Algebra & Number Theory

The congruence topology, Grothendieck duality and thin groups

Alexander Lubotzky and Tyakal Nanjundiah Venkataramana

Full-text: Access denied (no subscription detected)

However, an active subscription may be available with MSP at msp.org/ant.

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text

Abstract

This paper answers a question raised by Grothendieck in 1970 on the “Grothendieck closure” of an integral linear group and proves a conjecture of the first author made in 1980. This is done by a detailed study of the congruence topology of arithmetic groups, obtaining along the way, an arithmetic analogue of a classical result of Chevalley for complex algebraic groups. As an application we also deduce a group theoretic characterization of thin subgroups of arithmetic groups.

Article information

Source
Algebra Number Theory, Volume 13, Number 6 (2019), 1281-1298.

Dates
Received: 8 May 2018
Revised: 20 January 2019
Accepted: 8 March 2019
First available in Project Euclid: 21 August 2019

Permanent link to this document
https://projecteuclid.org/euclid.ant/1566353008

Digital Object Identifier
doi:10.2140/ant.2019.13.1281

Mathematical Reviews number (MathSciNet)
MR3994565

Zentralblatt MATH identifier
07103974

Subjects
Primary: 11E57: Classical groups [See also 14Lxx, 20Gxx]
Secondary: 20G30: Linear algebraic groups over global fields and their integers

Keywords
congruence subgroup thin groups

Citation

Lubotzky, Alexander; Venkataramana, Tyakal Nanjundiah. The congruence topology, Grothendieck duality and thin groups. Algebra Number Theory 13 (2019), no. 6, 1281--1298. doi:10.2140/ant.2019.13.1281. https://projecteuclid.org/euclid.ant/1566353008


Export citation

References

  • H. Bass and A. Lubotzky, “Nonarithmetic superrigid groups: counterexamples to Platonov's conjecture”, Ann. of Math. $(2)$ 151:3 (2000), 1151–1173.
  • H. Bass, J. Milnor, and J.-P. Serre, “Solution of the congruence subgroup problem for ${\rm SL}\sb{n}\,(n\geq 3)$ and ${\rm Sp}\sb{2n}\,(n\geq 2)$”, Inst. Hautes Études Sci. Publ. Math. 33 (1967), 59–137.
  • M. R. Bridson and F. J. Grunewald, “Grothendieck's problems concerning profinite completions and representations of groups”, Ann. of Math. $(2)$ 160:1 (2004), 359–373.
  • J. S. Chahal, “Solution of the congruence subgroup problem for solvable algebraic groups”, Nagoya Math. J. 79 (1980), 141–144.
  • K. Corlette, “Archimedean superrigidity and hyperbolic geometry”, Ann. of Math. $(2)$ 135:1 (1992), 165–182.
  • M. Gromov and R. Schoen, “Harmonic maps into singular spaces and $p$-adic superrigidity for lattices in groups of rank one”, Inst. Hautes Études Sci. Publ. Math. 76 (1992), 165–246.
  • A. Grothendieck, “Représentations linéaires et compactification profinie des groupes discrets”, Manuscripta Math. 2 (1970), 375–396.
  • A. Kontorovich, D. D. Long, A. Lubotzky, and A. W. Reid, “What is$\dots$ a thin group?”, Notices Amer. Math. Soc. 66:6 (2019), 905–910.
  • A. Lubotzky, “Tannaka duality for discrete groups”, Amer. J. Math. 102:4 (1980), 663–689.
  • G. A. Margulis, Discrete subgroups of semisimple Lie groups, Ergebnisse der Mathematik $($3$)$ 17, Springer, 1991.
  • M. V. Nori, “On subgroups of ${\rm GL}_n(\mathbb{F}_p)$”, Invent. Math. 88:2 (1987), 257–275.
  • V. Platonov and A. Rapinchuk, Algebraic groups and number theory, Pure and Appl. Math. 139, Academic Press, Boston, 1994.
  • V. P. Platonov and O. I. Tavgen, “On the Grothendieck problem of profinite completions of groups”, Dokl. Akad. Nauk SSSR 288:5 (1986), 1054–1058. In Russian; translated in Soviet Math. Dokl. 33:3 (1986), 822–825.
  • L. Pyber, “Groups of intermediate subgroup growth and a problem of Grothendieck”, Duke Math. J. 121:1 (2004), 169–188.
  • M. S. Raghunathan, “On the congruence subgroup problem”, Inst. Hautes Études Sci. Publ. Math. 46 (1976), 107–161.
  • M. S. Raghunathan, “The congruence subgroup problem”, pp. 465–494 in Proceedings of the Hyderabad Conference on Algebraic Groups (Hyderabad, 1989), edited by S. Ramanan et al., Manoj Prakashan, Madras, 1991.
  • P. Sarnak, “Notes on thin matrix groups”, pp. 343–362 in Thin groups and superstrong approximation (Berkeley, 2012), edited by E. Breuillard and H. Oh, Math. Sci. Res. Inst. Publ. 61, Cambridge Univ. Press, 2014.
  • J.-P. Serre, “Groupes de congruence (d'après H. Bass, H. Matsumoto, J. Mennicke, J. Milnor, C. Moore)”, pp. [exposé] 330 in Séminaire Bourbaki, 1966/1967, W. A. Benjamin, Amsterdam, 1968. Reprinted as pp. 275–291 in Séminaire Bourbaki 10, Soc. Math. France, Paris, 1995.
  • J. Tits, “Free subgroups in linear groups”, J. Algebra 20 (1972), 250–270.
  • T. N. Venkataramana, “A remark on extended congruence subgroups”, Int. Math. Res. Not. 1999:15 (1999), 835–838.
  • B. Weisfeiler, “Strong approximation for Zariski-dense subgroups of semi-simple algebraic groups”, Ann. of Math. $(2)$ 120:2 (1984), 271–315.