Subgroup properties of fully residually free groups (Q2750956)

A group \(G\) is called fully residually free (or sometimes \(\omega\)-residually free) if for any finite collection of nontrivial elements \(g_1,\dots,g_n\in G\) there exists a homomorphism \(\phi\colon G\to F\) onto a free group \(F\) such that \(\phi(g_i)\not=1\) for \(i=1,\dots,n\).NEWLINENEWLINENEWLINEIt is known [from \textit{O. Kharlampovich, A. Myasnikov}, J. Algebra 200, No. 2, 517-570 (1998; Zbl 0904.20017)] that a finitely generated group \(G\) is fully residually free if and only if \(G\) embeds into \(F^{\mathbb{Z}[x]}\), the \(\mathbb{Z}[x]\)-tensor completion of a free group \(F\) of rank two. The main results of the paper under review are:NEWLINENEWLINENEWLINETheorem A. The group \(F^{\mathbb{Z}[x]}\) has the Howson property.NEWLINENEWLINENEWLINETheorem B. Let \(G\) be a non-Abelian finitely generated subgroup of \(F^{\mathbb{Z}[x]}\). Suppose \(H\) is a finitely generated subgroup of \(G\) such that \(H\) contains a non-trivial subgroup \(N\) which is normal in \(G\). Then \(H\) has finite index in \(G\).NEWLINENEWLINENEWLINETheorem C. Let \(H\) and \(K\) be finitely generated subgroups of \(F^{\mathbb{Z}[x]}\) such that \(H\cap K\) has finite index in both \(H\) and \(K\). Then \(H\cap K\) has finite index in the subgroup generated by \(H\cup K\).NEWLINENEWLINENEWLINESimilar results are true for a fully residually free group \(G\).NEWLINENEWLINENEWLINETheorem D. Let \(G\) be a finitely generated subgroup of \(F^{\mathbb{Z}[x]}\). Then for any finitely generated subgroup \(H\) of \(G\) the group \(G\) has solvable membership problem with respect to \(H\).NEWLINENEWLINENEWLINEAgain a similar result is true for any fully residually free group \(G\).