Abstract
We show that for any non-virtually solvable finitely generated group of matrices over any field, there is an integer m such that, given an arbitrary finite generating set for the group, one may find two elements a and b that are both products of at most m generators, such that a and b are free generators of a free subgroup. This uniformity result improves the original statement of the Tits alternative.
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Arzhantseva, G.N., Burillo, J., Lustig, M., Reeves, L., Short, H., Ventura, E.: Uniform non-amenability. Adv. Math. 197(2), 499–522 (2005)
Benoist, Y.: Propriétés asymptotiques des groupes linéaires. Geom. Funct. Anal. 7, 1–47 (1997)
Borel, A.: Linear Algebraic Groups. Springer, New York (1991)
Borel, A.: On free subgroups of semisimple groups. Enseign. Math. 29, 151–164 (1983)
Borel, A., Tits, J.: Éléments unipotents et sous-groupes paraboliques de groupes réductifs I. Invent. Math. 12, 95–104 (1971)
Breuillard, E., Gelander, T.: On dense free subgroups of Lie groups. J. Algebra 261(2), 448–467 (2003)
Breuillard, E., Gelander, T.: A topological Tits alternative. Ann. Math. 166(2), 427–474 (2007)
Breuillard, E., Gelander, T.: Cheeger constant and algebraic entropy of linear groups. Int. Math. Res. Not. 2005(56), 3511–3523 (2005)
Carrière, Y., Ghys, E.: Relations d’équivalence moyennables sur les groupes de Lie. C. R. Acad. Sci., Paris, Sér I, Math. 300(19), 677–680 (1985)
Eskin, A., Mozes, M., Oh, H.: On uniform exponential growth for linear groups in characteristic zero. Invent. Math. 160(1), 1–30 (2005)
Gelander, T., Zuk, A.: Dependence of Kazhdan constants on generating subsets. Isr. J. Math. 129, 93–98 (2002)
Kuranishi, M.: On everywhere dense embedding of free groups in Lie groups. Nagoya Math. J. 2, 63–71 (1951)
Landvogt, E.: Some functorial properties of the Bruhat–Tits building. J. Reine Angew. Math. 518, 213–241 (2000)
Margulis, G.A.: Discrete Subgroups of Semisimple Lie Groups. Springer, New York (1990)
Platonov, V., Rapinchuk, A.: Algebraic Groups and Number Theory. Academic Press, New York (1994)
Raghunathan, M.S.: Discrete Subgroups of Lie Groups. Springer, New York (1972)
Raghunathan, M.S.: Discrete subgroups of algebraic groups over local fields of positive characteristic. Proc. Indian Acad. Sci., Math. Sci. 99(2), 127–146 (1989)
Schinzel, A.: Polynomial with Special Regard to Reducibility. With appendix by Zannier, U. Cambridge University Press, Cambridge (2000)
Serre, J.P.: Lie algebras and Lie groups. 1964 Lectures Given at Harvard University. Corrected fifth printing of the 2nd (1992) edn. Lect. Notes Math., vol. 1500. Springer, Berlin (2006)
Shalom, Y.: Explicit Kazhdan constants for representations of semisimple and arithmetic groups. Ann. Inst. Fourier 50(3), 833–863 (2000)
Tits, J.: Free subgroups of linear groups. J. Algebra 20, 250–270 (1972)
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Breuillard, E., Gelander, T. Uniform independence in linear groups. Invent. math. 173, 225–263 (2008). https://doi.org/10.1007/s00222-007-0101-y
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DOI: https://doi.org/10.1007/s00222-007-0101-y