On the residual $\pi$-finiteness of some free products of groups with central amalgamated subgroups

Let $\pi$ be a set of primes. A criterion of residual $\pi$-finiteness for free products of two groups with central amalgamated subgroups has been obtained for the case where one factor is a nilpotent finite rank group. Recall that a group $G$ is said to be a residually finite $\pi$-group if for every nonidentity element $x$ of $G$ there exists a homomorphism of the group $G$ onto some finite $\pi$-group such that the image of the element $x$ differs from $1$. A group $G$ is said to be a finite rank group if there exists a positive integer r such that every finitely generated subgroup of group $G$ is generated by at most $r$ elements. Let $G$ be a free product of groups $A$ and $B$ with normal amalgamated subgroups $H$ and $K$. Let also $A$ and $B$ be residually finite $\pi$-groups and $H$ be a central subgroup of the group $A$. If $H$ and $K$ are finite, then $G$ is a residually finite $\pi$-group. The same holds if the groups $A/H$ and $B/K$ are finite $\pi$-groups. However, $G$ is not obligatorily a residually finite $\pi$-group if we replace the requirement of finiteness of the groups $A/H$ and $B/K$ by a weaker requirement of $A/H$ and $B/K$ to be residually finite $\pi$-groups. A corresponding example is provided in the article. Nevertheless, we prove that if $A$ is a nilpotent finite rank group, then $G$ is a residually finite $\pi$-group if and only if $A/H$ and $B/K$ are residually finite $\pi$-groups.