On the residual n-finiteness of some free products of groups with central amalgamated subgroups

Let π be a set of primes. A criterion of residual π-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 π-group if for every nonidentity element x of G there exists a homomorphism of the group G onto some finite π-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 π-groups and H be a central subgroup of the group A. If H and K are finite, then G is a residually finite π-group. The same holds if the groups A/H and B/K are finite π-groups. However, G is not obligatorily a residually finite π-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 π-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 π-group if and only if A/H and B/K are residually finite π-groups.

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