On double coset separability and the Wilson-Zalesskii property

TL;DR

This paper explores the Wilson-Zalesskii property in residually finite groups, showing it holds for double coset separable groups and providing an example of a LERF group lacking this property.

Contribution

It proves that double coset separability implies the Wilson-Zalesskii property and constructs a LERF group without this property.

Findings

Wilson-Zalesskii property holds for all double coset separable groups

Constructs a LERF group that is not double coset separable and lacks the Wilson-Zalesskii property

Highlights the relationship between subgroup 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 $\bar{H} \cap \bar{K}=\overline{H \cap K}$, where the closures are taken in the profinite completion $\widehat G$ 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.