On the residual and profinite closures of commensurated subgroups

2017/06/21 by Caprace, Pierre-Emmanuel, Kropholler, Peter H., Reid, Colin D. +1
#20E08 #20E18 #20E26 #20F16 #20F65 #FOS: Mathematics #Group Theory (math.GR)

paper · doi:10.48550/arxiv.1706.06853

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.

Related

• Commensurators and Quasi-Normal Subgroups
• The profinite completion of a group localised at a subgroup
• Finite subgroups of the profinite completion of good groups
• On closure operations in the space of subgroups and applications
• Subgroup separability in residually free groups