 Thank you so thank you all for coming and first I would like to thank the organizers for inviting me to speak at this workshop it's a real honor and I'm very happy to be here. Okay so today I'll be telling you about some recent work that I've been doing with Giuseppe Matone who is a graduate student at USC and it's about these things we call positively ratio representations okay so this is not a real word I'm taking a past tense of a noun but this is the best you could do okay we are mathematicians so what are these these positively ratio representations are a particular kind of surface group representations okay and maybe to to to give you a more some motivation of why we consider this type of representations let me start by explaining to you a class of examples sorry and these are hidden representations okay so for the rest of today I'll let s be a closed orientable surface connected as well I should say that okay of genus at least two okay and I'm going to denote the fundamental group of the surface by gamma all right now the technical space of s right it's an object that we've already seen in this workshop and there many ways you can think about this space okay but I'll be thinking of this in two different ways today the first way is as the deformation space of hyperbolic structures on s okay or if you want the set of isotope classes of hyperbolic matrix you can put in your surface okay but you can also think of this from a representation theoretic point of view and this is the set of discrete and faithful representations row from gamma to the orientation preserving isometries of the upper half plane which is PSL2R okay consider up to conjugation by PGL2R right which are the isometries of each two okay now you can now next we consider this home morphism which I call yota n so this goes from PSL2R PSLNR and this is a home morphism induced the home morphism induced by the linear SL2R action on the n-1 tensor the n-1 symmetric tensor of R2 okay so SL2R acts naturally on R2 linearly and so it acts linearly on the n-1 symmetric tensor as well okay and this as a vector space is isomorphic to Rn okay so this gives you a way to put SL2R into the linear group of Rn right then you can check that this actually lands in SLNR and when you projectivize you get this homomorphism okay it turns out that this is also the unique up to conjugation irreducible representation from PSL2R into PSLNR okay meaning that if I want PSL2R to act on Rn without preserving any proper subspace then this is the only thing I can do okay but for us right what this gives us is a way to put tecno space into the space of home morphisms from gamma to PSLNR okay consider up to conjugation by PGLNR okay and what you do is you take every representation here and you just post compose it with this irreducible guy okay to get it into PSLNR right of course you need to check that this descends to a map on the conjugacy classes okay so now we come to our first definition the n-thitching component which I will denote by hit NS okay is the connected component of this space which I'll denote by X NS here that contains the image of the map iron okay so here it's well known that tecno space is connected thing right it's a cell in fact okay and this map iron here is continuous okay so the image of iron is some connected subset in here and you can just take the connected component of this that contains it okay in other words the hidden representations which are the representations in here are those that you can continuously deform until you get something in here okay so this space of representations was first studied by Nigel Higgin and he used the Higgs bundle techniques to understand the global topology of this space so for example he also shows that this guy is a cell okay just like protecting the space okay but the geometric properties of the representations in here right we're very mysterious until Fonso Laburi came along and prove this what I think is a remarkable theorem okay if you take any hidden representation row okay then row is going to be what we today call a north soft with respect to the power-bolic subgroup the minimal power-bolic subgroup okay so like everybody else I will not explain what a north soft means to you formally okay but you should think of this as saying that your representation has some very nice geometric properties with respect to some particular power-bolic subgroup okay and in particular right we know that this implies that row from gamma to G is a quasi-isometric embedding okay so here you put the word metric on gamma this is a gram of hyperbolic group and here you choose any left invariant metric you want remanent metric on your league group okay also if you take any non-identity element in your group okay then row of gamma is diagonalizable over R right with eigenvalues are having pairwise distinct absolute values okay and I would denote this this absolute values of the eigenvalues by lambda 1 of row of gamma to lambda n of row of gamma in descending order okay so lambda 1 of row of gamma is the eigenvalue of row of gamma with largest absolute value of the eigenvalue of row of gamma with largest absolute value and so on okay now choose a collection of non-negative numbers a1 to an-1 so that ai equals to an-i for i oh sorry okay from now on alright so here I'm just picking any collection of non-negative numbers with a particular symmetry okay so I'm really just choosing half of them and using this we can define this thing called a length function okay and this is a map from the space of closed geodesics on my surface okay so what is this this is defined to be the set of non-identity conjugacy classes in gamma okay consider up to the equivalence relation where the conjugacy class of gamma is equivalent to the conjugacy class of gamma inverse okay so this set here is naturally in bijection with the set of free homotopy classes of unoriented closed curves on your surface okay so if you choose a hydraulic structure on your surface then this becomes is naturally the the set of closed geodesics on your surface right because every free homotopy class contains a unique geodesic representative okay so I call this the set of closed geodesics but really this is a topological thing okay so the length functions are met from here to r plus and what you do is you just take any equivalence class containing the group element gamma so this is the closed geodesic corresponding to gamma and you send it to the sum from i equals to one to n minus one of ai times log of the ith eigenvalue of rule of gamma divided by the i plus one eigenvalue of rule of gamma okay so you can check that this map is well defined okay you need this condition to make it well defined okay so an example of yeah so maybe I should remark that this length function depends on the choice of my ai's okay but I will suppress that in the notation all right so an example of this right is in the case when n equals to 2 okay when n equals to 2 your second hitching component is exactly type in the space okay this is a consequence of the well-known fact that type in the space is a connected component of x2s okay so this is type in the space and in this case I only have one choice of the ai so I just choose a1 and say let's choose it to be one okay then the length then for any closed geodesic in cgs the row length of c is exactly the hyperbolic length of the closed geodesic c okay in the hyperbolic structure corresponding to my function representation row okay so these are the kind of things that I'm interested in studying right I would like to understand the length functions these guys that arise from a hitching representations or in general from these things we call positively ratio representations okay and if this is the kind of thing you're interested in one a natural quantity that you would like to understand is this thing called the entropy of the representation okay so definition for any hitching representation row and for any choice of again you need to choose some ai's right with this symmetry but I won't write that here okay the entropy of the representation is defined the following way okay first you count the number of closed geodesics on your surface okay whose row length is less than some number t okay and for any t this quantity turns out to be finite okay for hitching representations and in fact it grows exponentially with t okay so you take the exponential growth rate okay so this quantity is called the entropy now it's another well-known thing about a space that this quantity here is constant on technical space okay so this is kind of a boring quantity to look at if you're interested in technical space however once you change your lead group right to become a higher rank right for PSL nr where n is 3 and bigger then this quantity is no longer constant okay in fact it's a consequence of my thesis result that when n is at least 3 okay and and let's say you choose okay now you need to choose all the ai's to be 1 okay so in other words the length function I'm considering is the one that assigns to every close curve by the log of the top eigenvalue of row of gamma divided by the bottom eigenvalue of row of gamma okay so if you choose this particular function okay then there exists a sequence in the hitching component right along which your topological entropy your entropy would converge to zero okay and also a senior also produced them other examples of these things so produced other examples right where your entropy is going to zero in the case when n equals to 3 they're using other constructions so for me a natural question then is for these length functions right can I give a criterion on a sequence of representations to indicate when is the entropy along that sequence converging to zero okay so the question when is there a criterion for when this happens okay okay and so we we could not answer this question completely okay but we got pretty close okay if you restrict yourself to just sequences that stay in the epsilon thick part of the hitching component okay then it turns out that we have a criterion okay so first I need to tell you what the epsilon thick part is okay so this is a corollary of some theorems that I'm going to state later okay if so if for any epsilon larger than zero okay you define the epsilon thick part of the hitching component okay to be the set of hitching representations right where the row length of any closed geodesic is at least epsilon okay and of course this again depends on the choice of the ai's right but every time you choose an ai you can define this epsilon thick part okay okay then now you let a row i be a sequence in this epsilon thick part there will I be a sequence in here then the entropy along this sequence converges to zero if and only if for any subsequence of this sequence okay there exists the following things okay first a further subsequence which I will abuse notation and also denote by row i okay a possibly empty collection in the space of closed geodesics there are pairwise non intersecting and simple okay and finally a sequence of mapping classes okay such that alright so before I write the condition let me just say that the mapping class group acts on the hitching component just like how it acts on packing the space okay by remarking right and also for any length function you choose right the the the mapping class group preserves the epsilon thick part of the hitching component as well okay so you need these three things satisfying the following okay right bigger okay okay so first if you look at the supremum over all sequence all elements in your subsequence right this further subsequence okay and you look at the maximum overall curves in this possibly empty collection okay of the fi times row i length of your closed geodesics in here okay this has to be finite and second if you take the limit as I approaches infinity so move along your sequence and you look at the minimum okay over all closed geodesics okay they are in the complement in S of your collection of closed curves okay so s-c is a possibly disconnected union of surfaces with boundary okay and I've defined closed geodesics only for closed surfaces but obviously you can define it for surfaces with boundary as well and I require these guys to be non-parifero okay and then you look at the fi times row i lengths of all these guys okay this has to be infinite okay so this is the criterion so more informally what is this saying right this is saying that if I take a sequence in the epsilon thick part of the hitching component right then the entropy along this sequence is going to zero even only if up to taking subsequences and up to the mapping class group okay you can find some way to cut your surface into pieces maybe something like that okay so that the lengths of the curves they used to cut your surface have to remain finite along your sequence but on the complementary pieces right the shortest I mean the the systole length the shortest non-parifereal closed curve its length has to go to infinity okay so this is kind of a pretty easy criterion I think so yeah okay so maybe I should move on from now okay so this is a statement of a corollary of our main theorem in the setting of hitching representations okay after so this was initially the our target right this was something we wanted to prove and after we prove this realized that our techniques actually generalized to a much more general setting okay and that's how we came up with these things we call positively ratio representations okay so before I move on I would like to say a little bit about the the strategy of proving a corollary like this or theorem like this okay I mean if you try to work on this very quickly you realize that your life would be a lot easier if you knew that your length functions came from a negatively curved metric on the surface okay and sometimes that happens right in the case when n equals to 2 right your length functions I said you know they will arose as length functions of an of a hydraulic metric on your surface okay but in general right there is no canonical metric on your surface that gives rise to these length functions okay but we have something close it turns out that these length functions arise from geodesic currents right and that is good enough right for you to play a lot of this entropy counting this entropy counting game okay and so let me explain what geodesic currents are we kind of saw kind of saw this yesterday during Vivek's talk but in case you missed it I'm gonna there's gonna be a lot of overlap of what I do here with what she did okay so first what are these guys so a geodesic current right is a gamma invariant locally finite or L measure on the space of geodesics in the universal cover of s okay so this guy is defined to be the set of unordered pairs in the boundary of my group okay that are pair wise distinct okay so if you choose a hydraulic metric on your surface then this again is naturally identified with the set of geodesics in the universal cover okay but again this is a topological thing okay this has a I mean as a there's a natural gamma action on this space because this is pair of points in the boundary of the group there's a topology here as well okay because the boundary of the group has a natural topology and so this makes sense all right so and what's an example of this the easiest examples those that come from close geodesics on your those to the geodesic currents that you can associate to close geodesics on your surface okay so for any close geodesic my surface okay let eta be a group element that is primitive so that if you look at the equivalence class the equivalence the close geodesic corresponding to eta to the K this is C right for for some positive integer K okay so in the case when C is primitive then K is one okay but this kind of keeping track of how many times C winds around itself as well okay then in this case the med the geodesic current associated to the close geodesic C okay is sometimes I also abuse notation denote this just by C this is defined to be K times the sum over all elements in your group right of the DRock measure on gamma times eta minus gamma times eta plus right where eta minus and eta plus are the attracting and refilling fixed points of this group element in the boundary okay so let me kind of draw a picture of what I'm saying here alright so you have your surface downstairs okay and you choose this curve C this close curve this close geodesic I want to assign a geodesic current to this guy right so you just look in the universal cover okay you look at all the possible lifts of this geodesic upstairs okay something like this you know it goes the infinitely many okay so something like this is eta minus this is eta plus maybe this is gamma times eta minus gamma times eta plus and so on okay and on this infinite set of geodesics you just put the DRock measure on all of them you take the sum and you scale everything by K okay this turns out to be a geodesic current okay so what you've done is you've taken the space of close geodesics and put them inside of the space of geodesic currents okay now definition Bonheon define a very nice intersection pairing on the space of geodesic currents okay so there exists a continuous a symmetric by linear pairing I okay from CS times CS no negative that generalizes the geometric intersection number meaning that if I take this intersection pairing and I just restrict it to the geodesic currents that I that I get from the close geodesics then this is exactly the geometric intersection number okay so of course you know he gives a definition of how to you know actual proper definition of this but I won't do that in the interest of time okay so for us what this gives us is a way to define a length function given any geodesic current okay so this implies that for any geodesic current that you can define the length function our new and just like the length functions we define for hedging representations this is a map from the set of close geodesics on my surface r greater equals to 0 and you just take every close geodesic and you look at its intersection pairing right with this geodesic car okay now I'm gonna impose another condition on the geodesic currents I'm considering okay this one's important geodesic current new yes okay it's called period minimizing okay if for any positive number t the number of close geodesics whose lengths are less than t is finite okay so examples of period minimizing geodesic currents are the new view currents right those that come from negatively curved metrics on my surface where we saw that yesterday non-examples are measured lamination right for measure lamination you cannot there's no minimum for this I mean for and if you make t up really small this is still always infinite okay so for my purposes you know you should think of a little view currents as a as good measure lamination as bad okay so well what is the advantage of looking at the length functions of geodesic currents versus the length functions coming from representations right and the thing is turns out that when you are when you have a geodesic current that's period minimizing you can pretty much pretend right that you can that the that your length function comes from a length function on the offer met of a negatively curved metric on the surface okay so let me give an example of a proposition of something like this okay so let C be a closed geodesic on my surface okay and I'll add that is non-simple okay and then let C1 C2 C3 be close to the six right that I'll obtain right by performing surgery to my non-simple close geodesic C right at a self-intersection point okay so I'll draw a picture to explain what I'm doing okay so suppose that this is your non-simple close geodesic C okay and you have some self-intersection point right then there is a few ways you can cut C into pieces at this half intersection point right two ways you can cut it vertically like this and this will give you a two close geodesics like this okay I'm gonna call this C1 and C2 okay but you can also cut it horizontally like this right and this gives you a third close geodesic like this and this is C3 okay and the statement is that for any geodesic current without any assumptions at all okay the new length of C is always at least the new length of C1 plus the new length of C2 and it's also at least give me and also at least the new length of C3 if furthermore if the geodesic current new has the property that new of U is positive for every open set U in the space of geodesics in S-theta okay then the inequality is strict okay so if you have a negatively curved metric on the surface then something like this is obviously true right because once you cut your curves do not change length and then you pull it tight so they become shorter okay turns out that in the space if you work in the world of geodesic currents you can also prove something like this it's not so hard okay but proving a proposition like this in the world of representations right it's surprisingly tricky okay so that's why that's one advantage you have right when you think of length functions of geodesic currents versus length functions coming from representations okay and this is a point I would like to emphasize right for most of the other examples that say Vivica showed us yesterday about geodesic currents a lot of them right they arise you know when you really have a metric on your space or not on your surface right for which the length functions come from okay but for here you don't have a metric on your surface right and that's why you know proving something like this right it's easier to do in the world of geodesic currents yeah okay so that's that okay so this proposition gives us the following consequence right if you take a period minimizing geodesic current okay and say a subsurface okay a geodesic subsurface so let s prime in s be a geodesic subsurface okay meaning if you choose a negatively curved metric on s okay then s prime can be homotoped to a totally geodesic subsurface in the usual Riemannian geometry sense okay if you have something like this okay then there exists a pens decomposition okay let me not call it P and commit c1 to ck so it's a bunch of close curve simple close curves in s prime okay so that the new length of cj for any j is at least the new length of c for any close geodesic c in the complement of the previous curves in s prime okay so so what am I doing here right how do I build such a pens decomposition well you start off with your subsurface okay so let me draw a picture subsurface look something maybe this okay so your subsurface can be the whole thing and then you're close right but typically you have boundary okay and then now you look at all the non-peripheral close curves in here and you find the shortest one okay that exists because your period minimizing okay now this proposition right where did it go yeah this proposition ensures that okay so you ensures that either this is already a pair of pens or you can choose your shortest simple guy as your shortest close geodesic that's non-peripheral to be simple okay because if you if your shortest one is non-simple you just cut right to make it smaller okay unless you're a pair of pens then you cannot do it anymore right now you cut your surface here okay and then you get another geodesic subsurface you iterate okay until you get a pens decomposition I don't know this okay and this is called a minimal pens decomposition questions okay with this I can make several definitions some I'm going to define certain quantities associated to a period minimizing geodesic current okay so for this definition I need my geodesic current to be pure minimizing and s prime a geodesic subsurface okay then first I can define the cisto length right so this is L nu s prime this is defined to be the set of closed geodesics sorry is defined to be the minimum right of the new length of C such as C is a close geodesic s prime okay the shortest closed curve that the length of the shortest closed geodesic okay and you can define the crit the entropy right just like how we defined it for hitching representations okay so each nu s prime is defined to be the exponential growth rate right of the number of close geodesics right whose new length less than t okay so this is the entropy and finally we can define a third quantity using our minimal pens decomposition which we call the pented cisto length so third for any minimal pens decomposition P of s prime okay I define k nu of s prime right to be the length the new length of the closed geodesics s prime okay so that the C that is not let's just say this that is not Perry there is not a multiple of a simple closed curve P and non-peripheral okay this is called the pented cisto length so basically I look at my surface I have my minimal pens decomposition I look at all the close geodesic in here and I remove the guys that are just multiples of curves in my pens decomposition or multiples of curves in my boundary okay ignore those guys among the rest I find the shortest one and that's the pented cisto length okay and the theorem that we can prove is the following okay so that nu be a period of minimizing geodesic current I see now pure minimizing geodesic current okay s prime and s a geodesic subsurface right and P a minimize minimal pens decomposition for s prime a minimal with respect to this geodesic current this period minimizing geodesic current new okay this is true in this setting then you have the following inequality okay log 2 over 4 is at most the pented cisto length be scaled by the entropy of my subsurface okay and this is sorry I forgot I forgot the most important thing okay so there exists a constant C depending only on the topology of your surface okay such that for any period minimizing geodesic current and for any geodesic subsurface and for any minimal pens decomposition okay you have this inequality that the the pented cisto length renormalized by the entropy is bounded above by this constant times 1 times 5 sorry plus plus the log of 1 plus squared 5 plus 1 over 2 multiplied by the ratio of the pented cisto length to the cisto length okay so let me point out a few features of this inequality first this inequality right is a is invariant on the scaling okay so if I take my geodesic current I scale it right then the entropy let's say I scale my geodesic current by k the entropy will be scaled down by 1 over k and this guy will be scaled up by k right so nothing happens and of course these are ratios of certain lengths right so that when you scale nothing happens here as well okay so this is really an inequality on the space of projectivized geodesic currents okay so this might look kind of strange right but let me give you some consequences first if I don't care about the if I don't care about the cisto length at the pented cisto length I only care about the cisto length then you can actually make this a much simpler okay there exists a constant c depending only on the topology of my geodesic subsurface okay such that for any period minimizing geodesic current okay the cisto length I renormalize by the entropy bounded above by c okay so in particular I mean so yeah I should say this in the case when s prime is a closed surface and new is the new view current right then I've been told that this inequality can be abstracted from the current literature okay so something like this might not be entirely new but I've not seen anything close to this inequality before so if you have you should let me know as well okay what this implies right if is that if I have a subset of geodesic currents and I know I have a uniform lower bound away from zero of the entropy but then that would give me a uniform upper bound of the cisto length okay and there will have consequences for us later as well okay now a second corollary right is that is that the analog of the corollary I wrote down for hidden representations just now holds in this setting okay so first for any epsilon large and zero I need to define for you what is the epsilon thick part of the geodesic currents right so define a sub epsilon of s prime to be the geodesic currents the period minimizing geodesic currents okay so that the new length of c is larger than epsilon for any closed geodesic in my subsurface s prime okay and then okay I did not name that that corollary just now but I guess you guys know what I'm talking about so let's say the the Hitching corollary the H corollary so the corollary in that I first wrote down right in the setting of hedging representations right holds okay when I'm with let's say with the epsilon thick part of the hedging component it replaced with this guy here a epsilon s prime right and you just change all your sequences of representations with sequences of geodesic currents okay so with this corollary okay so maybe I should say a little bit of why these things are true okay so basically if you are in the epsilon thick part right of the of the geodesic currents right this ensures that l new s prime cannot be too small okay then you divide your inequality throughout by k new s prime right then on both sides right you're gonna have some expression in terms of k new s prime right that goes to zero if and only if k new s prime goes to infinity okay and then using that you can you can play around a little bit and you can show this corollary holds as well any questions so far okay so with this corollary right the only thing we need to do now right is to relate is to bring my representations into the picture okay everything is in geodesic current world so far right so how do you associate how do you relate the length functions you have from your representations to the geodesic currents right and this is where we realized that what we did actually works in a much more general setting okay and so these are positively ratioed representations okay so I'm going to give you a definition of this but to do that I need some basics from Lee theory okay so I'm gonna so if you're not so familiar with Lee theory right don't worry because I'll do a running example alongside what I'm gonna do okay so I'll let G be a semi-simple g group of non compact type okay and K is going to be a maximal compact subgroup okay and then a is going to be the maximal abelian subspace of the perp of the le algebra of K which lives inside of G okay so this guy is the le algebra of K this guy's the le algebra of G here I'm taking the perp with respect to the killing form okay so this guy is lies in here is a maximal abelian subspace okay and then I have a choice of simple roots which I call Delta right and on the set of simple roots I have the opposition in evolution right which I denote by Yoda okay so example you can take G to be PSL4R okay K to be PSO4 running out of chalk again okay and then a in this case is going to be the set of four by four diagonal matrices they are traceless okay so this is a three dimensional vector space alright and what else do I need so Delta right in this case you're gonna have three simple roots alpha 1 alpha 2 alpha 3 okay and each simple root is a map from it's a linear map from this vector space to R can what you do is you take your diagonal matrix and you send it to AI minus AI plus one okay so each route takes the difference in the adjacent entries along my diagonal okay and these are the simple roots particular linear functions on these guys okay and then the opposition in evolution in this case right takes each simple root and sends it to alpha 4 minus I okay so you take alpha 1 to alpha 3 alpha 3 to alpha 1 and alpha 2 is fixed okay so this is the running example now for any data subset of Delta okay you can get two things from data one is this thing called a parabolic subgroup okay in fact all parabolic subgroups are going to be conjugate to one of these guys right and you also get this projection pi data right from a to a data which is defined to be the subset of a right so that alpha of X is 0 for any alpha simple root that is not in data okay so let me do our example again so suppose that data is all of Delta okay then in this case a p data p Delta is my upper the upper triangular matrices yes l4r okay this is my parabolic subgroup and my projection is just the identity map in this case from a to a okay but suppose that in the case when Delta is alpha 2 then p of alpha 2 right is the the set of matrices right where the bottom two by two corner is 0 so this is in ps l4r okay and your projection right now it takes a a to a alpha 2 and what you do is you take your diagonal matrix and you send it to the following okay you you basically take the first two entries and you take the average and you put it there so a1 plus a2 over 2 a1 plus a2 over 2 okay and the last two entries you take the average of these two and put them there I'm sorry I think I wrote too small here okay but let me just in case you can read I'll just say that you take a1 a2 you take the average and you put the same thing in the first two entries here take a3 a4 you take the average and put the same thing here okay so this lands in a data a of alpha 2 right because if you evaluate alpha 1 and alpha 3 here you always get 0 right okay so one last thing before I well two last things before I I tell you the definition okay I need this thing called the Jordan projection and this is a map I denote by lambda from g to the positive valve chamber in my l e algebra in my in my maximal abelian subspace okay and in the example okay so this is given to you by the Jordan decomposition theorem okay and what this is in the example so let me make it simple okay suppose that a g in PSL 4r is diagonalizable over r okay with eigenvalues with absolute value lambda 1 to lambda 4 okay then lambda g right is going to be the the diagonal matrix which is the logarithm of the absolute value of lambda 1 the logarithm of the absolute value of lambda 4 okay so I mean PSL 4 the product of these guys is one so when you take the log the sum is zero okay so this is a traceless matrix as well last thing okay last one last one yeah I'm going to define for you a notion of cross ratio okay okay there are many definitions of cross ratios out there but the one I'm using right is the one by lat rapier okay so a cross ratio right is a continuous map okay from quadruples of points in the boundary of the group okay so that the first two points and the last two points do not share any common points okay so a can be b but a cannot be c and a cannot be d okay so it's a map from this set to r okay satisfying a few conditions the first one is a is a symmetry condition so b of a b c d is to be equal to b of c d a b okay for whenever this is defined and second you have a co-cycle condition so b of a b c d plus b of a b d e equals to b of a b c e so this is all the setup I need next I'm going to state a proposition which is probably the deepest thing I'm stating it right now and this is a I mean okay so it basically follows okay from the work of can I say that right okay some burino okay I'm and start and let up yeah from so let up yeah okay so the hardest part of this proposition is really to understand what these four people did and combine everything together okay and what it says is the following okay so let theta in Delta be a subset of the roots right set is such that is invariant under the opposition evolution okay and then also for any alpha in data I let the symmetrized alpha alpha sim right to be defined to be the sum of alpha and its image under the opposition evolution okay all right so we have this now then the statement is that if row from gamma to g is p data and also okay so you know having nice geometric properties with respect to this probably subgroup okay then there exists a cross ratio that you know by b data alpha okay with the property that be data alpha of such data let's say let's say for any gamma non-identity element the group a b data alpha of gamma minus gamma plus so this is the repelling attracting fixed point of my group element in the boundary of the group and then gamma x and x okay so x here is just any element in the boundary that is not gamma minus and gamma plus okay you can check that this quantity here doesn't depend on the choice of x okay this here is for example this is called the period of the cross ratio okay this quantity is equals to the following okay you take your group element you evaluate it using the representation okay then you hit it by the Jordan projection to let it land inside the positive valve chamber and then you hit it by the projection pi data okay and then you hit it by the root okay so this thing looks really scary okay but what is this really this is a length function okay if you work out the examples that I wrote down the example of PSL4 just now that I wrote down right you can see that in all those cases right this quantity this quantity here is just the logarithm of certain ratios and products of eigenvalues of row of gamma okay so this here is really a length function so what this is saying is that whenever you have you make these choices right choose a theta and alpha like this okay then the length function that you get right always comes from a cross ratio okay no it does not like if PSL nr for n even do you wait wait say that sorry aha no no so so theta is not a fixed point a fixed subset so you can take theta to be all of delta then that will be fixed right okay okay so with this proposition now I can write down my definition okay p data and not soft representation row from gamma to g okay it's a p theta positively ratio yes don't use the oh I'm sorry this is alpha sin yeah that's important here I'm I hit it with a symmetrized room okay thank you any other questions okay so if you have a not soft representation like this then this is called p theta positively ratio if for any alpha in data okay the cross ratio b theta alpha which is guaranteed by that proposition okay evaluated on any four points in the boundary of the group in the order so this is this has to be greater equals to zero for any a b c d in the boundary of the group in this order so the boundary of the group is topologically a circle right so I'm just saying that if you take a b c d four points like this and I evaluated using the cross ratio this always has to be non-negative okay and the upshot from this right is okay I mean this is a very fundamental principle it's you know something like if you have finite additivity right and you have non-negativity then you have countable additivity okay so let me just state state this as an observation by Ursula Hamunstad okay that if right is a cross ratio okay such that b of a b c d is non-negative for any quadruple a b c d in the boundary of the group okay then there exists a geodesic current okay so that when you do the intersection of this current right with any closed curve corresponding to the group element gamma okay this is the period of your cross ratio of your cross ratio is b of gamma minus gamma plus gamma x x for any non-identity element in your group okay so whenever you have a positively ratio representation right then the cross ratios you get you can always associate geodesic currents to them okay and this is how everything goes through everything you do in the world geodesic currents right will work here okay so I'm out of almost out of time but maybe I should say what the examples of this positively ratio representations are okay so you have hitching representations which are first introduced and these are positively ratio refers to any parabolic subgroup but you also have the maximum representations in there my these are the ones that Beatrice Posetti was telling about us telling us about on Tuesday okay those are all positively ratioed respect to the stabilizer of a Lagrangian subspace okay then you can build more by taking certain direct sums and tensors of these representations can the basic reason you can do that is because if I have two cross ratios with this property and then you add them you get another cross ratio of this property again okay so you can build many many more representations that way all right so but an important non-example quasi-fusion representations okay if you take a quasi-fusion representation that is a north soft respect to the line and hyperplane stabilizer and you can define the cross ratio as well okay but as once you're non-fusion right then this cross ratio cannot be positive right basically the the how bad your limit curve is right ruins the positivity okay and so this kind of trick doesn't apply for quasi-fusion representations okay so I should stop here thank you