(iii) is clear, since Jordan-Hölder decompositions of V into G-modules coincide with Jordan-Hölder decompositions into G/H-modules ,
The class P(G) is the smallest class of nite-dimensional representations of G containing all non-trivial irreducible representation of G is in P(G) and such that if W and V are in P(G), and 0 ? W ? V ? V ? 0 is a short exact sequence of G-modules, then V is in P(G) ,
Diophantine properties of nilpotent Lie groups, Compositio Mathematica, vol.63, issue.06, p.11571188, 2015. ,
DOI : 10.4171/CMH/102
Abstract, Mathematical Proceedings of the Cambridge Philosophical Society, vol.23, issue.01, p.3145, 2016. ,
DOI : 10.1112/S0025579313000077
A spectral gap theorem in simple Lie groups, Inventiones mathematicae, vol.20, issue.3, p.337361, 2016. ,
DOI : 10.1016/0021-8693(72)90058-0
Mesures stationnaires et fermés invariants des espaces homogènes, Ann. of Math, vol.174, issue.22, p.11111162, 2011. ,
DOI : 10.1016/j.crma.2008.11.001
Projections of Bodies and Hereditary Properties of Hypergraphs, Bulletin of the London Mathematical Society, vol.27, issue.5, p.417424, 1995. ,
DOI : 10.1112/blms/27.5.417
On the Erd®s-Volkmann and Katz-Tao ring conjectures, Geom. Funct. Anal, vol.13, issue.2, p.334365, 2003. ,
The discretized sum-product and projection theorems, Journal d'Analyse Math??matique, vol.24, issue.1, 2010. ,
DOI : 10.1007/978-3-0348-5438-2_19
Stationary measures and equidistribution for orbits of nonabelian semigroups on the torus, Journal of the American Mathematical Society, vol.24, issue.1, p.231280, 2011. ,
DOI : 10.1090/S0894-0347-2010-00674-1
On the spectral gap for nitely-generated subgroups of SU(2), Invent. Math, vol.171, issue.1, p.83121, 2008. ,
Uniform expansion bounds for Cayley graphs of SL 2 (F p ), Ann. of Math, vol.167, issue.22, p.625642, 2008. ,
A Spectral Gap Theorem in SU$(d)$, Journal of the European Mathematical Society, vol.14, issue.5, p.14551511, 2012. ,
DOI : 10.4171/JEMS/337
Exponential sum estimates over a subgroup in an arbitrary nite eld, J. Anal. Math, vol.115, p.5170, 2011. ,
A sum-product estimate in nite elds, and applications, Geom. Funct. Anal, vol.14, issue.1, p.2757, 2004. ,
Expansion in SL d (Z/qZ), q arbitrary, Invent. Math, vol.188, issue.1, p.151173, 2012. ,
Expansion in SL 2 $${(\mathbb{R})}$$ and monotone expanders, Geometric and Functional Analysis, vol.1, issue.4, p.141, 2013. ,
DOI : 10.1112/jlms/s1-26.4.256
Local spectral gap in simple Lie groups and applications, Inventiones mathematicae, vol.27, issue.5, 2016. ,
DOI : 10.1016/0022-1236(78)90013-7
URL : https://hal.archives-ouvertes.fr/hal-01449823
Lectures on approximate groups. IHP, Paris, 2011. ,
Approximate Subgroups of Linear Groups, Geometric and Functional Analysis, vol.2, issue.2, p.774819, 2011. ,
DOI : 10.1007/BF02684706
The structure of approximate groups, Publications math??matiques de l'IH??S, vol.58, issue.1, p.115221, 2012. ,
DOI : 10.2307/1969819
A polynomial bound in Freiman's theorem. Duke Math, J, vol.113, issue.3, p.399419, 2002. ,
Additive and Multiplicative Structure in Matrix Spaces, Combinatorics, Probability and Computing, vol.16, issue.02, p.219238, 2007. ,
DOI : 10.1017/S0963548306008145
SUBGROUPS OF FRACTIONAL DIMENSION IN NILPOTENT OR SOLVABLE LIE GROUPS, Mathematika, vol.221, issue.02, p.497511, 2013. ,
DOI : 10.1017/S0305004100072522
Trou dimensionnel dans les groupes de Lie compacts semisimples via les s??ries de Fourier, Journal d'Analyse Math??matique, vol.59, issue.1, p.311331, 2013. ,
DOI : 10.1112/S0025579313000077
A product theorem in simple Lie groups, Geometric and Functional Analysis, vol.21, issue.3, p.915941, 2015. ,
DOI : 10.1007/s00493-008-2271-7
Borelian subgroups of simple Lie groups. Duke Math, J, vol.166, issue.3, p.573604, 2017. ,
Calculation of the coecients in the Campbell-Hausdor formula. Doklady Akad, Nauk SSSR (N.S.), vol.57, p.323326, 1947. ,
Borel subrings of the reals, Proc. Amer, p.11211129, 2003. ,
On sums and products of integers, Studies in pure mathematics, p.213218, 1983. ,
Additive Gruppen mit vorgegebener Hausdorscher Dimension, J. Reine Angew. Math, vol.221, p.203208, 1966. ,
On uniform exponential growth for linear groups, Inventiones mathematicae, vol.2, issue.1, p.130, 2005. ,
DOI : 10.4310/jdg/1214428658
Sixty years of fractal projections. In Fractal geometry and stochastics V, p.325, 2014. ,
Hausdor dimension and the exceptional set of projections, Mathematika, vol.29, issue.1, p.109115, 1982. ,
Stiness of group actions, Lie groups and ergodic theory, p.105117, 1996. ,
Quasirandom Groups, Combinatorics, Probability and Computing, vol.6, issue.03, pp.363-387, 2008. ,
DOI : 10.1016/j.disc.2004.05.006
Principles of algebraic geometry. Wiley Classics Library, 1994. ,
Growth and generation in SL 2 (Z/pZ), Ann. of Math, vol.167, issue.22, p.601623, 2008. ,
Growth in SL 3 (Z/pZ), J. Eur. Math. Soc. (JEMS), vol.13, issue.3, p.761851, 2011. ,
Some connections between Falconer's distance set conjecture and sets of Furstenburg type, New York J. Math, vol.7, p.149187, 2001. ,
An introduction to harmonic analysis. Cambridge Mathematical Library, 2004. ,
Algebra, volume 211 of Graduate Texts in Mathematics, 2002. ,
Hausdor dimension and subgroups of SU (2), Israel J. Math, vol.209, issue.1, p.335354, 2015. ,
Ensembles semi-analytiques Notes from a course given in Orsay, 2006. ,
Some Fundamental Geometrical Properties of Plane Sets of Fractional Dimensions, Proc. London Math. Soc. (3), p.257302, 1954. ,
DOI : 10.1112/plms/s3-4.1.257
Hausdor dimension, orthogonal projections and intersections with planes, Ann. Acad. Sci. Fenn. Ser. A I Math, vol.1, issue.2, p.227244, 1975. ,
DOI : 10.5186/aasfm.1975.0110
Geometry of sets and measures in Euclidean spaces, volume 44 of Cambridge Studies in Advanced Mathematics, Fractals and rectiability, 1995. ,
Fourier analysis and Hausdor dimension, 2015. ,
Foundations of Lie theory. In Lie groups and Lie algebras I. Foundations of Lie theory. Lie transformation groups ,
A discretised projection theorem in the plane. ArXiv e-prints 1407, 2014. ,
New proofs of Plünnecke-type estimates for product sets in groups, Combinatorica, vol.32, issue.6, p.721733, 2012. ,
Associative algebras, volume 88 of Graduate Texts in Mathematics, Studies in the History of Modern Science, 1982. ,
Growth in nite simple groups of Lie type, J. Amer. Math. Soc, vol.29, issue.1, p.95146, 2016. ,
Bounds for multiplicities of automorphic representations, Duke Mathematical Journal, vol.64, issue.1, p.207227, 1991. ,
DOI : 10.1215/S0012-7094-91-06410-0
Applications algébriques de la cohomologie des groupes. ii : théorie des algèbres simples, pp.19-1950, 1950. ,
Lie algebras and Lie groups. 1964 lectures, given at Harvard University, 1992. ,
Projections of self-similar and related fractals: a survey of recent developments. In Fractal geometry and stochastics V, p.5374, 2014. ,
On Furstenberg's intersection conjecture, self-similar measures , and the L q norms of convolutions. ArXiv e-prints 1609, p.7802, 2016. ,
Product set estimates for non-commutative groups, Combinatorica, vol.129, issue.1, p.547594, 2008. ,
DOI : 10.1017/CBO9780511755149
The sum-product phenomenon in arbitrary rings, Contrib. Discrete Math, vol.4, issue.2, p.5982, 2009. ,
Additive combinatorics, volume 105 of Cambridge Studies in Advanced Mathematics, 2010. ,
Expansion in SL d (O K /I), I square-free, J. Eur. Math. Soc. (JEMS), vol.14, issue.1, p.273305, 2012. ,
A proof of Furstenberg's conjecture on the intersections of ×p and ×q-invariant sets. ArXiv e-prints 1609, p.8053, 2016. ,