Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Equations in a free $\mathbf{Q}$-group

Authors: O. Kharlampovich and A. Myasnikov
Journal: Trans. Amer. Math. Soc. 350 (1998), 947-974
MSC (1991): Primary 20E05, 20F10
MathSciNet review: 1389782
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: An algorithm is constructed that decides if a given finite system of equations over a free $\mathbf{Q}$-group has a solution, and if it does, finds a solution.

References [Enhancements On Off] (What's this?)

  • 1. Gilbert Baumslag, On free \cal𝐷-groups, Comm. Pure Appl. Math. 18 (1965), 25–30. MR 0177028
  • 2. Gilbert Baumslag, Some aspects of groups with unique roots, Acta Math. 104 (1960), 217–303. MR 0122859
  • 3. G. Baumslag, A. Myasnikov, and V. Remeslennikov, Residually hyperbolic groups, preprint, 1996.
  • 4. Leo P. Comerford Jr. and Charles C. Edmunds, Solutions of equations in free groups, Group theory (Singapore, 1987) de Gruyter, Berlin, 1989, pp. 347–356. MR 981853
  • 5. R. I. Grigorchuk and P. F. Kurchanov, On quadratic equations in free groups, Proceedings of the International Conference on Algebra, Part 1 (Novosibirsk, 1989) Contemp. Math., vol. 131, Amer. Math. Soc., Providence, RI, 1992, pp. 159–171. MR 1175769
  • 6. O. Kharlampovich and A. G. Myasnikov, Hyperbolic groups and free constructions, Trans. Amer. Math. Soc. 350 (1998), 571-613.
  • 7. G. S. Makanin, Decidability of the universal and positive theories of a free group, Izv. Akad. Nauk SSSR Ser. Mat. 48 (1984), no. 4, 735–749 (Russian). MR 755956
  • 8. G. S. Makanin, Equations in a free group, Izv. Akad. Nauk SSSR Ser. Mat. 46 (1982), no. 6, 1199–1273, 1344 (Russian). MR 682490
  • 9. A. G. Myasnikov and V. N. Remeslennikov. Exponential groups II: Extension of centralizers and tensor completion of CSA-groups, Internat. J. Algebra Comput. 6 (1996), 687-712.
  • 10. H. Presburger, Über die Vollständigkeit eines gewissen Systems der Arithmetik ganzen Zahlen, C. R. 1 Congr. Math. Pays Slaves, Warsaw, 1929, pp. 92-101.
  • 11. A. A. Razborov, Systems of equations in a free group, Izv. Akad. Nauk SSSR Ser. Mat. 48 (1984), no. 4, 779–832 (Russian). MR 755958
  • 12. E. Rips and Z. Sela, Canonical representatives and equations in hyperbolic groups, Invent. Math. 120 (1995), no. 3, 489–512. MR 1334482, 10.1007/BF01241140

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (1991): 20E05, 20F10

Retrieve articles in all journals with MSC (1991): 20E05, 20F10

Additional Information

O. Kharlampovich
Affiliation: Department of Mathematics and Statistics, McGill University, Montréal, P.Q., Canada H3A 2K6

A. Myasnikov
Affiliation: Department of Mathematics, City College (CUNY), Convent Avenue and 138th Street, New York, New York 10031-9100

Received by editor(s): April 5, 1996
Additional Notes: The first author was supported by grants from NSERC and FCAR; the second author was supported by the NSF Grant DMS-9103098
Article copyright: © Copyright 1998 American Mathematical Society