 Thank you very much for the renewed invitation to speak in this beautiful series of seminars. So I will essentially read my notes and start by defining, so I will speak on these bounded generation linear groups and exponential parametrizations since the object of essentially two papers with the authors that have written down, Pietro Corvaya, Julian de Mayo, Andrea Pinchuk and Gene Borelli. So I shall start by defining bounded generation, which is a notion that can be given in any abstract group and it may be seen as a strong form of finite generation and the precise definition is that the group gamma is said to be boundedly generated if there exists gamma 1, gamma r in gamma such that gamma is, so each element of gamma may be written as a product of powers in order powers of gamma 1 times the power of gamma 2 times the power of gamma r. Of course this coincides with the finite generation if the group is commutative but non-commutativity is very essential in this context. So the bounded generator, so let me remark they need not be distinct and the terminology is there is also an alternative one called finite cyclic width and this number r, the minimum such number is called the cyclic width sometimes. Let me give some examples. So every virtually a billion or virtually nilpotent group has bounded generation even only if it has finite generation. By virtually a billion for instance I mean that the group has an a billion subgroup of finite index, so virtually we'll denote this property of having a subgroup with the property and being of finite index. So here is a criterion to construct new examples of bounded generation from known ones. If we have an extension of the groups g, an extension of h by n, so an exact sequence like displayed there, if n and h have bounded generation then the same holds for g. But generally not the converse. And in particular a solvable group has bounded generation provided however all pieces in solvability filtration are finite regenerated. And however warning here there are solvable groups already inside sl2 which are finitely generated but not bounded regenerated. We give an example in one of the papers that I mentioned. Another example result by MULCHER it follows that every solvable subgroup of glmz has bounded generation so in the context of glmz the solvability suffices. Our context is indeed near this context of glmz because we shall be concerned with linear groups so called for me a linear group it means a subgroup of g of k so the k rational points of an algebraic group g for some field k. And that we shall always assume that the field is of a characteristic zero and usually it will be a number field. And typically the entries of the matrices so we have a group of matrices and that the entries shall be restricted to lying some subring of the field usually. For instance the ring of integers of the number field k. And this context is a context of the so called arithmetic groups. Let me give now some examples which show why we can be interested in bounded generation. This is because the bounded generation property has been found to have several relevant consequences toward many other independent questions on linear groups. For instance here are some theorems some results for instance are a pinchuk in 1990 found that the bounded generation has strong implications toward the so called SS rigidity which is an important property of linear groups which means the finalness of completely reducible complex representation in any given dimension. This also has implication in the dimension of the character so called character varieties I omit precise statements for time reasons for instance. And then we may mention a work of Lubotsky and the Platon of a pinchuk in 92 there are two series of independent works who proved that the congruence subgroup property which is a famous series of problems the congruence subgroup property regards for instance subgroup solve gl and z whether the every finite index subgroup is a congruent subgroup. And there is a generalization for arithmetic groups and this property this important congruent subgroup property was deduced from bounded generation in some cases by the works that I mentioned. Shalom Willis used the crucially this bounded generation for proving the so called Margolis Zimmer conjecture this is again technical it has to do with commensurability of subgroups of arithmetic groups of Chevrolet groups so but it was a completely independent problem and it was found that bounded generation had influence on this and the further implications appear work of Avni, Lubotsky and Mayer. Let me now give some simple examples but where it holds and not for instance bounded generation does not hold for the perhaps the simplest arithmetic group non commutative SL2Z in fact this group contains a noncyclic free group of finite index for instance the group gamma two which is a subgroup of index two of gamma of two and it is not difficult to see that this fact that it contains a noncyclic free group excludes the bounded generation. So, however, it is somewhat surprising that it does not hold for SL2Z but already for and at least three Carter Keller proved in 1983 for the bounded generation for all SLNO so the points with coordinates in a given ring of integers of number field and actually they prove the bounded generation using only elementary matrices in particular unipotent matrices this distinction will be relevant in the following in the sequel and let me remark that for n equal to anyway the property holds not for SL2Z but for other rings of integers and it holds as soon as the ring of integer has an infinity unit group so we may say almost always so as shown by Morgana Pinchuk and Suri in 2018. Now let me pause a bit on parametrizations which is another aspect of my talk and its connection with bounded generation so let us observe that bounded generation by unipotent matrices allows in particular to parametrize polynomially SLNO for n at least three. What do I mean with this in the sense that there is a polynomial map which is subjective from some power of the ring into two SLONTO SLNO namely we have a matrix in SLNO whose entries are polynomials in some number of variables so we may substitute integers in all in place of the variables and we obtain elements of SLNO and we may obtain we may find the polynomials so as to obtain every element this is easy after the theorem of Carter Keller because we observe that if gamma is a unipotent matrix the entries of the powers of this matrix are polynomials in the variable in the in the exponent and hence if we have bounded generation we need only a finite bounded number of variables to parameterize. This parametrization property is a kind of the euphantine property which can be of independent motivation and interest. Indeed for instance there was a question of Scholem going back to long ago who asked already for n equal to whether SL2z was parametrizable by polynomials and there were some negative answers in few small number of variables but surprisingly Wasserstein in 2010 approved that indeed gave a counter example to the expectation by proving that in sufficiently many variables we can parametrize SL2z by polynomials and then Lars and Gouyen recently they extended the thing to rings of integers other than z and this is remarkable because this shows in particular that whereas bounded generation implies the polynomial parametrization if the bounded generation occurs with unipotence the converse does not fold because we have remarked above that SL2z is not boundedly generated so there is there are connections between these parametrizations and the connections but not double implications so let us note now that what I say the work for unipotent matrices but if we also have some simple matrices in a in a bounded generation we still obtain parametrization but not anymore by polynomials but we have to introduce also exponential polynomials and the other extreme we have so-called purely exponential parametrizations if we have only semi-simple elements in a bounded generation we shall meet this restriction also below and these observations are easy in fact on diagonalizing a semi-simple matrix we see that the entries of the power are linear forms in the powers of the eigenvalues and if the matter is semi-simple so we find exponentials and not polynomials so let us go back to bounded generation further examples of bounded generation were found for linear groups the result of Carter-Teller on SLN was extended by Tavgen in 1991 to all Chevrolet groups of rank greater than one and to most so-called quasi-split groups which has to do with Borrel sub-groups I will skip the definitions so anyway these results and some other ones raised the expectation that bounded generation would hold in even greater generality let me comment again on the dichotomy unipotent and semi-simple bounded generation in many of the quoted results for instance for SLN bounded generation occurs with the unipotent but we would like also to allow semi-simple elements for other potential applications of the bounded generation property and with this in mind I shall call an isotropic any subgroup gamma of GLN k that contains only semi-simple elements and I shall for the moment think of the bounded generation for such a group this restriction is relevant because these groups there are very interesting groups which are an isotropic here are some examples that is first the most easy we can call it trivial case is constituted by virtually a billion groups generated by semi-simple matrices so I said this is trivial case one quaternions of norm one over a certain ring of s integers and more generally the same with quaternions replaced with other division algebras and the other series of interesting examples come from orthogonal groups of quadratic forms not represented zero these these forms are called an isotropic and this is the origin most probably of the terminology an isotropic for the groups that I that I am considering for me an isotropic group is a group I repeat containing only semi-simple matrices one gets many examples of orthogonal groups working in rings such as z allowing some denominators some ring ring of s integers we have remarked before that a group with bounded generation by semi-simple elements may be parameterized by purely exponential polynomials and let me give formally the definition of a purely exponential polynomial so you see it is a linear combination of with constant coefficients of exponentials of free where the bases are fixed and the exponents are linear forms in our variables supposed to take integer values so we have the dichotomy that I have already mentioned so polynomials on one side and exponential polynomials on the other side this dichotomy appears here in discussing this property but it appears in the euphanta in theory for instance it may remind of the Hilbert-Tent problem and Matiasiewicz made the last step for proving the undecidability raised by Hilbert by reducing a certain exponential day of hunting equation to usual day of hunting equation and the same dichotomy appears for instance already with the integral points on curves of genus 0 if we have an affine curve so where we seek for integral points if it has one point at infinity we may find infinitely many integral points parameterized by polynomials whereas if there are two points at infinity there are when there are infinitely many integral points they may be parameterized by exponentials and with three points we have only finitely many integral points with three points at infinity so this dichotomy appeared already in the euphanta in theory and it appears here again and despite the behavior so polynomials being quite different from the exponential polynomial there was expectations somewhat that some non-trivial finitely generated that anisotropic groups could be found over rings of S integers and satisfying the bounded generation however no such example was found and starting this problem with Corvaya-Rapinchuk and Reng in 2021 we found strong limitations to bounded generation in linear finitely generated anisotropic groups and so let me fix some notation so k any field of characteristic 0 and gamma in glnk be a linear group we have this theorem that gamma is boundedly generated by semi-simple elements so it need not be an isotropic but it may be boundedly generated by semi-simple elements then it is virtually solvable so it has a solvable subgroup of finite index let me make some remarks on this one may a little bit generalized the theorem by allowing not only semi-simple elements in a bounded generation but one may allow a single non-semi-simple generator and one keeps the same conclusion and this supplementary result costs some effort and indeed already allowing work there is some effort in the arguments and an open question which is the maximum number of such elements that we can allow in order to keep the conclusion the question is also motivated by examples showing that we can't allow more than four for instance a cell two over certain rings of s integers inside z can be generated by by by five non-semi-simple elements so and non-semi-simple elements lead to ordinary day of fronting equations so to usual polynomials so this explains somewhat the difficulty and they might even lead to undecidable issues so another remark is that there exists a virtually solvable finitely generated linear groups without bounded generation so a pure converse of the theorem does not hold so and a consequence regards another question which was open for some time regarding a profiled notion of bounded generation there is indeed a profiled version of the concept I skip the precise definition but you may imagine them it uses the topology of a profiled group and it works the same way it may be proved that if gamma hat denotes the profiled completion of a group gamma then if gamma is boundedly generated then gamma hat is boundedly generated in the profiled sense and it was an open question where whether the converse held and the above theorem leads to a negative answer providing examples of finitely generated groups gamma without bounded generation but such that the profiled bounded generation instead holds assuming moreover that gamma hat is so-called a residual finite which amounts to the topology of the completion being housed so to obtain this example this negative answer to the question I will not be able to do this because it uses a number of results by Platonov and Pinchuk and it uses the following corollary of the above theorem the above theorem I recall it works for every bounded generation by semi-simple elements without requiring that the group is an isotropic but if we also insist that the group is an isotropic then we have a stronger conclusion it has bounded generation if and only if it is finitely generated and it is virtually a billion this corollary is a quick consequence of the previous theorem it may be proved because a quick consequence given taken for granted some general theory because some general theory proved that a virtually solvable for a 9-9 isotropic group implies virtually a billion without disregarding bounded generation it is a general property so this virtually a billion is the most trivial case of bounded generation and I profit of this corollary for recalling we were calling zero characteristic but in positive characteristic indeed it was shown by Abert Lubbocki-Piber in 2003 that bounded generation implies virtually a billion without further conditions and they use completely different methods and conversely our methods would not apply without as they stand to their context let me add with further results in work also with Julian de Mayo and by means of a partially different method so related to the previous one but in fact different we obtained more precise conclusions concerning not only bounded generation but concerning also purely exponential parametrizations so which we have remarked that bounded generation implies either polynomial or exponential parametrization depending on the nature of the bounded generation either by important or semi-simple matrices but we have also remarked that the converse would not hold so there are groups which are like SL2Z which admit a polynomial parametrization but are not boundedly generated whereas for purely exponential parametrization the the situation is different is different we have the following theorem we Corvaya de Mayo, Rapinchuk and Ren and we have three equivalent condition that gamma has a purely exponential parametrization that gamma is an isotropic and has bounded generation so in this case purely exponential parametrization implies bounded generation and the third equivalence is that gamma is finitely generated and the identity component of its Zariski closure is a torus and this in particular essentially implies that gamma is virtually a P and I call a usual polynomial parametrization does not generate a bounded generation so this result this theorem which is more precise than the previous one is a consequence of sparseness of sets obtained from a purely exponential parametrization this feature of sparseness did not directly appear in the arguments for the former result let me mention a bit what I mean for sparseness you'll meet detail for time reasons and this sparseness is expressed by estimates related to heights so I use the capital according to paulaco and the redhoff notation the capital height means the exponential and the small lower case height means the logarithmic veil affine heights so the estimates have this shape we bounded the number of elements in gamma of height exponential height at most t and coming from a purely exponential parametrization we estimate them by a power of the logarithm of t and as is natural to expect somewhat but it's not clear a priori because we could have exponential polynomials in many variables producing small values so such estimates may be maybe even turned into an asymptotic formula so that the number of elements of height at most t are asymptotic to a constant time a power of the logarithm for some exponent less or equal than the number of variables these more precise information may be not entirely free of independent interest but it is not strictly needed for the present applications so the estimate looked natural but a priori we could have large in a large variables producing small values for the exponential polynomials polynomials so we could have many many values of the exponential polynomials even if the variables are large similar estimates were produced in the past by other authors let me mention just Everest and Linsky in 1999 but they worked with restrictions on the exponential polynomials and these restrictions would prevent the applications that we need the estimates are essentially derived from the following lower bound we let e be a purely exponential polynomial and I call a vector of integers minimal for the polynomial if its absolute value if it's normally clear the norm is minimal among all the vectors such that the polynomial assumes at those vectors the same value as at the vector x so the point is that the polynomial could assume the same value for a whole bunch of integral points and so we cannot expect to have a lower bound for the value the height of the value in terms of the height of the vector of variables but this lower bound exists if we restrict to minimal vectors there is a constant positive constant such that the logarithmic height of the value of the polynomial is greater than the constant times the norm of the vector for all minimal vectors except a finite set of values so not except the finite set of vectors but a finite set of values of the values of the polynomials so this condition of minimality is easily seen to be necessary for the conclusions and this estimate is the essential estimate to derive the previously mentioned estimates and let us compare these upper estimates with some lower bounds so by looking at the total elements number of elements in the group with the given bound on the height so the ones coming from a polynomial purely exponential polynomial parameterization there are at most the power of the logarithm of the height let me mention some results which say that on the contrary if we look at all the elements of the group we have many more so let us restrict to s arithmetic groups so of the shape s integral points of the of g where g is an algebraic subgroup of some glm and always is the ring of s integers in a number field in general with mild assumptions that I will skip but they really mild we have estimates that show that the number of elements of height at most t in our arithmetic group have polynomial growth so more than t to the delta for some delta greater than zero these estimates go back to seagull bail and others in some important cases related for instance to quadratic forms orthogonal groups of quadratic forms seagull wrote a number of paper and there was much research on this in our paper we have not really new results on this but we put together a number of separate the results obtained until recently and these lower bounds are not strictly needed for the above theorem but they illustrated the sparseness so they illustrated that the number of elements in the group of interest for us the number of elements which we may obtain from an exponential polynomial parameterization and so from a bounded generator from a bounded generating set are really really very few compared to the whole and so the bounded generated subsets are very small and unfortunately the group is not boundedly generated so let me employ the last few minutes of the talk by so serving to some aspects of some of the proofs let me think of the first theorem one so that i restated bounded generation by semi simple elements implies that the group is virtually solvable and we have to prove this for all linear groups in over any field of characteristic zero and we assume by contradiction that gamma is not virtually solvable but it has bounded generation with semi simple elements and the first point is to reduce to linear groups of a number field so from a field of characteristic zero this is a common practice here there are some features that do not appear in other in other contexts so by specialization it is easy to find the specialization which maintain the second property so which maintain the semi simplicity of the elements this is easy it is a little bit more intricate to maintain the fact that gamma is not virtually solvable because the the derived series could be a priority very long but one instead uses the uniform boundedness of the length of the derived series which produces solvability and and it is known this is known inside the glm and the ones that we have this uniform boundedness one can prove the existence of good specializations which preserve this effect so this kind of reduction for theorem one has alternatives indeed because then we use other theorems of the euphantine type that are known to hold in for arbitrary fields of zero characteristic and that they are even easier to prove over fields which are truly not a number fields but this reduction becomes more more essential for the second theorem because the second theorem depends for number fields it depends on estimates involving heights and we haven't very easy notions of heights in fields which are not number fields there are notions of course even in function fields but these notions do not have the good properties that we have on number fields and so to work with heights in general fields would be difficult probably and so the reduction to number fields is to be preferred especially for the second theorem so another feature is existence of multiplicatively independent eigenvalues so to state this we let gamma one gamma are bounded generators we assume that they exist and we assume that gamma is not virtually solvable and then to use this hypothesis we take the quotient g prime g over the radical so called and the radical of g of the component identity component of g and the working in g prime it is not difficult to see that we may further assume that g is g zero is the non-trivial semi-simple group so with trivial radical and at this point we use the theory of Prasad and Rappinchuk of generic elements and one that uses the existence of an element of the group such that the eigenvalues of the element are multiplicatively independent from those of the given hypothetical bounded generators and we mean that the respective eigenvalues generate subgroups of q star with trivial intersection or q bar star with trivial intersection and here we use for this the theory of Prasad and Rappinchuk and I would like to remark that this multiplicative independence of values of rational functions because after all these eigenvalues may be seen as a rational functions perhaps not on the group itself but on a finite cover of the group on a variety which is a finite cover and in a completely independent context especially of unlikely intersections this multiplicative independence of values has been studied and in this view the result of Prasad and Rappinchuk is not unrelated to theorems obtained jointly with Bombardian master and later refined by Moran and by other authors and so I wanted to mention this because presumably that could be a useful and interesting interplay between the theory of Prasad and Rappinchuk and the unlikely intersection theory of multiplicative independence I hope that someone perhaps undertakes this analysis so to sum up we have obtained the matrix having an eigenvalue not a root of unity and such that denoting by mu 1 mu s the eigenvalues of the hypothetical bounded semi-simple generators we have that the corresponding groups have trivial intersections and we may assume that all of these eigenvalues lie in a given number field and now the conclusion at least for the first theorem is a simple application of the theory of integral points on sub-varieties of Torai gm to the air we have just to so this theory is the so-called mortal land context in the somewhat easiest case of Toric which was known before the general mortal land and we exploit this theory and we want to prove that not all the powers of the matrix gamma that we have constructed lie may be expressed as products of powers of the given matrices gamma 1 gamma r products of powers in this order however and assuming the contrary for each m in z there would exist exponents depending on m such that the displayed equation holds and by diagonalization we derive from this equation certain exponential equations like the one which is displayed where q is a fixed polynomials and the news are fixed monomials and the bias are integers and I can now conclude and skip the rest this equation these kind of equations give give a variety a variety defined in the in the torus gm to the t plus one defined by the displayed equation y equal q of x1 x t and the point belongs to a finitely generated subgroup of the torus generated by these points and now we use the theorem of Laurent which is essentially the mortal land statement for the toric case if we have a finitely generated subgroup of some power of gm and if we have sigma be any subset then that's the risk closure of this subset has a very special shape it is a finite union of translates of algebraic subgroups of gm to dm so this result goes back to Laurent but it goes back actually also to produce a result by Everts of underportage liquify and everything depends on the subspace theorem of Schmidt which in turn is a higher dimensional version of the theorem of Laurent and the algebraic subgroups of gm to dm are defined by finitely many binomial equation as of the type x1 to the a1 xn to the am equal to 1 in integers and this yields a very explicit application of the theorem and to conclude with this result in practice we apply the result with the variety that we have obtained and the points in it and we deduce a contradiction with the multiplicative independence of the eigenvalue which we had found with the theory of Prasadra Finchuk so this concludes the argument and I would stop here.