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On the residual and profinite closures of commensurated subgroups

Part of: Structure and classification of infinite or finite groups

Published online by Cambridge University Press:  30 July 2019

PIERRE–EMMANUEL CAPRACE ,
PETER H. KROPHOLLER ,
COLIN D. REID  and
PHILLIP WESOLEK
PIERRE–EMMANUEL CAPRACE
Affiliation:
Université Catholique de Louvain, IRMP, Chemin du Cyclotron 2, bte L7.01.02, 1348 Louvain-la-Neuve, Belgique. e-mail: [email protected]
PETER H. KROPHOLLER
Affiliation:
Mathematical Sciences, University of Southampton. e-mail: [email protected]
COLIN D. REID
Affiliation:
University of Newcastle, School of Mathematical and Physical Sciences, Callaghan, NSW 2308, Australia. e-mail: [email protected]
PHILLIP WESOLEK
Affiliation:
Binghamton University, Department of Mathematical Sciences, PO Box 6000, Binghamton, New York 13902-6000, U.S.A. e-mail: [email protected]
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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.

MSC classification

Primary: 20E26: Residual properties and generalizations; residually finite groups
Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 169 , Issue 2 , September 2020 , pp. 411 - 432
Copyright
© Cambridge Philosophical Society 2019

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Footnotes

†

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.

References

REFERENCES

Alexoudas, T., Klopsch, B. and Thillaisundaram, A.. Maximal subgroups of multi-edge spinal groups. preprint arXiv:1312.5615.Google Scholar
Bader, U., Furman, A. and Sauer, R.. On the structure and arithmeticity of lattice envelopes. C. R. Math. Acad. Sci. Paris 353(5) (2015), 409–413.CrossRefGoogle Scholar
Belyaev, V. V.. Locally finite groups containing a finite inseparable subgroup. Siberian Math. J. 34(2) (1993), 218–232.CrossRefGoogle Scholar
Burger, M. and Mozes, S.. Groups acting on trees: from local to global structure. Inst. Hautes Études Sci. Publ. Math. (92) (2000/01), 113–150.CrossRefGoogle Scholar
Burger, M. and Mozes, S.. Lattices in product of trees. Inst. Hautes Études Sci. Publ. Math. (92) (2000/01), 151–194.CrossRefGoogle Scholar
Caprace, P.–E. and Monod, N.. Decomposing locally compact groups into simple pieces. Math. Proc. Cambridge Philos. Soc. 150(1) (2011), 97–128.CrossRefGoogle Scholar
Caprace, P.–E. and Monod, N.. A lattice in more than two Kac–Moody groups is arithmetic. Israel J. Math. 190 (2012), 413–444.CrossRefGoogle Scholar
Detinko, A. S., Flannery, D. L. and O’brien, E. A.. Algorithms for linear groups of finite rank. J. Algebra 393 (2013), 187–196.CrossRefGoogle Scholar
Ershov, M. and Jaikin–zapirain, A.. Groups of positive weighted deficiency and their applications. J. Reine Angew. Math. 677 (2013), 71–134.Google Scholar
Garrido, A.. Abstract commensurability and the Gupta–Sidki group. Groups Geom. Dyn. 10(2) (2016), 523–543.CrossRefGoogle Scholar
Grigorchuk, R. I. and Wilson, J. S.. A structural property concerning abstract commensurability of subgroups. J. London Math. Soc. (2) 68(3) (2003), 671–682.CrossRefGoogle Scholar
Jeanes, S. C. and Wilson, J. S.. On finitely generated groups with many profinite-closed subgroups. Arch. Math. (Basel) 31(2) (1978/79), 120–122.CrossRefGoogle Scholar
Kropholler, P. H.. On finitely generated soluble groups with no large wreath product sections. Proc. London Math. Soc. (3) 49(1) (1984), 155–169.CrossRefGoogle Scholar
Kropholler, P. H.. An analogue of the torus decomposition theorem for certain Poincaré duality groups. Proc. London Math. Soc. (3) 60(3) (1990), 503–529.CrossRefGoogle Scholar
Lennox, J. C. and Robinson, D. J. S.. The theory of infinite soluble groups. Oxford Mathematical Monographs (The Clarendon Press, Oxford University Press, Oxford, 2004).CrossRefGoogle Scholar
Mal’cev, A. I.. On homomorphisms onto finite groups. Ivanov. Gos. Ped. Inst. Ucen. Zap. 18 (1958), 49–60.Google Scholar
Meskin, S.. Nonresidually finite one-relator groups. Trans. Amer. Math. Soc. 164 (1972), 105–114.CrossRefGoogle Scholar
Neumann, B. H.. Groups with finite classes of conjugate elements. Proc. London Math. Soc. (3) 1 (1951), 178–187.CrossRefGoogle Scholar
Pervova, E. L.. Maximal subgroups of some non locally finite p-groups. Internat. J. Algebra Comput. 15(5–6) (2005), 1129–1150.CrossRefGoogle Scholar
Robinson, D. J. S.. On the cohomology of soluble groups of finite rank. J. Pure Appl. Algebra, 6(2) (1975), 155–164.CrossRefGoogle Scholar
Tits, J.. A “theorem of Lie-Kolchin” for trees. Contributions to algebra (collection of papers dedicated to Ellis Kolchin), pages 377–388 (Academic Press, New York, 1977).Google Scholar
Tits, J.. Sur le groupe des automorphismes d’un arbre. Essays on topology and related topics (Mémoires dédiés à Georges de Rham), pages 188–211 (Springer, New York, 1970).Google Scholar
Wesolek, P.. Commensurated subgroups in finitely generated branch groups. J. Group Theory 20(2) (2017), 385–392.CrossRefGoogle Scholar
Wise, D. T.. Non-positively curved squared complexes: Aperiodic tilings and non-residually finite groups. (ProQuest LLC, Ann Arbor, MI, 1996. PhD. thesis–Princeton University).Google Scholar
Wise, D. T.. Subgroup separability of the figure 8 knot group. Topology 45(3) (2006), 421–463.CrossRefGoogle Scholar
Wise, D. T.. Complete square complexes. Comment. Math. Helv. 82(4) (2007), 683–724.CrossRefGoogle Scholar