On the residual and profinite closures of commensurated subgroups

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.

Limits of contraction groups and the Tits core

The Tits core G^+ of a totally disconnected locally compact group G is defined as the abstract subgroup generated by the closures of the contraction groups of all its elements. We show that a dense subgroup is normalised by the Tits core if and only if it contains it. It follows that every dense subnormal subgroup contains the Tits core. In particular, if G is topologically simple, then the Tits core is abstractly simple, and if G^+ is non-trivial then it is the unique minimal dense normal subgroup. The proofs are based on the fact, of independent interest, that the map which associates to an element the closure of its contraction group is continuous.

Locally normal subgroups of simple locally compact groups

We announce various results concerning the structure of compactly generated simple locally compact groups. We introduce a local invariant, called the structure lattice, which consists of commensurability classes of compact subgroups with open normaliser, and show that its properties reflect the global structure of the ambient group.