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Quantifying residual finiteness of arithmetic groups

Part of: Structure and classification of infinite or finite groups Discontinuous groups and automorphic forms Other groups of matrices Lie groups

Published online by Cambridge University Press:  19 March 2012

Khalid Bou-Rabee  and
Tasho Kaletha
Khalid Bou-Rabee
Affiliation:
Department of Mathematics, University of Chicago, 5734 University Ave., Chicago, IL 60637, USA (email: khalid@math.uchicago.edu)
Tasho Kaletha
Affiliation:
Department of Mathematics, University of Chicago, 5734 University Ave., Chicago, IL 60637, USA (email: tkaletha@math.uchicago.edu)
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Abstract

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The normal residual finiteness growth of a group quantifies how well approximated the group is by its finite quotients. We show that any S-arithmetic subgroup of a higher rank Chevalley group G has normal residual finiteness growth ndim (G).

Keywords

arithmetic groupsnormal residual finiteness growthresidual finiteness

MSC classification

Secondary: 20E26: Residual properties and generalizations; residually finite groups 11F06: Structure of modular groups and generalizations; arithmetic groups 22E40: Discrete subgroups of Lie groups 20H05: Unimodular groups, congruence subgroups
Type
Research Article
Information
Compositio Mathematica , Volume 148 , Issue 3 , May 2012 , pp. 907 - 920
Copyright
Copyright © Foundation Compositio Mathematica 2012

References

[Alp87]Alperin, R. C., An elementary account of Selberg’s lemma, Enseign. Math. (2) 33 (1987), 269–273.Google Scholar
[Bou10]Bou-Rabee, K., Quantifying residual finiteness, J. Algebra 323 (2010), 729–737.CrossRefGoogle Scholar
[SGA3]Demazure, M. and Grothendieck, A., Schémas en groupes I, II, III, Lecture Notes in Mathematics, vols 151, 152, 153 (Springer, New York, 1970).Google Scholar
[His84]Hiss, G., Die adjungierten Darstellungen der Chevalley-Gruppen, Arch. Math. 42 (1984), 408–416.CrossRefGoogle Scholar
[Hog82]Hogeweij, G. M. D., Almost classical Lie algebras: I, II, Indag. Math. 44 (1982), 441–460.CrossRefGoogle Scholar
[KM11]Kassabov, M. and Matucci, F., Bounding residual finiteness in free groups, Proc. Amer. Math. Soc. 139 (2011), 2281–2286.CrossRefGoogle Scholar
[LL]Larsen, M. and Lubotzky, A., On linear groups of polynomial normal subgroup growth; the (G2 F4 E8) theorem, in Algebraic groups & arithmetic, Tata Institute of Fundamental Research Studies in Mathematics, eds S. G. Dani and G. Prasad (TIFR, Mumbai), 441–468.Google Scholar
[LMR00]Lubotzky, A., Mozes, Sh. and Raghunathan, M. S., The word and Riemannian metrics on lattices of semisimple groups, Publ. Math. Inst. Hautes Études Sci. 91 (2000), 5–53.CrossRefGoogle Scholar
[LS03]Lubotzky, A. and Segal, D., Subgroup growth (Birkhäuser, Basel, 2003).CrossRefGoogle Scholar
[MP94]Moy, A. and Prasad, G., Unrefined minimal K-types for p-adic groups, Invent. Math. 116 (1994), 393–408.CrossRefGoogle Scholar
[PR94]Platonov, V. P. and Rapinchuk, A. S., Algebraic groups and number theory, Pure and Applied Mathematics, vol. 139 (Academic Press, Boston, MA, 1994), Translated from the 1991 Russian original by Rachel Rowen.Google Scholar
[Ste68]Steinberg, R. G., Lectures on Chevalley groups (Yale University Press, 1968).Google Scholar