 And so what I want to talk about today is some piece of work I obtained in collaboration with Mark Boerger, and it's roughly one year old. And so I want to tell you something that we proved about ultra limits of maximal representations, but I actually want to mostly give you some context in which this result takes place, and namely I want to start telling you what are for me maximal representations, and why I do consider that interesting object to be studied. And I will motivate that by telling you some geometric properties of those, maximal representation, and then only after that I'll move to ultra limits, and so actions on asymptotic codes, so that I can describe you a result about elements acting with fixed points. But as I said, I hope I will try slow for known experts, and I'm sorry for experts who know all this stuff already, and maybe have even heard me talking about this already, but please ask me things if you don't understand. So what are maximal representations? So maximal representations are, for me, families of homomorphisms, from rho from gamma to g, where I have to tell you what's gamma and what's g for me, where gamma is a discrete group that's the fundamental group of a surface, that's a hyperbolic surface that's either compact or finite volume. Actually I only care that gamma being a marked fundamental group of a surface, I will end out this surface with a hyperbolic metric so that I can talk about geodesic, but that's not needed at all, I mean it's just to make some statement easy to say. And g is going to be the g group sp to nr, which is the matrix group preserving a symplactic form, namely matrices gl to nr, such that g transpose time the matrix 0 identity minus identity 0, g is equal to 0, is equal to these matrix J. So of course I could do this in a more general term, but I want to be concrete and I mean it's an easy linear algebra object, I mean this group of matrices. So I have those homomorphism, I will restrict to some class of nice homomorphisms of surface group in this, I mean which will give me subgroups of this league group that are isomorphic to either fundamental group of a surface or free groups. And I want to think of those at least for the beginning as actions of this group here on to the symmetric space for this league group of gamma. So what is this guy here? So symmetric spaces are examples of cat zero spaces and I like the symmetric space for the symplactic group because it's really in high, I mean it's a lot of a high rank generalization of the uproar of plane. So the symmetric space, so sp to nr is a simple league group with which has maximal compact subgroups that are, it's known compact maximal compact subgroups that are all homomorphic to un and the question space, the symmetric space can be realized as first x plus Iy of matrices such that x is a symmetric matrix and by n matrix and y is the positive definite and by n matrix. So you clearly see the analogy of this with the uproar of plane model. In the uproar of plane n would be 1 and indeed sp2 is sl2 and this is really the same model, the cat zero space and this is a model inside symmetric and by n matrices with complex coefficients that I think of in a fine chart of the Lagrangians of c to the 2n where Lagrangians are maximal isotropic subspaces for this symplactic form I wrote here. So this is really, you should really think of this as the uproar of plane where both the horizontal and vertical direction are symmetric matrices but actually the analogy goes much further. So I want to tell you why this is really like this, the uproar of plane. So this is a complex manifold and indeed in this business unless, I mean I know that Jeff is not going to be happy but the picture is going to be drawn in the uproar of plane model or the Poincare model but not straight line, I mean lines are not straight but moreover it has, I mean this symmetric space is a nice property so it's a cat zero space, so it's a geodesic metric space but in a symmetric space not all geodesic are made the same I mean that's the main difference between high rank symmetric spaces and rank one spaces. So, but however if you take a geodesic, a specific geodesic that's singular in some specific direction then its parallel set is a half-dimensional totally geodesic that they can use to play the role of geodesics in the standard hyperbolic setting so these are going to be parallel set of some singular geodesic and they have the property that have an intersection pattern that is really similar to the one of geodesic and indeed one way you can picture this so an example of those is the set of I, Y such that Y is symmetric positive definite that would be what you would draw here, right? that's an example of a geodesic and moreover you have also the generalization of the semi- the orthogonal semicircle that are obtained by translating this guy so this is a feature that it's an analog of what happens to the hyperbolic plane and another important feature of this symmetric space is that on the boundary, so there is some boundary that is endowed with a partial cyclic order S1 right that's the I rank generalization of S1 so let me tell you what I mean by that so now another difference of higher rank symmetric space as opposed to rank one space is that the boundary is not as easy as in rank one in rank one you have the visual boundary that's an easy well-defined object that coincide to the boundary to the homogeneous space for this group here and it's also the topological boundary in all models you draw whereas in higher rank it's more complicated both the visual boundary and the topological boundary stratify in different homogeneous spaces but both in the visual boundary and the topological boundary of this realization you see a specific closed G orbit so those are the Lagrangians of R2N so the isotropic subspaces such that the symplactic pairing restricted to W is identically 0 is a closed G for G's SP to an R orbit which lives both in the boundary so in the topological boundary of this realization of the symmetric space inside these Lagrangians and as a stratum and in the visual boundary as a cut zero space so of course you should notice that the Lagrangians are in N times N plus one half dimension on subspace as a real manifold whereas this is twice that dimension so it's not going to be the topological boundary in this realization however it's useful because you have this property that G acts on pairs of Lagrangians transitively we're here when I write two I mean transverse subspace so it's an open subset of the pairs of points and has only N plus one orbits in the Lagrangians on R2N to the third power I mean triples of transverse Lagrangians so in the circle you have that SL2R acts transitively on pairs of points and you have the orientation of triples of points that is complete invariant for the action on triples and in these case you have what is called the Maslow co-cycle classify I mean dermis this N plus one orbit and allows us to talk about positively oriented orbits okay are there questions on this so I want to look at actions of fundamental group on surfaces on symmetric on this specific kind of symmetric space it's actually slightly more general than this but it's going to be enough for today and I have three properties of these symmetric spaces that are the same as I see in SL2 and allow me to redraw many picture that I see for SL2 in this higher rank setting namely the fact it's a complex manifold that I have this half dimensional subspace and I have this partial cyclic ordering on triples okay and now this is going to allow me to tell you what maximal representations are and I'll do that by slightly cheating in the interest of time namely I give you a definition slash theorem this is due to burger yotse laburi and Wienerd which tells me that a normomorphism row from gamma into SP to NR is maximal or is a maximal representation if there exists a boundary map, a map phi from the boundary S1 into the Lagrangian which is row, equivariant and monotone monotone means sending maximal triples positively oriented triples to maximal triples positively oriented so here I'm saying that it's a definition theorem because actually you can define in a more conceptual way maximal representation using homology and overbounded homology but I want to sweep that over under the rag and just take this that would be a theorem proven using that fact as a definition for today because that's what's going to be sufficient for the rest that's what I said at some point I said that if it was a free group I was fixing a finite volume hyperbolization so that it's a quotient of the topological boundary I mean you can also do that by taking the actual boundary but then it's not completely true here that it needs to send every positively oriented triple to a positive oriented triple I mean it's all stuff that you can do but it's more annoying to write things correctly and indeed that's the first example so if n is equal to 1 then maximal representation so if n is equal to 1 as p2 is also 2 and maximal representations are precisely in this setting are exactly lifts of holonomies of hyperbolizations what I mean by this is that so if I fix an hyperbolic metric on my surface then I can identify the universal cover of my surface with h2 and this is going to give me a map from I mean given two different hyperbolization at that point I will have a quasi isometry between h2 to h2 that extends to a positive boundary map right so in particular now I didn't want to quotient by the PSL2R action by the SL2R action but this identifies with the tachymor space essentially I mean if you mod out conjugation so but there are two other important examples direct sum of maximal representations and this is easier to see with the homological characterization of maximal representations but also you can build up this boundary I mean if you have boundary maps you can build them together to get the boundary map on a bigger space and by these I mean that if I have a number of maximal representation in different symplectic group I can take the direct sum of those symplectic the orthogonal symplectic sum of those vector spaces and put them in blocks to get the maximal representation in the bigger group and the other example is that the image so if I denote by yota from SL2R to SP to an R the reducible representations then yota composed with rho is maximal for any hyperbolicization rho and this gives me a lot of examples of maximal representations in various symplectic groups and those are going to be in a sense of functional locus in a sense of functional locus or generalization of functional locus because it's going to be much smaller than the ambient group so any questions on this? So now, ok, so that brings me to part two in which I want to tell you why these are interesting objects and tell you some nice geometric properties of those objects that make them interesting to be studied for me so the first one is again, you do the same paper where the definition appeared so again for a symplectic group burger you also see laburi vener or with or without laburi depending on whether you want to do the open or closed surfaces which give me many very important features of these representations and the first one that's very stunning is that is a union of components so this is but more, those are all discrete and injective homomorphisms and say in the case of closed surfaces those are a nose of representations and if you don't know what a nose of representations are probably maybe Andreas will talk about it at some point during this week or someone but anyway this implies that they are QI and bad things and probably you know what that is that makes them interesting and in a sense all these properties together make them very interesting because I mean it's a very unusual phenomenon that whole representation in a component are very well behaved normally you would expect that you're able to deform a representation to get something that's non-geometric but in this setting the positivity forces nice behavior throughout the whole component but now I want to tell you a more geometric feature of these representations that makes them kind of interesting and it's one of the motivation what this fits into the higher tecneural theory setting which is the fact that these representations satisfy color lemma so they're really remembering a lot of the geometry of the hyperbolic of the tecneural space and let me start recalling you what color lemma means because it's a very useful fact in hyperbolic very old and very useful fact in hyperbolic geometry that maybe not all students know at this point so I fix an hyperbolic metric on my favorite surface that's the surface of genus G and I assume that there is a geodesic gamma in my hyperbolic in my hyperbolicization which is very short then Linda Key improved that any so that this geodesic gamma has a very wide embedded color namely that it has a gamma has a long embedded of length roughly log of 1 over L of gamma so if gamma is long L of gamma and L of gamma goes to 0 then this goes to infinity logarithmic in number and in particular if I have gamma and eta geodesics on the surface with intersecting axis then if I take e to the length of gamma minus 1 then e to the length of eta minus 1 this is bigger equal than 1 and you see that this is kind of correct because if L of gamma is very close at least when gamma is very close to 0 L of gamma so this tells me that L of eta is bigger equal than the log of 1 over L of gamma and this is roughly the same picture and this is very useful in many places in hyperbolic geometry and we prove that the same is true for maximal representations namely that if I have rho from gamma in sp to an r, maximal and I take gamma and eta elements of gamma that I think is close geodesic on my surface with intersecting axis then we get that e to the L of rho of gamma divided by root n minus 1 times e to the L of rho of eta divided by root n minus 1 is bigger equal than 1 and this is kind of a surprising I mean this is kind of a surprising fact because for maximal representation these isometries need not even be hyperbolic meaning that they do not necessarily have a mean set but even if the mean set exist this might be very far away so there are example of maximal representation for which of rho of gamma and rho of eta can be far away in the symmetric space namely that now so in hyperbolic geometry the dimension was so low that crossing axis mean that the mean set need to intersect and you can apply local analysis there and that use this from Margulis Lemma kind of argument whereas here I can produce for you examples of maximal representation so that these guys have mean sets or even sets in which the translation is close to optimal that are very far away but nevertheless this representation remembers a lot of the geometry of hyperbolic of my hyperbolic hyperbolizations and also I should definitely say that this was due to I mean more refined version was due to Gison Li and Tangren Zhang for representation which in this case is things that can be deformed from this guy here I said that maximal representation are a union of components so I start with one of those I deform them I still get the maximal representations for all those they prove better estimate for this because this is not achieved not even mean this is a not as I mean this is a corollary but it's not sharp as opposed to the color lemma and hyperbolic setting and their version here is slightly better than ours and also this is really a non-general phenomenon because for example it's not true I mean nothing similar for other classes of nicely behaved representation of surface groups like the quasi-fuction space but that was something that was pointed out by Tangren and Gison any question up to this point but I hope that these persuaded you that those are interesting objects because of some unexpected behavior right? I mean there are unexpectedly nice for many reasons so we should want to understand what really forces this and how I mean understand more of the geometry of this object what we are going to do by looking at ultra limits the idea of ultra limits is a very old idea that if you want to understand phenomena or large scale phenomena of objects you should rescale to blow this phenomenon so that they become more they become easier to see let me tell you what what I mean by that and what I do what I mean in general so I take rho k from gamma to g an unbounded sequence of representations so if I take a bounded sequence that converges in the representation variety and since maximal representation for components I will still get a maximal representation so I want to go to infinity and see what new I can discover there but if I go to infinity I lose information of my representation and if I see something in the limit I need to rescale and what is a good way to rescale I fix x0 remember that I want to think of those as actions on the symmetric space and I fix a base point x0 and I look how much say that gamma is generated by a1 b1 and maybe c1 with some relations product in the product of ci and I look at how much my generating set translate my base point and I define the displacement of this base point to be the sum of those translations so I define lambda k to be the sum of the distance of x0 and s times x0 squared where s in s is so this is s my generating set I choose to be standard but not necessarily and if I want to see something in the limit I need to rescale at least by this constant because I want that all my generators move my base point somewhere where I can sit so I rescale the metric by lambda k at each factor and I get let's fix the omega and ultra filter let me comment a bit on it later so I get the action rho omega an isometric action rho omega from gamma into the isometry of the asymptotic cone of x with the distance rescale by lambda k so this is a metric so an ultra filter I'm not going to get into detail what an ultra filter is something that is provided by logic and allows me to take limits of any sequence in a coherent way respecting all kind of nice properties of my objects and it's not so hard to say that since I rescale so that all my generators will still act on this limit I will get an isometric action from the sequence of isometrics so and c omega is actually the limit of this metric space set of sequences in x to the n sequences of points in my symmetric space such that the distance y k x 0 divided by lambda k is more than infinity but if I wanted the distance induces a distance on this space I need to portion so I want to define the distance between two sequences as the limit of the distances but if I want to do that I need to get the metric quotient so this is going to be mod out by the equivalence relation given by y k is equivalent to z k if the limit of the distance of y k and z k divided by lambda k goes to 0 so this is the way of looking at my symmetric space from far away from farther and farther away and the idea should be that this object I get at the end is in a sense more combinatorial and easier to study than the symmetric space I was starting with so it should be easier to spot properties in that context than it was in the symmetric space so idea that was the philosophy I was telling you before some aspects of c e omega of x are more combinatorial and easier to study than in the symmetric space and so it's well known what this asymptotic cone is and it was proven by Claren Leib that c e omega is in a fine building what do I mean by this I'm not going to define you what is in a fine building but I'm going to give you an example that is very relevant to this picture so the first example that's very relevant to this picture is what happens in SL2R that should be the motivating example so for SL2R I would get T that's the universal R3 which is an object that is a zero hyperbolic space that for each triple of points there's determinant tripod but it's an R3 meaning that instead of the simplicial three we're more used to I'm allowing branching in all possible points and moreover at each point I would have uncountably many different direction so that's the first example to keep in mind which is complicated because it's branching in every point in infinitely many direction but still is the whenever you look at finitely many points it's very easy to understand what's going on much easier than in the hyperbolic plane and another very important example is that one should think of is the product of N of those T of N is equal to the Cartesian product of N R3s it's not true that these asymptotic cone we're looking at is that object but it is true that it's covered by families of sub-symmetric spaces that are of this form so in this symmetric space I have more so in an R3 I have branching at points if I take a product of R3 this is covered by Rn so R2 and that branching in the horizontal vertical direction that corresponds to factors whereas here I'm also allowing branching on diagonals for this space here but again if you want to understand configuration of up to four points the boundary then you can relate your picture in this object here so you can find the sub-symmetric space of that form in which everything happens so I was telling that these are these that are easier to study than symmetric spaces why is this true it's because well for example we drop the dimension a lot so the symmetric space was an open subset of symmetric N by N matrices that has dimension N times N plus 1 whereas this is covered by Rn so it has dimension N as opposed to N times N plus 1 but moreover all the negative curvature phenomenon has been split apart from the flat phenomenon right so the feature of high rank symmetric spaces is that you have some flat sub-spaces and some directions in which you have negative curvature and here I'm concentrating the negative curvature in the branching and keeping the flat part that way so but moreover we created sets of minimal displacement so for example elements my isometries of our buildings have zero translation length if and only if they have an actual fixed point and this makes the study of those objects much easier in hyperbolic geometry you have parabolic for example zero translation length but I have a fixed point at infinity but not in the space whereas here all fixed points are removed within the space so let me get to the theorem that we proved for this action so the context is I have an unbounded sequence of maximal representation I get an action on this our building and I want to study the elements of my group that can act in this space with a fixed point so I get rho omega from gamma into the isometries of the asymptotic cone a limit of maximal representations then we find the topological the composition of the surface along simple closed curves so let me draw a picture to make this more concrete so at my surface that was important the very first definition of my maximal representation and we can find the canonical set of curves C i's that are simple and pairwise disjoint with the property that each of those of C i has rho of C i that I think of a conjugacy class of isometries of elements in my fundamental group as a fixed point at least one fixed point in C omega moreover for each gamma crossing the composition namely for each other element gamma that intersects non-trivial at least one of the elements in my decomposition rho of gamma has positive translation length and moreover for each complementary piece sigma v there is a dichotomy so either each I mean the translation length of rho omega of gamma is positive for each gamma contained in sigma v so you take a complementary piece like this one whole torus and either all curves inside that piece have positive translation length or there exist a point v in the asymptotic cone which is fixed by rho omega of p1 pi1 of sigma v with respect to any base point and what I mean by this is that I have a conjugacy class of subgroups of gamma that are conjugated to the fundamental group of my piece and each such a conjugacy class will have a fixed point somewhere that are going to be distinct and actually we can use these fixed points in some cases to build an invariant convex sub tree but first maybe I should ask if it's clear if it was too fast c could be empty empty and in that case they could either all have positive length or they could be a global fixed point because when I'm doing this procedure I'm fixing a base I was looking at representation not up to conjugation but fixed representations so at the fixed choice of the base point that could be the very wrong one and could get more and more wrong within the sequence and in that case that would be a global fixed point for that action we can prove that that's the only case and actually we're saying that we can use these fixed points to produce invariant sub trees in some cases so that's the second part that if all complementary subsurface I mean correspond to short geodesic then if I want to say this is that then the collection of fixed points extends in quasi-isometric embedding to the decomposition the I given by the composition given by my collection of curve curves of cutting curves into the asymptotic cone what I mean by this I mean that I add my collection of curves I lift them up the universal cover and I add the dual tree to this the composition of H2 each point corresponds to a conjugacy class conjugated it to one subsurface and so I can map it into the building with that corresponding fixed point canonical fixed point and this extends to a map to these three which is a quasi-isometric embedding and moreover is geodesic with respect to a suitable finseler metric in general it's not going to be geodesic with respect to the metric on this object induced by the Riemannian metric on the symmetric space but if I decided to endow this mysymetric space with the finseler metric then that would be geodesic with a specific finseler metric that I'm not going to find here unless you ask for it are there questions on this? all of them are going to be realized I mean for any topological data I mean you choose and that is kind of I mean you can do it easily by cheating already in the technical space you realize all of this I mean you can also get Zariski dance representation that realize all the composition by getting a sequence by twisting with suitable elements in the mapping class group so now I have some choice to be made in this in my five but let me for maximal representation themselves that goes into my idea that looking at this object it's easier than looking before and I can deduce information on the object I was interested at the beginning so something that is kind of interesting is that I take S in gamma a connected generating set and by this I mean that S is a generating set for my group and the union of the axes project to a connected graph in my surface then if each S in S as a fixed point somewhere in the building then the limiting action has a global fixed point and another and this is because if I have when on trivial the composition then I have at least one long element and at least one of the elements in my generating set is going to cross one of the curves in my decomposition but the important feature is that in general when isometries of these asymptotic cones are going to have fixed points they're going to have tons of fixed points I mean at least in axes of fixed points but they might have big subset they might fit big convex subset of the building and so the fixed point I was starting from can be very far away one from the other but I can also find another fixed point that's fixed by all of them and another property is that there are at most 8G minus 8 plus 3P if I didn't mess up my computation yesterday this thing growth rates longer sequences row K of gamma so if I take an element in my fundamental group I can look at the translation length of the element under the various representations and in total I'll have a bounded number of distant growth rates because of that so I wanted to tell you something about the tools and the proof but I guess I'm in one minute I'm not going to be able to say anything