Hostname: page-component-cd9895bd7-jkksz Total loading time: 0 Render date: 2024-12-25T06:17:11.713Z Has data issue: false hasContentIssue false
  • English
  • Français

∀-free metabelian groups

Published online by Cambridge University Press:  12 March 2014

Olivier Chapuis
Olivier Chapuis*
Affiliation:
Institut Girard Desargues, CNRS - Université Lyon I, Mathematiques, Bât. 101, 43, BD DU 11 Novembre 1918, F-69622 Villeurbanne Cedex, France, E-mail: chapuis@jonas.univ-lyonl.fr
Get access Rights & Permissions [Opens in a new window]

Extract

In 1965, during the first All-Union Symposium on Group Theory, Kargapolov presented the following two problems: (a) describe the universal theory of free nilpotent groups of class m; (b) describe the universal theory of free groups (see [18, 1.28 and 1.27]). The first of these problems is still open and it is known [25] that a positive solution of this problem for an m ≤ 2 should imply the decidability of the universal theory of the field of the rationals (this last problem is equivalent to Hilbert's tenth problem for the field of the rationals which is a difficult open problem; see [17] and [20] for discussions on this problem). Regarding the second problem, Makanin proved in 1985 that a free group has a decidable universal theory (see [15] for stronger results), however, the problem of deriving an explicit description of the universal theory of free groups is open. To try to solve this problem Remeslennikov gave different characterization of finitely generated groups with the same universal theory as a noncyclic free group (see [21] and [22] and also [11]). Recently, the author proved in [8] that a free metabelian group has a decidable universal theory, but the proof of [8] does not give an explicit description of the universal theory of free metabelian groups.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 62 , Issue 1 , March 1997 , pp. 159 - 174
Copyright
Copyright © Association for Symbolic Logic 1997

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] Baumslag, G., Wreath products and extensions, Mathematische Zeitschrift, vol. 81 (1963), pp. 286–299.CrossRefGoogle Scholar
[2] Baumslag, G., Groups with the same lower central sequence as relatively free groups I, Transaction of the American Mathematical Society, vol. 129 (1967), pp. 308–321.Google Scholar
[3] Baumslag, G., Groups with the same lower central sequence as relatively free groups II, Transaction of the American Mathematical Society, vol. 142 (1969), pp. 507–538.CrossRefGoogle Scholar
[4] Baumslag, G., Some reflexions on finitely generated metabelian groups, Contemporary Mathematics, vol. 109 (1990), pp. 1–9.CrossRefGoogle Scholar
[5] Baumslag, G., Cannonito, F. B., and Robinson, D. J. S., The algorithmic theory of finitely generated metabelian groups, Transaction of the American Mathematical Society, vol. 344 (1994), pp. 629–648.CrossRefGoogle Scholar
[6] Chang, C. C. and Keisler, H. J., Model theory, North-Holland, Amsterdam, 1973.Google Scholar
[7] Chapuis, O., On the theories of free solvable groups, submitted for publication.Google Scholar
[8] Chapuis, O., Universal theory of certain solvable groups and bounded Ore group-rings, Journal of Algebra, vol. 176 (1995), pp. 368–391.CrossRefGoogle Scholar
[9] Ershov, Yu. L., Lavrov, I. A., Taimanov, A. D., and Taitslin, M. A., Elementary theory, Russian Mathematical Survey, vol. 20 (1965), pp. 35–105.CrossRefGoogle Scholar
[10] Gaglione, A. M. and Spellman, D., The persistence of universal formulae in free algebras, Bulletin of the Australian Mathematical Society, vol. 36 (1987), pp. 11–17.CrossRefGoogle Scholar
[11] Gaglione, A. M. and Spellman, D., More model theory of free groups, Houston Journal of Mathematics, vol. 21 (1995), pp. 225–245.Google Scholar
[12] Gupta, N., Free group-rings, American Mathematical Society, Providence, 1987.CrossRefGoogle Scholar
[13] Karpilovsky, G., Commutative group algebras, Dekker, New York, 1983.Google Scholar
[14] Ledlie, J. F., Representation of free metabelian Dπ-groups, Transaction of the American Mathematical Society, vol. 153 (1971), pp. 307–346.Google Scholar
[15] Makanin, A. G., Decidability of the universal and positive theories of a free group, Mathematical USSR Izvestia, vol. 25 (1985), pp. 75–88.CrossRefGoogle Scholar
[16] Malcev, A. I., On free solvable groups, Soviet Mathematical Doklady, vol. 1 (1960), pp. 65–68.Google Scholar
[17] Mazur, B., Questions of decidability and undecidability in number theory, Journal of Symbolic logic, vol. 59 (1994), pp. 353–371.CrossRefGoogle Scholar
[18] Mazurov, V. D., Merzlyakov, Yu. I., and Tchurkin, V. A. (editors), The Kourovka notebook (unsolved problems in group theory, seventh augmented edition), vol. 121, 1983, American Mathematical Society translation.Google Scholar
[19] Neuman, H., Varieties of groups, Springer-Verlag, Berlin, 1967.CrossRefGoogle Scholar
[20] Pheidas, T., Extentions of Hilbert's tenth problem, this Journal, vol. 59 (1994), pp. 372–397.Google Scholar
[21] Remeslennikov, V. N., ∃-free groups, Siberian Mathematical Journal, vol. 30 (1989), pp. 998–1001.CrossRefGoogle Scholar
[22] Remeslennikov, V. N., ∃-free groups as a group with a length function, Ukranian Mathematical Journal, vol. 44 (1992), pp. 733–738.CrossRefGoogle Scholar
[23] Robinson, D. J. S., A course in the theory of groups, Springer-Verlag, New York, 1982.CrossRefGoogle Scholar
[24] Roman'kov, V. A., Equation in free metabelian groups, Siberian Mathematical Journal, vol. 20 (1979), pp. 469–471.CrossRefGoogle Scholar
[25] Roman'kov, V. A., Universal theory of nilpotent groups, Mathematical Notes, vol. 25 (1979), pp. 252–258.CrossRefGoogle Scholar
[26] Simonetta, P., Décidabilité et interpretabilité dans les corps et les groupes non commutatifs, Thèse de doctorat, Université Paris VII, 1994.Google Scholar
[27] Timoshenko, E. I., Preservation of elementary and universal equivalence under the wreath product, Algebra and Logic, vol. 7 (1968), pp. 273–276.CrossRefGoogle Scholar