Linear approximate groups

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2010, 17: 57-67. doi: 10.3934/era.2010.17.57

Linear approximate groups

Emmanuel Breuillard ^1, , Ben Green ^2, and Terence Tao ^3,

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

Received January 2010 Published September 2010

Abstract
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Under a Creative Commons license

This is an informal announcement of results to be described and proved in detail in [3]. We give various results on the structure of approximate subgroups in linear groups such as ${\rm{S}}{{\rm{L}}_n}(k)$. For example, generalizing a result of Helfgott (who handled the cases $n = 2$ and $3$), we show that any approximate subgroup of ${\rm{S}}{{\rm{L}}_n}({\mathbb{F}_q})$ which generates the group must be either very small or else nearly all of ${\rm{S}}{{\rm{L}}_n}({\mathbb{F}_q})$. The argument is valid for all Chevalley groups $G(\mathbb{F}_q)$. Extending work of Bourgain-Gamburd we also announce some applications to expanders, which will be proven in detail in [4].

Keywords: expander graphs., Approximate groups, growth.

Mathematics Subject Classification: 20G40, 20N9.

Citation: Emmanuel Breuillard, Ben Green, Terence Tao. Linear approximate groups. Electronic Research Announcements, 2010, 17: 57-67. doi: 10.3934/era.2010.17.57

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, to appear in Quart. J. of Math. arXiv:0907.0927
[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, to appear in Quart. J. of Math. arXiv:0907.0927
[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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