×

zbMATH — the first resource for mathematics

Irreducible affine varieties over a free group. II: Systems in triangular quasi-quadratic form and description of residually free groups. (English) Zbl 0904.20017
This paper completes the programme started in the first paper (see the preceding review Zbl 0904.20016) by proving the theorem: Theorem. A finitely-generated group is fully residually free if and only if it is isomorphic to a subgroup of \(F^{\mathbb{Z}[x]}\). Theorems are also proved which describe the algebraic structure of finitely generated subgroups of \(F^{\mathbb{Z}[x]}\) in terms of free constructions.

MSC:
20E26 Residual properties and generalizations; residually finite groups
20E05 Free nonabelian groups
20E06 Free products of groups, free products with amalgamation, Higman-Neumann-Neumann extensions, and generalizations
20F05 Generators, relations, and presentations of groups
20E22 Extensions, wreath products, and other compositions of groups
20E07 Subgroup theorems; subgroup growth
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Adian, S.I., Burnside problem and identities in groups, (1975), Science Moscow · Zbl 1343.20040
[2] H. Bass, Groups acting on non-archimedian trees, 1991
[3] G. Baumslag, A. Myasnikov, V. Remeslennikov, Algebraic geometry over groups, 1996 · Zbl 0938.20020
[4] G. Baumslag, A. Myasnikov, V. Remeslennikov, Residually hyperbolic groups and approximation theorems for extensions of centralizers, 1996
[5] Bestvina, M.; Feighn, M., A combination theorem for negatively curved groups, J. differential geom., 35, 85-101, (1992) · Zbl 0724.57029
[6] Cohen, D., Combinatorial group theory: a topological approach, London math. soc. stud. texts, (1978), Cambridge Univ. Press Cambridge · Zbl 0389.20024
[7] Gildenhuys, D.; Kharlampovich, O.; Myasnikov, A., CSA groups and separated free constructions, Bull. austral. math. soc., 52, 63-84, (1995) · Zbl 0838.20025
[8] Guba, V., Equivalence of infinite systems of equations in free groups and semigroups to finite subsystems, Mat. zametki, 40, 321-324, (1986) · Zbl 0611.20020
[9] O. Kharlampovich, A. Myasnikov, Irreducible affine varieties over a free group. 1. Irreducibility of quadratic equations and Nullstellensatz, 1996 · Zbl 0904.20016
[10] O. Kharlampovich, A. G. Myasnikov, Hyperbolic groups and free constructions, Trans. Amer. Math. Soc. · Zbl 0902.20018
[11] Lyndon, R.C.; Schupp, P.E., Combinatorial group theory, (1977), Springer-Verlag · Zbl 0368.20023
[12] Makanin, G.S., Decidability of the universal and positive theories of a free group, Math. USSR-izv., 25, 75-88, (1985) · Zbl 0578.20001
[13] Makanin, G.S., Equations in a free group, Izv. akad. nauk SSSR ser. mat., 46, 1199-1273, (1982) · Zbl 0511.20019
[14] Makanin, G.S., The problem of solvability of equations in a free semigroup, Math. USSR-sb., 32, (1977) · Zbl 0379.20046
[15] K. V. Mikhajlovskii, A. Yu. Olshanskii, Some constructions relating to hyperbolic groups, in, Proc. Int. Conf. on Cohomological and Geometric Methods in Group Theory, Durham, 1994
[16] Myasnikov, A.; Remeslennikov, V., Length functions ona, Proc. inst. appl. math. Russian acad. sci., 26, 1-33, (1996)
[17] Myasnikov, A.G.; Remeslennikov, V.N., Exponential groups 2: extension of centralizers and tensor completion of csa-groups, Internat. J. algebra comput., 6, 687-711, (1996) · Zbl 0866.20014
[18] Razborov, A., On systems of equations in a free group, Math. USSR-izv., 25, 115-162, (1985) · Zbl 0579.20019
[19] A. Razborov, On Systems of Equations in a Free Group, Steklov Math. Institute, Moscow, 1987 · Zbl 0632.94030
[20] Razborov, A., On systems of equations in free groups, Combinatorial and geometric group theory, Edinburgh, 1993, (1995), Cambridge Univ. Press Cambridge, p. 269-283 · Zbl 0848.20018
[21] Remeslennikov, V., ∃-free groups and groups with length function, Second international conference on algebra, barnaul, 1991, Contemp. math., 184, (1995), Amer. Math. Soc Providence, p. 369-376 · Zbl 0856.20002
[22] Remeslennikov, V.N., ∃-free groups, Siberian math. J., 30, 153-157, (1989) · Zbl 0724.20025
[23] Remeslennikov, V.N., ∃-free groups as groups with length function, Ukrainian math. J., 44, 813-818, (1992) · Zbl 0784.20015
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.