Irreducible affine varieties over a free group. I: Irreducibility of quadratic equations and Nullstellensatz. (English) Zbl 0904.20016

A. G. Myasnikov and V. N. Remeslennikov [Int. J. Algebra Comput. 6, No. 6, 687-711 (1996; Zbl 0866.20014)] showed that Lyndon’s group \(F^{\mathbb{Z}[x]}\) (the free exponential group over the ring of integral polynomials) can be described using very special HNN extensions (extensions of centralisers). Lyndon’s group is fully residually \(F\), and Myasnikov and Remeslennikov conjectured in 1992 that every finitely generated fully residually free group is a subgroup of Lyndon’s group. The aim of these two papers is to prove this conjecture.
The first paper concentrates on setting up the machinery required, by making an intensive investigation of quadratic extensions. Let \(G\) be a group and \(F(X)\) the free group on \(\{x_1,\dots,x_n\}\). Let \(G[X]\) denote the free product of \(G\) and \(F(X)\). Let \(s\in G[X]\), then \(s=1\) is an equation over \(G\). A solution \(\{a_1,\dots,a_n\}\) for a set \(S=1\) of equations is a subset of \(G\) whose substitution for the variables in the elements of \(S\) yields the identity of \(G\). \(S\) is quadratic if no variable occurs more than twice in \(S\). Let \(V(S)\) denote the set of all solutions of \(S\) and \(\text{Rad}(S)\) the set of all \(s\in G[X]\) which have \(V(S)\) as solutions, then \(G_{R(S)}\) denotes the quotient group \(G[X]/\text{Rad}(S)\).
The main result of the first paper is: Theorem. Let \(G\) be a fully residually free group and let \(S=1\) be a consistent quadratic equation over \(G\). Then \(G_{R(S)}\) is \(G\)-embeddable into \(G^{\mathbb{Z}[x]}\).
See also the following review Zbl 0904.20017.


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
Full Text: DOI


[1] G. Baumslag, A. Myasnikov, V. Remeslennikov, Algebraic geometry over groups, 1996 · Zbl 0938.20020
[2] Comerford, L.P.; Edmunds, C.C., Quadratic equations over free groups and free products, J. of algebra,, 68, 276-297, (1981) · Zbl 0526.20024
[3] Comerford, L.P.; Edmunds, C.C., Solutions of equations in free groups, (1989), de Gruyter Berlin · Zbl 0663.20023
[4] B. Fine, A. M. Gaglione, A. Myasnikov, G. Rosenberger, D. Spellman, A classification of fully residually free groups, J. Algebra · Zbl 0899.20009
[5] Grigorchuk, R.I.; Kurchanov, P.F., Some questions of group theory related to geometry, Sovremennye problemy matematiki. fundamental’nye napravlenia, Itogi nauki i techniki, 58, (1990), VINITI Moscow · Zbl 0781.20023
[6] Grigorchuk, R.I.; Kurchanov, P.F., On quadratic equations in free groups, Contemp. math., 131, 159-171, (1992) · Zbl 0778.20013
[7] 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
[8] Lyndon, R.C., Groups with parametric exponents, Trans. amer. math. soc., 96, 518-533, (1960) · Zbl 0108.02501
[9] Lyndon, R.C., Equations in groups, Bol. soc. brasil. mat., 11, 79-102, (1980) · Zbl 0463.20030
[10] Lyndon, Roger C.; Schupp, Paul E., Combinatorial group theory, (1977), Springer New York · Zbl 0368.20023
[11] Makanin, G.S., Equations in a free group, Izv. akad. nauk SSSR, ser. mat., 46, 1199-1273, (1982) · Zbl 0511.20019
[12] Myasnikov, A.G.; Remeslennikov, Y.N., Exponential groups 2: extension of centralizers and tensor completion of csa-groups, Intern. J. algebra comput., 6, 687-711, (1996) · Zbl 0866.20014
[13] Razborov, A., On systems of equations in a free group, Math. USSR-izv., 25, 115-162, (1985) · Zbl 0579.20019
[14] A. Razborov, On Systems of Equations in a Free Group, Steklov Math. Institute, Moscow, 1987 · Zbl 0632.94030
[15] Schutzenberger, M.P., C.R. acad. sci. Paris, 248, 2435-2436, (1959)
[16] Stallings, J.R., Finiteness of matrix representation, Ann. math., 124, 337-346, (1986) · Zbl 0604.20027
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.