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Linear approximate groups
1. | Laboratoire de Mathématiques, Bâtiment 425, Université Paris Sud 11, 91405 Orsay, France |
2. | Centre for Mathematical Sciences, Wilberforce Road, Cambridge CB3 0WA, United Kingdom |
3. | Department of Mathematics, UCLA, 405 Hilgard Ave, Los Angeles, CA 90095 |
References:
[1] |
L. Babai and A. Seress, On the diameter of permutation groups, European J. Combin., 13 (1992), 231–-243.
doi: doi:10.1016/S0195-6698(05)80029-0. |
[2] |
E. Breuillard and B. J. Green, Approximate groups II : The solvable linear case,, preprint, ().
|
[3] |
E. Breuillard, B. J. Green and T. C. Tao, Approximate subgroups of linear groups,, preprint. \arXiv{1005.1881}, ().
|
[4] |
E. Breuillard, B. J. Green and T. C. Tao, Expansion in simple groups of Lie type,, preprint., ().
|
[5] |
J. Bourgain and A. Gamburd, Uniform expansion bounds for Cayley graphs of $SL_2(F_p)$, Ann. of Math. (2), 167 (2008), 625-642.
doi: doi:10.4007/annals.2008.167.625. |
[6] |
J. Bourgain and A. Gamburd,, Expansion and random walks in $\SL_d(\Z/p^n\Z)$ I, J. Eur. Math. Soc. (JEMS), 10 (2008), 987-1011. |
[7] |
J. Bourgain and A. Gamburd,, Expansion and random walks in $\SL_d(\Z/p^n\Z)$ II, With an appendix by Bourgain, J. Eur. Math. Soc. (JEMS), 11 (2009), 1057-1103. |
[8] |
J. Bourgain, A. Gamburd and P. Sarnak, Affine linear sieve, expanders, and sum-product, Invent. Math, Springeronline (2009). |
[9] |
J. Bourgain, A. Glibichuk and S. Konyagin, Estimates for the number of sums and products and for exponential sums in fields of prime order, J. London Math. Soc. (2), 73 (2006), 380-398.
doi: doi:10.1112/S0024610706022721. |
[10] |
M.-C. Chang, Convolution of discrete measures on linear groups, J. Funct. Anal., 253 (2007), 303-323.
doi: doi:10.1016/j.jfa.2007.03.008. |
[11] |
M.-C. Chang, Product theorems in $\SL_2$ and $\SL_3$, J. Math. Jussieu, 7 (2008), 1-–25. |
[12] |
M.-C. Chang, On product sets in $\SL_2$ and $\SL_3$,, preprint., ().
|
[13] |
M.-C. Chang, Some consequences of the polynomial Freiman-Ruzsa conjecture, C. R. Math. Acad. Sci. Paris, 347 (2009), 583-588. |
[14] |
L. E. Dickson, "Linear groups with an exposition of Galois Field Theory," Chapter XII, Cosimo classics, New York, 2007. |
[15] |
O. Dinai, Expansion properties of finite simple groups,, preprint. \arXiv{1001.5069}, ().
|
[16] |
A. Eskin, S. Mozes and H. Oh, On uniform exponential growth for linear groups, Invent. Math., 160 (2005), 1-30.
doi: doi:10.1007/s00222-004-0378-z. |
[17] |
A. Gamburd, S. Hoory, M. Shahshahani, A. Shalev and B. Virag, On the girth of random Cayley graphs, Random Structures Algorithms, 35 (2009), 100-117.
doi: doi:10.1002/rsa.20266. |
[18] |
N. Gill and H. Helfgott, Growth of small generating sets in $\SL_n(\Z/p\Z)$,, preprint. \arXiv{1002.1605}, ().
|
[19] |
W. T. Gowers, Quasirandom groups, Combin. Probab. Comput., 17 (2008), 363-387.
doi: doi:10.1017/S0963548307008826. |
[20] |
B. J. Green, Approximate groups and their applications: Work of Bourgain, Gamburd, Helfgott and Sarnak,, preprint. \arXiv{0911.3354}, ().
|
[21] |
M. Gromov, Groups of polynomial growth and expanding maps, Inst. Hautes Études Sci. Publ. Math. No., 53 (1981), 53-73. |
[22] |
H. Helfgott, Growth and generation in $SL_2(Z/pZ)$, Ann. of Math. (2), 167 (2008), 601-623.
doi: doi:10.4007/annals.2008.167.601. |
[23] |
H. Helfgott, Growth in $\SL_3(\Z/p\Z)$, preprint (2008). arXiv:0807.2027 |
[24] |
J. E. Humphreys, "Linear Algebraic Groups," Springer-Verlag GTM 21, 1975. |
[25] |
E. Hrushovski, Stable group theory and approximate subgroups, preprint (2009). arXiv:0909.2190 |
[26] |
M. Larsen and R. Pink, Finite subgroups of algebraic groups, preprint (1995). |
[27] |
C. Matthews, L. Vaserstein and B. Weisfeiler, Congruence properties of Zariski-dense subgroups, Proc. London Math. Soc, 48 (1984), 514-532.
doi: doi:10.1112/plms/s3-48.3.514. |
[28] |
M. V. Nori, On subgroups of $\GL_n(\F_p)$, Invent. Math., 88 (1987), 257-275.
doi: doi:10.1007/BF01388909. |
[29] |
L. Pyber and E. Szabó, Growth in finite simple groups of Lie type, preprint (2010). arXiv:1001.4556 |
[30] |
I. Z .Ruzsa, Generalized arithmetical progressions and sumsets, Acta. Math. Hungar., 65 (1994), 379-388.
doi: doi:10.1007/BF01876039. |
[31] |
T. C. Tao, Product set estimates in noncommutative groups, Combinatorica, 28 (2008), 547-594. |
[32] |
T. C. Tao, Freiman's theorem for solvable groups,, preprint., ().
|
[33] |
T. C. Tao and V. H. Vu, "Additive Combinatorics," Cambridge University Press, 2006.
doi: doi:10.1017/CBO9780511755149. |
[34] |
J. Tits, Free subgroups in linear groups, Journal of Algebra, 20 (1972), 250-270.
doi: doi:10.1016/0021-8693(72)90058-0. |
[35] |
P. Varjú, Expansion in $\SL_d(\mathcalO_K/I)$, $I$ squarefree,, preprint., ().
|
[36] |
V. H. Vu, M. Wood and P. Wood, Mapping incidences,, preprint., ().
|
show all references
References:
[1] |
L. Babai and A. Seress, On the diameter of permutation groups, European J. Combin., 13 (1992), 231–-243.
doi: doi:10.1016/S0195-6698(05)80029-0. |
[2] |
E. Breuillard and B. J. Green, Approximate groups II : The solvable linear case,, preprint, ().
|
[3] |
E. Breuillard, B. J. Green and T. C. Tao, Approximate subgroups of linear groups,, preprint. \arXiv{1005.1881}, ().
|
[4] |
E. Breuillard, B. J. Green and T. C. Tao, Expansion in simple groups of Lie type,, preprint., ().
|
[5] |
J. Bourgain and A. Gamburd, Uniform expansion bounds for Cayley graphs of $SL_2(F_p)$, Ann. of Math. (2), 167 (2008), 625-642.
doi: doi:10.4007/annals.2008.167.625. |
[6] |
J. Bourgain and A. Gamburd,, Expansion and random walks in $\SL_d(\Z/p^n\Z)$ I, J. Eur. Math. Soc. (JEMS), 10 (2008), 987-1011. |
[7] |
J. Bourgain and A. Gamburd,, Expansion and random walks in $\SL_d(\Z/p^n\Z)$ II, With an appendix by Bourgain, J. Eur. Math. Soc. (JEMS), 11 (2009), 1057-1103. |
[8] |
J. Bourgain, A. Gamburd and P. Sarnak, Affine linear sieve, expanders, and sum-product, Invent. Math, Springeronline (2009). |
[9] |
J. Bourgain, A. Glibichuk and S. Konyagin, Estimates for the number of sums and products and for exponential sums in fields of prime order, J. London Math. Soc. (2), 73 (2006), 380-398.
doi: doi:10.1112/S0024610706022721. |
[10] |
M.-C. Chang, Convolution of discrete measures on linear groups, J. Funct. Anal., 253 (2007), 303-323.
doi: doi:10.1016/j.jfa.2007.03.008. |
[11] |
M.-C. Chang, Product theorems in $\SL_2$ and $\SL_3$, J. Math. Jussieu, 7 (2008), 1-–25. |
[12] |
M.-C. Chang, On product sets in $\SL_2$ and $\SL_3$,, preprint., ().
|
[13] |
M.-C. Chang, Some consequences of the polynomial Freiman-Ruzsa conjecture, C. R. Math. Acad. Sci. Paris, 347 (2009), 583-588. |
[14] |
L. E. Dickson, "Linear groups with an exposition of Galois Field Theory," Chapter XII, Cosimo classics, New York, 2007. |
[15] |
O. Dinai, Expansion properties of finite simple groups,, preprint. \arXiv{1001.5069}, ().
|
[16] |
A. Eskin, S. Mozes and H. Oh, On uniform exponential growth for linear groups, Invent. Math., 160 (2005), 1-30.
doi: doi:10.1007/s00222-004-0378-z. |
[17] |
A. Gamburd, S. Hoory, M. Shahshahani, A. Shalev and B. Virag, On the girth of random Cayley graphs, Random Structures Algorithms, 35 (2009), 100-117.
doi: doi:10.1002/rsa.20266. |
[18] |
N. Gill and H. Helfgott, Growth of small generating sets in $\SL_n(\Z/p\Z)$,, preprint. \arXiv{1002.1605}, ().
|
[19] |
W. T. Gowers, Quasirandom groups, Combin. Probab. Comput., 17 (2008), 363-387.
doi: doi:10.1017/S0963548307008826. |
[20] |
B. J. Green, Approximate groups and their applications: Work of Bourgain, Gamburd, Helfgott and Sarnak,, preprint. \arXiv{0911.3354}, ().
|
[21] |
M. Gromov, Groups of polynomial growth and expanding maps, Inst. Hautes Études Sci. Publ. Math. No., 53 (1981), 53-73. |
[22] |
H. Helfgott, Growth and generation in $SL_2(Z/pZ)$, Ann. of Math. (2), 167 (2008), 601-623.
doi: doi:10.4007/annals.2008.167.601. |
[23] |
H. Helfgott, Growth in $\SL_3(\Z/p\Z)$, preprint (2008). arXiv:0807.2027 |
[24] |
J. E. Humphreys, "Linear Algebraic Groups," Springer-Verlag GTM 21, 1975. |
[25] |
E. Hrushovski, Stable group theory and approximate subgroups, preprint (2009). arXiv:0909.2190 |
[26] |
M. Larsen and R. Pink, Finite subgroups of algebraic groups, preprint (1995). |
[27] |
C. Matthews, L. Vaserstein and B. Weisfeiler, Congruence properties of Zariski-dense subgroups, Proc. London Math. Soc, 48 (1984), 514-532.
doi: doi:10.1112/plms/s3-48.3.514. |
[28] |
M. V. Nori, On subgroups of $\GL_n(\F_p)$, Invent. Math., 88 (1987), 257-275.
doi: doi:10.1007/BF01388909. |
[29] |
L. Pyber and E. Szabó, Growth in finite simple groups of Lie type, preprint (2010). arXiv:1001.4556 |
[30] |
I. Z .Ruzsa, Generalized arithmetical progressions and sumsets, Acta. Math. Hungar., 65 (1994), 379-388.
doi: doi:10.1007/BF01876039. |
[31] |
T. C. Tao, Product set estimates in noncommutative groups, Combinatorica, 28 (2008), 547-594. |
[32] |
T. C. Tao, Freiman's theorem for solvable groups,, preprint., ().
|
[33] |
T. C. Tao and V. H. Vu, "Additive Combinatorics," Cambridge University Press, 2006.
doi: doi:10.1017/CBO9780511755149. |
[34] |
J. Tits, Free subgroups in linear groups, Journal of Algebra, 20 (1972), 250-270.
doi: doi:10.1016/0021-8693(72)90058-0. |
[35] |
P. Varjú, Expansion in $\SL_d(\mathcalO_K/I)$, $I$ squarefree,, preprint., ().
|
[36] |
V. H. Vu, M. Wood and P. Wood, Mapping incidences,, preprint., ().
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