 So, thank you very much for this invitation and true to what has gone on before, I felt like a young post-doc during the introduction. Okay. So, let's see. So, my title is Limits of Limit Sets. I think the groundwork was set by Caroline during her talk. So, let's sort of recapitulate that. So, what's the setup? You have S. So, for the purposes of this talk, S is going to be a closed surface of genus greater than one. Rho is a representation from pi 1S into PSL2C equal to isom H3. And rho of pi 1S, we denote as G. H3 mod G is a hyperbolic free manifold M. Okay. Then there's an inclusion from the surface S into M, which is a homotopy equivalence. I tilde from the universal cover to the universal cover is a lift of I. And this is a theorem that was referred to during Caroline's talk. Theorem I tilde from S tilde to M tilde extends to a continuous map I hat from D2, which is the same as S tilde, which is the same as H2. Union the circle at infinity to D3, which is the same as M tilde union the sphere at infinity. I hat is called the Canon Thurston map. Why? Because the most famous examples of this are due to Canon and Thurston. Okay. So, let's try parameterizing the problem. So, here we had G. Now you have Gn, converging to G. This is a convergence of clangent groups. I'll say a little more about that later on. Does CTN equal to I hat in previous notation? Let's call this G infinity. Does CTN converge to CT infinity? Okay. Yeah. I is this inclusion of S into M. M is a three-manifold, which is a quotient of hyperbolic free space by this representation, discrete faithful representation. So, M, that's right. No, no. I'm not thinking of this. This is a closed surface. This guy is just a closed surface. This is a topological inclusion. It's just a homotopy equivalence. There's no geometry here. M is, okay. Here's a fact. Here's a fact. No, no, no. Just a moment. M is H3 mod G. It's homomorphic to the surface cross R. So, you take the surface S and include it as surface cross zero, where zero is chosen at random. Zero is any number, okay? So, that's the inclusion that we're talking about. And then the inclusion lifts to the universal curve. Okay. So, yeah. Maybe I should sort of say this explicitly. There's a fair bit of work. I mean, this is not trivial, okay? But to prove that it's a homotopy equivalence is easy. It's a manifold, right? Whose fundamental group is pi 1S, and the universal curve is contractable. So, it's a k pi 1S1. So, there is an inclusion of S into that manifold, which is a homotopy equivalence, yeah? That's all. That's all. That's all we're talking about. Yeah? So, it's just a homotopy equivalence. There's no geometry here, okay? And in a certain sense, what I'll be talking about is largely topology today, okay? Okay. So, does this sequence of Kanan-Thurston maps, I hat, converge, well, CTN, let's say, I hat is not called the Kanan-Thurston map. Boundary I is called the Kanan-Thurston map. Okay. There's some notation, yeah? That's right. This is, so you have an inclusion of S into M that gives rise to a continuous extension. You call the boundary map the Kanan-Thurston map. Oh, oh, I see. Boundary I sub n. Is that what you meant? Okay, yeah, thanks, yeah. So, I've changed that notation here. Okay, fine. So, does this sequence, if you have a sequence of convergent planning groups, does the sequence of corresponding Kanan-Thurston maps converge? This is the question, okay? We have to make explicit the meaning of these arrows. Okay? Because convergence could be one of many things. The way, I think this is the sort of vague formulation of the problem that is sort of second part of Thurston, Bulletin AMS 82, question 14, okay? Yeah, this paper has this whole bunch of problems which is guided research in hand. There's just one problem which is still left, which I think came up in terms of dialogarithms. Okay, so this is the vague sense in which this question was formulated by Thurston and we sort of try to make this question precise. Pardon me? If it does converge, it will converge to CT infinity. There's no choice. So, sub questions, so question one, GN converges to G strongly, which means algebraic plus geometric. Let's say a little more about this in a moment. It implies, question mark, CTN converges to CT infinity uniformly. Okay? So, if you have strong convergence, does it imply, I mean, see, these are now maps from a circle to a sphere. This is sort of, I mean, you can ask very basic questions about them. I mean, these are sequence of maps from a circle to a sphere. You can ask whether they converge uniformly, whether they converge point-wise, yeah? So, uniform is the strongest sense of convergence. So, we sort of formulate it in this strong incarnation. If the convergence of groups is strong, which means that the convergence is both algebraic and geometric. So, what's algebraic convergence? I'm going to say this, not write this. It means you take generators for your group and look at row N acting on those generators. That sequence of elements in PSL2C converges to row infinity of that element, okay? That's algebraic convergence. What's geometric convergence? Look at the entire image of the representation as a close subgroup of PSL2C. That sequence of subgroups in the compact open topology converges to G infinity. That's geometric convergence, okay? So, there are different ways of saying that. So, algebraic plus geometric is what's strong convergence? Question two, GN converges to, this is the weak incarnation of the question. GN converges to G algebraically. I suppose once upon a time, this used to be called weak convergence. I have no idea. Does this imply that CTN converges to CT infinity point-wise? Okay? So, if this arrow is strong, is this arrow strong? If this arrow is weak, is this arrow weak? These are the two questions we are asking. All right. And theorem, this is not a theorem I'll be talking about. Can everybody see here? Okay, let's try it out, yeah. Theorem, joint work with Caroline series is that answer to question one is yes. I'll have very little to say about the proof of this, the positive answer to question one, but I shall use this particular thing in some context. Yes, is there a question? No? Okay. So, the strong version actually, there's a reason why I'm not going to dwell on this because the techniques for proving question one has an affirmative answer, essentially boil down to the techniques that went into this theorem. So, it's, I mean, you look at GN to be the constant sequence G, right? Then it boils down to this question here. You have a fixed group. Does there exist a canister map, yeah? And essentially the techniques that went into proving this can be souped up a little bit to give you a positive answer to this strong implies strong question. Yeah, okay, okay, okay. So, maybe I'll write this down. So, what's geometric convergence? Convergence is rho n pi one s. This is contained in PSL2C, converges to rho infinity of pi one s contained in PSL2C in the compact open topology on PSL2C, okay? So, I think, so you just look at this as a set, okay? This is a set, this is a closed set, this is a discrete set, this is a discrete set, yeah? The convergence of these discrete sets is in the compact open topology. So, you take any large ball inside h3 of PSL2C or h3. So, you look at the orbit, look at the orbit of fixed point, yeah? And then the orbits converge, yeah? So, the orbit of the first fellow converges to the orbit of the last one, okay? So, that's right. Yeah, so for instance, yeah? So, for instance, here's one example to bear in mind. Suppose you have a loxodromic, okay? And then you start, so you have a sequence of representations of Z into your PSL2C that has an attracting fixed point and a repelling fixed point. You start bringing your attracting and repelling fixed points together, yeah? You're going to converge to a parabolic, right? But if there's a loxodromic, there's a twist around that geodesic and this collection of rotations becomes dense in the circle dual to this guy, then you're going to pick up another Z corresponding to the linking guy, yeah? Yeah, so you're going to, so the algebraic limit is going to be a parabolic, a single parabolic, the geometric limit is going to be a Z plus Z, okay? All right. So that's not strong convergence. That's an example where the convergence is not strong, yeah? However, this point-wise convergence does hold in that particular case, yeah? All right. So now we'll talk about two. So there are two examples which I'll focus on in the context of question two. A, these are due to Kirchhoff and Thurston from the mid-80s. GN, well, I'll write an equal to here. It's what Caroline called X, let's see, yeah? X tau power N of X. What's X? X is the surface here. This is genus. This is X. There's this curve, sigma here. Tau equal to drain twist along sigma, okay? Single drain twist along sigma. You take tau power N, that means the N-fold drain twist along sigma. Take the uniformization, this Q here means simultaneous uniformization of X and drain twist power N of X, okay? Yeah, good, is there a question? It's, you take a compact, large compact ball, you look at the orbit intersect that ball, that collection of compact sets converges, okay? For every ball, okay? And then they take larger and larger balls, yeah, that's it. I mean, I think it's, well, okay. So you're from Grenoble. So it's also called the Shaboti topology. Okay, so now you have this drain twist along sigma and you look at GN, that's the simultaneous uniformization of these two guys. And then geometric limit, okay? So let's call, so this was sort of the first sort of non-trivial example where you have algebraic limit and geometric limit differing. So what's the algebraic limit of GN? What's this? It's the surface here and this guy becomes a cusp. So this is not a very nice picture. What's happening is that this guy is becoming thinner and thinner by drain twisting. So it's the neck is becoming longer. When it reaches the limit, this becomes a rank one cusp, okay? Geometric limit is different. It's still geometrically finite. What it has is a surface at the bottom and a surface at the top and so this is surface cross i and you have this curve here, this curve here which corresponds to sigma, but here inside there's this rank two cusp, okay? So it's a souped up, it's a version of this last thing where you have a loxodromic element converging to a parabolic element and the geometric guy. So what's happening here locally at this place is precisely that. So the drain twist gives you the loxodromic component. It's a twist around this geodesic. So the geometric limit, the algebraic limit is this parabolic here, the geometric limit is this rank two cusp here, okay? So you've taken the surface cross i, take sigma, small neighborhood of that, that's an annulus, take an annulus cross i, that's a solid torus, inside that you remove the core, okay? The core geodesic is removed, okay? So there's a rank two cusp here. So this is a rank one cusp, which means a rank one parabolic and this is a rank two cusp corresponding to a z plus z. This corresponds to just a z, okay? So that's the first example. The second example, this is due to Jeff Brock and his thesis and we were grad students together, I also sort of had a ringside view of his battling it out with these examples. So what are these examples? This is Brock's thesis actually. So what's this? You have now alpha, this was sort of, I mean, it was said, yeah, I think these examples came up in Caroline's talk. So this is what's called a partial pseudo annus half. What's that? So you have again, favorite surface and you have, this is your left half and this is your right half. Alpha restricted to the left side is the identity, okay? Alpha restricted to the right half is a pseudo annus half, pseudo annus half, diffio or homeo half, just this part, fixing, what's that? Fixing sigma. So this is sigma, this guy is fixed. So you can think of this sort of archetypal pseudo annus half, this is a punctured torus, the map at the level of homology or fundamental group is 2111, okay? Just look at, think of that, just that one matrix, yeah? So that's the map on the right side, on the left side, it's the identity, okay? What's GN? GN is again, well, it's the simultaneous uniformization with the group corresponding. That's why there's this quote. The simultaneous uniformization gives rise to either something on the boundary or a quasi-Fuxian group. So I'm going to call this group, the simultaneous uniformization of X, which is this guy, this whole guy is, this is X and alpha power n of X, okay? Yeah, what can happen? Yeah? If you look at it locally, then it's this example that I just illustrated, you have this, you have hyperbolic, yeah? Converging where the endpoints converge, right? So then you're going to converge to a parabolic along that, yes? But if there's a twist, it does not happen in dimension to it, has to happen in higher dimensions, has to happen in dimension three or above, okay? So look at dimension three, there's a loxodromic. So what is the twist along this parabolic? So there's a rotation along the parabolic, yes? So now if you take power n, yeah? And you translate a very small distance along the parabolic, but you've gone round many times, yeah? That is going to give rise to a parabolic in the transverse direction, okay? That's sort of vaguely what's going on, yeah? All right, so now what's the algebraic limit here? Sort of similar to this Kirchhoff-Thurston case. So the algebraic limit on this side, it's sort of fixed, it's SL, but on this side, so this becomes a rank one cusp, and so there's this, yeah? This is a simply degenerate end. What do you mean by a simply degenerate end? You look at the pseudo-anosoph monodromy acting on this, yeah? So then this cross-eye glued by the pseudo-anosoph monodromy is going to give rise to a hyperbolic free manifold of fibering over the circle, yes? So that's well, yes means that's a double limit theorem of Thurston, it's one third of his sort of major work prior to 83, but yeah, I mean it's sort of now part of the culture, so I'm just sort of stating that. It's by no means trivial, but so if you have this convergence, what you have is a sequence of these punctured tori, I mean this is a puncture. So this guy does become a rank one cusp, okay? And you have the sequence of punctured tori exiting the end, okay? That's the algebraic limit. What about the geometric limit? That's harder to draw. Oh boy, the first time I need the best. Okay, both space has been exceptionally badly managed because I have the algebraic limit on the right extreme and I have the geometric limit on the left extreme. I don't know a way of correcting this. Okay, that's right, that's right. So imagine an endless board. Okay, so geometric limit in rock examples, what is this? Well, again on the left side, nothing really is happening, yeah? You have a surface here and you have a cusp here, okay? So this is simply degenerate end. Now, let me use some colors, okay? All right, let's not draw this, yeah? So this is our simply degenerate end. And there's another piece at the top, this is our sigma. So as a first approximate, think there's this surface here, there's a surface at the bottom, puncture torus, there's a third surface at the top, yeah? All these three, there's a book of, I mean, okay, so there are these three pieces. One corresponds, I mean, each of them, say, each of them corresponding to a puncture torus glued along a common meridian, yeah? So this is not a manifold, yeah? You're thickening it up, then you have, what do you have? You have a genus to surface, cross it with i on the annulus corresponding to sigma, you glue in another half surface cross i, okay? So that's what it is topologically. So this is, yeah? And this side was degenerating towards the bottom, towards the top, and this top surface is also simply degenerate, but it degenerates downwards, okay? So you have half surface cross zero infinity, half surface cross zero infinity, all of them glued along this common single rank one parabolic sigma to the surface where nothing is happening, okay? So I'm saying that this is the geometric limit. I mean, it requires, I mean, it requires about three years of work, I mean, three or four years of work. I mean, that's just thesis. So anyway, so I won't go into saying why this is the case, but yeah, just, yeah, I think it's appeared in 2001, thereabouts, all right. So these are the two examples that I'll be dwelling on, hope not to erase them till they serve the purpose, okay? Good, and let me tell you the answer in case I don't have time towards the end of the talk. Here in this Kirchhoff-Thurston case, this limit is geometrically finite, okay? So answer to question two is yes for the Kirchhoff-Thurston examples, okay? This is again joint work with Caroline. And answer to question two is no, all right? So what's question two? It's the weak incarnation of the question. GN converges to G weakly or algebraically. Does it imply that CTN converges to CT infinity, yeah? And the answer in this case is yes, and the answer in the second case is no, okay? But I should say that the answer in the second case is barely no, okay? Just about, I mean, the only place where it could possibly break down, it does break down. So there's Mr. Murphy at work there, okay? All right, so now let's try to say how we are going to attack these questions, okay? So these are the examples, I've given you the answers, now let's say something about the proof, all right? So digression, electric geometry. Sturminology is a little unfortunate. Sturminology is due to Thurston, I mean, it's due to Farb really, but he attributes it to Thurston, okay? So this is again Benson-Farb's thesis that we are talking about. So capital X is a geodesic metric space, H is a collection of subsets. E X H, this is equal to X union, H belongs to this collection, H cross closed interval zero one. This is given the product metric, let's call this and each H cross one is given the zero metric, which means any path lying on H cross one has length zero. So this is not quite a metric, it's really a pseudo metric, okay? It's really the geometric counterpart of doing quotient spaces, okay? No, no, no, no. So what you should think about is, I mean, the way to think about this and I think this is how this terminology came up, you have the space capital X, inside that you have this collection of subsets, yes? And you make those subsets infinitely conducting, okay? So it's sort of the way lightning travels. So this is called, yeah, it's given the, and this, so this is called the electric space and so this metric D electric is called the electric metric. It's really a pseudo metric, okay? Alternately, so what you're doing really is on the subsets corresponding to elements of H, you're just retaining the topology, yeah? So it's just making a product with zero one, yeah? Topology means that the notion of open sets and closed sets makes sense, no distance. But on the rest, you have the distance, okay? So I mean, just as topology takes sort of us, topological space forgets everything up, I mean forgets the topology on subsets, it just makes it just, it gives it just a set structure, right? I mean the stuff that you quotient out in the equivalence relation, it's just a set after the quotient out operation, right? There's no topology there. So here, what you're doing is sort of categorically shifting that construction, you're retaining the geometry of the rest of the space on this part, you're just retaining the topology, yeah? Yeah, that's right, that's right. So each H cross one is given and H cross zero is identified with H contained in X. That's right, that's exactly right. That's exactly right. So you could think of, I mean that's, I think the way Benson sort of thought of it first, you look at H and you construct a core, yeah? But that destroys the topology, somehow I want to retain the topology, okay? So it's just take the product metric and the top part you just give, okay? All right, so what's this good for? And this is really what we'll be needing. Proposition, X is now going to be H three for us, H is a collection of convex subsets such that for all H one not equal to H two in H, product, the projection of H two onto H one has uniformly bounded diameter, okay? So what's the hypothesis? You have H three, you have a ball here, you have a ball here, it's easy to see that the projection of this guy is some small set, okay? These guys are disjoint and so if they are balls, then the projection is disjoint. So there's a generalization, the hypothesis is just now, the collection of convex subsets such that the projection of one onto the other has bounded diameter, uniformly bounded diameter. Then electric, so what's going to be an electric geodesic? So now I think we have a chance to draw a picture. So suppose you have, and you have a very small ball somewhere very far away, yeah? So now you want to go, you have another ball here. You want to go from say a point outside this ball to a point here, okay? You can travel for free inside these balls, yeah? So it's like it's going from here and then it hits this convex set perpendicularly. It travels from one convex set to the next by traveling along a mutual perpendicular, then along this, yeah? Then along this and then here. I'm drawing the electric geodesic. So the electric geodesic when it travels from one, so think of these are sort of huge conducting spheres, yeah? How does electricity travel between infinitely conducting spheres? It travels perpendicular to them, right? So this guy, this guy, this guy, and so on, yeah? So it sort of zips across all these spheres and goes from one point to the other. So that's the electric geodesic, yeah? So then electric geodesics and hyperbolic geodesics track each other outside H, okay? So what's the saying? Is that if you look at the electric geodesic and look at this, look at, oh, I wish, yeah? No, this is all white. There was some small little piece of chalk which I found somewhere. You look at the hyperbolic geodesic joining this point to this point and what this proposition says is that outside these infinitely conducting spheres, the red line and the white line stay close, okay? All right. So this is what we'll be using. I won't prove this, this is, so this is really all that we'll be using from electric geometry. Now let's get to criteria for criteria for convergence. So what do you have? GN converges to G algebraically, yeah? Which means what? Which means that GN is equal to rho N pi 1S, G is equal to okay. G infinity is equal to rho infinity pi 1S. Gamma, this is the, so this is the notation I'll use. This is the Cayley graph of pi 1S and this is quasi-isometric to the universal cover of S with the hyperbolic metric, okay? Good. Sufficient criteria, convergence of CTN to CT infinity. These turn out to be also necessary. One, ah, it's not allowed to go back there, I'll erase this. Uniform UEP, uniform embedding of points. What does this say? It says that rho N, there exists F from natural numbers to natural numbers, a proper function, which means going off to infinity as N goes to infinity, such that the distance in the Cayley graph of one from gamma is greater than equal to N implies the distance in H3 from, so let's fix a base point O and rho, rho, let's call this something else, let's call this M, zero, rho N, G is greater than equal to F of M for all N equal to one to infinity, okay? Infinity is included, all right? So which means that all these Cayley graphs are uniformly embedded in your hyperbolic space. That's necessary, that's not good enough. Let's just erase this. Criterion two is UEPP, Uniform Embedding of Pairs of Points, which says the following, for all G, H belonging to gamma, belonging to gamma, such that distance in gamma from one to G, H, look at the geodesic, pairs of points in a hyperbolic space determine uniquely a geodesic, sort of uniquely, okay? So you look at the geodesic joining G and H and gamma, if this is greater than equal to M, this implies that the distance in H3 between the base point and you look at rho NGO, I should have said rho NG acting on O. O is a base point, right? So rho NG acting on O, rho NH acting on O, and now you look at the, oops, look at the geodesic in H3 joining these, yeah? The geodesic in H3 joining these two points need have nothing whatsoever to do with this geodesic, okay? That's important, yeah? That's the whole point of proving the existence of Canon Thurston maps. It says that pairs of points embed properly, okay? If you do this for one group, if you can prove this for one group, then you would have proved the existence of a Canon Thurston map, okay? So it seems sort of a naive thing. I mean, you have proper embedding automatically. If you have a group acting properly discontinuously on a space, the embedding of the group is proper. So all this work is to upgrade something that's trivial, in the case of a single point, to something that's a little less trivial for pairs of points, okay? Took about 30 years to do that, all right. So, pardon me? I didn't finish that statement, that's right. So this is greater than equal to F of N. So you have, okay, here's your gamma, which I, with my bad eyesight, I identify gamma with H2. I have G here, I have H here. I join these by a geodesic, yeah? This lies outside a little n-ball. Map it over to H3, okay? So, yeah. So now you have, this is rho NgO, okay? So let's just call this G.O. And you have some point H.O here. Join these by a hyperbolic geodesic, okay? All right, this is a geodesic in H3, this is a geodesic in H2, yeah? And this is your O. This ball here has radius greater than equal to F of N. So this, if the, if this pair of points defines the geodesic lying outside a large ball, you map the end points over, then join. After joining, the geodesic also lies outside a large ball, yeah? I should say that just last year, or maybe the year before that, Baker and Riley, Tim Riley and Owen Baker, have given a counter-example to show that if this guy is a hyperbolic group and this guy is a hyperbolic group, not H3, then you do have a counter-example. So can the first and maps do not always exist, right? So it's sort of counter-intuitive, but I mean, how could it go wrong? Well, it does go wrong, okay? So if, I mean, so there's something very specific about H3 going on here. It's not enough space to do crazy stuff. All right, so this is our criterion. And in the last 10 minutes, I will try to say why these two criteria, these are sufficient criteria for convergence, and then we'll, okay, so that's fine. So now I can remove this and try applying this criterion to our examples. So now the last 10 minutes, apply convergence criteria examples. All right, that is the criterion for uniformness. That is the criterion for uniformness, and I'm just going to say what is the criterion for point-wise, yeah? Yeah, so that is the criterion for uniform convergence. From uniform convergence to point-wise convergence, we use this profound observation that uniform convergence on a point is the same as point-wise convergence, okay? So you restrict uniform convergence to a single point. That's all you need, and so what's the, so how are you going to do this? How are you going to get point-wise convergence? You take gamma, again, let's draw this picture. This is gamma, this is boundary gamma, and you have a point, okay? A point defines a geodesic ray, yeah? A geodesic ray is just R plus. It's an honest to God hyperbolic space, right? And you apply this uniform convergence to the space, and then you have point-wise convergence, that's all, okay? So uniform convergence, uniform embedding of points on a ray, uniform embedding of pairs of points on a ray is the same as point-wise convergence, all right? Okay, so basically all that you're doing is you're taking your boundary, and you're reducing it to a single point, and then you have convergence point-wise criterion. All right, so now we'll apply this convergence criteria to our examples. Step one, all right. So now it's time to draw this example a little better, and you have this guy, and you have, so you have this is x, this is alpha power n of x. So which means what? There's one copy of this is SR cross zero, SR cross one by, well, let's call this one, and this is SR cross n, okay? So there are these n blocks from zero to n, pasted on the right. So we zoom in here, this becomes a Margolis tube, which means what? It becomes a picture like this. It's a piece here, piece here, piece here, and there are n pieces, and then there's this one guy, which has length one. So this Margolis tube here blow up, and you have a picture like this. So what's the section? The section is sort of a disk in the hyperbolic space. This corresponds to the left, and then there are sort of n pieces, and then there are, there's stuff hanging off from there. So there's blocks all around this, and there's this Margolis tube. It has length n along the right and one along the left. This is our model for GN, okay? Good. So now how are we going to apply the criteria? Step one, large pieces electrocuted. So now we are going to use our electric geometry. Large pieces electrocuted implies convergence. So this is a mnemonic. What this says is that you have this stuff on the right side, on the left side you have nothing. When you lift to the universal cover, you have this hyperbolic two-plane. This is corresponding to S tilde, and inside that you have lifts of S R tilde, S R tilde one, S R tilde two, and so on, yeah? What you do is you put, these are all going to be convex sets, which will satisfy this criterion in this electric thing, that the projection of each of these pieces onto each of the rest is a bounded diameter. So electrify or electrocute, depending on your love for violence. Electrocute lifts of S R tilde cross zero n, all right? And the electric geodesics for rho n and rho infinity are the same. Why? Because the only change is happening on top of this SR, yeah? So if you have a guy, if you have a geodesic ray, you have this point, right? Remember this profound observation of ours, yeah? So there's a point on the boundary and you have this ray on the bottom sheet, yeah? If it cuts infinitely many of these blocks, then these blocks are convex guys, so stuff that's outside the end will remain outside the end. They can't backtrack, okay? So step one is saying that if you electrocute these large pieces and the ray that you started off with has infinite length with respect to this electrocution, then you have point-wise convergence at the end of such rays, yeah? Step two, as usual with such talks, I'll have to give the entire proof in about two minutes. And this comprises about two-thirds of the paper. All right. Step two, so what does this reduces? So this construction reduces the problem to looking at geodesic rays which end on the boundary of a copy of SR2. So infinite length is done. We are left to deal with things which end here, okay? So on the boundary of SR2. So enough to look within an SR2. So you can sort of shift your base point. It's finite length, so you shift your base point and you think of looking at just one SR2. But then we can use this theorem that we've said before. Uniform, so on SR tilde, rho n converges to rho infinity strongly, okay? So this implies that CTn restricted to SR tilde converges to CT infinity, restricted to SR tilde uniformly. This was our theorem which I have said nothing about, yeah? Strong convergence implies uniform convergence. So what does that mean? Are we really done? I mean, I said that there was going to be a negative answer. We seem to have arrived at a positive answer, yeah? Not quite. I have a couple of minutes left. That's fine. So the crucial thing is, here you look at rho n restricted to S, phi 1S. Here you're looking at rho n restricted to just the right side, okay? So what happens is the following. If you have, now you have this canter SR with its canter set boundary. And you have a geodesic, you have your base point somewhere. It enters into this, yeah? And ends somewhere here. So this is X infinity, yeah? So step two furnishes a positive answer to CTn converges to CT infinity, provided SR tilde with cusps corresponding to sigma tilde provided in SR tilde with cusps corresponding to sigma tilde X, OX has infinite length, infinite length in the electric metric corresponding to cusps, okay? See what's happening here, in order to go from step one to step two, you have to sort of ignore information at the places where it cuts across the cusp, yeah? Ignoring that information is really tantamount to saying that the cusps have electric length, yeah? So you electrocute the cusps here, so this is, you can think of this as a step one as large electricity implies convergence, step two as small electricity implies convergence, okay? So there you electrocuted large blocks, here we are being less violent, we are just electrocuting the cusps, yeah? And if it has infinite length there, then this strong convergence theorem does apply and we can conclude convergence, yeah? What's left? So, and this is in the last minus one minutes, I'll just say what's left, step three, left are points X belonging to boundary SR tilde which are identified by C T infinity with boundary of sigma tilde, okay? So you had these guys, the sigmas here, these were becoming cusps, yeah? And if you had a ray which got identified with this cusps, you had a ray and if you had a ray which got identified with this cusp here, then you would not be able to apply the small electricity, right? And such points, again there's a theorem here are exactly the end points of crown domains or what are crown domains? You look at SR, you have, this is Phi pseudoanus of acting on this, so there's a stable lamination on the right side. So if you open it up in the universal cover, then there's this cusp here, this is sigma tilde at infinity and then there's this crown domain and then there's this crown, this inverted crown corresponding to leaf of a lam, of a domain of a lamination which contains the cusp, okay? And I think Caroline said something about saying that it just so turns out that exactly at these points we have proved almost everything that it converges everywhere and the only place where it could break down are these crown tips, okay? This is an inverted crown, yeah? And it turns out that CTN converging to CT infinity does break down at these crown tips, okay? So what we have, I mean the convergence of planning groups is fiercely democratic or anti-royalty depending on your point of view, it disrespects the crown tips and respects everybody else, okay?