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The profinite topology on any abstract group $G$, is one such that the fundamental system of neighborhoods of the identity is given by all its subgroups of finite index. We say that a group $G$ has the Ribes-Zalesskii property of rank $k$, or is RZ${}_k$ with $k$ a natural number, if any product $H_1{H}_2\cdots H_k$ of finitely generated subgroups $H_1,H_2,\cdots ,H_k$ is closed in the profinite topology on $G$. And a group is said to have the Ribes-Zalesskii property or is RZ if it is RZ${}_k$ for any natural number $k$. In this paper we characterize groups...