Hostname: page-component-8448b6f56d-c47g7 Total loading time: 0 Render date: 2024-04-23T07:27:33.947Z Has data issue: false hasContentIssue false
  • English
  • Français

Decision problems concerning S-arithmetic groups

Published online by Cambridge University Press:  12 March 2014

Fritz Grunewald  and
Daniel Segal
Fritz Grunewald
Affiliation:
Mathematical Institute, University of Bonn, Bonn, West Germany
Daniel Segal
Affiliation:
Mathematical Institute, University of Bonn, Bonn, West Germany
Get access Rights & Permissions [Opens in a new window]

Extract

This paper is a continuation of our previous work in [12]. The results, and some applications, have been described in the announcement [13]; it may be useful to discuss here, a little more fully, the nature and purpose of this work.

We are concerned basically with three kinds of algorithmic problem: (1) isomorphism problems, (2) “orbit problems”, and (3) “effective generation”.

(1) Isomorphism problems. Here we have a class of algebraic objects of some kind, and ask: is there a uniform algorithm for deciding whether two arbitrary members of are isomorphic? In most cases, the answer is no: no such algorithm exists. Indeed this has been one of the most notable applications of methods of mathematical logic in algebra (see [26, Chapter IV, §4] for the case where is the class of all finitely presented groups). It turns out, however, that when consists of objects which are in a certain sense “finite-dimensional”, then the isomorphism problem is indeed algorithmically soluble. We gave such algorithms in [12] for the following cases:

= {finitely generated nilpotent groups};

= {(not necessarily associative) rings whose additive group is finitely generated};

= {finitely Z-generated modules over a fixed finitely generated ring}.

Combining the methods of [12] with his own earlier work, Sarkisian has obtained analogous results with the integers replaced by the rationals: in [20] and [21] he solves the isomorphism problem for radicable torsion-free nilpotent groups of finite rank and for finite-dimensional Q-algebras.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 50 , Issue 3 , September 1985 , pp. 743 - 772
Copyright
Copyright © Association for Symbolic Logic 1985

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

[1]Ax, J., Solving Diophantine equations modulo every prime, Annals of Mathematics, ser. 2, vol. 85 (1967), pp. 161–183.CrossRefGoogle Scholar
[2]Ax, J. and Kochen, S., Diophantine problems over local fields. II, American Journal of Mathematics, vol. 87(1965), pp. 631–648.CrossRefGoogle Scholar
[3]Ax, J. and Kochen, S., Diophantine problems over local fields I, American Journal of Mathematics, vol. 87 (1965), pp. 605–630.CrossRefGoogle Scholar
[4]Borel, A., Introduction aux groupes arithmétiques, Hermann, Paris, 1969.Google Scholar
[5]Borel, A., Density and maximality of arithmetic subgroups, Journal für Se Reine und Angewandte Mathematik, vol. 224 (1966), pp. 78–89.CrossRefGoogle Scholar
[6]Borel, A., Some finiteness properties of adele groups over number fields, Institut des Hautes Études Scientifiques, Publications Mathématiques, no. 16 (1963), pp. 5–30.Google Scholar
[7]Borel, A. and Mostow, G. D. (editors), Algebraic groups and discontinuous subgroups, Proceedings of Symposia in Pure Mathematics, vol. 9, American Mathematical Society, Providence, Rhode Island, 1966.CrossRefGoogle Scholar
[8]Borel, A. and Serre, J.-P., Théorèmes de finitude en cohomologie Galoisienne, Commentarii Mathematici Hehetici, vol. 39 (1964), pp. 111–164.CrossRefGoogle Scholar
[9]Borel, A. and Serre, J.-P., Cohomologie d'immeubles et de groupes S-arithmétiques, Topology, vol. 15 (1976), pp. 211–232.CrossRefGoogle Scholar
[10]Borevich, Z. I. and Shafarevich, I. R., Number theory, Academic Press, New York, 1966.Google Scholar
[11]Bruhat, F. and Tits, J., Groupes réductifs sur un corps local. I, Institut des Hautes Études Scientifiques, Publications Mathématiques, no. 41 (1972), pp. 5–252.Google Scholar
[12]Grunewald, F. J. and Segal, D., Some general algorithms. I: Arithmetic groups. Annals of Mathematics, ser. 2, vol. 112 (1980), pp. 531–583.CrossRefGoogle Scholar
[13]Grunewald, F. J. and Segal, D., Résolution effective de quelques problèmes diophantiens sur les groupes algébriques linéaires, Comptes Rendus des Séances de l'Acadedémie des Sciences, Serie I: Mathématique, vol. 295 (1982), pp. 479–481.Google Scholar
[14]Hijikata, H., On the structure of semi-simple algebraic groups over valuation fields. I, Japanese Journal of Mathematics, New Series, vol. 1 (1975), pp. 225–300.Google Scholar
[15]Hochschild, G., The structure of Lie groups, Holden-Day, San Francisco, California 1965.Google Scholar
[16]Kneser, M., Konstruktive Lösung p-adischer Gleichungssysteme, Nachrichten der Akademie der Wissenschaften in Göttingen. II: Mathematisch-Physikalische Klasse, 1978, no. 5, pp. 67–69.Google Scholar
[17]Odoni, R. W. K.A proof by classical methods of a result of Ax on polynomial congruences modulo a prime, Bulletin of the London Mathematical Society, vol. 11 (1979), pp. 55–58.CrossRefGoogle Scholar
[18]Ono, T., Arithmetic of algebraic tori, Annals of Mathematics, ser. 2, vol. 74 (1961), pp. 101–139CrossRefGoogle Scholar
[19]Platonov, V. P.. The problem of strong approximation and the Kneser-Tits conjecture for algebraic groups, Mathematics of the USSR—hvestija, vol. 3 (1969), pp. 1139–1147; addendum, Mathematics of the USSR—hvestija., vol. 4 (1970), pp. 784–786.CrossRefGoogle Scholar
[20]Sarkisjan, R. A., Galois cohomology and some questions in the theory of algorithms, Mathematics of the USSR-Sbornik, vol. 39 (1981), pp. 519–545.CrossRefGoogle Scholar
[21]Sarkisjan, R. A., On a problem of equality in Galois cohomology, Algbra and Logic, vol. 19 (1980), pp. 459–472.CrossRefGoogle Scholar
[22]Tarski, A., A decision method for elementary algebra and geometry, 2nd ed., University of California Press, Berkeley, California, 1951.CrossRefGoogle Scholar
[23]Bruhat, F. and Tits, J., Groupes réductifs sur un corps local. II, Institut des Hautes Études Scientifiques, Publications Mathématiques (to appear).Google Scholar
[24]Collins, D. J. and Miller, C. F. III, The conjugacy problem and subgroups of finite index, Proceedings of the London Mathematical Society, ser. 3, vol. 34 (1977) pp. 535–556.CrossRefGoogle Scholar
[25]Grunewald, F. J. and Segal, D., How to solve a quadratic equation in integers, Mathematical Proceedings of the Cambridge Philosophical Society, vol. 89 (1981), pp. 1–5.CrossRefGoogle Scholar
[26]Lyndon, R. C. and Schupp, P. E., Combinatorial group theory, Springer-Verlag, Berlin, 1977.Google Scholar
[27]Miller, C. F. III, On group-theoretic decision problems and their classification, Annals of Mathematics Studies, no. 68, Princeton University Press, Princeton, New Jersey, 1973.Google Scholar
[28]Segal, D., Polycyclic groups, Cambridge Tracts in Mathematics, no. 82, Cambridge University Press, Cambridge, 1983.CrossRefGoogle Scholar
[29]Serre, J.-P., Trees, Springer-Verlag, Berlin, 1980.CrossRefGoogle Scholar
[30]Tits, J., Travaux de Margulis sur les sous-groupes discrets des groupes de Lie, Séminaire Bourbaki 1975/76, Lecture Notes in Mathematics, vol. 567, Springer-Verlag, Berlin, 1976, pp. 174–190.Google Scholar
[31]Tits, J., Reductive groups over local fields, Automorphic forms, representations, and L-functions, Proceedings of Symposia in Pure Mathematics, vol. 33, Part 1, American Mathematical Society, Providence, Rhode Islans, 1979, pp. 29–69.CrossRefGoogle Scholar