## 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 $$G^{\mathbb{Z}[x]}$$.

### 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

### Citations:

Zbl 0866.20014; Zbl 0904.20017
Full Text:

### References:

  G. Baumslag, A. Myasnikov, V. Remeslennikov, Algebraic geometry over groups, 1996 · Zbl 0938.20020  Comerford, L.P.; Edmunds, C.C., Quadratic equations over free groups and free products, J. of algebra,, 68, 276-297, (1981) · Zbl 0526.20024  Comerford, L.P.; Edmunds, C.C., Solutions of equations in free groups, (1989), de Gruyter Berlin · Zbl 0663.20023  B. Fine, A. M. Gaglione, A. Myasnikov, G. Rosenberger, D. Spellman, A classification of fully residually free groups, J. Algebra · Zbl 0899.20009  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  Grigorchuk, R.I.; Kurchanov, P.F., On quadratic equations in free groups, Contemp. math., 131, 159-171, (1992) · Zbl 0778.20013  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  Lyndon, R.C., Groups with parametric exponents, Trans. amer. math. soc., 96, 518-533, (1960) · Zbl 0108.02501  Lyndon, R.C., Equations in groups, Bol. soc. brasil. mat., 11, 79-102, (1980) · Zbl 0463.20030  Lyndon, Roger C.; Schupp, Paul E., Combinatorial group theory, (1977), Springer New York · Zbl 0368.20023  Makanin, G.S., Equations in a free group, Izv. akad. nauk SSSR, ser. mat., 46, 1199-1273, (1982) · Zbl 0511.20019  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  Razborov, A., On systems of equations in a free group, Math. USSR-izv., 25, 115-162, (1985) · Zbl 0579.20019  A. Razborov, On Systems of Equations in a Free Group, Steklov Math. Institute, Moscow, 1987 · Zbl 0632.94030  Schutzenberger, M.P., C.R. acad. sci. Paris, 248, 2435-2436, (1959)  Stallings, J.R., Finiteness of matrix representation, Ann. math., 124, 337-346, (1986) · Zbl 0604.20027
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