On the finite presentation of subdirect products and the nature of residually free groups. (Q2845585)

A subgroup of a direct product of groups is said to be a subdirect product if its projection to each factor is surjective. A subdirect product is said to be full if it intersects each of the direct factors non-trivially. A subgroup \(S\) of the direct product \(G_1\times G_2\times\cdots\times G_n\) is said to be virtually surjective on pairs (VSP) if for all \(i\neq j\) the projection \(p_{ij}(S)\subseteq G_i\times G_j\) has finite index. In [\textit{M. R. Bridson} and \textit{C. F. Miller III}, Proc. Lond. Math. Soc. (3) 98, No. 3, 631-651 (2009; Zbl 1167.20016)] it is proved that the VSP condition is necessary for a full subdirect product of non-Abelian free and surface groups to be finitely presented.NEWLINENEWLINE In the present paper the authors begin with a generalization of the above mentioned result and prove: Theorem A. Let \(S<G_1\times G_2\times\cdots\times G_n\) be a subgroup of a direct product of finitely presented groups. If \(S\) is VSP, then it is finitely presented and separable.NEWLINENEWLINE (A subgroup of a group is said to be separable if it is closed with respect to the profinite topology).NEWLINENEWLINE The proof of the above Theorem A is based on the following asymmetric version of a result of \textit{G. Baumslag} et al., [Comment. Math. Helv. 75, No. 3, 457-477 (2000; Zbl 0973.20034)].NEWLINENEWLINE Theorem B. Let \(f_1\colon\Gamma_1\to Q\) and \(f_2\colon\Gamma_2\to Q\) be surjective group homomorphisms. Suppose that \(\Gamma_1\) and \(\Gamma_2\) are finitely presented, that \(Q\) is of type \(F_3\), and that at least one of \(\ker f_1\) and \(\ker f_2\) is finitely generated. Then the fibre product of \(f_1\) and \(f_2\), \(P=\{(g,h)\mid f_1(g)=f_2(h)\}\subseteq\Gamma_1\times\Gamma_2\) is finitely generated.NEWLINENEWLINE Here the authors show an effective version of this result where it is provided an algorithm that, given natural input data, constructs a finite presentation for the fibre product. Using this result it is obtained an effective version of Theorem A. Namely it is proved that there is a uniform partial algorithm that, given finite presentations for the factors \(G_i\) and a finite generating set for \(S\) satisfying VSP, will output a finite presentation for \(S\).NEWLINENEWLINE By definition, a group \(G\) is residually free if it isomorphic to a subgroup of an unrestricted direct product of free groups. In general, it are required infinitely many factors in this direct product. However in [\textit{G. Baumslag, A. Myasnikov} and \textit{V. Remeslennikov}, J. Algebra 219, No. 1, 16-79 (1999; Zbl 0938.20020)] it is proved that a residually free group can be embedded in a direct product with finitely many factors of limit groups. Moreover in [\textit{O. Kharlampovich} and \textit{A. G. Myasnikov}, Contemp. Math. 378, 87-212 (2005; Zbl 1093.20019)] it is described an algorithm to find such an embedding. Here, using Theorem A, it is obtained a stronger result (Theorem C in the paper) and it is obtained a new algorithm for the embedding of a finitely presentable residually free group into a finite direct product of limit groups.NEWLINENEWLINE This result provides effective control over the finitely presented residually free groups. The multiple conjugacy problem and the membership problem for finitely presented subgroups of finitely presented residually free groups is solvable. Moreover it is proved that there is a uniform partial algorithm for finding presentations of finitely presentable subgroups of finitely presented residually free groups.NEWLINENEWLINE For the isomorphism problem of finitely presentable residually free groups are obtained partial results. In this direction it is proved the followingNEWLINENEWLINE Theorem. (Theorem G in the paper) The class of finitely presentable residually free groups is recursively enumerable. More precisely, there is a Turing machine that will output a list of finite group presentations \(\mathcal P_1,\mathcal P_2,\ldots\) such that:NEWLINENEWLINE (1) the group \(G_i\) presented by each \(\mathcal P_i\) is residually free.NEWLINENEWLINE (2) Every finitely presented residually free group is isomorphic to at least one of the groups \(G_i\).NEWLINENEWLINE The paper concludes with a question concerning the isomorphism problem. Question. Given finitely presented full subdirect products \(G,H\) of a collection of limit groups \(\Gamma_1,\Gamma_2,\ldots,\Gamma_n\) (at most one of which is Abelian), can we find automorphisms \(\vartheta_i\) of \(\Gamma_i\) for each \(i\), such that \((\vartheta_1,\ldots,\vartheta_n)(G)=H\)?NEWLINENEWLINE A partial answer is given.NEWLINENEWLINE Proposition. There is a solution to the isomorphism problem in the case when at most 2 of the \(\Gamma_i\) are non-Abelian.