On the linearity of HNN-extensions with Abelian base. (Q436080)

From the introduction: The purpose of the present note is to investigate the linearity of HNN-extensions of the form \(G=\langle t,K\mid tat^{-1}=\varphi(a),\,a\in A\rangle\) with \(K\) a finitely generated Abelian group and \(\varphi\colon A\to B\) an isomorphism between subgroups of \(K\). Our main result shows that those groups are \(\mathbb Z\)-linear if and only if they are residually finite.NEWLINENEWLINE We conjecture that fundamental groups of trees of groups with finitely generated Abelian vertex groups are always linear and even that fundamental groups of graphs with finitely generated Abelian vertex groups are linear if and only if they are residually finite, but we have not been able to prove such a result with the techniques at hand.