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Subgroup properties of Demushkin groups†

Published online by Cambridge University Press:  05 October 2015

ILIR SNOPCE  and
PAVEL A. ZALESSKII
ILIR SNOPCE
Affiliation:
Universidade Federal do Rio de Janeiro, Instituto de Matemática, 21941-909 Rio de Janeiro, RJ, Brasil. e-mail: [email protected]
PAVEL A. ZALESSKII
Affiliation:
Universidade de Brasília, Departamento de Matemática, 70910-900 Brasília, DF, Brasil. e-mail: [email protected]
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Abstract

We prove that a non-solvable Demushkin group satisfies the Greenberg–Stallings property, i.e., if H and K are finitely generated subgroups of a non-solvable Demushkin group G with the property that H ∩ K has finite index in both H and K, then H ∩ K has finite index in 〈H, K〉. Moreover, we prove that every finitely generated subgroup H of G has a ‘root’, that is a subgroup K of G that contains H with |K : H| finite and which contains every subgroup U of G that contains H with |U : H| finite. This allows us to show that every non-trivial finitely generated subgroup of a non-solvable Demushkin group has finite index in its commensurator.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 160 , Issue 1 , January 2016 , pp. 1 - 9
Copyright
Copyright © Cambridge Philosophical Society 2015 

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Footnotes

†

Research partially supported by CNPq

References

REFERENCES

[1]Abrashkin, V. A.A group-theoretic property of ramification filtration. Izv. Ross. Acad. NaukSer. Mat. 62 (1998), no. 6, 3–26; translation in Izv. Math. 62 (1998), no. 6, 1073–1094.Google Scholar
[2]Gordeev, N. L.Infiniteness of the number of relations in a Galois group of maximal p-extensions with a bounded ramification of a local field. Izv. Ross. Acad. NaukSer. Mat. 45 (1981), no. 3, 592–607; translation in Izv. Math. 18 (1982).Google Scholar
[3]Greenberg, L.Discrete groups of motions. Canad. J. Math. 12 (1960), 415–426.CrossRefGoogle Scholar
[4]Hillman, J. A. and Schmidt, A.Pro-p groups of positive deficiency. Bull. London Math. Soc. 40 (2008), 1065–1069.CrossRefGoogle Scholar
[5]Kochloukova, D. and Zalesskii, P.On pro-p analogues of limit groups via extensions of centralisers. Math. Z. 267 (2011), 109–128.CrossRefGoogle Scholar
[6]Kochloukova, D.Subdirect products of free pro-p groups and Demushkin groups. Int. J. Algebra Comput. 23 (2013), 1079–1098.CrossRefGoogle Scholar
[7]Lubotzky, A.Combinatorial group theory for pro-p groups. J. Pure Appl. Algebra 25 (1982), 311–325.CrossRefGoogle Scholar
[8]Neukirch, J., Schmidt, A. and Wingberg, K.Cohomology of Number Fields. Second edition. (Springer-Verlag. Berlin, 2008)CrossRefGoogle Scholar
[9]Rosset, Sh.Finite index and finite codimension. J. Pure Appl. Algebra 104 (1995), 97–107.CrossRefGoogle Scholar
[10]Serre, J. P.Sur la dimension cohomologique des groupes profinis. Topology 3 (1965), 413–420.CrossRefGoogle Scholar
[11]Serre, J. P.Local Fields (Springer–Verlag, Berlin, 1979)CrossRefGoogle Scholar
[12]Serre, J. P.Galois Cohomology. (Springer-Verlag, Berlin, 2002)Google Scholar
[13]Snopce, I. and Zalesskii, P. A.Subgroup properties of pro-p extensions of centralisers. Sel. Math. New Ser. 20 (2014), 465–489.CrossRefGoogle Scholar
[14]Stallings, J. R.Topology of finite graphs. Invent. Math. 71 (1983), 551–565.CrossRefGoogle Scholar
[15]Wilton, H. and Zalesskii, P. Distinguishing geometries using finite quotients. arXiv:1411.5212v1.Google Scholar