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