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 \(H,K \leqslant G\), one has \(\bar{H} \cap \bar{K}=\bar{H \cap K}\), where the closures are taken in the profinite completion \(\hat G\) 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.