On the residual polynilpotence of tree products of groups with locally cyclic edge subgroups

Let $\mathcal{N}$ be a non-empty class of nilpotent groups closed under taking subgroups, quotient groups, finite direct products, and unrestricted direct powers of the form ${\prod }_{x\in X}{Y}_x$, where $X\in \mathcal{N}$ and $Y_x$ ($x\in X$) are isomorphic copies of a group $Y\in \mathcal{N}$. Let also the symbols ${\mathcal{N}}^{\mathfrak{t}\mathfrak{f}}$ and ${Poly}_k{\mathcal{N}}^{\mathfrak{t}\mathfrak{f}}$ denote the class of all torsion-free $\mathcal{N}$-groups and the class of polynilpotent groups such that $X\in {Poly}_k{\mathcal{N}}^{\mathfrak{t}\mathfrak{f}}$ if and only if $X$ has a subnormal series of length at most $k$ whose factors belong to ${\mathcal{N}}^{\mathfrak{t}\mathfrak{f}}$. We prove that if all free groups are residually ${\mathcal{N}}^{\mathfrak{t}\mathfrak{f}}$-groups and $2⩽n<\mathrm{\infty }$, then any tree product of $n$ residually ${\mathcal{N}}^{\mathfrak{t}\mathfrak{f}}$-groups with non-trivial torsion-free locally cyclic edge subgroups is residually a $({Poly}_{n+1}{\mathcal{N}}^{\mathfrak{t}\mathfrak{f}})$-group. In particular, all cyclically-pinched one-relator groups are residually finitely generated $({Poly}_3\bar{\mathcal{N}}\text{mbox}{}^{\mathfrak{t}\mathfrak{f}})$-groups, where $\bar{\mathcal{N}}\text{mbox}{}^{\mathfrak{t}\mathfrak{f}}$ is the class of all torsion-free nilpotent groups. We prove also that any tree product of $n$ $\mathcal{N}$-groups with non-trivial torsion-free locally cyclic edge subgroups has a homomorphism onto a $({Poly}_{n-1}{\mathcal{N}}^{\mathfrak{t}\mathfrak{f}})$-group that is injective on all edge subgroups.