 Yeah, so thank you very much to the organizers for inviting me to this conference to give this talk. It's a pleasure to come here and give, present my work. So now it has been a long day. So if I was in the audience, I might have liked to skip my own talk. But since, since still I can see lots of, but still I see lots of people around. So thank you in advance. So this is my last talk. The talk is over. But anyway, so let's go ahead and see what, yeah, pardon. So I think this talk looks like I am, I shall be talking about something about conjugacy classes in these groups. But there is a geometric motivation behind the study. That's a classical result by Frick and Bogd. So this result is a incremental result in the development of Tejmuller theory, for example. So the result says that you take any non-elementary two-generator subgroup of SL2R or more generally SL2C, then they are determined up to conjugacy by the traces of the generators and their product. And this theorem actually gives you the length parameters of the Fenchelians and coordinates on the Tejmuller space. Now this theorem holds more generally for polish table pairs and I will briefly describe what are they. Now as I have mentioned that this result is incremental in giving Fenchelians and type coordinates on the Tejmuller space and I will say briefly that there have been some generalization of this result in some groups to give Fenchelians and type coordinate on more general geometric spaces. Now in general we can ask to generalize this result in a purely group theoretic sense for other groups. Now the question can be reformulated in terms of representation of pregroup of two-generator into a given group. That means we want to find out the minimal generating set for the character variety of pregroup of two-generators into a given group G of your choice. But G should be nice geometric I mean it should come from a geometric motivation I mean one can ask this question also for fundamental group I mean for finite groups as well but that is a different story all together I mean I would not get into that. Now there have been some work for finding minimal set of generators when the target group is the special linear group over the complex numbers. Now there are works by Florentino, Lawton, Sikora and Rensky who has got some progress in this direction. Now in this talk our interest is the case when n equal to 4 and this case is related to the complex hyperbolic geometry in dimension 3. So my main motivation on this problem was to give Fenchelians and type coordinate on the representation space of fundamental group of a surface into SU 3 1. Now in geometric terms if we can give such coordinate that will give us some insight about the topology of complex hyperbolic quasi-proximal spaces in dimension 3. Now so this talk is related to two aspects one is the special linear group of 4 by 4 matrices over the complex numbers and the other is character variety of SU 3 1 which is related to complex hyperbolic geometry. So I shall briefly describe the complex hyperbolic space I am sure I mean John has described it in the last week but let me quickly recall it. So you take a vector space of signature of diamonds and n plus I am sorry I mean there is a typo here okay. So I mean you can replace 4 by n I mean so we start with a complex vector space equipped with the Hermitian form of signature n comma 1 then there are these three sets again I have restricted to n equal to 3 the set of all negative vectors I can say I mean vectors of negative length positive length and zero length and then the complex hyperbolic space is the projectivization of the negative length vectors and its boundary can be identified with the 2n minus 1 dimensional sphere and the complex hyperbolic space can be identified with the 2n dimensional disk I am sure John have described all these. So excuse my typos in this slide. Now the isometric group of the complex hyperbolic space of dimension n is SU n comma 1 which is the I mean which is a group that satisfy you can so GL you can take this matrix. So in linear algebraic terms this group SU n comma 1 is given by elements from GL n plus 1c which satisfy this relation and we are assuming that they have determinant one so that is the group SU n comma 1. Now as in the real hyperbolic geometry there is a fixed point classification for isometries of the complex hyperbolic space as well and isometry is called elliptic if it has a it has a fixed point on the complex hyperbolic space it is hyperbolic if it fixes exactly two points on the boundary and it is parabolic if it fixes exactly one point on the boundary. Now in the morning Pierre describe one result by Goldman which classifies these isometries using some polynomial function. Now with Parker and Prasad we have generalized Goldman's result for SU p comma q where p q are arbitrary number and as a special case we have given a complete algebraic classification for SU 3 1 that is the group of our interest in the stock. So the result Pierre describe in the morning we have a generalization for that result in SU 3 1 I would not get into the detail of that result because that will not be used here but now let me give some background on what do I mean by a character variety which is the main focus of the stock. So you start with the finitely generated group with relations R and generator gamma 1 up to gamma R and take a connected Lie group G. Now the set of all homomorphisms from gamma into G naturally sits inside R copy even Cartesian product of R copies of G and this embedding is given by this evaluation map you map Rho onto its images Rho images under the generators. Then home gamma G in here is the subspace topology. So here subspace topology by subspace topology I mean the topology of the Lie group. So I am considering G as a Lie group so it is a smooth differentiable manifold and it has a natural topology Euclidian topology you can think of. So here I am we are giving home gamma G the subspace topology from G to the power R and then I define another subset of home gamma G that is the subset of all elements in home gamma G whose conjugation orbit is closed. Such points whose conjugation orbits are closed in home gamma G are called polystable points. So by a polystable representations I mean what has been written here that means the conjugation orbit is closed. So the theorem of Freak and Bogd is actually holds for any polystable representations of F2 into SL2C and we define the G character variety of gamma to be the conjugation orbit space home star gamma G mod G that means you take the yeah no there is no restriction no I mean Sebel means the conjugation orbit is closed so there is no I mean it is a I mean if you take parabolic yeah then yes right loxodermic or even if you take elliptic I think yeah that will be closed. So essentially we are taking semi-simple representation I mean for if you take the representation has to be semi-simple I think it will be always conjugation orbit will be closed. So by character variety we mean the orbits of the polystable I mean I mean orbits under the polystable representations alright. Now in this talk our interest is when gamma is the free group of two generators and G is SL4C or SU31. Now there are some known results about the character variety of any arbitrary finitely generated group gamma into a group G so when G is a complex reductive affine algebraic group then home gamma G is a affine variety that means it is a that means it is a subset of C to the power n for some n with some polynomial relations on it. So the polynomial relations are obtained just by cutting out the product variety G to the power r by the words which represents the relations in the finitely generated group gamma. Now it is a theorem by Florentino and Lawton and also it is implicit in a previous work by Luna that the character variety is homeomorphic to the geometric points of the geometric invariant theoretic quotient. So now here there are several things involved in this statement. So here we are considering the character variety with respect to the subspace topology coming from the Lie group or the Euclidean topology I mean when you take G as a affine variety there is a Euclidean topology on it. So we are taking chi gamma G with respect to that topology and the geometric invariant theoretic quotient is by definition is the set of all prime ideals in this coordinate ring. Now given any variety V the coordinate ring is given by the underlying polynomial ring modulo the ideal that defines the variety. So here this is the coordinate ring of this variety and the GIT quotient is the set of all prime ideals of this coordinate ring and by geometric so generally in geometric invariant theory this set has the Zariski topology on it. But since it has affine variety structure there is a natural Euclidean topology that is coming from the affine space C to the power n. Now here instead of the Zariski topology they have considered the Euclidean topology and with respect to that topology this character variety is homeomorphic to this GIT quotient. Now there is they have further proved that the GIT quotient with this topology is homotopic to this non-house drop quotient phase from gamma G mod G. So there is no restriction about polystability here. So in general this space may not be nice this may be I mean mostly this is non-house drop but it is proved that this character variety is actually homotopic to this space under the Euclidean topology. Now let me briefly recall what are the ring of invariance from invariant theory. So you start with a free non-commutative monoid generated by R symbols X1 from XR and let MR plus be the monoid generated by matrices R matrices which are matrices in R n square determinants. Now there is a suggestion from the free monoid FR plus onto the free monoid generated by these matrices the natural projection map. Now if we take any word W in this matrix monoid then it will be image of a word in FR plus under this map. Now suppose this norm be the function that takes a cyclically reduced word onto its word length that means you just take a cyclically reduced word mod maps the word onto its word length. Then it is a classical result by processing that the ring of invariance of this coordinate ring is generated by the traces of the matrices. And this is the result that is sort of a weak generalization of the result of Fricke and Gauld in some sense but we will come back to it later. Now note that here that here the ring of invariance we are taking the group GL the general linear group actually it is the Lie algebra of the general linear group and SL and C acts on product of R copies of this group. But actually there is a natural map that is the determinant map using that we can actually see that the coordinate ring of the character variety is equal to the coordinate ring of the coordinate ring over this group which is a special linear group which is a group of our interest. And this is essentially coming from the fact that the determinant is a conjugacy invariant of a matrix I mean of this polynomial ring I would say and this is how it has been obtained. So essentially you mod out this coordinate ring by the determinant and you can say that and because the characteristic polynomial allows the determinant to be written in terms of the traces it follows that the character the coordinate ring of the character variety is also generated by the same traces that is in processes theorem. So now we will use some notations and I will show you one diagram that has been heavily used to obtain the minimal parameters for the character variety in the case when n equal to 3. So you take you construct this matrix X k star which is essentially the sorry cofactor of the matrix X k. So cofactor means you just chop out the jth row and ith column and multiply it by minus 1 to the power i plus j. So this is that matrix and write M R star with a monoid generated by X 1, X 2, X R and these elements X 1 star, X 2 star and X R star. Now you have another mon I mean this monoid has a normal sub monoid generated by the determinant of these matrices here. Using that you define this set which is a quotient of the previous monoid by N R. Now as you can see I mean it is easy to see that in this quotient X k star will be actually equal to X k inverse and naturally this M R will be a group that has been obtained from that monoid. Now let C M R denote the group algebra with respect to C. Now since you have matrix addition and scalar multiplication it will give you a group algebra structure and correspondingly to the monoid M R star you have another group algebra obtained in the same way. Now this is a diagram that relates this non-commutative group algebras to the character variety and why they are important in finding the minimal generating set of the character variety. Now this diagram is you can see it in Plotton's thesis and he actually use this relationship. So there is this natural projection map from this monoid to the free group of R generators and from this group algebra to the group algebra over the group M R and then there is this natural projection map and this diagram commutes. And then using process E you can have a trace function that maps this group algebra onto the coordinate ring of the character variety. So you can if you want to obtain minimal generating set here you can use this non-commutative diagram in order to do some combinatorial manipulation to obtain the desired minimality. Yeah, yeah, yeah. So FR plus means the I say you then monoid generated by R symbols X 1 to X R where is that? Yeah, here right. So this is the free non-commutative monoid generated by the symbols X 1 up to X R and M R plus means the free non-commutative of course monoid generated by R matrices and the R matrices you can choose like with R n square variables. But these are monoid they are not groups. Now if I want to make them group we have to chop them by the determinant function and that will give you that group. So this is the relationship that is implicit in our work as well but I would just describe what is the theorem of Lotton I mean Pierre has described it in the beginning in the morning I think. So you take any representations, polystable representations of F2 into SL3C then that is generated by these traces. So you take trace of X trace of Y trace of X Y inverse X inverse Y inverse. So 1, 2, 3, 4, 5, 6, 7, 8. So these 8 traces they determine a polystable pair completely into the in the character variety of a pregroup of 2 generator into SL3C. X Y is missing you think yeah maybe I mean about I have X Y inverse here. No this is not symmetric generator yeah but I think this is what is written in Lotton's thesis perhaps maybe I have to check and I think this is correct. No I think the approach you used you got a symmetric generator right X Y and but I have oh yeah yeah so I have X inverse Y inverse but you need 1, 2, 3, 4, 5, 9 yeah maybe I am missing X Y here right because yeah but anyway I mean the point is that using this result Pierre and he mentioned in his paper that independently when has proved this theorem that you take any pair of elements in this character variety then they are generated by this 5 traces. And John has described one relation among these 5 traces in one of Pierre has who Pierre okay has described one relation. So this is a nice result in the sense because these 5 traces gives you 10 dimensions and one relation one complex relation right that reduce 2 dimension. So this character variety any element here is parameterized by a 8 dimensional parameters. So that is very nice and ideal situation just like the classical Tejmuller theory. Now Parker and Platis proved a yeah yeah say it again now these results include the trace parameters which are needed to classify a pair of elements in this character variety but they that is not given a relation the relation you have to find out separately but in this case yeah if you know these traces that is what yeah. So here John and Platis they proved a special case of this result using a different method that method says that I mean I think this is more explicit what Caroline is asking you take a pair of loxodromic elements in SU21 then up to conjugacy this pair of elements are determined by their traces and they point on the cross ratio variety corresponding to these elements. So this essentially gives you a family of parameters that determines your pair of elements up to conjugacy. So this is more like the classical setup but the only sort of eyes I would say disadvantage is that it does not gives you how the embedding looks like but anyway it gives quite a lot of information and this result actually was the starting point of our investigation because we wanted to generalize this result for SU31 but till late we have only got partial generalization. So before stating the partial generalization let me define some technical term. So a pair AB of loxodromic elements in SU31 is called non-singular if what happened if these loxodromics do not have a common fixed point and the fixed points of A and B do not lie on a common two dimensional total geodesic subspace of the complex hyperbolic space. And the second condition is a bit more complicated but it essentially says that the fixed point set of A is disjoint from so given an element A in SU31 it has two eigenvectors whose lengths are positive. So the second condition says that the fixed point set of A is disjoint from at least one of those copies of total geodesic subspaces those are orthogonal to these eigenvectors and the same is true for B as well. So these two technical conditions are needed in order to prove this following theorem it is in a joint work with Siprasad. So this theorem states that that you take any non-singular representation of F2 into SU31 then there exists two non-zero complex parameters alpha i and beta j such that along with these coefficients the traces of the image under this representations and the point on the cross ratio variety completely determine the representation up to conjugacy. Now these two condition essentially so what are alpha i's and beta j they are the cross ratios but I mean have you discussed cross ratio no not really okay so I mean maybe if I have time I can I mean so cross ratio essentially has been defined for four points. So on the boundary you take four points then the cross ratio is defined corresponding to four points by this formula I mean sorry what is x4, x1, x3, x2 and what is what is the x2, x4 right I hope I am correct in defining the cross ratios am I or whatever so I mean essentially you take the take four null vectors and take this fraction of inner products so that defines the cross ratio. Now Parker and Platis they and more generally recently Platis they have proved that this cross ratio actually change if you permute these four vectors so up to permutation there will be three cross ratios remaining and they determine a affine space in the and that is called the cross ratio variety. Now here alpha i and beta j are the cross ratios but instead of four null vectors you replace one of the null vectors by positive eigenvectors. So the same definition goes through but the point here is that when you replace this four vectors four null vectors by a positive vector then this may not be well defined and even if it is well defined it may be 0. So just to make just to ensure that this term is nonzero for at least one alpha i and one beta j we have to impose these two conditions. So the theorem says that under these two conditions you will always get at least one nonzero alpha i and one nonzero beta i such that they along with traces and one point on the cross ratio variety will completely determine the non singular representations. So this is a partial generalization of the result by Parker and Platis but I mean it gives something but the general case I mean I do not know how to do that in this setting. So that is why we need to look into the geometric invariant theoretic approach in order to understand the general case. Now the starting point as in the work of Lawton is to classify the polystable representations of a free group of two generators into SL4C. Now in this case there has been already some work by the by Drenski and Sadiqova and also Djokovic. So now I give here two tables so in these tables we have listed the traces according to their work length. So this is one table which we called G1 and there is one more table that is called G2. So we have this 10 families of traces corresponding to several work lengths. So as you can see I mean this is quite symmetric set of generators and the theorem of Drenski and Sadiqova and also Djokovic. Here we have essentially used the Djokovic's list. Drenski and Sadiqova has slightly different list in this setup. So this says that G1 union G2 gives you a minimal system of 32 generators for the I mean for an element in the character variety when the target group is SL4C. Now here G1 is a system of maximum system of maximal algebraic parameters. That means none of these parameters in the table G1 they satisfy one polynomial equation. Nothing is known about how these parameters behave and furthermore no polynomial relation is known among these 32 parameters. Now in our joint work we could improve this number by 2. So that means we have proved that G1 union G2 minus 2 elements gives you a minimal system of 30 generators for the character variety. So this is our one of our result. So if you omit the traces of X to the power 4 and Y to the power 4 in the previous tables, then the rest of the elements is a minimal system of 30 generators for the co-ordinated ring of the character variety. That means you take any polystable representation in this character variety they are determined up to conjugacy by these 30 generators. And you can see X to the power 4 and Y to the power 4 are in the first list. So I mean and so it should be 13. There should be a maximal set of 13 algebraically independent elements because we have we are omitting 15 I think let us count. So how many elements are there? So 2 plus 2 plus 3 plus 4, so 5, 9, 1, 2, 3, 4, 5, 6, 14, right 1, 2, 3, 4, 5, 6, 7, 8, 6 plus 4, 10 plus 3 plus 2. So 15, 17. So we have 17 traces here and we are omitting X to the power 4 and Y to the power 4. So what I have written is correct. We have a maximal set of 15 algebraically independent elements in this generating set. Yeah, so there is no relationship between G1 and G2 but the point is that the traces of G1 Dresky and Sarikov and Djokovic they prove that it is a algebraically independent maximal subset of the generating set. So these are so there is no polynomial relation among these traces but what are the relations between these traces or if you take the generating set as a whole that is not known. So what we know only here minimal means they are not I mean there cannot be any smaller maximally algebraic set. Yeah, I mean if you take any of these traces then they will satisfy polynomial equation that is what it means. So this is a yeah it is still minimal yes. No, it is a maximal sorry it is a maximal set it is not minimal. No, I am not saying minimal right it is a maximally algebraically set. No, no maximal is it where no, no, no there are two things here. So G1 union G2 is a minimal system of generators for the character variety and G1 is a maximal algebraically independent set. So they are different thing yes exactly and we do not know what are the relations. What we could prove that we can take we can remove two elements from the list of Djokovic and Sadikova and we still have 15 algebraically independent sets. So we have improved the number by Djokovic and Genskiy and Sadikova by 2 but you do not know what are the relationship between them we have no idea. So now essentially we have used this lemma to prove the result. So you take a generating set for the coordinate ring and suppose trace of ux cube v is in the generating set. So you start with the characteristic polynomial and then you multiply the characteristic polynomial on the left by word u and on the right by x inverse phi and then that gives you this polynomial equation. So that means if you take traces of this latter equation you can see that this remains a generating set as long as the rest of the traces are in the subring generated by G with this trace omitted. So this is a small technical lemma that we have used in our proving this result and this is the kind of technique that has also been used by Lawton. I mean this is a there is a normal geometric invariant theoretic technique called partial polarization. So this is essentially sort of that technique that has been used in a more smaller scale. Now you take a involutive outer automorphism that permutes x and y so that means that maps x onto y and y onto x and let there is another involutive automorphisms that maps x to its inverse. So these two maps are in the coordinate ring of the character variety and what as a corollary to the previous result we get a more symmetric set of 30 generators which is minimal. Here s is this set. So you see here we could get rid of the generators which correspond to words of length 10. In G1 union G2 we have elements up to word length 10 but here we have reduced it to traces of elements up to word length 9. So now build upon this result this result is needed in order to give the minimal set of generators for SU31. So as a consequence of this corollary what we have obtained is the following. So you take the following 22 traces then these 22 traces determine any polystable pair up to conjugacy. So here in this list you can see I mean there are trace x trace y x square x y y square x inverse y these are all word length of 2 then you have traces of corresponding to word length 3 x y square y x square and then word length 4 x square y square x y x y x inverse y inverse x y and then 5 then this bunch is the majority corresponding to word length 6 and then you have 2 traces corresponding to word length 7. So total 22 traces and the theorem says that these 22 traces determine any polystable representation or a from the free group into SL4C or equivalently any polystable pair. So I think that is it.