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Izv. RAN. Ser. Mat., 1992, Volume 56, Issue 3, Pages 483–508 (Mi izv936)  

This article is cited in 10 scientific papers (total in 11 papers)

Abstract properties of $S$-arithmetic groups and the congruence s problem

V. P. Platonov, A. S. Rapinchuk

Abstract: Suppose $G$ is a simple and simply connected algebraic group over an algebraic number field $K$ and $S$ is a finite set of valuations of $K$ containing all Archimedean valuations. This paper is a study of the connections between abstract properties of the $S$-arithmetic subgroup $\mathbf\Gamma=G_{O(S)}$ and the congruence property, i.e. the finiteness of the corresponding congruence kernel $C=C^S(G)$. In particular, it is shown that if the profinite completion $\Delta=\widehat\Gamma$ satisfies condition $(\mathbf {PG})'$, (i.e., for any integer $n>0$ and any prime $p$ there exist $c$ and $k$ such that $|\Delta/\Delta^{np^\alpha}|\leqslant cp^{k\alpha}$ for all $\alpha>0$, then $C$ is finite. Examples are given demonstrating the possibility of effectively verifying $(\mathbf {PG})'$ .

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English version:
Russian Academy of Sciences. Izvestiya Mathematics, 1993, 40:3, 455–476

Bibliographic databases:

UDC: 512.743
MSC: 20G30, 11E57, 20H05
Received: 13.12.1991

Citation: V. P. Platonov, A. S. Rapinchuk, “Abstract properties of $S$-arithmetic groups and the congruence s problem”, Izv. RAN. Ser. Mat., 56:3 (1992), 483–508; Russian Acad. Sci. Izv. Math., 40:3 (1993), 455–476

Citation in format AMSBIB
\by V.~P.~Platonov, A.~S.~Rapinchuk
\paper Abstract properties of $S$-arithmetic groups and the congruence s problem
\jour Izv. RAN. Ser. Mat.
\yr 1992
\vol 56
\issue 3
\pages 483--508
\jour Russian Acad. Sci. Izv. Math.
\yr 1993
\vol 40
\issue 3
\pages 455--476

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    This publication is cited in the following articles:
    1. Lubotzky A., “Subgroup Growth and Congruence Subgroups”, Invent. Math., 119:2 (1995), 267–295  crossref  mathscinet  zmath  adsnasa  isi
    2. Gopal Prasad, Andrei S. Rapinchuk, “Computation of the metaplectic kernel”, Publications Mathématiques de l’Institut des Hautes Études Scientifiques, 84:1 (1996), 91  crossref  mathscinet  zmath
    3. S. I. Adian, E. I. Zel'manov, G. A. Margulis, S. P. Novikov, A. S. Rapinchuk, L. D. Faddeev, V. I. Yanchevskii, “Vladimir Petrovich Platonov (on his 60th birthday)”, Russian Math. Surveys, 55:3 (2000), 601–610  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi
    4. Farb B., Lubotzky A., Minsky Y., “Rank-1 Phenomena for Mapping Class Groups”, Duke Math. J., 106:3 (2001), 581–597  crossref  mathscinet  zmath  isi
    5. Abert M., “Symmetric Groups as Products of Abelian Subgroups”, Bull. London Math. Soc., 34:4 (2002), 451–456  crossref  mathscinet  zmath  isi
    6. Abert M., Lubotzky A., Pyber L., “Bounded Generation and Linear Groups”, Int. J. Algebr. Comput., 13:4 (2003), 401–413  crossref  mathscinet  zmath  isi
    7. Seress A., “A Product Decomposition of Infinite Symmetric Groups”, Proc. Amer. Math. Soc., 131:6 (2003), 1681–1685  crossref  mathscinet  zmath  isi
    8. Alexander Lubotzky, Benjamin Martin, “Polynomial representation growth and the congruence subgroup problem”, Isr J Math, 144:2 (2004), 293  crossref  mathscinet  zmath  isi
    9. Igor V. Erovenko, Andrei S. Rapinchuk, “Bounded generation of S-arithmetic subgroups of isotropic orthogonal groups over number fields”, Journal of Number Theory, 119:1 (2006), 28  crossref
    10. Pyber L., Segal D., “Finitely Generated Groups with Polynomial Index Growth”, J. Reine Angew. Math., 612 (2007), 173–211  crossref  mathscinet  zmath  isi
    11. Proc. Steklov Inst. Math., 292 (2016), 216–246  mathnet  crossref  crossref  mathscinet  isi  elib
  • Известия Российской академии наук. Серия математическая Izvestiya: Mathematics
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