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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.



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F.R.S.-FNRS senior research associate, supported in part by EPSRC grant no EP/K032208/1.

supported by EPSRC grants no EP/K032208/1 and EP/ N007328/1.


ARC DECRA fellow, supported in part by ARC Discovery Project DP120100996.



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On the residual and profinite closures of commensurated subgroups



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