The intersection of the subgroups of finite $p$-index in a multiple HNN-extension of an infinite cyclic group

Let $G$ be a multiple HNN-extension of an infinite cyclic group. We will calculate the intersection $(N_p){}_{\omega }(G)$ of the normal subgroups of finite $p$-index in $G$ thus generalizing the result of Moldavanskii for Baumslag-Solitar groups. As a corollary we give necessary and sufficient conditions for $G$ to be residually finite $p$-group.