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On conjugacy growth of linear groups†

Published online by Cambridge University Press:  31 October 2012

EMMANUEL BREUILLARD ,
YVES CORNULIER ,
ALEXANDER LUBOTZKY  and
CHEN MEIRI
EMMANUEL BREUILLARD
Affiliation:
Laboratoire de Mathématiques, Bâtiment 425, Université Paris-Sud 11, 91405 Orsay, France. e-mail: [email protected], [email protected]
YVES CORNULIER
Affiliation:
Laboratoire de Mathématiques, Bâtiment 425, Université Paris-Sud 11, 91405 Orsay, France. e-mail: [email protected], [email protected]
ALEXANDER LUBOTZKY
Affiliation:
Einstein institute of Mathematics, Hebrew University, Jerusalem 91904, Israel. e-mail: [email protected], [email protected]
CHEN MEIRI
Affiliation:
Einstein institute of Mathematics, Hebrew University, Jerusalem 91904, Israel. e-mail: [email protected], [email protected]
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Abstract

We investigate the conjugacy growth of finitely generated linear groups. We show that finitely generated non-virtually-solvable subgroups of GLd have uniform exponential conjugacy growth and in fact that the number of distinct polynomials arising as characteristic polynomials of the elements of the ball of radius n for the word metric has exponential growth rate bounded away from 0 in terms of the dimension d only.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 154 , Issue 2 , March 2013 , pp. 261 - 277
Copyright
Copyright © Cambridge Philosophical Society 2012

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Footnotes

†

The authors are grateful for grants from the ERC and the NSF.

References

REFERENCES

[1]Breuillard, E.On uniform exponential growth for solvable groups. Pure Appl. Math. Q. 3, Margulis Volume (2007), no. 4, part 1, 949–967.CrossRefGoogle Scholar
[2]Breuillard, E.A height gap theorem for nonvirtually solvable subgroups of GLn. Ann. of Math. (2) 174 (2011), no. 2, 1057–1110.CrossRefGoogle Scholar
[3]Breuillard, E. A strong Tits alternative, preprint arXiv:0804.1395.Google Scholar
[4]Breuillard, E. and Cornulier, Y.On conjugacy growth for solvable groups. Illinois J. Math. 54 (1) (2010), 389–395.CrossRefGoogle Scholar
[5]Breuillard, E. and Gelander, T.Uniform independence for linear groups. Invent. Math. 173 (2008), no. 2, 225–263.CrossRefGoogle Scholar
[6]Breuillard, E., Green, B. J. and Tao, T. C.Linear approximate groups.. Electron. Res. Announc. Math. Sci. 17 (2010), 57–67.Google Scholar
[7]Breuillard, E., Green, B. J. and Tao, T. C.Approximate subgroups of linear groups. Geom. Funct. Anal. 21 (2011), no. 4, 774–819.CrossRefGoogle Scholar
[8]Eskin, A., Mozes, S. and Oh, H.On uniform exponential growth for linear groups. Invent. Math. 160 (2005), no. 1, 1–30.CrossRefGoogle Scholar
[9]Hardy, G. H. and Wright, E. M.An Introduction to the Theory of Numbers, 5th ed (Oxford Science Publishing 1979).Google Scholar
[10]Grothendieck, A.Étude locale des schémas et des morphismes de schémas, Troisième partie, in Éléments de géométrie algébrique. IV. Publ. Math. Inst. Hautes Études Sci. 28 (1966), p. 5–255.CrossRefGoogle Scholar
[11]Guba, V. and Sapir, M.On the conjugacy growth functions of groups. Illinois J. Math. 54 (1) (2010), 301–313.CrossRefGoogle Scholar
[12]Hull, M.Conjugacy growth in polycyclic groups. Arch. Math. (Basel) 96 (2011), no. 2, 131–134.CrossRefGoogle Scholar
[13]Lang, S. and Weil, A.Number of points of varieties in finite fields. Amer. J. Math. 76, (1954). 819–827.CrossRefGoogle Scholar
[14]Larsen, M. and Pink, R.Finite subgroups of algebraic groups. J. Amer. Math. Soc. 24 (2011), no. 4, 1105–1158.CrossRefGoogle Scholar
[15]Lubotzky, A. and Larsen, M.Normal subgroup growth of Linear groups, the G 2, F 4, E 8 case. In Algebraic Groups and Arithmetic, Raghunathan volume, (Tata Institute Publ. 2004), 440–468.Google Scholar
[16]Lubotzky, A. and Mann, A.On groups of polynomial subgroup growth. Invent. Math. 104, 521–533 (1991).CrossRefGoogle Scholar
[17]Magnus, W., Karass, A. and Solitar, D.Combinatorial Group Theory. (Dover Publishing New York 1976).Google Scholar
[18]Mann, A.How Groups Grow. London Math. Soc. Lecture Note Series, 395 (Cambridge University Press, 2012), 199 pp.Google Scholar
[19]Matthews, C. R., Vaserstein, L. N. and Weisfeiler, B.Congruence properties of Zariski-dense subgroups. I. Proc. London Math. Soc. (3) 48 (1984), no. 3, 514–532.CrossRefGoogle Scholar
[20]Milnor, J.Growth of finitely generated solvable groups. J. Diff. Geom. 2 (1968), 447–449.Google Scholar
[21]Neukirch, J.Algebraic Number Theory, 2nd ed. (Springer-Verlag, Berlin, Heidelberg 1999).CrossRefGoogle Scholar
[22]Ol'shanskii, A. Yu.Geometry of Defining Relations in Groups. Translated from the 1989 Russian original by Yu. A. Bakhturin. Mathematics and its Applications (Soviet Series), 70 (Kluwer Academic Publishers Group, Dordrecht, 1991), 505 pp.CrossRefGoogle Scholar
[23]Osin, D.Small cancellations over relatively hyperbolic groups and embedding theorems. Annals of Math. 172 (2010), 1–39.CrossRefGoogle Scholar
[24]Pink, R.Strong approximation for Zariski-dense subgroups over arbitrary global fields. Comment. Math. Helv. 75 (2000), 608–643.CrossRefGoogle Scholar
[25]Pyber, L. and Szabó, E. Growth in finite simple groups of Lie type of bounded rank. preprint (2010), arXiv:1001.4556.Google Scholar
[26]Rivin, I. Growth in free groups (and other stories). arXiv math/9911076 (preprint 1999).Google Scholar
[27]Rosen, M.Number Theory in Function Fields. (Springer-Verlag, New-York 2002).CrossRefGoogle Scholar
[28]Tits, J.Free subgroups in linear groups. J. Algebra. 20 (1972), 250–270.CrossRefGoogle Scholar
[29]Weisfeiler, B.Strong approximation for Zariski-dense subgroups of semi-simple algebraic groups. Annals of Math. 120 (1984), 271–315.CrossRefGoogle Scholar
[30]Wolf, J.Growth of finitely generated solvable groups and curvature of Riemannian manifolds. J. Diff. Geom. 2, 421–446 (1968).Google Scholar