Geodesic
Growth in finite simple groups of Lie type
Pyber, László1 ; Szabó, Endre1
1 A. Rényi Institute of Mathematics, Hungarian Academy of Sciences, P.O. Box 127, H-1364 Budapest, Hungary
Journal of the American Mathematical Society, Tome 29 (2016) no. 1, pp. 95-146

Voir la notice de l'article provenant de la source American Mathematical Society

We prove that if $L$ is a finite simple group of Lie type and $A$ a set of generators of $L$, then either $A$ grows, i.e., $|A^3| > |A|^{1+\varepsilon }$ where $\varepsilon$ depends only on the Lie rank of $L$, or $A^3=L$. This implies that for simple groups of Lie type of bounded rank a well-known conjecture of Babai holds, i.e., the diameter of any Cayley graph is polylogarithmic. We also obtain new families of expanders. A generalization of our proof yields the following. Let $A$ be a finite subset of $SL(n,\mathbb {F})$, $\mathbb {F}$ an arbitrary field, satisfying $\big |A^3\big |\le \mathcal {K}|A|$. Then $A$ can be covered by $\mathcal {K}^m$, i.e., polynomially many, cosets of a virtually soluble subgroup of $SL(n,\mathbb {F})$ which is normalized by $A$, where $m$ depends on $n$.
DOI : 10.1090/S0894-0347-2014-00821-3

Pyber, László 1 ; Szabó, Endre 1

1 A. Rényi Institute of Mathematics, Hungarian Academy of Sciences, P.O. Box 127, H-1364 Budapest, Hungary 
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Pyber, László; Szabó, Endre. Growth in finite simple groups of Lie type. Journal of the American Mathematical Society, Tome 29 (2016) no. 1, pp. 95-146. doi: 10.1090/S0894-0347-2014-00821-3

[1] Annexstein, Fred, Baumslag, Marc On the diameter and bisector size of Cayley graphs Math. Systems Theory 1993 271 291

[2] Babai, L., Kantor, W. M., Lubotsky, A. Small-diameter Cayley graphs for finite simple groups European J. Combin. 1989 507 522

[3] Babai, Lã¡Szlã³, Seress, ÁKos On the diameter of permutation groups European J. Combin. 1992 231 243

[4] Blanco, Cristina, Jeronimo, Gabriela, Solernã³, Pablo Computing generators of the ideal of a smooth affine algebraic variety J. Symbolic Comput. 2004 843 872

[5] Bourgain, Jean, Gamburd, Alex Uniform expansion bounds for Cayley graphs of 𝑆𝐿₂(𝔽_{𝕡}) Ann. of Math. (2) 2008 625 642

[6] Bourgain, Jean, Gamburd, Alex Expansion and random walks in 𝑆𝐿_{𝑑}(ℤ/𝕡ⁿℤ). II J. Eur. Math. Soc. (JEMS) 2009 1057 1103

[7] Bourgain, Jean, Gamburd, Alex, Sarnak, Peter Affine linear sieve, expanders, and sum-product Invent. Math. 2010 559 644

[8] Bourgain, J., Katz, N., Tao, T. A sum-product estimate in finite fields, and applications Geom. Funct. Anal. 2004 27 57

[9] Breuillard, Emmanuel, Green, Ben, Tao, Terence Linear approximate groups Electron. Res. Announc. Math. Sci. 2010 57 67

[10] Breuillard, Emmanuel, Green, Ben, Tao, Terence Approximate subgroups of linear groups Geom. Funct. Anal. 2011 774 819

[11] Breuillard, Emmanuel, Green, Ben Approximate groups, II: The solvable linear case Q. J. Math. 2011 513 521

[12] Carter, Roger W. Simple groups of Lie type 1972

[13] Carter, Roger W. Finite groups of Lie type 1985

[14] Cohen, Arjeh M., Seitz, Gary M. The 𝑟-rank of the groups of exceptional Lie type Nederl. Akad. Wetensch. Indag. Math. 1987 251 259

[15] Chevalley, Claude Classification des groupes algébriques semi-simples 2005

[16] Dinai, Oren Growth in 𝑆𝐿₂ over finite fields J. Group Theory 2011 273 297

[17] Doerk, Klaus, Hawkes, Trevor Finite soluble groups 1992

[18] Elekes, Gyã¶Rgy, Kirã¡Ly, Zoltã¡N On the combinatorics of projective mappings J. Algebraic Combin. 2001 183 197

[19] Feit, Walter, Tits, Jacques Projective representations of minimum degree of group extensions Canadian J. Math. 1978 1092 1102

[20] Freä­Man, G. A. Groups and the inverse problems of additive number theory 1973 175 183

[21] Freiman, Gregory A. On finite subsets of nonabelian groups with small doubling Proc. Amer. Math. Soc. 2012 2997 3002

[22] Fulton, William Intersection theory 1984

[23] Gill, Nick, Helfgott, Harald Andrã©S Growth of small generating sets in 𝑆𝐿_{𝑛}(ℤ/𝕡ℤ) Int. Math. Res. Not. IMRN 2011 4226 4251

[24] Salehi Golsefidy, Alireza, Sarnak, Peter The affine sieve J. Amer. Math. Soc. 2013 1085 1105

[25] Gowers, W. T. Quasirandom groups Combin. Probab. Comput. 2008 363 387

[26] Hartshorne, Robin Algebraic geometry 1977

[27] Helfgott, H. A. Growth and generation in 𝑆𝐿₂(ℤ/𝕡ℤ) Ann. of Math. (2) 2008 601 623

[28] Helfgott, H. A. Growth in 𝑆𝐿₃(ℤ/𝕡ℤ) J. Eur. Math. Soc. (JEMS) 2011 761 851

[29] Hrushovski, Ehud Stable group theory and approximate subgroups J. Amer. Math. Soc. 2012 189 243

[30] Hrushovski, E., Pillay, A. Definable subgroups of algebraic groups over finite fields J. Reine Angew. Math. 1995 69 91

[31] Humphreys, James E. Linear algebraic groups 1975

[32] Humphreys, James E. Conjugacy classes in semisimple algebraic groups 1995

[33] Isaacs, I. Martin Character theory of finite groups 2006

[34] Khukhro, E. I., Klyachko, Ant. A., Makarenko, N. Yu., Melnikova, Yu. B. Automorphism invariance and identities Bull. Lond. Math. Soc. 2009 804 816

[35] Kleidman, Peter, Liebeck, Martin The subgroup structure of the finite classical groups 1990

[36] Ellenberg, Jordan S., Hall, Chris, Kowalski, Emmanuel Expander graphs, gonality, and variation of Galois representations Duke Math. J. 2012 1233 1275

[37] Larsen, Michael Exponential generation and largeness for compact 𝑝-adic Lie groups Algebra Number Theory 2010 1029 1038

[38] Landazuri, Vicente, Seitz, Gary M. On the minimal degrees of projective representations of the finite Chevalley groups J. Algebra 1974 418 443

[39] Liebeck, Martin W., Seitz, Gary M. On the subgroup structure of exceptional groups of Lie type Trans. Amer. Math. Soc. 1998 3409 3482

[40] Lubotzky, Alexander Expander graphs in pure and applied mathematics Bull. Amer. Math. Soc. (N.S.) 2012 113 162

[41] Mumford, David Varieties defined by quadratic equations 1970 29 100

[42] Nori, Madhav V. On subgroups of 𝐺𝐿_{𝑛}(𝐹_{𝑝}) Invent. Math. 1987 257 275

[43] Nikolov, N., Pyber, L. Product decompositions of quasirandom groups and a Jordan type theorem J. Eur. Math. Soc. (JEMS) 2011 1063 1077

[44] Nikolov, Nikolay, Segal, Dan On finitely generated profinite groups. I. Strong completeness and uniform bounds Ann. of Math. (2) 2007 171 238

[45] Olson, John E. On the sum of two sets in a group J. Number Theory 1984 110 120

[46] Pyber, Lã¡Szlã³ Asymptotic results for permutation groups 1993 197 219

[47] Ruzsa, I. Z., Turjã¡Nyi, S. A note on additive bases of integers Publ. Math. Debrecen 1985 101 104

[48] Tao, Terence Product set estimates for non-commutative groups Combinatorica 2008 547 594

[49] Varjãº, Pã©Ter P. Expansion in 𝑆𝐿_{𝑑}(𝒪_{𝒦}/ℐ), ℐ square-free J. Eur. Math. Soc. (JEMS) 2012 273 305

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