Hostname: page-component-586b7cd67f-dlnhk Total loading time: 0 Render date: 2024-11-26T09:38:27.752Z Has data issue: false hasContentIssue false
  • English
  • Français

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: ilir@im.ufrj.br
PAVEL A. ZALESSKII
Affiliation:
Universidade de Brasília, Departamento de Matemática, 70910-900 Brasília, DF, Brasil. e-mail: pz@mat.unb.br
Get access Rights & Permissions [Opens in a new window]

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 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

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