The solubility of sets of equations over groups (Q1076804)

Let G be a group. By a finite set S of equations over G it is meant a set \(S: w_ 1=1\), \(w_ 2=1,...,w_ n=1\) where \(w_ i\), \(i=1,2,...,n\) is an element of the free product G*F of G and the free group F freely generated by the variables \(z_ 1,...,z_ r\) entering into the equations. If H is a group containing G, the equations S can be solved in H if the inclusion map from G to H extends to a homomorphism from G*F to H such that the kernel contains all the \(w_ i\), \(i=1,2,...,n\). The equations can be solved over G if they can be solved in some group H containing G. Let N be the normal subgroup of G*F normally generated by \(w_ 1,w_ 2,...,w_ n\). Then the system S is soluble over G if and only if the intersection \(G\cap N\) is trivial, because then the natural map of G into G*F/N is an embedding and this happens only if S is soluble over G. A group H containing G is called finitely presented over G if it is isomorphic to G*F/N for some finite set S of equations over G which is soluble over G. The main result of this paper is the following: If the finitely generated group G is non-trivial there does not exist a group H, finitely presented over G, such that every finite set of equations over G which is soluble over G is soluble in H. The proof of this theorem occupies much of the paper. Among the corollaries is that no existentially closed group is embeddable in a finitely presented group.