Narrow Results

Here we characterize the residually finite groups G which are the fundamental groups of a finite graph of finitely generated torsion-free nilpotent groups. Namely we show that G is residually finite if and only if for each edge group of the graph of groups the two edge monomorphisms differ essentially by an isomorphism of certain subgroups of the Mal'cev completion of the corresponding vertex groups.

In this paper the concept of local embeddability into finite structures (being LEF) is examined for the class of semigroups. The established results include the connections to the previously studied class of LEF groups, residual finiteness, linear semigroups and the preservation of being LEF under certain semigroup constructions, such as adjoining zero or identity and taking direct, semidirect and wreath products.

Free products of two residually finite groups with amalgamated retracts are considered. It is proved that a cyclic subgroup of such a group is not finitely separable if, and only if, it is conjugated with a subgroup of a free factor which is not finitely separable in this factor. A similar result is obtained for the case of separability in the class of finite p-groups.

Let $A$ be a completely decomposable homogeneous torsion-free abelian group of rank $n$ ($n\ge 2$). Let ${\mathrm{\Theta }}_m(n)=A⋊⟨\alpha ⟩$ be the split extension of $A$ by an automorphism $\alpha$ which is a cyclic permutation of the direct components twisted by a rational integer $m$. Then ${\mathrm{\Theta }}_m(n)$ is an infinite soluble group. In this paper, the residual finiteness of ${\mathrm{\Theta }}_m(n)$ is investigated.