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Subgroup separability in residually free groups. (English) Zbl 1158.20013

A group \(G\) is called residually free if, given an element \(g\in G\setminus 1\), there exists a homomorphism \(f\colon G\to F\) (\(F\) free) with \(f(g)\neq 1\). It is proved that the finitely presentable subgroups of residually free groups are separable and that the subgroups of type \(\text{FP}_\infty\) over \(\mathbb{Q}\) are virtual retracts. Remind that a subgroup \(H\) of a group \(G\) is a virtual retract if and only if it is contained in a subgroup \(V\) of \(G\) of finite index for which \(H\) is a retract.
A uniform solution to the membership problem for finitely presentable subgroups of residually free groups is given.

MSC:

20E26 Residual properties and generalizations; residually finite groups
20E05 Free nonabelian groups
20E07 Subgroup theorems; subgroup growth
20F10 Word problems, other decision problems, connections with logic and automata (group-theoretic aspects)
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References:

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