# On the residual and profinite closures of commensurated subgroups

## TL;DR

This paper proves that in finitely generated groups, commensurated subgroups have virtually normal residual closures, leading to broad implications for separable subgroups and various classes of groups.

## Contribution

It establishes that the residual closure of a commensurated subgroup in certain groups is virtually normal, a significant generalization in subgroup theory.

## Key findings

- Residual closure of commensurated subgroups is virtually normal in finitely generated groups.
- Separable commensurated subgroups are virtually normal.
- Applications include results on separable subgroups, polycyclic groups, and groups acting on trees.

## Abstract

The residual closure of a subgroup $H$ of a group $G$ is the intersection of all virtually normal subgroups of $G$ containing $H$. We show that if $G$ is generated by finitely many cosets of $H$ and if $H$ is commensurated, then the residual closure of $H$ in $G$ is virtually normal. This implies that separable commensurated subgroups of finitely generated groups are virtually normal. A stream of applications to separable subgroups, polycyclic groups, residually finite groups, groups acting on trees, lattices in products of trees and just-infinite groups then flows from this main result.