 Okay, so now I want to Somehow start approaching the third point so we have a surface We know we can we know how to put a hyperbolic structure on it So now we want to understand all the different ways we can take this one Oriental topological surface and endow it with a hyperbolic hyperbolic structure and I mean you can now think of What the hyperbolic structure is in whatever terms you want to think about it So we really want to look at this space of all hyperbolic structures and there's one way To make this a bit simpler to look at that is to look at not just a hyperbolic structure on the surface But at the same time look at what people call a marking So what we really will look at is at marked hyperbolic structures so For this we you think of We have our topological surface Oriented topological surface of genus G here and We have the same surface with a hyperbolic structure and we will denote it by a different letter So whenever I have a surface with a hyperbolic structure given on it. I denote it by X and Now I want that this My topological surface and my surface X and they are topologically the same surface, but I want a bit stronger relationship. I want to Have a way of comparing this surface with this hyperbolic surface by Taking in addition homeomorphism from this topological surface to this hyperbolic surface But I mean I can forget about the hyperbolic structure So I want to think of a hyperbolic structure not just as the surface X Which carries a hyperbolic structure, but as a pair of the surface X with this homeomorphism so we Look at the space of all hyperbolic structures on signal gene. So this is a set of pairs X and F where X is a hyperbolic surface and F From S to X is a homeomorphism But now we want to We will actually mod out by some equivalence relation which I want to explain now So if we didn't have the marking the equivalence relation we would take it just well we have two hyperbolic surfaces We identify them if there's an isometry from one surface to the other right because if we have an isometry We can't really distinguish the surfaces from their geometric from the geometric point of view So if we have this marked Hyperbolic structure, we take a slightly finer Equivalence relation. So assume we have Two marked hyperbolic structures X and X prime with their markings F and F prime We say that they are equivalent if we find an isometry here which Has the following property if we compare this homeomorphism from this surface SG to this Hyperbolic surface X prime and this homeomorphism which we get by going on the upper Taking the other marking and then taking the isometry. We want that they are homotopic or isotopic people such that so if there's so we identify them if there's an isometry such that F composed of I is isotopic to F prime And so this turns out that I mean not every isometry will identify the marked Hyperbolic structure, so it's we have a we have a space which Looks I mean basically the space of hyperbolic structures, but it's a bit finer than the space of hyperbolic structures So let me let's look a bit at what What this marking gives us so one thing this marking gives us and this is one of the key things is that if I have this Topological surface right and I pick a curve on the surface so for example this curve Gamma then having this marking I can Say on X which curve is the curve gamma So because I can just take f of gamma and this will be a curve on X and this is the curve with the name gamma So it gives me a way to label curves and identify curves on X And I could take an isometry of X which move this curve to a completely different one Which is not homotopic and this would be an isometry, but this would not give me an isometry Which gives me a man here my equivalence relation. So so markings give us a way to Name curves and this is very useful Markings that we have this marking also tells it that we have a nice group acting on this space namely the There's the mapping class group on The surface which acts on the space, so what's the mapping class group? So this is I take all homeomorphism and since I have an orientation I want orientation preserving homeomorphisms and I mod out by orientation preserving homeomorphisms of Sigma G which are homotopic to the identity so If I act with this group I can act on this space. So how do I act? I just take my marking and I pre-compose With a homeomorphism in the mapping class group and to check that this action is actually well-defined You have to I mean you see that if you act with something which is homotopic to the identity actually will get to equivalent marked Surfaces and if you act with something which is not homotopic to the identity it will get actually two different Marked surfaces, but as a if you forget if you would forget about the markings So just think about the hyperbolic structure. They are they're the same surface right, so So if you are interested in just a space of hyperbolic structures without marking You can look at this space of marked hyperbolic structures with the action of the mapping class group and take the quotient Okay, so this One other having this label for Names for for curves on the surface allow us to give a very nice parameterization of this space of marked hyperbolic structures and We will see that this is a very nice in some sense simple Simple manifold so we have nice parameters and I want to this this is the first thing I want to describe on this space and we basically have Everything we need for that already, so there's a nice parameterization of the space which are called It's called fentanyl reason parameter parameterization or fentanyl reason parameters, so for this We use this the idea we already used to put a hyperbolic structure on the surface We look at pants decomposition, so we fix a pants decomposition fix a Pants decomposition of sigma of sd. So just topologically And now if we have if we if we have a hyperbolic structure and a marked hyperbolic structure want to Parameterize this hyperbolic structure by somehow a set of parameters which we associate to the surface Given this pants decomposition, so we actually fix Something slightly more than a pants decomposition. So we also fix Some kind of perpendicular. I mean we're not perpendicular, but transverse ox here, but I don't want to go into that much detail for each of the Curves so we have this three Draw them orange so we have Curve C1 Curve C2 and The Curve C3 so we have in general three g minus three simple closed curves and What do we know so we know that if we if we had a hyperbolic structure on the surface and we cut this hyperbolic surface along these Simple closed curves we can measure for each hyperbolic structure The length and we get for each curve we get the number so we get We have pair X F we get The length of these Three curves if we realize them as simple closed geodesics with respect to the hyperbolic structure, so we have Three g minus three positive real numbers and We basically know that if we cut our surface open on each of these pairs of pants We have a unique hyperbolic structure, which is determined by these numbers So now we have all these pieces and now want to understand. Well, if we try to take these pieces and glue our Surface X F back together What what can go wrong, right? So if we take two pairs of pants, let's say with length L1 Could make this L1 it doesn't matter so L2 L2 and three Three Right when you want to glue the hyperbolic structure here and here I really have to make sure that the length is the same because I want to glue the curve but then I mean I glue and I Can somehow have a degree of freedom which I can glue so I have just two circles and they can rotate them Around each other before I glue so there's addition to the length parameter when I want to Re-glue my surface out of the pieces there's for each curve among which I glue I have what's called a twist parameter which tells me how do I glue and how much I have to twist so so I get For each curve a twist parameter tau 1 up to tau 3g minus 3 and This is a positive Real number and the one important thing and this is again because we have the marking the twist parameter is actually a real number it's not just in the circle because They take a pair of pants two pairs of pants and now I twist a full turn around the circle I mean for the hyperbolic metric it doesn't change but for my marking it changes. So if I look at such a curve, which is transversal and I wrap them once around The point and then for example I close them up to be a loop then I change the homotopic class of that loop So if I have this I mean this marking the twist parameters are really real numbers and now If I if I give you these numbers Right, you can construct for me marked Hyperbolic surface so what do you do you take these numbers you construct 2g Pears of pants where you somehow take the you have the building plan out of a dear topologic pants decomposition of your surface So you take you produce the hyperbolic pairs of pants and then you want to put them together and the twist parameters tightly precisely How to put them together so this way you can really associate to any mark type of electric structure the set of 6g minus 6 numbers and go uniquely back To construct the mark surface. Yes Where do I read the twist parameter? Okay, so so for this it's important to have this transversal. So this is one thing which is not so Nice about the twist parameters or the twist the length parameters are completely Canonical so I I mean once I fix my pants decomposition and I have say this curve I say okay, this is this curve in the hyperbolic surface. I take the Geodesic representing this homotopy class and this has a certain length and I can't do anything wrong So for the twist parameter, there are different ways of writing known twist parameters Depending on a choice and depending on a normalization if you want so one thing I can do is I can I so if I have this Transversals here, right? I can build my reference surface So if I if I have these length parameters I build these two pair of pants and now I glue the pair of pants and let these curves glue together Right so that they are really glued together now if I compare this with another hyperbolic surface having the same length Then I measure I mean how If I look at these two this transversal curve, how did I? How was it glued in the other hyperbolic? So I prefer not I mean we can discuss it after a bit It's a bit annoying with the twist parameter that it's we have to be a bit more careful to properly Define them really as real number not just modulo the the length of the corresponding to the second, okay Okay, so what does this? Title so if we do fentilnesian coordinates, we give you get the permutation of this space and we can permit rise it as just a Space a Cell of dimension 6g minus 6 So I mean if you want you could take the log of the length if you want to read a number and not the positive Positive real number. Okay, so this is this is in some sense very nice because it gives you a very nice understanding of how the space of marked hyperbolic structures looks like Let me make a couple of remarks. There's a one remark is It's actually Not so easy It's really difficult to Write down how the coordinates change if you change the pants decomposition So if you just think of the simplest example of having a Just looking at two pairs of pants glued along one curve and now I say well I take this this thing which is this four-hole sphere and I glue Want to glue it in this way into two pairs of pants And I want to understand how the fentilnesian parameters change. It's very difficult to write down So it's not Somehow not nice and so in some sense you get a very nice Parametrization, but you break a lot of the symmetry so For example, if you are interested in how the mapping class group acts, right? So if you act with the mapping class group will change the pants decomposition You don't see it at all in these parameters Right or I mean it's you can't so on the other hand they are also nice in other respects and I mean one thing I want to Mention without Really explaining What I say or going into detail. So if you look at this space of marked hyperbolic structures It as a space in itself has a lot of interesting Structure so for example, it's in a very natural way Sympathetic manifold in a completely in a way For which you don't need to fix this pair of pants. You can put in a write down the parameters But if you look at the symplectic structure and you will want to try to express it in this parameters There's a beautiful formula of boy part which says that it's actually very easily expressed in these parameters so If you look at these parameters you can think of them as functions on this space, right? The parameter gives is you take your point how marked hyperbolic structure It gives you you have this 6g minus 6 functions And if you have a function on a smooth manifold you can look at its differential and this gives you one form and So if you take the one form which corresponds to the length function of the curve Ci and you take the one form which corresponds to the twist parameter Then if you take the the wedge of it so and the two form Would you get by summing up this form one two three? G minus three this is a Sympathetic structure gives you the symplectic structure on the space of marked hyperbolic structures And if you think about it one thing this tells you this symplectic structure is completely canonical So in particular, it's invariant under the action of the mapping class group So I told you I mean it's really difficult to understand change of coordinate for pants decomposition so that it's it's a mess to understand the mapping class group action in these coordinates, but Actually once you do that so taking the one form taking the word looking at this two form It's invariant on the mapping class group. So that's one of the miracles of I mean of this formula So another thing which is nice about Fentiel Nielsen coordinates is that it gives you Immediately certain ways to navigate around in the space of marked hyperbolic structures because I can Take one of these curves. So I don't even need the full pants decomposition So I just take one curve here Well, perhaps it's better to take the I mean it's easier to explain with this curve here So I take one curve. I cut my surface into two pieces. I have the hyperbolic structure on the two piece I don't change it, but now I just reglu with a different twist And I can vary the twist as real parameter So I get something like a flow on the space of marked hyperbolic structures, which there's a name of Fentiel Nielsen twist flow and this again has a very nice Interpretation in terms of this symplectic Structure namely this flow is the Hamiltonian flow associated to the corresponding length function. So there there's tons of interesting nice structure in here and I'm not going to Tell you more about that. So what I want to What I want to now is do Stay looking at this space of marked hyperbolic structures But look at it from a slightly different Point of view. We'll get more in terms of This way which I shortly described of thinking of a hyperbolic structure given by a symmetry group of a tiling and we get to that From this space as well and for this the marking is again very very important so If we look at this Space of marked hyperbolic structures, right? So X is a hyperbolic surface and as a hyperbolic surface and I explain one way to think about it is as Some are having a tiling of the hyperbolic plane taking the symmetry group of this in this tiling and taking the quotient of the Hyperbolic plane by this symmetry group. So one way of thinking of X is to think of it as the quotient of the hyperbolic plane by a Group gamma where gamma is a discreet subgroup of PSA to R and then we have this Okay, so If I know and this is this is something I explained to a few of you in the in the break So if I have a discreet subgroup of PSA to our so there's the group of isometries of the hyperbolic plane Acting on the hyperbolic plane it acts always properly discontinuously There's just a small exercise of if you have a discreet group subgroup of a group of isometries That's always the case I get this quotient and this quotient has the in the surface has the property Because this is a contractable space that this group gamma is actually isomorphic to the fundamental group of the surface right Having this marking we actually Get another way in another way in an isomorphism with the fundamental group of the of our fixed topological surface SG namely we can look at this homeomorphism and look at what is the map it induces on the level of fundamental group So it sends curves to curves and it sends homotopic class of curves to a homotopic class of curves So the marking we did it gives us a way to associate this hyperbolic surface Map f star which goes from the fundamental group of this surface as Into I mean well it goes to the fundamental group of x at which we identified with this discreet group gamma So it goes to gamma which is a discreet subgroup So Well Let me write it in a different way. So I have here a map from marked hyperbolic structures to group homomorphisms of the fundamental group of sigma g which is some fixed abstract group into the group of isometries of the hyperbolic plane and One thing you have to know Understand and what happens with the equivalence relation and then I mean this is a short exercise that I mean the equivalence relation actually Tied to that. I mean what you get is actually well defined up to the action of PSL to R because you can act by if you have this homomorphism you can act on PSL to R by conjugation Right, and if you act by conjugation on PSL to R you don't change the the the equivalence class of the Of the mark hyperbolic structure, so it's like adding a isometry taking a so As defined there yes So I want to so I have xf and I send it. So this is the map So So I so I can think of I mean this is having a mark type of an extractor I can think of forgetting about part of the Part of the information I have and just remembering the homeomorphism the homomorphism I get on the level of fundamental groups right, so one Let me that's right one word here. So this Such things actually work in also more general context of geometric structure and what do you yeah I mean if you have a geometric structure you associate to the geometric structure its holonomy, which is basically this group homomorphism, so One thing I want to address now a bit is how do we understand the image of this map? so the first thing is That this map actually doesn't forget anything so this map is Injective, so we can actually think of of the space of marked hyperbolic structures on SD just as a subset as The subset of this Space of more algebraic nature of conjugacy classes of representations of homomorphisms, so So but now we so we if we think of this and here's a one other we we saw two other properties One other property of what the image is so if we have a hyperbolic structure I mean it's given by the quotient of a discrete subgroup of PSI to are not just any subgroup of PSI to are and we had already the question do we find surface? I mean groups which are isomorphic to surfaces which are densest they exist and so this is non-trivial information Here that I mean this if we think of this set of marked hyperbolic structures Just as a subset of group homomorphisms. It has one It has actually two nice structures, so No, so if I take a Homomorphism I'll just abstract homomorph and write it as row in the image of my holonymy map then row has discrete image and Because it it was a map which wasn't used from a homeomorphism. I mean it's also an inject It's really an identification of of the two groups. So it's it's also injective and This homomorphism itself Injective and so vice versa, but what I said before if I have if you give me such a group homomorphism So forget about all geometry. So we just have a surface. We have its fundamental group and We have the isometry group over the hyperbolic plane Then if you give me a group homomorphism, which is Injective and has discrete image you are actually giving me a hyperbolic surface Because I can take this This group homomorphism I get the if I look at row of my fundamental group of SG This now sits in PSA To R as a discrete subgroup Right so I can make it act on the hyperbolic plane by isometries Since it acts by isometries and this discreet dissection is Nice property discontinued. So I have a well-defined quotient manifold and this quotient manifold is actually my surface But endowed with a hyperbolic structure so so this set is Really almost equal to the set of group homomorphisms, which are injective with discrete image for just one slight subtlety that I was looking at Oriented surfaces and I wanted my hyperbolic structure to somehow be compatible with the orientation and so basically they are just two ways and I Can I can either take the surface with the orientation? I have but I can look at it also with the opposite orientation and In both case I would get discreet injective group homomorphism So if you just look at the set of discreet injective group homomorphism This will actually have two parts and one correspond to orientation Preserving hyperbolic structures and one corresponds to orientation reversing hyperbolic structures. Okay, so now I Want to address a bit the question. How do you? If you how do you how can you somehow re recover this set here of hyperbolic structures when you're really Looking at the space of homomorphism and okay So I said you can recover it if you give me an injective homomorphism with discreet image But if you give me Homomorphism, how would you give me a group homomorphism? So one way to give a group homomorphism of an abstract group is if you have a finitely generated group is to take the finite Generators and say okay, so I give you an element in PSA to R for each of the generators Right, then you have to check whether the relation is satisfied. So you have to multiply certain matrices and If the relation is satisfied, then this is a group homomorphism But if you do then and this is one way to give a group homomorphism, it's incredibly difficult to decide whether this is injective and It's also difficult to decide whether this is discreet except if you already give me so for example, if you would give me just matrices with integer coefficients, then of course it's discreet because the Matrix the group of P PSI to C so of a two by two matrices with integer coefficients is itself a discreet group But if you give me them to me in a somehow More difficult way, it's really very hard to check when there are whether group homomorphism given on generators is Has discreet image and whether it's injective. So there's so this is so I want to Give you a result and tell you about some invariant which actually in this case gives you in some sense a computable way to understand when you are given a group homomorphism in terms of generators whether this will be Injective with discreet image or if you want to be look at it more geometrically with that This comes really from a hyperbolex structure on your surface Okay, so for this let's start here. So let me just Very calm because because I erase it the so we look at the fundamental group of the surface and This fundamental group of surface has a very nice Presentation as a finitely generated group. So we have this a1 b1 up to a g Bg so these are my 2g generators and I have just one relation. I have to satisfy which is that the commutator Product of all commutators is one Okay, so now I am given a homomorphism of this group into p s to R so Row also PSI To our and for so it's basically given by taking a I and sending it to a matrix I and I tell you that so homomorphism so you know that if you take the product of commutators you get the identity here so now there's a very nice in In variant for this for such a representation, which is called the Euler number and let me give you the Let me tell you what the Euler number is or how you define it. I can think of it. So You can look at this group and let me so Perhaps let me digress a bit So the PSI to R acts on the upper half plane or if you want also on the poroncite disk or it acts on R2 but it also acts Not just on R2 and acts also on the circle So it acts on the circle and now you can think of it in different ways as the boundary of the upper half plane So this is r times infinity or if you look at the actual now to it acts on the space of lines and R2 So it's actually one in the circle. So this actually acts on the circle and It acts on the circle in a nice way by a homomorphism. So this is Actually sits in the group of mumboffin and now This is one of the reason why we looked at orientation preserving homomorphism We think of this action for example on the poroncite disk preserving the orientation Then the induced action on the boundary of the poroncite disk So on S1 also preserves the orientation on S1 so it acts in it by orientation preserving homomorphisms on S1 Okay, so now if you look at these Two groups or if you look at I mean this is Now I'm not going to explain this in too much detail, but these these groups are actually as I mean, this is these are legal so they are topological spaces as well topological space. They are not simply connected But they actually have a fundamental group So for PSA to R what's the basic rule what you could I mean if you think about it And what's the fundamental group on basically you have to understand and you can go from PSA to R to SO2 Which you can think of I mean if you think of the hyperbolic plane and look at the stabilizer of point It's SO2. So this is a contractable space. So this is really the same as SO2 and I mean SO2 is just a circle So is it topologically this is Homotopic equivalent to the circle and so you can look at not this space, but you can look at its universal cover and If you look at the universal cover together with the group Structure you get a group what is usually called PSA to R twiddle which projects here So this is just taking the universal cover and this is again just as a digression You can think of also lifting this in the action of orientation preserving homomorphism on S1 to Homie homomorphisms of the realign which preserve the order on the realign so Let me not not write it here. So this is not that important But I think Katie Mann next at the end of this week or next week will talk a bit about actions on the circle So I want to make this this connection here. Okay, so we have this This universal cover of the group and now if I have a homomorphism I can ask well Why do I have to look at it as a homomorphism into PSA to R? Why can't I look at it as a homomorphism into PSA to R twiddle or On the way said can I take my homomorphism and can I lift it to PSA to R twiddle? So can I write this map R as a map? R twiddle here and then just the projection down to PSA to R Okay, and now when we when you try to do that. So whenever I take one generator I Can just lift it So if I have the image of one generator this AI I can just lift it to PSA to R So I can do that with every generator and if I want to lift the representation I have to do it in such a way and I preserve the relation, right? So if I can look at the product of all these commutators a I twiddle Be I Twiddle or I mean if I write it in terms of row. This is row twiddle of AI Row twiddle of BI I have all these commutators So now the first Observation or short exercise one Can make is that if you choose two different lifts? I mean these lifts are I mean they are generally different, but if you take two different lifts and you Compute the commutator Actually, the commutator will not depend on the lift And this has to do because this if you look at it as a group level and what people call a central extension So if you look at the kernel here, what's the kernel? So we basically said this is like the circle So this is like the cover or universal cover of the circle So it's like R over the circle and the kernel is Then and if you look at that in here It's it's in the center of the group so it commutes with all elements and if I take two different lifts They differ by some element in that but since this commutes with all elements. I can take it out of the commutator so When I do that actually this commutator it doesn't depend on how I lift But it's not the commutator of these elements, but it's some Lifted commutator map one, but you have to compute this thing So take you can choose whatever lift you want Make it easier. I'll just keep this choose whatever lift you want you compute this relation So what do you know about this relation? You know about this relation if you project back to PSA to R It goes to the identity right because that's where it came from it came from a rapid from really group so, you know if it's If you look at it here, you don't know what it is But you know it's some element in this kernel of this map So it's some element in Z and this is so Whatever you get out of here. This is the Euler number. So it's a It allows it to take any group homomorphism of the fundamental group surface into PSA to R and associate to it an integer Right, so there is actually a very nice Thing that this you can't get any integer, but you can only get integers in a In a bounded interval, so it actually takes values in 2 minus 2g and 2g minus 2 so this We have seen this number. It's the Euler characteristic of your surface. So it takes values in this interval and this is a Result by I mean depending on which context you look at by millen or wood and so it's usually called the millen wood Inequality that this Euler number takes values in this in this interval and so here's a Here's a very nice Theorem of Bill Goldman Which tells you that this number actually tells you a lot about your representation. So this is 89 I mean basic most many of the things were proven in his PhD thesis And then I think it's published 92 So what he shows is that if you look at this number it actually Distinguishes for you the all the connected component of the space of homomorphism. So if you look at the the connected components of the space of homophisms PSI to our and I mean you can check that doesn't depend on the act by conjugation and Matters of the connected component of this space are distinguished by the Euler number so each level set of the Euler number gives you one connected component of the space of homomorphisms and row is faithful with discreet image and only if The Euler number of row in absolute value is equal to 2g minus 2. So there it is they the two extremes Give you discrete Faithful representation with discreet image and one extreme and will be those which preserve the orientation and one extreme will be those which reverse the orientation but this Tells us that the this if we we several really just start with the space of homomorphism. So without thinking about any Geometric structure the Euler number allows us to find in this space of homomorphisms all those homomorphism which come Which arises amount from a hyperbolic structure on the surface. So so I E so the Euler the level set of say Euler number 2g minus 2 as I love hyperbolic structures and this is This is In my opinion very nice Very nice result because it's it gives you in some I mean in principle a computable way To given the representation of your surface on generators to check whether it's Injective with discreet image. Yes Is there an example of a row which is injected but does not have discreet image? Yes, I mean both ways you can I mean so so for example, you can have a dense representation which is injective and And you could also have some which have discreet image Which are not injective this perhaps easier to imagine because you can just collapse handles and Okay, so let me just make one Side remark here without going into detail So you you saw then we what we really used to define the Euler number is that our group was topologically As a topological space at non-trivial fundamental group and we could take go to this Universal cover and then you could do the same thing as well. So there there are other groups There are Other groups So for example the what is called the symplectic group sp2 and R. So this is the group of linear automorphisms of a vector space so R to the end with a skew symmetric non-degenerate by linear form And which have the same Which have the same features same feature namely that you have a central extension by Z Z of that group so by a universal covering space and so you get You get for these groups if you look at group homomorphisms you get generalizations of the Euler number which go under the name of to lead a number and one of the things One of the nice properties one has for them is that Some of this to lead a number again is In integral invariant in some bound and if you are at the extremist of the bound You can ensure that your representation is faithful with discrete image But you will also have faithful and just a faithful representation with discrete image for certain other values So it's not the one-to-one correspondence, but just in one one direction and this gives you so it goes Meets to one people call Maximum representations and there are some Sometimes nice things one can show about them Some are making sure showing to what extent they are similar to hyper these Repentations into PSA to are coming from hyperbolic structures and to other extent which they are different so a lot of active research going on there and now I want to take the last 10 minutes to go a bit beyond Hyperbolic structures and I mean this was the first comment beyond hyperbolic structures, but I want to Make some I mean look at some other things where we see We go and go beyond hyperbolic structures beyond this group PSA to arm, but when we see some Also interesting geometry attached to that Let me get down the board okay, so let's So basically want to just shortly meant to you to two classes of With one group homomorphism or geometric structure. So one are quasi Function representations and the other are convex Protective representations okay, so and To this describe them in both cases some sense we get to these representation in corresponding geometric structures, which are not hyperbolic structures by taking so we have our X F Hyperbolic structure right mark hyperbolic structure and then we just look at the Homomorphism from the fundamental group into P That's And this is a nice group homomorphism, but now we can think of PSA to R is naturally sitting in other groups So for example in the group of Mobius transformations, which we saw so I think it's naturally a subgroup of the group of Mobius transformations PSA to C and I can think if we look at the The group of Mobius transformations, I mean we already on some sense from the the definition that it acts on CP1 Over this and actually has a nice Other and I'm not going to explain this a little bit It also rises as they are so much a group of a space with a hyperbolic metric on it, but just one dimension higher so this is a Group of isometry again orientation preserving of the Hyperbolic space of dimension 3 and I didn't tell you what this is But you can take the hyperboloid model and just Add a dimension a positive dimension to your Minkowski vector space and basically get them get the model you can also have you also have a ball model where this h3 sits as the interior of a two-sphere so at the interior of the CP1 and the way here we have a hyperbolic structure, which you could think of as This as the pouring create punk a disk embedded in this hyperbolic three space Sitting in here. So if you start with the group of models in which come from a hyperbolic structure, it preserves this Punk a desk and it also preserves the circle in CP1 And now you could start deforming this representation in PSA to see so you have your representation again given by a 2g Matrices with real coefficients and now would you just start making some of the coefficients complex in such a way that you still keep the relation True so that it's a it's a representation And then what happens so in the if you do that you will not preserve any more this copy of the punk a disk where you will just act on the full hyperbolic three space But we'll get that in the for small deformations in the boundary you preserve a fractal Circle so a fractal drawdown curve so a homomorphic image of the circle, but fractal one and So there's a neighborhood of all these homomorphic homomorphisms which come from hyperbolic structures Which have this property and these are called quasi-fusion quasi-fusion Representations and there's one Let me just point out two nice things about them if you look at this Picture so you still have the circle in CP one and I mean by Since you are subgroup of PSA to see you act on the entire CP one you preserve this quasi circle So this fractal circle and the action on this fractal circle is very bad So it's like the action of your surface group on on the boundary of the punk a disk When you take a hyperbolic structure, but if you throw out this Quasi-circle you have two things which are like discs right there topologically discs and the action on this topologically discs are nice so this is now not by a summit trees, but it's by Möbius transformations and The actual is properly discontinued so you can look at the quotient manifold So when you do that when you start with a quasi-fusion representation You get these upper disc and the lower disc and you get the quotient and get two surfaces which are quotients of An open subset of CP one so they still have a conformant structure They don't have a metric anymore, but a conformant structure until you get from a quasi-fusion representation to conformant structures and there's a Famous theorem which can't bear simultaneous uniformization theorem which tells you that the converse is also true So if you give me one topological surface and you give two different conformant structures, or it could be the same But I mean give two conformant structures on the surface then there's one way of Simultaneously getting those by realizing your fundamental group of the surface into in PSA to see as such a quasi-fusion representation, okay, so that's all I want to say and Dick Canary I mean you have one of the experts on quasi-fusion representations here So when you have questions I'm sure that you're happy to answer on the questions Okay, so now let me Just Shortly say something here. So these convex product was structures is in some sense the same philosophy, so you take your Holonomy representation of your hyperboleic structure. So this is or if you want any faithful homomorphism with discrete image in PSA to R now we embed it in a bigger group, but We take a different group. So we embed it in The group P as 3R But in a in a special way So now we have different ways of embedding it and the way we embed it is in the following way So if you think again of the very first model of the hyperboleic plane, we looked at the hyperboleic model then we have this Minkowski form in R3 and it's of signature one two and What I didn't say when I introduce it But basically you can think of the groups of asymmetries of this hyperboleic model as basically the group of linear transformation preserving this form we want to take the orientation preserving you have the form so you take the those with Determinant one and you might want to put a connected component here so this this group is actually isomorphic to this group and this group naturally sits in Although I have to PSA to R if you want put it first in SL to R and put it in there so so we and and the picture we which we should draw here is so PSA 3R X naturally on RP2 and we had the Klein model right So be the Klein model which was in an FN chart like a like a disc in in RP2 and this was preserved by By this I mean by SO 1 2 right so now when we deform what happens When you deform You deform this circle so you get and what happens that no matter how one deforms one still gets a domain which is strictly convex but Not anymore in the lips or circle and in general it will have as soon as you deform and you don't preserve this thing This will have C1 boundary, but nowhere C2 boundary, so it's it's not a fractal curve It's a nice differentiable Curve at C1, but it's nowhere C2 and then if you so there there two There's one one thing which makes this case very different from this one from for quasi-focusing representations if you if you Continue deforming at some point You will leave the set of quasi-focusing representation. So this fractal circle will actually break open and you don't have You don't have a frack quasi-circle anymore So here no matter how far you deform you will always have this strictly convex strictly convex set and this leads to so if you look at the set of representation that's Also, what is called the Hitchin component for? as 3r and there are Related things for if you go with n higher and let me just make one last comment. So here you have another nice Geometry, so there we saw a Conformant structure, so here if you have your surface group preserving this strictly convex set you can Define a distance function precisely the same way as we defined it here for the Klein model and this gives you a Distance function, which does not come from a Romanian metric anymore, but it comes from a Finzler metric On the surface and so you get this space of deformations You can get a nice geometric structure which corresponds to it. Okay, so sorry for going over time by three minutes