Abstract
Let Γ be anS-arithmetic group in a semisimple group. We show that if Γ has the congruence subgroup property then the number of isomorphism classes of irreducible complexn-dimensional characters of Γ is polynomially bounded. In characteristic zero, the converse is also true. We conjecture that the converse also holds in positive characteristic, and we prove some partial results in this direction.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
[BLMM02] H. Bass, A. Lubotzky, A. R. Magid and S. Mozes,The proalgebraic completion of rigid groups. Geometriae Dedicata95 (2002), 19–58.
[CR91] C. W. Curtis and I. Reiner,Methods of Representation Theory. Vol. I. Wiley, New York, 1981.
[DdSMS99] J. D. Dixon, M. P. F. du Sautoy, A. Mann and D. Sega.Analytic Pro-p Groups, second edn., Cambridge Studies in Advanced Mathematics, Vol. 61, Cambridge University Press, Cambridge, 1999.
[Erd41] P. Erdös,On some asymptotic formulas in the theory of the “factorisation numerorum”, Annals of Mathematics42 (1941), 989–993.
[Hum72] J. E. Humphreys,Introduction to Lie Algebras and Representation Theory, Graduate Texts in Mathematics, Vol. 9, Springer-Verlag, New York, 1972.
[JZ03] A. Jalkin-Zapirain,The zeta function of representation of pro-p groups, preprint, 2003.
[KL90] P. Keidman and M. Liebeck,The Subgroup Structure of the Finite Classical Groups, London Mathematical Society Lecture Note Series, Vol. 129, Cambridge University Press, Cambridge, 1990.
[Lan93] S. Lang,Algebra, third edn., Addison-Wesley, Reading, MA, 1993.
[LL] M. Larsen and A. Lubotzky,Normal subgroup growth of linear groups: The (G2, F4, E8)-Theorem, inAlgebraic Groups and Arithmetic (Mumbai 2001) (S. G. Dani and G. Prasad, eds.), Tata Institute of Fundamental Research Studies in Mathematics, Narosa Publishing House, New Delhi, 2004, pp. 441–468.
[LM85] A. Lubotzky and A. R. Magid,Varieties of representations of finitely generated groups. Memoirs of the American Mathematical Society58 (1985), no. 336, xi+117.
[LS94] A. Lubotzky and A. Shalev,On some A-analytic pro-p groups, Israel Journal of Mathematics85 (1994) 307–337.
[LS03] A. Lubotzky and D. Segal,Subgroup Growth, Progress in Mathematics, Vol. 212, Birkhäuser Verlag, Basel, 2003.
[Lub91] A. Lubotzky,Lattices in rank one Lie groups over local fields, Geometric and Functional Analysis (GAFA)1 (1991), 406–431.
[Lub95] A. Lubotzky,Subgroup growth and congruence subgroups, Inventiones Mathematicae119 (1995), 267–295.
[Mar91] G. A. Margulis,Discrete subgroups of semisimple Lie groups, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], Vol. 17, Springer-Verlag, Berlin, 1991.
[PR93] V. P. Platonov and A. S. Rapinchuk,Abstract properties of S-arithmetic groups and the congruence problem, Russian Academy of Sciences. Izvestiya. Mathematics40 (1993), 455–476.
[PR94] V. Platonov and A. Rapinchuk,Algebraic Groups and Number Theory, Pure and Applied Mathematics, Vol. 139, Academic Press, Boston, MA, 1994.
[PR96] G. Prasad and A. S. Rapinchuk,Computation of the metaplectic kernel, Publications Mathématiques de l’Institut des Hautes Études Scientifiques84 (1996), 91–187.
[Pra77] G. Prasad,Strong approximation for semi-simple groups over function fields, Annals of Mathematics105 (1977), 553–572.
[Rag76] M. S. Raghunathan,On the congruence subgroup problem, Publications Mathématiques de l’Institut des Hautes Études Scientifiques46 (1976). 107–161.
[Ros02] M. Rosen,Number Theory in Function Fields, Graduate Texts in Mathematics, Vol. 210, Springer-Verlag, New York, 2002.
[Seg99] Y. Segev,On finite homomorphic images of the multiplicative group of a division algebra, Annals of Mathematics149 (1999), 219–251.
[Ser70] J.-P. Serre,Le problème des groupes de congruence pour SL 2, Annals of Mathematics92 (1970), 489–527.
[Ser92] J.-P. Serre,Lie Algebras and Lie Groups, second edn., Lecture Notes in Mathematics, Vol. 1500, Springer-Verlag, Berlin, 1992.
[Ven88] T. N. Venkataramana,On superrigidity and arithmeticity of lattices in semisimple groups over local fields of arbitrary characteristic, Inventiones Mathematicae92 (1988), 255–306.
Author information
Authors and Affiliations
Corresponding author
Additional information
Dedicated to Andy Magid on the occasion of his sixtieth birthday
Rights and permissions
About this article
Cite this article
Lubotzky, A., Martin, B. Polynomial representation growth and the congruence subgroup problem. Isr. J. Math. 144, 293–316 (2004). https://doi.org/10.1007/BF02916715
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF02916715