model transformations and I said that they were, every model transformation was contributed to one of the forms. Either set goes to set plus one or set goes to lambda times set. And then, yes, yes, yes. Well, stopping in L2, you already can grant that the action on the complement is properly discontinuous. Okay? Now, if you put more conditions, you get different sets. No. Your question is very, very, very appropriate because we will see by the end of today's lecture that this concept of limit set gives you a very, very interesting limit set, which is very good in many cases. But there are some cases in which it is clear that we need something more refined. Okay? This is still a concept that applies always. So if it applies always, you are likely to be missing something. Okay? So we will see that in certain cases you need something more refined. I'll come to that later. Okay? Yeah. There is not such an extension, but there are natural ways of embedding PSL and C in PSL, N plus 1C, and so on, which is somehow the equivalent of the extension. One natural way is what we call a construction that we call suspension. Yeah. Yeah. Yeah. Yeah. I mean, that's a very important point. Many of the properties we have in real hyperbolic geometry and so on are proved by using that the boundary sphere is the boundary of the hyperbolic space. And then you use the inside to prove things about the outside. Here you cannot use that. And for an auth, it is a boundary, but there is not something like the sphere bound in the hyperbolic space. Okay? So in this case, you can go from automorphism of CPM to automorphism of CPM plus 1. But there is not really one something canonical as Poincare's extension. Example, sorry. In that example, L1 is... So somehow, L0 union L1 is the usual limited, the set of accumulation points of orbits. L0 is the closure of the points with infinite isotropy. Then you remove that. You look at the complement and you look at the orbits of points in the complement and you see where they accumulate. Okay? So those two together would be the usual set of accumulation points of the orbits. But then you have to add L2 in this definition. Okay? Yeah. Okay? Okay, so now let's see this example. So in PSL2 C, we have this classification. There are many ways. So we know that elliptic elements have two fixed points and none of them, them is neither attractive nor repelling and one is conjugate to the other. These ones have only one fixed point. These ones have two fixed points, one repelling, one attractive. The limit set. Right. Okay? And we also know that in this case, every matrix A, B, every element here has a lifting to SL2C and every such matrix can be taken to a Jordan form of type 001 or lambda, lambda. Sorry? Yes. Thank you. Yeah. Okay? In this case, you have parabolic. In this case, you have either elliptic or loxodromic depending if the norm of lambda is one, you have elliptic, otherwise, you have loxodromic. Okay? Well, then you can go in general. Cyclic groups in P, SL, and C. You can define elliptic if it g, element g is elliptic if it has a lifting to SL and C, such that which is diagonalizable and all eigenvalues norm. Then you can say it's parabolic if it has a lifting which is non-diagonalizable and eigenvalues have norm one. And loxodromic otherwise. That means it has a lifting which is diagonalizable or not. But at least one eigenvalue has norm different to one. Okay? And then this is a definition which extends the classical definition for elements in P, U, N, 1. And this was given for a Mexican colleague Navarrete for N equals three and by uncle Cano and Loessa for N bigger than three. And then you can prove theorem is elliptic if and only if the concarni limit set is either empty or the group is non-discreet. Yes, yes, exactly, exactly. Then parabolic if and only if the concarni limit set of the group generated by this element consists one single projective sub-space CPM. Loxodromic if and only if the limit set consists projective sub-spaces of CPM which can be of different dimensions. CPMs, Cpk is smaller. Yeah, sub-space. So, for example, in the case of PSL3C this guy can consist of either two lines or one line and one point. In principle, it could be two points but that doesn't happen. Okay? So, we have this. Let me give another family of very interesting examples that will bring us closer to the next lecture. John. So, another family is complex hyperbolic. So, we are looking at subgroups G containing P, U and N1. So, this also brings us closer to those lectures. He was considering the real case. That's very quick. Which is naturally as a group of PSL N plus 1C. So, these are without the P. These are the automorphisms of projective space that preserve a quadratic form of signature N comma 1. Okay? So, what you have in that setting is you have projective space, CpN. And then, if you look at the null vectors in Cm plus 1. So, if you look at... So, here you are preserving the quadratic form Z naught square plus Z1 plus ZN square minus ZN plus 1 square. So, you are preserving... You are looking at linear maps which preserve this quadratic form. Or an equivalent one. Let me take this one, this model. So, if you look at the points where this is equal to zero in null vectors, and you projectivize what you get is a sphere. So, you have a sphere of dimension 2N minus 1. Now, if you look at the negative vectors where this is less than this, you get the inside of the ball. And that is the model for complex hyperbolic geometry. And then the transformations, these elements are the holomorphic isometries of this ball with the bergman metric, the complex hyperbolic metric. Okay, but they are... So, you are acting on the whole complex space preserving this ball. But here is my isometries, outside is not my isometries. In predictive space, you have a canonical metric which is the Fubini study metric. But these maps do not preserve that metric. Okay? What about the limit set? We can forget about the rest. We can just look at complex hyperbolic space. We have a group of transformations there. And you can just focus your eyes on complex hyperbolic space. That's very, very interesting, very rich. And you can do, as in the classical case, you look at points inside. You look at orbits of points. You are acting by isometries in the inside. So, the orbits will accumulate in the boundary. And you will have a limit set on the boundary defined in the usual way, at the set of accumulation points of the orbits. Okay? That is somehow the usual limit set. We call it the chain limber limit set of the group. Because they start this first in this complex case. Okay? And that limit set has all the properties of the limit set in complex hyperbolic geometry. It either consists of one or two points, or infinitely many points. It's a minimal set. It's closed invariant. The complement is the largest open set where the action is properly continuous. It has all the nice properties if you focus your eyes on the complex hyperbolic space and its boundary. But now you remember that you are actually acting in a complex hyperbolic space, complex projective space, and you want to see the action on the whole space, then perhaps you are missing something. Okay? And in fact, for example, when n equals 2, and when n equals 3, n equals 2. So, you are acting on CP2. And you consider, for example, a group here which is a lattice. And so this limit set is the whole three sphere. Then the action in the whole complement is minimal. Okay? So the action is very far from being properly discontinuous on the complement. As far as saying that every orbit is dense in the complement. Okay? So this notion, when you look at the action on the whole complex projective space, is not such a good notion. Okay? So here you can consider, for example, what about the pulcarni limit set? Okay? And to describe that setting, this setting is very beautiful. You can do the following. Look at one point in the sphere. This is a coordination one sphere here. Okay? At each point, if you could take one point and try a point, you look at the tangent space of the sphere. It has real dimension to n minus one. And so there is one unique projective hyperplane of complex dimension n minus one which is tangent to the sphere at this given point. Okay? So at each point you have a unique projective hyperplane tangent to the sphere at the given point. If you now look at the limit set, and you look at that tangent hyperplane at each point, this set is invariant under the action. The action is by holomorphic transformations. So the tangent space at each point moves to the corresponding tangent space, tangent derivative hyperplane. So the union of all these projective hyperplanes at points in the tangent limit set is a closed invariant set. Theorem? That is the Kulkarni limit set. Okay? Here we have g, it's a group of p, u, n, one, discrete, lambda, c, j, contain, the sphere, the chain, the inverse limit set for each x in this guy. L, x, the projective minus one plane the sphere. The Kulkarni limit set is the union of all these x's for x in this unit set. And here in this case you have all the nice properties. The complement lambda-cool equals cpn lambda-cool is the largest reaction properly. It's continuous. So in this case there's a unique action and a unique largest in such region and it is this one. Yes? Yes? Yes, yes, sorry. Sorry, sorry, sorry, sorry. Yeah, yeah, yeah, yeah. Please, you're right. This set, this group has cardinality so that the group is non-elementary. Yes, thank you. And finally this region is the continuity region. And you have more interesting properties in this example. So just let me say that what do I mean by what? The omega-cool is the set of all points in projective space where the action is equicontinuous regarded as a family of transformations of projective space. Okay, so it's the equivalent of the factor set. Now let me say that this theorem this theorem again was proved by Navarrete about 10 years ago for n equals 2 then Cano and myself proved something in higher dimensions. We could improve this part. We proved this that the complement of this set was the region of equicontinuity that was proved by Anker Cano and myself. And now there's a recent article by Cano, Liu Bing Juan Marlon, I think proving this result and Anker Cano will speak next week about the proof of this result. Okay, so this is fairly recent. And now let me come to another type of examples which is where we started this something I did with Alberto Berhovsky and myself in 90s late 90s which is here we use something which has a big board two store theory but what I'll do is very, very elementary. We don't need two store theory for this but it's actually very, very beautiful geometry. So remember yesterday we saw that so remember that Cp1 this S2 okay and how we proved it we made a drawing we put C C and we put a copy of the complex here numbers and then we took a point here there was a line through the origin that line meets this complex line in one point and then you define a map that to this element here you have to say this element here okay and then you have to add one point when you come to okay and I'll give you this well-known diffeomorphism now let's keep the drawing but let's make a minor change now we put the quaternions so that is C2 with some additional structure or R4 with more additional structure okay and now you consider quaternionic lines now watch it you have to what is a quaternionic line if you are in R8 this R8 a real line means you take one vector and you multiply by all real numbers and you multiply by left or right it doesn't matter complex line you take one vector and you multiply by all complex numbers quaternionic line you take one vector and you multiply by all quaternions and you have to choose you multiply either by the left always or by the right always you like what? politics well, Mexico is more mixed no? what is going on? okay so let's suppose we consider left quaternionic lines okay let's see if it works it's not going to change to right okay so now take a one quaternionic line this is a four dimensional real plane you have another four dimensional real plane they meet transversally in exactly one point and you do this for all quaternionic lines and you get something which is what? on the one side you have P H 1 the quaternionic projective space you get what? H P 1 sorry see there's a problem with left okay so you have the quaternionic projective line and what do you have in this side? what is this? you are having union one point and what is that? so we'll get the quaternionic projective line is nothing but the four dimensional sphere okay but notice that we have more each quaternionic line and the quaternionic line means that you have one vector and you multiply by all quaternions okay now what happens if you have the same vector for you multiply only by the complex numbers you have a complex line containing that quaternionic line okay and if before that you rotate slightly you move your vector in the quaternionic space and you multiply by the quaternions you get another line by complex numbers you get another line okay so if you see this quaternionic line is actually a copy of C 2 and if you remove the origin it's actually filled by complex lines through the origin okay now if in C 4 if you think of this as being C 4 and you collapse each complex line is containing a unique quaternionic line okay now if you can collapse all complex lines do each complex line collapse to a point you get C p 3 okay if each quaternionic line is collapsed to a point you get S 4 and you have a natural projection okay each point here is in a unique complex line is containing a unique quaternionic line so you project the corresponding point here well this is the so-called twist of fibrations or also called the Calabi-Pentrose fibrations for different reasons okay now it's a local attribute of fibrations which is the fiber of each point one point here represents a quaternionic line which is the fiber the space of all complex lines in that quaternionic line so it is the space of all complex lines which is the property of C2 so the fiber is Cp1 so we have this beautiful fibrations so P3 fibers of S4 with fiber S2 okay now now here now look at the conformal orientation preserving conformal automorphisms of S4 this is the same if you want as isometrists symbolic I don't want to think of it in this way I want to think of this in this way with a plus okay well then we can show that this is the same the space of all maps of the form AQ plus B all the quaternionic logistic transformations that we like to write them as quotients but since this is not commutative if you were you don't know what to do so you better write it like this CQ plus D to the minus 1 and we'll A, B, C and D quaternionic satisfy something okay so this is something which is well known Alphors also in Cliffor Algebra so all the conformal automorphisms of the forest sphere can be expressed in terms of quaternionic mobius transformations but now see that each of these guys is a quaternion so if you think that the quaternions are C2 at the end each of these actually is a 2 by 2 complex matrix okay so you can write this at the end as a matrix A B, C, D where each of these is a 2 by 2 complex matrix so you have a natural embedding of this group into P, S, L 4 C okay in other words every conformal automorphism of the four-dimensional sphere is canonically a holomorphic transformation of Cp3 and not not only every transformation but as a group we have a canonical embedding so whenever you have a group of isometries of hyperbolic five-space you have canonically have a group of holomorphic transformations in Cp3 okay now you don't know how it behaves up there a priori okay so you have S4 you have Cp3 you have a twist of vibration you have the corresponding fiber which is called the twist of line and if you have a group acting here this group has a canonical lifting here which is taking fibers into fibers okay here the action is conformal so if you come here you have an horizontal each point you have a horizontal space horizontally the action is conformal what about in the vertical direction then that's something we prove this with an exercise where we prove horizontally you are taking Cp1's into Cp1's and then we prove that that action on the fibers is by isometries with respect to the usual ground metric okay so the action lifted here is conformal in this direction isometries in this direction then using that you can show easily the Kulkarni limit set of the action here is just the limit set here and you take the inverse image okay and you can say more so the Kulkarni limit set here is just the pullback of the usual limit set here and you can interesting properties for example this happens always this is always for example if the group is non-elementary the action on the limit set here is minimal when you lift it here it may or may not be minimal it is minimal if and only if up to conjugation so you have conformal automorphisms of let's say S1 here I mean the group of inversions here let me put it this way you have isometries of H2 which is conformal automorphisms of S2 by Pancras extension they embed the isometries of H3 then isometries of H4 so this goes to isometries of S3 and then you go to isometries of S4 if the action on the limit set here is minimal if and only if up to conjugation the group you have is a risky then here or here otherwise the action is not minimal okay so that allows you to see even more than what you see here so all the conformal dynamics embeds here and here you can see even more so for example a corollary of this is 30 fundamental group of a compact hyperbolic five dimensional manifold then it acts canonical in CP3 with dense orbit, every orbit is dense in CP3 okay so it's now to one more example which is very simple yes probably yes yeah yes yes yeah yeah very good that's very close to what I will say now thank you now again coming to this setting we have the projected space, CP3 we have S4 and we have the piston lines follow me, this is the inverse image of one part if you remove this point and you remove this line what you get is a locally trivial dimension over R4 which is actually trivial because R4 is contractible so if you remove one line and the fiber what you get left is nothing but R4 times S2 okay in other words take one fiber and take a neighborhood of it what is that neighborhood S2 times a four-dimensional disc and what is the complement S2 times a four-dimensional disc okay so it is very very interesting so if you take one twist line and you took a tubular neighborhood of it that's like a mirror that's splitting the projected space in two equal halves okay so that makes you think in short key groups okay and in fact this is for crystal lines but now if you take any projective line you can find an element in PSL4C taking this projective line into this line so whenever you have a projective line in CP3 you take any tubular neighborhood of it it's boundary splits projected space in two parts which are diffeomorphic okay so that's very interesting because now for example if we go here we have a group of here and we are lifting it in a projective space so two store lines going to two store lines but now perhaps you can just tilt the two store lines a little and somehow you have to have something similar well that's a construction you can do always so that's also something I did with Alberto now take in CP3 take two projective lines a retired projective lines which are very wise disjoint this corresponds to a two plane in C4 this corresponds to a transversal two plane in C4 it's easy to find a linear transformation taking this one into the interchanging these two lines okay to take a neighborhood of this one into a neighborhood of this one a larger neighborhood into a larger neighborhood there's going to be something in the middle which splits which is an invariant set okay and then on the this transformation the inside goes to the outside and the outside comes to the inside and you have in fact many choices there is not one map it's not like in the spheres of if you have a collimation one sphere if you have something like inversions you have freedom you have many choices but I can choose this two with a neighborhood as thin as I want and I can find a linear transformation taking this line into this one and the inside into the outside leaving the boundary invariant not fixed point invariant but as I said the boundary is invariant so now we can play ping-pong I'll give you a collection of pairs of lines all of them for each of them we choose a linear transformation interchanging them and we choose the linear transformations so that the corresponding mirrors are very thin very very close to the pink lines if we do that we can assure that the corresponding group you get you can control you can you have a natural from the mental domain you can control almost everything and then in that case you have a limit set which is just the set of accumulation points the lines that define the group okay now this limit set defined in this way is a cantor set times Cp1 and the action in that space in that limit set is transversely minimal I mean it is minimal in the set of lines okay you have many many many many interesting properties about this thing now we know that it is not always minimal because that contains this particular case and in this case so it is not always minimal but we also so the natural question we have was whether or not this limit set was the cool kernel limit set okay and I sometimes a few months ago I suggested to some of my younger colleagues that they should find such type of examples in pu31 because then you will have very nice groups of complex type of only groups the problem is with these young people who have no respect for the elders and instead of proving what I told them to prove they did just exactly the opposite they prove that such groups do not exist okay all such groups are in pu22 okay this is a very recent theorem by Angelcano Vanessa Alderete Mendes and Carlos Cabrera there is going to be a poster about this next week okay and something very very remarkable they also prove that in many cases this limit set is the cool kernel limit set but they find explicit examples where it is not this is examples where this is a close a proper subset of the cool kernel limit set is smaller but containing the other one the action in the complement is properly discontinuous but in those examples the complement is not the region of the continuity so there are things to be understood and I think I will stop here