Residual properties of HNN-extensions with base group an Abelian group (Q1122652)

As the title indicates, this paper deals with residual properties of the HNN-extension \(G=<t,K|\) \(t^{-1}At=B,\phi >\), where K is a finitely generated abelian group. It is proved that G is residually solvable. The authors give also necessary conditions for the residual nilpotence of G. In the special case where \(K=A\) or \(K=B\), or in the special case where \(A=B\), they give a characterization for G to be residually nilpotent. More precisely they define the map (endomorphism) f, by \(\alpha f=\alpha^{-1}(\alpha \phi)\) and prove that if \(K=A\), then G is residually nilpotent iff the only f-invariant subgroup of A is trivial, if \(A=B\) then G is residually nilpotent iff the only f-invariant subgroup is trivial and the torsion part of K/A is a finite p-group, for a prime p.