 Okay. Thank you. Well, this talk is largely going to be a review of examples that some people, maybe quite a few of you may have seen before, they get successively a bit more complicated, but it relates as you'll see to some work I've been doing with Mahan who's here and at the end I just want to try and explain ond we found something rather unexpected and surprising in the course of our investigation. I'm not really going to prove much and then I shall leave it to Mahan in his talk to do some more, you know, I'll try and tell you what we're talking about and then Mahan will tell you the mathematics behind it. So let me write some of this on the board because I don't know, I can't ever remember what was on the previous slide. So we're going to always in this talk have a compact surface and it may just as well be Junas 2 for all what I want to say. And we're going to have a representation from the fundamental group of S into PSL to C. So these are hyperbolic isometries and so this group I'm going to, actually what I'm going to do is fix, fix a hyperbolic structure. So S I'm going to think about as hyperbolic two space modded out by some focusing group and then this group is isomorphic to gamma and it's going to be mapped by an isomorphism to some subgroup here so G is rho gamma and right so that's the basic setup and so if I look at H3 and I'm going to assume throughout the talk that this representation is both free sorry faithful so one to one and discrete so H3 over G is going to be a hyperbolic three manifold which we'll call M and it's a theorem which actually requires some quite serious topology really to prove that this manifold I'm just going to tell you this it's always homomorphic to the surface well if it's an open manifold it's the surface cross R okay. So that's our setup and I'm going to be talking about limit sets which I know was topic of one of last week's lecture courses and was mentioned again this morning so the way I'm going to think about it is I've got my focusing group gamma which is acting here in H2 which I'll just draw as the hyperbolic disc and the different ways of defining the limit set one way is to take all the attracting fixed points of all the elements in gamma and they all around here and in fact if this is a compact surface then the limit set here so lambda gamma plus is the attracting fixed points and then lambda gamma itself is just the closure of that which is the whole unit circle in the case I what I'm talking about extends to other cases but let's just stick with this and same definitions for G but now the limit set of G G is acting on hyperbolic three space in its boundary so lambda G which is going to be the closure of the so G may contain some parabolic elements so I throw in the fixed points of those and take the closure of that and this set is contained in S2 which is the boundary of S3 and I shall usually think of it as the Riemann sphere okay so what we're interested in is mapping the limit set of gamma to the limit set of G so you can obviously map from the limit set the attracting points of something here you can send it to the attracting point of its image and then you can ask does this map extend continuously to the closure and it's a theorem that as you can see has been going along for best part of a hundred years in various degrees and it probably still not finished this theorem but this map extends to a continuous map which by its definition really has to be an equivariant from lambda gamma to lambda G so in the case I'm talking about when S is a compact surface you come right and say mahan finally that for all cases of G right all right mahan yeah yeah okay so but in yeah sorry yeah okay so the the most famous case well actually I mean actually the Nielsen stuff already is very interesting so even when G is a focusing group this has real content this theorem but that's the most famous case it's possible to create groups G which images like this of surface groups for which the limit set is the whole Riemann sphere probably many people have seen this I'm actually not going to talk about this case in this talk but I'm going to talk about something closely related to it so in the case here this limit set is a circle this limit set is the whole sphere so in this example you've actually got a space filling curve as this map so it gives you a hint this map has to be pretty nontrivial to do that okay so I'm going to run through a whole bunch of examples of successive yeah I'm not telling you all I'm saying is it's got to be a faithful discrete representation so in my different examples I'll show you what the group may be no it yeah won't be a shocky group because we're representing a compact surface yeah so topologically the manifold will always look like this and in my successive examples I'll talk about what it looks like geometrically so first example I expect many people are familiar with our quasi foxy in groups and that's the case from this viewpoint so we have this map I sub row going from the limit set of gamma which is the circle to the limit set of g which is contained in the two sphere so the first case could be supposing I'm going to take I'm just going to assume that this map exists and it does in all the cases I'm talking about so this map exists suppose this map is i row is one to one so it's a bijection from this to this then the image of this is obviously a Jordan curve and it separates the sphere into two halves it is not at all obvious that this Jordan curve is what's called a quasi circle here are pictures of quasi circles so that means something about how bad badly behaved this map is on I don't think I'm going to take the time to explain that but these are various images forget all the lines inside just look around the edge these that's a round circle and then these are quasi circles and a Jordan curve always separates the plane or the sphere into two parts so we get two simply connected g invariant parts in the compliments of this curve yeah no it inclusion into what this is this is a homomorphism to this set but this set typically is not equal to this how do I define it no no no I didn't say it was obvious at all what I said it's obvious okay if I have a gamma in my foxy and group right and then this gamma has an attracting fixed point too so you know I've got a here's h2 and here's I've got an axis of something and it's moving along this axis from a negative one to a positive one yeah okay so I take this and then I look at row of gamma so that's something in SL2C and let's make life simple assume it's loxodromic so it also has two fixed points on the boundary one repelling one attracting take the attracting one okay and so what's i row is going to take gamma plus to row of gamma plus okay so that's a subset of here to a subset of here and the big question is can you extend this map so I'm saying you can obviously write this down and it's obviously g equivarian the question is can you extend it to a continuous map that's to the whole of the limit that's the question okay so the first case I want to look at is the case when so I'm saying you it's a big theorem that you can always extend this map let's assume you can extend it and it's a homeomorphism and it's another non-trivial theorem that in that case you always well the image of a circle is a Jordan curve it's injective right so it separates the sphere into two invariant parts and the if you take either of the two complementary regions so to say the region inside this black line that's invariant under the group and in that part the group acts properly discontinuously it's a piece of complex plane so the quotient has a complex structure so it's a Riemann surface so we get two Riemann surfaces associated to this and the famous theorem of bears bears simultaneous uniformization theorem he says you can choose any conformal that's complex structures any two on this pair of surfaces and there is a group for which the complementary pieces have exactly those two structures okay and moreover that up to conjugation that uniquely determines the group so I'm going to call the group that you get by this procedure q of x y now if you ask me actually to find the group that's a very difficult problem to solve some PDEs but abstractly it always exists and we're just going to take that so here is a or maybe I should draw a picture of the group like this because the manifold what's the manifold look like so the manifold looks like this so in the middle there's a kind of compact part that contains all the closed geodesics in the manifold so these aren't holds every every closed loop in the manifold has a geodesic representative inside some bounded piece here and then the thing kind of flares out no closed geodesic penetrates outside here and then at infinity down here you get you lose the hyperbolic part you're on the boundary at infinity you get a conformal structure and here we have say omega minus over g which I'm calling x and up here is infinity you have a piece of complex plane whose quotient is y y that's called y so that's what this manifold is like and I can think of m yeah that's the picture of it okay so there is a well-known result what do I mean by continuous motion of limit sets so supposing now instead of one group here for g I've got a whole family of them and they depend on some complex parameter that I'm calling lambda and that means just that I have certain generators of this group and all the entries are just holomorphic analytic functions of lambda okay and supposing I vary the lambda and I ask what happens to the movement of this limit set lambda g and the theorem is that it moves sort of jointly continuously in the two variables and actually it's really better than that this if I fix the size so I fix a point here and I let the lambda vary I get a function which is a holomorphic function of lambda which is a very useful thing and out of this joint continuity you deduce that if you have a sequence of these quasi-fuxian groups converging to another one then the maps that we got from the s1 to subset of s2 move well they convert uniformly as maps from s1 to s2 okay so this is so how do we know this theorem there's a theorem called the lambda lemma in holomorphic dynamics which some of you may have come across of which this is a particular application of course you can prove this directly and in fact it's one of the consequence of one of the theorems sort of simple case of one of the theorems I proved with mohan that we can sort of rederive this result but anyway this is a well-known result so what I want to do for the rest of the talk is so up here my groups were staying in the class of quasi-fuxian groups where this map is one to one and my question is what happens or our question was what happens as you allow the quasi-fuxian group that you get here to degenerate in some way sort of go to the boundary of quasi-fuxian groups what happens to this map so there's the question right so that's what I'm talking about so and I'm going to make life simple by just looking at sequences where I'm going to fix the bottom surface and allow this top surface to vary in various different ways and in the different examples you see all the different phenomena that I want to talk about okay so I don't know if I can make this work I'm going to have a little movie break here see if I can make it work yeah right we're going to have a movie so this is moving limit sets and I was very proud of myself it's the first time I've made a movie cut and pasted a movie right so Peter Leaper well I'm watching it come up Peter Leaper is a computer graphics person who has a Vimeo website that has beautiful pictures on it so these are pictures of quasi-fuxian groups for those who know what it is these are isometric circles and they're moving around landing on complicated things and now especially you want you to watch this one so you see the plane separated into the inner colored part see something happened and these circles got pinched off and then they sort of twisted and unpinched and if you watch carefully they're going to twizzle around again more and more and more see there's sort of some kind of twisting going on there do it like this better there's some twisting here and then it pinches off and it goes quite fast here another pinching happens and I think that is yeah that's the end right okay that's a movie break yes yeah this is all in a bear slice right I just to make the talk simple I just fix a bear slice but it doesn't we don't our results hold out more generally but just yeah okay I think I have to do this by hand yeah okay so um the first couple of examples in a way explain what I was just showing you there so we're going to talk about pinching so what I'm going to do for the rest of the talk I'm going to fix a curve around the top well really I should draw this on s but I'm going to choose a curve there that I call sigma that's just separating the surface into two halves and the first sequence of things I'm going to do is I'm going to make Riemann surfaces in which this curve gets shorter and shorter and shorter so really what I want to say is that I'm talking about extremal length but let's just think I'm making it shorter and shorter I mean you have to say what length am I talking about but in any case we can make a sequence of groups so we're going to make q x down here is fixed and y n is going to be pinching okay and we so basket has a theorem that says he wrote down exactly the formulas of quasi conformal deformations that implement this he sort of wrote down what you have to do to this region y to to contract this curve and it always converges to something and it converges to a representation in which the row n of any particular element in the group converges to a definite mobius transformation in the limit so what you have to prove you have to prove that you don't just kind of go away to nothing or to infinity this thing is a definite matrix for the limit and so here's what happened to the limit sets here is the kind of unpinched one and that's what this movie did it went like from this and then this point and that point were fixed points of say row n of gamma and as n gets larger those two points come closer and closer together until here they've merged into one point and it's pinched so that's the kind and you know kind of obviously this picture converges to this picture who what who's who okay q of x y is the manifold the quasi the quasi-fuxing group so that the bottom surface is x and the top surface is y okay so you could say g n I just wrote q for quasi-fuxing so I'm going to fix this and using bears theorem find a sequence of groups where this top surface changes okay and the limit sets all move around somehow in some very complicated way and they land on this surface on this picture right so the um what happened to this map it can't any more be one to one in fact we saw two points came together to one point it's a theorem of bill floyd that in fact the only place where this map in this example fails to be one to one is exactly at the pinch points so there's really one pinch thing and then all its images around the place under the group right and here's the manifolds which I've already I've already drawn the one on the left so of course the curve this this sort of thing goes really all the way through the manifold it's geodesic representative is somewhere there in the middle of the manifold but as you approach the limit this gets very short and what the distance between here and here has to get very very large for that to happen and in the limit the top bit sort of goes way on up up there to infinity infinitely far away and you get a limit manifold that looks like this so let me not dwell on all this I mean there's many things behind what I'm saying but I want to give some more pictures so that's my first example and what happened to the limit set I'll state a theorem in a minute but what happened to the limit sets is quite what you would expect but there is another example which is quite similar to this this one I've done now we're going to change what the y n is and what we're going to do instead of just making this curve shorter we are going to start twisting the right hand side of this thing relative to the left hand side so we're going to do a dang twist which involves if I had a curve that was transverse to here then in the next picture it will be wrap around and go like that and then it will in the next picture it will wrap around twice and go like that and so on and you can believe me that okay so we're going to look at the limit set of these groups okay I think this bit in blue the people who are experts will know the people who are semi-experts can read this I'm not going to explain it you might say to me what do I mean by phi to the n of y so you have to talk about marked structures and taigmonus space and stuff but let's not dwell on it okay but what happens is that if I took a curve that crossed this sigma I took a curve beta on this top surface it kind of gets longer and longer and longer and the only way a curve can get longer and longer and longer is if this curve gets very short by hyperbolic geometry so this is a slightly misleading argument because really we're doing something up here but anyway it's what happens so the only way that this can happen is that this curve actually gets shorter at least the geodesic representative of this in the three manifold has to get short and so there I've written l to the n of y of the sigma curve has to go to zero so again this curve sigma is pinched and again the groups converge to each other element wise algebraic convergence means kind of element wise um as you would expect what about the limit sets so here are pictures of the limit set so this was like in the movie here is one of the sort of y n's for large n and you see instead of these two points just straightforwardly coming together a whole lot of spiralling has been introduced which is somehow coming from this dain twisting here and it it's very fascinating to go through the detail of that so you can either read the paper of um Kirkhoff and Thurston which has got lots in it or you can read Al Mardin's great book out of circles he goes through like an enormous detail how this can happen but look what happens to the limit set in the limit you suddenly this thing if you just look at a kind of um house dwarf limit so a limit of the pictures you get this so what has happened here is that this curve that's getting very short it's not just getting a short translation it's acquiring a lot of twisting at the same time for this to happen and the shorter it gets it twists and twists and in the limit if you just look at the kind of accumulation points of the sequence of group elements that you get as you look at the limits of the group as a kind of geometric object sitting inside Athol 2c you acquire an extra parabolic transformation that commuted with the row infinity of sigma that you had in this t sort of appears out of nowhere an extra thing comes in and this limit set is really the limit set not of g infinity but the algebraic group that we had before with this one pinched together with another element that is producing all this extra effect so so here's again a sort of comparison here was the just plain pinching and here is what happens if instead of just pinching you twist and then if you look at the house dwarf limit of these limit sets you get a picture like this so this is really very interesting example so here are now a few definitions and theorems so a group sequence of groups converges algebraically if it does element by element that's just what you would think it converges geometrically however there's a bunch of different ways of describing it but I thought this one might be the easiest to understand so you take a fundamental polyhedron for gn and supposing you can find a sequence of these or a sequence of Dirichlet domains with a certain centre which instead of converging to a polyhedron for g infinity they converge to a fundamental polyhedron for a bigger group and they converge uniformly on compact subsets of age three so that's called converging geometrically and if it happens that the two results give you the same thing the convergence is called strong and sort of as a general principle if things converge strongly it's you can expect to prove good theorems if they just converge algebraically lots of strange things can happen and to state the next theorem let me just another definition I've talked about a fundamental polyhedron in 2d having a finite sided fundamental domain is equivalent to having a finite set of generators in 3d that's no longer true you can have groups with a finite number of generators but you can't find a finite sided fundamental domain you have to use infinitely many sides because of lots of spiraling going on so a group is called geometrically finite if there is a finite sided fundamental region and this again these kind of groups are way easier to handle than other ones so we'll come to the other ones in a minute so here's a theorem so this theorem was originally proved by Jorgensen and Marden in around 1990 and then it wasn't pushed to the the opposite of geometrically finite is geometrically infinite it would push to the geometrically infinite case in the light of a great many further developments by Evans later on so here's a theorem suppose we have a sequence of these groups and you can think that these guys are all quasi-fuxian they don't have to be but let's suppose they are in my setup and suppose algebraically we'd go like this so point element wise we like this but geometrically we converge to something which might be different then they prove that the limit sets converge in the sense of pictures in the house of metric on subsets right so the limit sets converge to the limit set of the geometric limit which is what we saw in the pictures and actually this gives us a condition for strong convergence the two g infinity equals h if and only if the limit set of the algebraic limit is equal to the limit set of the geometric limit so in my first if i can go back quickly in my first whoops in this case the convergence is not strong because this limit set is different from this limit set so this limit set is sort of sitting inside this one but it's not equal this set this one the convergence is strong okay so ah okay um i think i'm not going to have time to go through this so let me this is an attempt to explain topologically the difference between algebraic and geometric limits this is really what's explained in this paper of kercoff and thirst and it really repays a lot of study this paper so if you're a tool interest in this subject you know this is a really powerful example this is so we have one so this is a very schematic picture of the manifolds um with the top surface y infinity and actually this should be called this is called sigma right this is the thing we're going to pinch in the geometric limit we take two copies of this thing and we stick them together along one boundary you can think of it using van campens theorem this extra thing t that came in has the effect of gluing on another copy and we get sort of another copy of the whole the whole thing kitten caboodle if i'm not allowed to say that whole stuff up here same as down here and the reason the limit set so much more complicated is you're kind of seeing whatever you had here and then you see it reflected and reflected and reflected around um everywhere else and it sort of doubles up so this is what's going on here right so now let me state some theorems of myself and my hands so the first theorem is suppose we have strong convergence let me be a little bit vague what the exact conditions are but certainly in the examples i'm showing this this is true supposing these groups converge strongly then these maps on limit sets not only do the limit sets converge house to off actually the maps converge uniformly so this is kind of like the best quasi foxy in case limit sets move continuously uniform convergence great but now suppose the convergence was not strong so the limit groups are different and so we know from what i've said the limit set so the groups are different the limit sets are different if it was true that this converged to that uniformly it would be a little exercise to prove that this limit set converge to that limit set in the house to off metric something about diagonal convergence so that but we know that's not true so this thing can't converge uniformly so what's the theorem all right what can we say so here is the first thing that we proved so we've got sequence of groups converging the two things are not the same and let's suppose that this geometric limit is geometrically finite which actually implies so is g infinity but let's suppose so we're kind of in the geometrically finite case then we proved that actually these things do converge but only point wise we lose the uniformity because all this twisting up somehow you know near the the parabolic points it messes up the potential uniformity of convergence and actually to prove this we partly made use of the results again of Jorgensen and Marden which says as long as you stay in the class of geometrically finite groups then actually this dain twisting business is the worst that can happen so really if you can handle this dain twist example in the geometrically finite case then you're done so that's what we had to very carefully analyse what the effect of all this twisting on the on the limit sets so I'm not going to go into how we did that but what I want to try and say in my remaining time a bit more for people who knew more about this background I'm talking about so I'm going to we we then went on to say well what happens if this limit group is not geometrically finite it's geometrically infinite which is infinitely much more difficult to handle I would say so these are the kind of the canon thirst and map that I mentioned a space filling curve would come into this category so I'm going to show you two more examples and one is going to be very light the canon thirst an example of strong convergence and try and give you well not prove that there may applies but just give you a flavour of how that works but the thing I really want to get to in my second example we'll have an example where we don't have strong convergence a famous example and to our surprise we found out that the second theorem also fails so it's no longer true that you have point wise convergence there are certain very special points where the limit of the limit points is not the limiting limit point okay so that's what I want to try and explain so my third example so all the examples are kind of in the same vein somehow so instead of dain twisting well I've got the same picture I've got a similar sequence of quasi-fuxian groups but instead of doing a dain twist so you know about the classification of diffeomorphisms of surfaces you can dain twist you can have finite ordered things or you can have pseudo anosof things so we're going to have a map that I'm going to call five from s to s which is going to be pseudo anosof and I'll remind you in a second what a little bit what that means basically it means it it stretches in one direction and contracts another transfers direction and we're going to again just again look at this same sequence but instead of dain twisting to the n so yn in this case it's going to be q of x and I may as well stick with x I don't need to have a different surface I'm going to look at this sequence of manifolds so I've just done something which is a kind of more drastic thing on the top surface basically okay and here are some pictures of here is a picture of what happens when n is large so I'm showing you a bunch of pictures I didn't draw any of them except the really bad ones these ones were done by David Wright in our book Indra's pearls well I why or where why because this and then McMullen proved it okay not obvious not obvious it's kind of okay it's kind of the whole of Thurston's theory about ending laminations is sort of let me when I go a bit further I'll show you more about the manifolds but I'm not going to prove the limits except but it does so here is a group which is a q of x phi to the n of x for some large value of n and the inside part the inner component omega minus is coloured yellow so you can see it and the outer part is white you can see the limits that is very scrunched up and here is the limiting thing so don't ask how you managed to draw this but you managed to draw it and what's happened is the yellow part has completely disappeared away so the sort of upper components here you don't see anything and in fact remember I said in the middle of the manifold this quasi focusing thing there's a kind of compact part which contains all closed geodesics as we approach this limit these two this compact region gets larger and larger and larger and in the limit the top side of it just goes off a way to infinity yeah uh yes okay yeah this is actually sorry this is a bunch of tourist groups out of Endersfields yeah okay right yeah because we've got a lot of parabolic points here so I don't know if there are any pictures of the other kind maybe Jeff has some I don't know okay um so it is a theorem non-trivial theorem that the convergence in this exact that there is a limit and it is strong and the theorem of myself Mahon says that therefore these maps converge uniformly and you look at the picture well you know this thing looks like it's converging to this everything looks true right so um let me just say a little bit so so we can understand this and so we can understand the final example better let me say a little bit about Sidhu and Ossoff maps so the Sidhu and Ossoff map is a sort of generalisation to a hyperbolic surface of an Ossoff map I think an Ossoff map on a tourist which is something which has an expanding direction and a contracting direction so it's kind of like that but you can't do it quite there's nicely but for the present purposes let me see if I've got yeah all right this is a really crummy drawing but all right so for people in the know you have two you have two measured laminations which sort of so this is meant to be a picture of a measured lamination and you should think of these are the directions in which things are expanding say and there's another so they don't fill up the whole surface they just fill up a bit of it but the bits in between you can reconstruct the dynamics in a systematic way and then there is another transverse set of things like this and this is another lamination and along these green lines everything contracts and I've just for what our purpose is really we can just pretend that we've got two directions so I'm going to draw y up here and x here and I'm going to do what you might think is kind of unintuitive but anyway that's what does this map look like so it takes a kind of square box here and it maps it to a thing which has contracted this way and expanded that way like that so here is what my phi does here is y now if you really want to follow what I'm saying you have to figure out which is stable and which is unstable I'm going to make these guys be unstable leaves because stuff's expanding and here we're going to have stable leaves so these leaves are stable so these guys are the contracting direction then I a measured lamination goes with so we have a transverse measure here which is transverse to the unstable leaves so it's the unstable lamination so it's lambda u and so lambda u if I got that right according to my picture here yeah lambda u is a measure that measures distance across here which you can think is the y coordinate and across here you have the other one which is lambda s so right we don't want to I don't have time to do a whole lecture on sidon loss of maps but basically think expanding and contracting in these opposite directions and now let me give you a picture of this manifold so the convex core of a three manifold hyperbolic three manifold is the smallest convex set that contains all the closed geodesics what I've been talking about before and I didn't say it before a group is geometrically finite if and only if the convex core has finite volume if we don't have any parabolic elements in sight if there are no cusps you can think compact so in our case you can think compact right so probably right let me try so somehow the key to all these after a great deal of expanded effort the key to understanding these geometrically infinite manifolds is to try and make a model of what they look like it's sort of coarse metric or picture of them so here is my rough picture of the convex core of the limit manifold well the mn and the m infinity so here we're going to have the surface x and so I think really this is a genus two thing and then we're going to have a lot of levels in the manifolds and then at the top we're going to come up with the surface phi n of x so this is the same surface hyperbolically but we've changed the marking so which curve is labelled as which curve is changed drastically by this pseudo enos of map and so the convex core is roughly this bit and then it flares off here and it flares off here so this is mn and this level here looks like phi to the jx and the distance from here to here is kind of one roughly and it turns out that in this particular case you can put an approximate metric on this thing which looks like this so um you can read this or not as you like it's explaining how we decide whether which of these x and y directions is contracted in which so it's kind of counterintreative you you you contract in the along the unstable leaves when you change the marking so here's a metric so if I measure a sort of t up here in that direction and on this surface I'm going to measure x along this direction of the unstable leaves y along the stable leaves and the stuff that's kind of not on any leaf I just fudge it somehow in amongst it doesn't really matter what I do I just sort of interpolate somehow so it turns out that the manifold has a model like this and what's the limit manifold well instead of this it just goes on to infinity in the same way right so here is a model of m infinity and what happens so here's a picture so there's my m infinity going off there and um let's take a curve that was down here if I apply this pseudo so we take a curve on f if I fix this surface this hyperbolic surface and I apply any pseudo in itself map over and over again this map whatever the curve was it gets longer and longer and longer and eventually it limits on I guess the unstable lamination but it gets longer is that right unstable one yeah it gets longer and longer and longer in the metric hyperbolic metric on this surface but if I go up to this surface then the labels that the surface has got the same metric but the labels have all changed by phi so the thing that was labeled gamma down here is now labeled phi inverse gamma but and so okay so you work out precisely which is phi and phi inverse and you discover that the length of the curve phi to the j of gamma on this surface here is the same as the original length of the original curve down here so a curve that down here was got longer and longer on this surface it's just got a fixed length and it's roughly what what this model means is that the geodesic representative of that sort of short short curve is roughly at this level and then at this level it'll be phi to the j plus one and so on so the upshot is we've got a sequence of curves whose geodesic representatives are kind of roughly higher and higher up the picture and they're all going out this way down here they'd all be getting longer and longer but here they're staying of the same length or approximately the same length and that is what Thurston defined as an ending lamination or the limit of these curves is an ending lamination and these curves if you know about the theory of projective measured laminations these curves converge to a lamination and if you know about laminations and you work it through you discover that indeed this sequence of curves converges to this is proving that in the projective measured laminations it converges to the unstable lamination so the ending lamination of this thing is the unstable lamination okay so now I'm going to here's a proposition and I'm going to do this because in my final example this is important but let's look let's look in this manifold m sub n so what does this say so lambda u is a is a lamination and these curves phyton of gamma are converging to it so in the focusing group I have a leaf l that belongs to the unstable lamination and here I've got my manifold m sub n and on every level I've kind of got a copy of this x but on every level the way I label everything has changed the marking has changed so on every level not only can I think I can find copies of all the curves but really these are sort of pleated surfaces in here so on the top level I can find you can think that there's a copy of this leaf of lamination sitting so there's sort of a copy of l upon this thing which is supposed to be like phyton bn of x now what I claim is what this proposition is saying is actually there are lots of copies of l just as there were lots of copies of the curved gamma all the way up but their lengths all got different the copy that's up here is actually nearly geodesic so the nearest bit things have been geodesic for a copy of l is up here so what that means is if I would pick two points on l and in the three-dimensional manifold try to go on the shortest path so who knows what the shortest path is it's like maybe it's like this right so actually what this proposition is saying that's wrong actually this shortest path can't come down so far it's got to stay quite close to here and how do I prove that so I haven't written the proof but you first of all on every level of this project onto the y coordinate the unstable leaves and you project onto a you project onto an unstable you project onto a hyperbolic geodesic you contract length but then you project upwards and remember the metric was something like dT squared plus c to the minus 2t I'm going to put it the wrong way around dy squared plus c to the 2t the x squared I've got that the wrong way around I think it's dx of y anyway you finally sort out which you think it is the point is as you contract the length along the unstable leaves is measured by dx and that contracts so you contract distances you go up here and if you know anything about quasi geodesics in hyperbolic geometry all this contracting is enough to prove you did better to just go near to l so this thing is realised up here and that kind of shows you why in the limit the leaf of this lamination so it had two endpoints here but in hyperbolic three space the image of this thing is going further and further away from the middle so it's two endpoints are kind of going out to infinity and in the limit the two endpoints come together so that's a theorem I guess originally due to Jaya Minsky that this all right that these two points come together so if I take a leaf of this then the endpoints map to the same point in the limit limit set so maybe that gives you an idea what all this talk about ending lamination being not realised in empathy just means there's no geodesic that represents it because the two endpoints of it map to the same point so you can't connect them via geodesic that's all realised means and here is a theorem which in various degrees of generality is due to these people so this map on limit sets is one to one except at the point where you've collapsed because it's the end of the leaf of the lamination so this limit set is so complicated because you've scrunched up all these points right now I'm going to take a little bit longer to explain the last example so the last example is a famous example of Jeff Brock we've got almost all the ingredients now so instead of taking sudo on us off I'm going to go back to my curve sigma and I'm going to find myself a map which we've got kind of the left hand surface and the right hand surface and this map restricts it to the left hand surface it's just going to be the identity the map restricted to sigma is going to be the identity but the map restricted to the right hand surface is going to be some sudo on us off map of the right hand surface okay so you can concoct maps like this and we're going to do just the same thing look at the sequence of groups so now what I'm doing instead of dain twisting here I'm scrunching up this side in some very complicated way what happens okay well here sorry here are oops here are pictures of the limit set these pictures were done by Jeff Brock so Jeff Brock's thesis is all about these examples so here we're very near I confess I don't completely understand this but anyway we're completely very near the limit and here we've reached the limit so again there's a kind of complicated procedure from here to here and what happens in this example is Brock showed just like in this dain twist case as you approach the limit this guy has to get shorter and shorter and twist more and more and in the limit a new parabolic element appears that commutes with this one and it means that the geometric limit and the algebraic limit are different sorry is degenerate yeah yeah partially degenerate I don't fully understand this picture but it's so this is what it's it's almost this way you've not done anything this bit nothing happened but this one is getting very short so they're getting almost tangent to something else and they're scrunched up ones that SR okay so here's our theorem that surprised us so the theorem is oh sorry that's the wrong gamut there are points in the limit set of the focusing group for which this sequence of images does not converge to this so if it was geometrically finer we said this converges point wise to this we're saying now there are some points where just doesn't converge and we can say precisely which points these are I think my time is getting short so we can actually say not only do we say there are some points we show that sort of I guess it's right to say almost all points converge so most points converge but there are some exceptional points which when we draw the lamination you always have certain complementary regions of laminations work it's a little complicated because we've got the sigma but this is kind of a complementary region these boundary things are the ones where the endpoints of one of these leaves don't converge to the end point of the limit thing which has collapsed so let me try and explain yeah yeah yeah so really I should be drawing it really I should be drawing like this I should be drawing a picture like this so I should have sigma here and then on the right hand surface on one side of this I have a lamination which actually it's boundary in the cover it would look like this wouldn't it so this is kind of all carried to it limits on sigma so I'm talking about a leaf like this or a leaf like this um so roughly what's going on is this side of the surface this side of the whole thing behaves like the previous pseudo anosa for example and this side behaves like a nothing example this and this stay bounded distance apart this and this get infinitely far apart this side goes up and up and the only way that can happen is if this becomes shorter and shorter so there's a kind of very sort of long thin bit here so this surface gets kind of distorted but if you chop off somewhere it looks like it always did and you have to go a very long way to get across here and then this side gets goes off to infinity and gets scrunched up if I take a leaf of this on the unstable lamination somewhere on here then just as in my previous example it goes up and up and up and up and up and in the end it's not there anymore it's quasi geodesic just like in the previous case so it goes up and up and up and up and up and in the limit it's not there anymore it's endpoints come together and that's it but in the geometric limit what happens um okay so that's here's an approximating manifold one side sort of stays put the other goes up and up and up here is the algebraic limit one side's gone off to infinity but in the geometric limit we take two copies of this just like we did in the kercoff thirsting case turn the other one upside down glue them together along the side that's kind of good here using a map t that mysteriously appears and we sort of get two copies of this and this side of the thing is fine and here we get an end going off to infinity so one of the ends going off to infinity is here and then we get another end going off to infinity here and this little leaf that I cared about is here and although down here in the algebraic limit it's nowhere in sights just disappeared actually because this half of the surface sort of stayed put in the geometric pictures you see this side as much as you see this side there's a remarking trick that says you see this side so this little leaf stays put here and so it kind of is within you know finite space so in all the examples it's a uniform quasi geodesic at bounded distance from the origin so its endpoints stay a uniformly bounded distance part so it it it doesn't collapse well I've suppressed the reason why I to make sure it stays a bounded distance I have to be very careful about my lift of this leaf I have to make sure I can get to it going up this left side which means I have to lift the pseudo nos off to the cover so that the leaf is fixed so if somebody cares about this you but but it has to be a boundary leaf to be able to make the lift properly to make this argument really right but that's why this thing so that's how we find our country example