 Okay, so this is the last in my talk about complex hyperbolic things, so I'm going to do what I was supposed to do yesterday, and then maybe I will try to do what I'm supposed to do today, and well, if we don't finish, we don't finish. So let's just remember that we've got A in- and I'm really wanting today to think of SU to start with, a ddwy'r Prifysgol, dwi'n gwneud o'r dechrau a ddim yn gwneud o bwysig o'r troi o ddwylai'r ddynami ac o'r ddwylai'r ddwylai. Mae'n ddwylai dechrau, a mynd i'n ddim bod ddim yn ddwylai diwyllwg, byddai'n ddwylai diwyllwg, ddwylai diwyllwg. A oedd yr wych yn cymdeilio arall i gael fuddiol, fynd am ydych chi'n ei ddefnyddio ar gyfer siaradol i'r ddwylai. Ac ydy'r thysg yw'r hynny yw'r ffordd yma? Y cyfnoddiad yma yw'r hirthaf o'r rhanau o bobl yn ymlaen o'r adnoddau cyfnoddau i'w cyfnoddau a'r cyfnoddau cyfnoddau, os ydych am yr adnoddau'n cyfnoddau a'r cyfnoddau i'w rhanau o heddiw, oherwydd hyfforddau a'r adnoddau, oherwydd yna nhw'n ddweud. I think I'm going to credit this to Chennon Greenberg, it possibly is older, but that's a fantastic paper from I think 1974. It aims to give a unified treatment of real complex and quaternionic hyperbolic geometry in all dimensions, Yn cael ei wneud o Octoenionic, ond mae'n meddwl i chi oeddiwch y pepe yma, chyn Greenberg Limit Set. Mae'n dda i amser o oeddiwch chi'n meddwl yma yma. A sy'n Loxodromic, mae'r ddweud wedi cael rhan o ffordd fwyntiau, Mae'r bwysig ar y bwysig yma o'r ffordd. Mae'n dweud o ddweud o'r ddimension. Rhywbeth y cerddol yn bwysig yn iddirian, ond mae'n bwysig. Rhaid i'r ffordd. Yn ei ddweud. Rhaid i'r ffordd. Mae'n ddweud o'r parabolic hawdd yn bwysig ar y bwysig a oeddaeth yn bwysig ar y bwysig, pan oedd o'r cyflwynt cyhoedd hwylwgol hiredrwyr. A i'w ddod o'r elyptiwch, i'w ddod o'r cyflwynt cyhoedd, o'r ddod y ffordd o'r hwylwgol. A'r ddod o'r ddod o'r ddod. Hwylwch yn adrygiad y ddod o'r ddod o'r elyptiwch a'i ddod o'r ddod o'r ddod o'r ddod. Felly mae'r rhaid i'w ddweud o'r blaenau, o'r ffordd o'r ddechrau F-L2R, oherwydd oedd yn ei wneud i ddweud i F-L2C, i'r ddweud o'r ddweud o'r tricotome. Felly ydych chi'n meddwl y gallwn i ddweud o ddweud o'r ddweud o'r ddweud o'r ddweud o'r ddweud o'r ddweud o'r ddweud o'r ddweud. ac mae'r ddweud y ddweud y gallwch chi'n gweithio'r ffordd yma yn ymgyrchol. Mae'n bwysig yn ni'n gweithio'r algebrau lineir, ac mae'n ddweud yma yn ymgyrchol. Mae'n ddweud yma yn ddweud yma yn ymgyrchol, ac mae'n ddweud yma yn ddweud yma yn ymgyrchol. Here's a nice lemur. If lambda is an eigen value of A, then so is lambda bar inverse. That looks really, really useful when you're dealing with the standard positive definite unitary matrices. All of your eigen values have the form e to the i theta, Ac y gallwch yn ddegon iawn, ystod sef yr eitau, dydy'r euithetaeth a'r euithetaeth yn ystod, roedd hi'n gwneud. Llywodraeth rho'r coch fynd yw'r wneud erbyn bod Aamda erifrwyr o'r rhaid. So bod oeddwn yn dda i ddigon yw'r cyffredig. Gwrs eu bod yn thyf i ddim yn drwsbyddio o quwitarniaeth a ddwy kitwch gyda'r wahanol ymlaenau euithetaeth, ac gallwch yn gweithio'r wahanol am erioed – I'm really wanting to stick to the complex case and because I'm going to be dealing with traces and there's no sensible definition of trace for the, so you work a little bit harder, but the same sort of thing will work. I'm just going to stick to the complex case today. So let's see if we can prove this lemur because it's first thing in the morning, let's kind of see if that coffee is kicked in and see if we can prove this. Well, it's really easy to prove, which is good news. So lambda is an eigenvalue of A, this is what we're assuming. This means that lambda bar is an eigenvalue of A star. A star is the conjugate transpose. The eigenvalues of a matrix and its transpose are the same, so all I would do would be complex conjugating the characteristic polynomial, so lambda bar is an eigenvalue of A star. What do we know? A inverse is conjugate to A star and if lambda bar is an eigenvalue of A inverse, then lambda bar inverse is an eigenvalue of A. So now, if lambda is an eigenvalue of A with mod lambda strictly bigger than 1, so is lambda bar inverse, which is not equal to lambda, and if we're in the case of 3x3 matrices, so we're just going to assume that n equals 2 here, so we're in su21, that means that the product of the 3 eigenvalues has to be equal to 1 because that's the determinant, and so that means which would be, which is, and notice that this has got absolute value 1. So that's immediately told me if I just have somebody, a single eigenvalue whose modulus is different from 1, that immediately tells me what the 3 eigenvalues are. Now let's suppose then that we've still got this same lambda, let's suppose that v is an eigenvector for lambda, then I immediately claim that v is in v0. Remember v0, this was the, it was a null vector, so it's on the light cone if you want to use the relativistic terminology. Well that's again not so difficult to see because if I think of vv, A is supposed to preserve this Hermitian form, so that's av av. Well av is just lambda v because it's an eigenvector. I can, I know how to take constants outside of here. The first term is linear, I take a lambda out, the second term is conjugate linear, I take a lambda bar out, so I'm going to get mod lambda squared vv, and the only way this can happen is if either mod lambda is equal to 1, which it isn't, or vv is equal to zero. You know, it's a sort of thing that's quite good to do just after breakfast because it doesn't take much work, but we actually feel like we're achieving something. So a similar argument is going to show that, so we've shown that this has a null eigenvector, similarly this one's going to have a null eigenvalue because its absolute value is strictly less than one. So we've immediately seen that if I have an eigenvalue with modulus bigger than one, then I have two null eigenvectors, that means two fixed points on the boundary. The dynamics tells me that one of them is attractive and one is repulsive because that's the absolute value of the eigen, absolute value of the eigenvalue tells me that, and so I've immediately concluded I'm loxodromic in that definition I've just erased. So a has two null eigenvectors for lambda and lambda bar inverse, one expanding because mod lambda is bigger than one or contracting mod lambda bar inverse is less than one, and so we're loxodromic. If I was in higher dimensions you might then ask, well can I have more of these things? Could I have maybe four fixed points on the boundary? In fact you can't because, and again if you go through the notes there's pages and pages of elementary arguments that tell you everything that's going on, but what's happening here is these two eigenvectors span a subspace where h restricted to this subspace has signature one one, again that's not so difficult to see, and then on the complement of that subspace or the orthogonal complement that means all vectors are positive definite, and so once I've chosen one pair of eigenvectors like this then all others would have to have, all other eigenvectors would have to be not on the null cone and so their absolute value would have to be one by that argument down there. I didn't say that very well so let me say it again. If I was in higher dimensions and I had a pair of eigenvalues like this then their corresponding eigenvectors span a subspace where the form's got signature one one, that means that all other eigenvectors would have to, would have to be positive definite then have to have positive norm, if they've got positive norm vv norm then the eigenvalue would have to have modulus one. So you keep on going like this. I could spend the whole hour and a quarter going through elementary linear algebra to give you this classification, it's very very similar. So what would happen now is we would restrict ourselves to where all our eigenvalues have got absolute value one, then we would go into the diagonalizable and the non-diagonalizable case, in the non-diagonalizable case there's different possibilities for the Jordan normal form that would then give me a finer classification of things being parabolic and so on. So I'm going to just cut to the chase and tell you, I had those three pieces of classification, there's some slightly finer classification in there as well. So that more or less deals with the whole loxodromic issue. So when I'm parabolic there are essentially three cases, in fact technically four cases, so let's do the cases, well we could have, we could have eigenvalues e to the i theta e to the i theta e to the minus two i theta, I must have a repeated eigenvalue and they must have norm one so I can always take it to be like this. In which case the eigenspace corresponding to this eigenvector is not diagonalizable so this has to be one-dimensional, a similar argument shows that the vector has to be null just using the Jordan normal form and so what I'm going to do is I'm going to get something where in my Heisenberg picture from yesterday if this eigenspace, well the eigenvectors there but there's a, well okay if it would, if it looked like that, where's my color's gone, it's over here, then you would have something that would be a translation upwards together with a rotation around and so the orbit would look like that and so we would call this a screw motion or a screw parabolic. A long time ago Bill and I flirted with the term ellipto parabolic but I decided that I preferred screw parabolic because it's sort of a bit more visual. Of course if I conjugate this by one of these translations, all of these contact planes are going to tilt so I could get something that's like a very tilted rotation like that. Okay, otherwise I would have eigenvalues, well one, one, one or equivalently I could have omega, omega, omega or omega bar, omega bar, omega bar where omega is e to the two pi i over three because of course all three of these are equivalent because I have this, I can always multiply by a cube root of unity because I'm on this three-fold cover but normally we would just say that this is unipotent and actually I classified these for you yesterday. These are going to be these Heisenberg translations so that that would be Heisenberg translation and so that would be by something like tor t. So if tor is non-zero we call this non-vertical and if we have zero t we would call this vertical. This is in the centre of the Heisenberg group and I should be really careful if I'm doing conjugacy classes then I ought to put plus or minus there because upward translation is not conjugate to downward translation in exactly the same way that in SL2R left translation is not conjugate to right translation unless I introduce orientation reversing or in this case anti-holomorphic isometry. So that gives me a pretty good classification of what's going on here. The final case is where I'm elliptic so again we have some sub cases we would say that this is regular elliptic if all the eigenvalues are different and this is going to fix a unique point in H2C corresponding to the negative eigenvalue to eigenvector v minus. So given a set of three eigenvalues there distinct eigenvalues there are three choices for conjugacy classes because there are three different possibilities for which one could have its eigenvector in negative space. That's a slightly subtle point which I think Bill Goldman misses in his book so there's a little point there where he's just slightly inaccurate but it's one moment's thought you can fix that but that's just to warn you in case you use those results. If it's not regular elliptic well you could just call it special elliptic and sometimes that's used. In fact at least for two complex dimensions there's we can call this a complex reflection either in a point or in a complex line. So complex reflection in a point that just means that on the entire tangent space you do the same thing. You have a multiple of the identity perhaps by a rotation of some sort whereas complex reflection in a line that line is complex line is fixed and you would rotate around that. These ones are also sometimes called boundary elliptic because they fix a bad point on the boundary. So here we have a unique fixed point again but the dynamics are different than the regular elliptic case and here we have something that fixes a whole complex line's worth of points to go faster. One of the fantastic things about SL2R and SL2C is that we can determine the conjugacy class or up to a little bit about slight ambiguity we can determine the conjugacy class only using the trace. Of course we always have the ambiguity of parabolic or identity there and it's the same thing here. So if A is in SU21 then the characteristic polynomial of A is going to be x cubed minus tall x squared plus tall bar x minus one. We know that the trace should be this coefficient and the determinant which is one should be that coefficient. It's not so difficult to determine what this coefficient is because what is it? It's the sum of the products of pairs of eigenvalues because I've only got three that would be it's the sum of the one over eigenvalues but we've already seen that one over an eigenvalue means that I could then take a conjugate of a perhaps a different one but among the set of them that would be preserved. So in fact we get this value and we want to know when does that have a repeated root because that's the sort of thing that we would use to classify what's going on. So this has a repeated root when chi A of x and its derivative share a root that's kind of the first elementary thing you learn in calculus I guess and then there's a great thing that tells me when two polynomials share a root and that is the resultant of those polynomials and to cut a long story short that means so so Tor here I should have said I didn't tell you there Tor is the trace of A I always it's yes there's an ambiguity there because Tor was my translation part so don't there's only a finite number of letters in the world and if I cut down to the the chase then then we have this polynomial so we have a discriminant or resultant polynomial this is a resultant so how do you calculate the resultant it's it's an m plus if you've got two polynomials p and q of degree m and n it's an m plus n determinant where you put the coefficients of your first polynomial in your first entry is fill up with zeros next row down put a zero do the same thing fill up with zeros all the way through until you've got a bunch of zeros then your coefficients are your first one then you start with the coefficients of the second one you put a whole load of zeros you shift across you shift across and so in this case we're going to have a five by five determinant to calculate um and there's a great theorem that says that those share a common root if okay and um in general for unitary matrices is shiv around no okay there's a paper I wrote with shiv and with prishnendu when we did the same thing in general we had some nice formulae that shiv produced um to tell me what's going on and then we we looked at the case of four by four matrices so so this function here if I do the calculation it's going to be I get this this f of tor which is mod tor to the fourth minus eight real part of tor cubed plus 18 mod tor squared minus 27 it's unfortunately slightly more ugly than the analogous thing for sl2c which is mod tor squared minus four but this is the price you pay and the locus where that is zero is a is a classical curve called a deltoid so it's a very bad drawing of a deltoid I need françois to draw me a deltoid because he can always draw things much better than me let's try again so I'm going to have three points there so this is three this is three omega and this is three omega bar that should be minus one so I can come down like that like this deltoid so in the old fashioned days people learned all about these special curves I talked about lemniscates yesterday and what happens is in here we're regular elliptic out here we're loxodromic here we would be unipotent so that's Heisenberg translation or identity and also there and also there because remember I have cube roots of unity and on there I would either be screw parabolic or boundary elliptic or sorry or complex reflection right so when when we go to the next dimension up the characteristic polynomial is no longer determined by the trace it's because it's a fourth order polynomial the first and last terms of the trace in the conjugate of the trace the middle term is real and so now I could draw something in three space where I have the trace down here and the middle term going up there and that's a beautiful surface called called the holy grail it's not wasn't our name that it's called the holy grail it's a something by called by chillingworth along and I can attempt to draw the holy grail for you if you want a proper picture you should look in my paper with shiv and krishnendu what you get is goodness me I hadn't prepared this so let's see if I can do this I might get this off by 90 degrees you have something that looks like a sort of a curvy tetrahedron that in the middle on the inside that curvy tetrahedron you get all of the elliptic people so it is going to be bounded in the same way and then outside here you get something which is like a bowl but which has been squashed flat at the bottom so it's like a parabolic type bowl but then you would squeeze it along the bottom so you would have these two corners and here you would have the same thing going the other way but it's so cunningly designed that it's also a ruled surface and so there are actually lines that would come down and down and down and other lines that would come down and down and down like this okay so this is actually a ruled surface which I can't possibly draw properly yeah and also so so it's that is this surface and then there are four space curves which are called the whiskers which they kind of come up and they sit inside the top bowl here and inside the bottom bowl there right and what happens in this case is inside that here you would have elliptic inside the top bowl and the bottom bowl you would have things that had were were doubly loxodromic so you would have two pairs of so this would only work in u22 wouldn't work in u31 so you would have two pairs of loxodromic six points so ie only in u22 and out here would be my standard loxodromic where I have one pair and then one pair of unit modulus things the things on the whiskers they're the ones out there but where both of these loxodromic eigen values are the same so we get a repeated pair of like we've got lambda lambda bar inverse lambda lambda bar inverse and then all of the other things they're all going to happen in different places like that and I can't draw the picture in higher dimensions than that because it's that would mean I would have two complex dimensions and well it's a Thursday so Todd would be able to do it for us right okay so that's that's how you do some classification right I mean there's all sorts of other things I can do I can if I have a loxodromic element it's going to have a translation along a vector together with a rotation I can read this off from the trace I don't get a nice function like cosh anymore but I can still do it there's all sorts of you know I can so on and so on so all the things that you might hope for in from sl2c of using trace to get to determine geometry or geometry to determine trace and more or less work in this case but what I wanted to do I want to now switch gear this was the end of yesterday's talk so the beginning of today's talk is I wanted to to talk about an analog of these volumes of representations and and to see how you might represent surface groups inside pu21 so are there any questions about these traces and eigen values and eigen vectors before we move fantastic it's all so confusing that nobody can formulate a question so I would normally call a surface sigma but in the in the in the honored tradition of this conference s is now going to be a surface of genus g with p punctures so it's finite type Riemann surface and I'm going to assume that chi is negative so okay I'm going to let pi1 be pi1 of s the fundamental group and I'm going to be interested in representations of rho from pi1 into su21 the ones I'm going to be particularly interested in so I'm going to look at all representations but then I'm going to look at what I would think of as interesting representations and so rho is called complex hyperbolic quasi fuxian if rho is discrete faithful geometrically finite I'll tell you what some of the I'll give references to some of these words in a moment and type preserving that just means that loops that are homotopic to boundary components are represented by parabolic elements and everything else other than the identity is represented by loxodromic elements peripheral loops parabolic when I say loop I should mean homotopy class of loops because I'm dealing with the fundamental group and I possibly also want to think of base points but I'm just going to call them loops and otherwise loxodromic let's just briefly talk about what geometrically finite is the classical definition for geometrically finite is that there exists a finite no is that all Dirichlet polyhedra should have finitely many sides or perhaps all fundamental polyhedra should have finitely many sides there are then four other equivalent definitions and there's a beautiful pair of papers by Brian Bodich where he first deals with the with the constant curvature case and then the variable curvature case and there are five equivalent definitions of geometrically finite except in the variable curvature case number three which is finite sided fundamental polyhedra doesn't work in fact in higher dimension real hyperbolic dimensions I guess that doesn't work either and so actually there are four equivalent definitions and maybe in the spirit of this conference the one that you could think about because Pepe's been telling us all about limit set so I'm now going to think of Chen Greenberg limit set is that every point of the Chen Greenberg limit set is either a conical limit point or about a bounded parabolic fixed point now a conical limit point means that when all of these orbits accumulate they're going to accumulate in a cone and of course cone now is going to have to be some sort of paraboloid because because of the geometry here but there's some something that we can think of as a cone and bounded parabolic fixed point means that either it's got full rank or we can we can just put neighbourhoods of the domain of discontinuity in a suitable way around it I'm not going to go into that so geometrically finite is just you can think of it as nice so now I want to think about how I might start to construct such representations while the limit set could be a wild knot yeah which is exactly the things that we were talking about on the problem session a couple of days ago so I don't think I would like to call it a wild not a Jordan curve although you can there is a homo there's a there's a there's something you can map the circle into it in a sort of a sensible way but so how might I construct such representations well let's just I mean my philosophy which I think you're hopefully getting to grips with now is whenever I think of a problem like this I think what would I do in the case of SL2R and SL2C if I want to construct a representation of a surface group into SL2C the easiest place to start is to put it into SL2R and then embed SL2R into SL2C in the obvious way so that's good we can do that because what that would be really saying is I've got a copy of the totally geodesic copy of the hyperbolic plane and my representation is going to preserve that that okay well we can do that we have two different sorts of totally geodesic copies of the hyperbolic plane either complex lines or totally or Lagrangian totally real planes so if rho is fuxian preserving a complex line we we're very imaginative with our names we we say that this is we say it's c fuxian and if rho is fuxian preserving a Lagrangian plane we say that it's r fuxian another way to say this is that up to conjugation we could so c fuxian means that we would have rho going from pi1 into well we've got u11 which is the same as as um I guess what SL2R but and I want to sit this inside SU21 so how do I do this well I I can think of having here's my SU21 think of the of the the ball model I put my u11 piece here put some zeros there but I've still got somebody up in that top left hand corner so I write this as u1 cross that and then I take s of the whole lot so obviously that's isomorphic to to u11 because I could just um the the the first entry is always determined by the is the conjugate of the determinant of the second one and it's r fuxian if and and this is supposed to be a fuxian representation right we're not going to deal with these non-maximal volume representations right pi1 into SO21 sitting inside SU21 so in the first case this one would preserve the complex line preserve z1 equals zero so if if my picture is here there's my z1 axis there's my z or z2 axis maybe up here then it's going to preserve this complex line so we're going to have some sort of fundamental domain in here and then I'm also going to be allowed to rotate around that complex line because I've got an extra complex dimension and so in fact this is going to be a belian representations cross fuxian representations the second case this is going to be this is going to be to preserve uh r2 intersect h2c which again looks like a disk but it's a different sort of disk right I want to now define an invariant of this representation which is going to be very similar to the volume of the representation that we've already seen and this is going to be the teledo invariant and guess what teledo invariant is also called tor so it now had three different tors so you should really shoot the speaker right but this one's tor of row so hopefully it's not going to get confused with tor of an element or tor the translation length I'm going to give you two ways to define this um let's let's do the first way which is the proper way and then I'll tell you how to think about it and they're both of them going to be related very closely to what Bertrand would say I have uncomplex hyperbolic space a caliform and that's so caliform is you you take your your metric and you but you plug in uh the j operator which is the square root of minus one in the second and then that's going to give you your caliform which I again is going to call omega this is nothing to do with the cube root of unity and I'm I'm going to take s down here which is my surface and here is its universal cover I want to take a map here which is f into h2c which is going to be a row equivariant and um immersion I guess of the universal cover into that so so f is supposed to be row equivariant so let's remind ourselves what that means it means that if I've got alpha in pi one then if I take x and an x in s tilde if I first apply alpha to x and then apply row no apply f of f so that it should be the same as first applying f to x and then applying row to alpha okay and we saw over the last couple of days about how we might construct row equivariant things in um in the real hyperbolic setting and um you can do a similar thing in this case so how do I define my my Toledo invariant I take my caliform on complex hyperbolic space I pull it back under the the the equivariant map f I integrate over my surface s I always like to divide by two pi for good luck and then that is going to be my Toledo invariant okay so very much like the definition of the volume that we saw before except I'm using the caliform rather than the the the hyperbolic metric so properties of of Tor Tor varies continuously with row in the obvious topology on the representation space which you could think of as being continuously varying all the matrix entries if you if you want to be very low tech it's independent it's independent of of my equivariant map f it lies in the interval from the Euler characteristic to the minus the Euler characteristic notice I'm being I'm not writing 2 minus 2g because I'm possibly thinking about punctures sorry if I I'm going to tell you what the the different cases in just a second so okay so see fuchsian if and only if mod mod Tor is is minus chi our fuchsian implies that Tor is equal to zero and now we're going to start to get into the the different cases whether we have punctures or whether we don't yes row that's this is supposed to be an alpha in here it's just that my alpha kind of kind of sank and then yeah yeah so I mean over here I've got a caliform so I'm pulling it back under here and then I've got a fundamental domain in my universal cover because because it's equivariant I could just integrate over my surface down here for exactly the same reason that in the case we've been talking about before I guess that that you could have non-trivial things with volume zero can you have none yeah exactly and if you just hold on to that question actually yeah I haven't told you the second way to think about this so well it would just hold properties for a moment the second way to think about it is the following and again this is very much like what we've seen I'm going to take my surface possibly with punctures and I'm going to triangulate it and let's just see what happens to a given triangle so let's call these points p1 p2 and p3 and when I'm in my complex hyperbolic space let's just suppose that p1 appears here and p2 appears there and this is the complex line spanned by those two now some p3 is somewhere up there and what I'm going to do is I'm going to take the orthogonal projection of p3 down into this complex line that gives me a Poincaré triangle so this is pi12 of p3 gives me a triangle I know what its Poincaré area is the Bergman metric on this complex line is the same as the Poincaré metric and so I take its Poincaré area signed because I want to make sure I'm going around consistently in the same way so we take the signed Poincaré area and then I sum this up over all triangles in my triangulation and I guess because I've I've also got this area business because I want to therefore still divide by two pi and I'm going to get my Toledo invariant so so one over two pi times the sum of all the areas of the projected triangles gives me the Toledo invariant and again you it's very much like what we've seen about these triangles yes it's this unique line spanned by p1 and p2 I mean you then have to check that if I took the line spanned by by p2 and p3 and projected onto there I get the same thing so there's there are many things to to check if you use this directly the way I like to think about this is the the caliform or rather the complex structure which is the caliform is trying to measure is something like sunlight coming down on a on a nice sunny day like we have outside here and I have my my my surface sitting somewhere in in my space and and the sunlight comes down and I get a shadow downstairs and if my surface is exactly lined up with respect to the to the the the complex structure then the area of my shadow is the same as the area of my surface if it's tilted like that its area is going to be zero and I could also get zero by this is what this is what I was going to do this is my clever trick is if I were to triangulate it with triangles like that so that all my positive area and my negative area cancelled out right so that's not lined up like that but I would get lots of positive triangles and negative triangles all cancelling because I've got an orientation on here and I've got an orientation down here and if this point ends up on the wrong side of that line I would say I get a negative one because if I go around my triangle and it should always have been p1 p2 p3 if I get p1 p2 p3 I get positive p1 p3 p2 I get negative yeah if it lands exactly on this line I get a degenerate triangle and the area is zero and that's exactly what's going to happen in the aphuxian case so perhaps I just draw another picture there in the aphuxian case then I'm going to get all three points are going to lie on the same line so I get area zero this is aphuxian and in the cphuxian case it's going to all lie in the same line anyway and so I get the full area of the triangle so that's this is how you should define the thledo invariant and that's how you should think about it at least if you are a visual thinker like me so now we split between the cases where we have punctures and where we do not have punctures so assume p equals zero i.e s is closed without boundary and my representation is going to su21 then my thledo invariant is an even integer it's not quite what we've been seeing there because you were allowed all integers for your volume here we have an even integer if I on the other hand I went to pu21 then it would be two thirds of an integer but I don't I don't want to think about that case so let's we've learned lots of things let's put them together in something which is called the goldman thledo rigidity theorem so we're going to suppose that p equals zero and that rho is cphuxian or rho nought is cphuxian and we're going to say that rho is is a nearby representation in other words something I can continuously deform rho rho nought into rho inside my representation variety so let's think about what this means c nought c rho nought cphuxian so the thledo invariant is plus or minus the Euler characteristic p is equal to zero so the thledo invariant has to be an even integer and it's a continuous function on my representation variety so as I continuously deform rho nought I get a new thledo invariant which also must be minus the Euler characteristic because it's a continuous integer valued function on the representation variety thledo invariant of rho is minus plus or minus the Euler characteristic and so that means that rho is also cphuxian so if we want to develop a theory of quasi conformal represent quasi quasi quasi fuxian representations it looks like we've got off to a very bad start because we want you know the classical thing is you take a fuxian representation and you wiggle it a little bit and then you get something new unfortunately we've discovered that we can't get anything new so that's either good news or bad news but the the rest of the news is that after that things change so let's just suppose that p is p is equal to zero still sorry and rho nought is now a fuxian so let's just remind ourselves we've got a whole copy of tychimolo space here which is going to be six g minus six dimensional my my representation space is going to be two g minus two times the dimension of the group so that's going to be 16 g minus 16 dimensional and so my tychimolo space is cutting out a really quite a low dimensional subspace of my representation variety near that point so the theorem is that there exists an open neighbourhood of chqf representations about rho nought and i know three completely and utterly different proofs of this fact and i think that any mathematical fact that way you have three proofs or three different approaches that actually shows that something is interesting is happening the i'm not good i don't know what what i think the historical order is probably that geisha proved in fact that the space of chqf representations is open and that's he's what he's doing there is he's comparing the the the bergman metric on the surface with the word metric and as a hyperbolic group and so he's looking at things there okay so this is actually a special case of a very general result this would be geisha so this is via let's just say gromov hyperbolic it's a yeah it's a sort of geometric group theory more specific than that i think secondly in historical order but at about the same time as this it's a result of of me and platys and we really built fundamental domains so what we did was we took we've got an aphroxian group so this is a this is a lagrangian plane and we have some fundamental domain in there if i look at the pre-image under orthogonal projection i'm going to get a sort of a tube over that which is fibred by other lagrangian planes and then we show that if i wiggle my representation sufficiently small neighbourhood then then this same picture persists so it's just so this is really using the complex hyperbolic geometry and the third proof is due to loft in and macintosh and that uses cubic differentials i could just basically say analysis and what they do actually is something even more beautiful we know we had our row equivalent embedding of the universal cover what they're able to do is they're able to to produce a row equivalent lagrangian minimal surface inside um so i know this is not the way they do it not the way they tell me i should think about it but i still think about it as if i think of my limit set as being a wiggly actually in this case it is going to be a a a Jordan curve a fractal Jordan curve then they're somehow solving Dirichlet problem to get a a lagrangian minimal surface which or whose boundary is that actually which is also a quick variant so they're sort of thinking of dipping their limit set in soap film and then seeing what produce what happens well the connection is well i think yes we're only a very indirect one um that we're we're dealing with cubic differentials because i mean it's the next it's the next dimension up so i think there's the cubic differentials is because we're dealing dealing with three by three objects i think and um and then the reason why that thing was triangular was because i was dealing with three by three matrices but it's it's the right so that there there's a beginning and subsequently Lofton has a paper with a couple of other people who i forget who they are why he's developing this theory further so one of the things that we're really missing in this complex hyperbolic quasi-fuxian is a good analysis treatment because it was alphas and bears really got the classical theory going using analysis um and so um so so they're solving a particular differential equation and one of the the the things that comes out of this solution is that they produce a Lagrangian minimal surface yeah that that that you right okay can i can i can i quickly finish my story and then i'm going to can i'm going to treat this can you hold this over to the questions because i've got about two minutes um and i if i do that then i'm going to go over time um so what happens when p is not equal to zero well in this case my Toledo invariant does not have to be an integer this is exactly why i was asking this question yesterday about what happened when you were having domination with these um um punctured surfaces and you said well we get cone angles um we no longer have an integer invariant but what happens is remember these screw motions my cusps get represented by screw motions and this angle of screw together with um the Toledo invariant has to be constant so when i when i when i change my Toledo invariant this angle somehow picks up the difference but this is a result of of Nikolaj Guzefski and me is that there exists a continuous path of um c h q f representations interpolating between r fuxian and c fuxian so this goldman Toledo rigidity theorem doesn't fails and so actually i don't just have an isolated component of of c fuxian things i can go all of the way down to the r fuxian things and perhaps uh to end with a somehow a little nice little story about this um in 2012 i was at a conference Pepe was there as well uh Ravi was there in Almora in India and i we had a question session like we had yesterday except we were having it out on the terrace because it was the only place that was was warm enough and um and Ravi was asking me about this construction and how we did this and at one point i was using a theorem of somebody called millington and all i knew was this person was called mh millington and so i said he and Ravi said no millington was a she okay that's fine i learned something that day i then went back and i looked her up on the web and it turned out that she was a student in Durham in the early 60s um and she got the best exam mark of any student over a long period of time which was in those days you could get as many it wasn't just limited to 100 you could just do as many questions as you like and you would get the mark for that and her mark was double the mark of any other student and unfortunately she died very young she wrote two papers um but um so i learned about about you know very very good student at my own university by traveling halfway around the world and listening to Ravi so i have still have a lot to learn okay that's all i'm going to talk about now so that's it