On double coset separability and the Wilson–Zalesskii property

Abstract

A residually finite group G has the Wilson-Zalesskii property if for all finitely generated subgroups H, K ⩽ G, one has H ∩ K = H ∩ K, where the closures are taken in the profinite completion G of G. This property played an important role in several papers, and is usually combined with separability of double cosets. In the present note we show that the Wilson-Zalesskii property is actually enjoyed by every double coset separable group. We also construct an example of a LERF group that is not double coset separable and does not have the Wilson-Zalesskii property. 2020 Mathematics Subject Classification. 20E26, 20E18.