On the residual and profinite closures of commensurated subgroups

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. Various applications to separable subgroups, polycyclic groups, residually finite groups, groups acting on trees, lattices in products of trees and just-infinite groups are described.

Lattice envelopes

“…This implies (wNbC) for linear groups. (3') As mentioned before, this is [20,Corollary 18]. (4') See Theorem 2.12.…”

“…The statement about (NbC) for the groups of class (3') is due to Caprace-Kropholler-Reid-Wesolek [20,Corollary 18] and the fact that the class (3') is closed under passing to quotients by finite normal subgroups.…”

Distal actions on coset spaces in totally disconnected locally compact groups

“…By Proposition 4.4(i), we see that H is codistal in Comm G (H); hence byLemma 4.7, there is an open subgroup E of Comm G (H) such that H is cocompact in E and H is an intersection of finite index open subgroups of E.By[5, Main Theorem], it follows that given a finite subset X of E, there is a finite index subgroup K X of H that is normal in H, X . We can clearly replace K X with its closure in H, and so ensure that K X is closed, hence open in H. Thus E = K∈K N E (K) where K is the set of open normal subgroups of H of finite index.…”

“…second-countable groups in the sense of Wesolek (see [17]): given a group G of decomposition rank α, we show that, subject to some obvious restrictions, every possible decomposition rank of a compactly generated closed subgroup of G occurs as the rank of a compactly generated open subgroup of G. (See §4. 5…”