Hostname: page-component-78c5997874-lj6df Total loading time: 0 Render date: 2024-11-09T20:38:44.717Z Has data issue: false hasContentIssue false
  • English
  • Français

Growth of Homology of Centre-by-metabelian Pro-$p$ Groups

Part of: Connections with homological algebra and category theory

Published online by Cambridge University Press:  09 January 2019

Dessislava H. Kochloukova  and
Aline G. S. Pinto
Dessislava H. Kochloukova
Affiliation:
Department of Mathematics, State University of Campinas (UNICAMP), 13083-859 Campinas-SP, Brazil Email: [email protected]
Aline G. S. Pinto
Affiliation:
Department of Mathematics, University of Brasília, 70297-400 Brasília, Brazil Email: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

For a centre-by-metabelian pro-$p$ group $G$ of type $\text{FP}_{2m}$, for some $m\geqslant 1$, we show that $\sup _{M\in {\mathcal{A}}}$ rk $H_{i}(M,\mathbb{Z}_{p})<\infty$, for all $0\leqslant i\leqslant m$, where ${\mathcal{A}}$ is the set of all subgroups of $p$-power index in $G$ and, for a finitely generated abelian pro-$p$ group $V$, rk $V=\dim V\otimes _{\mathbb{Z}_{p}}\mathbb{Q}_{p}$.

Keywords

pro-p groupmetabelian grouphomology

MSC classification

Primary: 20J05: Homological methods in group theory
Type
Article
Information
Canadian Journal of Mathematics , Volume 72 , Issue 1 , February 2020 , pp. 203 - 224
Copyright
© Canadian Mathematical Society 2018 

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

Author D. H. K. was partially supported by “bolsa de produtividade em pesquisa” 303350/2013-0 CNPq, Brazil and “projeto de pesquisa regular” FAPESP 2016/05678-3; author A. G. S. P. was partially supported by “Projeto Universal 482658/2013-4” from CNPq, Brazil.

References

Baumslag, G., Subgroups of finitely presented metabelian groups . J. Austral. Math. Soc. 16(1973), 98–110. https://doi.org/10.1017/S1446788700013987.Google Scholar
Bieri, R., Homological dimension of discrete groups . Queen Mary College Mathematical Notes, London, 1981.Google Scholar
Bieri, R. and Strebel, R., Valuations and finitely presented metabelian groups . Proc. London Math. Soc. (3) 41(1980), no. 3, 439–464. https://doi.org/10.1112/plms/s3-41.3.439.Google Scholar
Bridson, M. R. and Kochloukova, D. H., The torsion-free rank of homology in towers of soluble pro-p groups . Israel J. Maths. 219(2017), no. 2, 817–834. https://doi.org/10.1007/s11856-017-1499-6.Google Scholar
Bryant, R. M. and Groves, J. R. J., Finitely presented centre-by-metabelian Lie algebras . Bull. Austral. Math. Soc. 60(1999), no. 2, 221–226. https://doi.org/10.1017/S0004972700036352.Google Scholar
Corob Cook, G., Bieri-Eckmann criteria for profinite groups . Israel J. Math. 212(2016), no. 2, 857–893. https://doi.org/10.1007/s11856-016-1311-z.Google Scholar
Corob Cook, G., On profinite groups of type FP∞ . Adv. Math. 294(2016), 216–255. https://doi.org/10.1016/j.aim.2016.02.020.Google Scholar
Dixon, J. D., du Sautoy, M. P. F., Mann, A., and Segal, D., Analytic pro-p groups . Second edition. Cambridge Studies in Advanced Mathematics, 61. Cambridge University Press, Cambridge, 1999. https://doi.org/10.1017/CBO9780511470882.Google Scholar
Groves, J. R. J., Finitely presented centre-by-metabelian groups . J. London Math. Soc. (2) 18(1978), no. 1, 65–69. https://doi.org/10.1112/jlms/s2-18.1.65.Google Scholar
Groves, J. R. J. and Kochloukova, D. H., Embedding properties of metabelian Lie algebras and metabelian discrete groups . J. London Math. Soc. (2) 73(2006), no. 2, 475–492. https://doi.org/10.1112/S0024610705022581.Google Scholar
King, J., Embedding theorems for pro-p groups . Math. Proc. Cambridge Philos. Soc. 123(1998), no. 2, 217–226. https://doi.org/10.1017/S0305004197002181.Google Scholar
King, J., Homological finiteness conditions for pro-p groups . Comm. Algebra 27(1999), no. 10, 4969–4991. https://doi.org/10.1080/00927879908826743.Google Scholar
King, J., A geometric invariant for metabelian pro-p groups . J. London Math. Soc. (2) 60(1999), no. 1, 83–94. https://doi.org/10.1112/S0024610799007693.Google Scholar
Kochloukova, D. H., Metabelian pro-p groups of type FPm . J. Group Theory 3(2000), no. 4, 419–431. https://doi.org/10.1515/jgth.2000.033.Google Scholar
Kochloukova, D. H. and Mokari, F. Y., Virtual rational Betti numbers of abelian-by-polycyclic groups . J. Algebra 443(2015), 75–98. https://doi.org/10.1016/j.jalgebra.2015.07.005.Google Scholar
Kochloukova, D. H. and Pinto, A. G. S., Embedding properties of metabelian pro-p groups . J. Group Theory. 9(2006), no. 4, 455–465. https://doi.org/10.1515/JGT.2006.029.Google Scholar
Kochloukova, D. H. and Pinto, A. G. S., Centre-by-metabelian pro-p groups of type FPm . Math. Proc. Cambridge Philos. Soc. 145(2008), no. 2, 305–309. https://doi.org/10.1017/S0305004108001163.Google Scholar
Kochloukova, D. H. and da Silva, F. S., Embedding homological properties of metabelian discrete groups: the general case . J. Group Theory 10(2007), no. 4, 505–529. https://doi.org/10.1515/JGT.2007.041.Google Scholar
Kochloukova, D. H. and Zalesskii, P., Homological invariants for pro-p groups and some finitely presented pro-C groups . Monatsh. Math. 144(2005), no. 4, 285–296. https://doi.org/10.1007/s00605-004-0269-9.Google Scholar
Pletch, A., Profinite duality groups II . J. Pure Appl. Algebra 16(1980), no. 1, 285–297. https://doi.org/10.1016/0022-4049(80)90034-1.Google Scholar
Remeslennikov, V. N., Imbedding theorems for profinite groups . Izv. Akad. Nauk SSSR Ser. Mat. 43(1979), no. 2, 399–417, 480.Google Scholar
Ribes, L. and Zalesskii, P., Profinite groups . Ergebnisse der Mathematik und ihrer Grenzgebiete, 40. Springer-Verlag, Berlin, 2000. https://doi.org/10.1007/978-3-662-04097-3.Google Scholar
Weibel, C. A., An introduction to homological algebra . Cambridge Studies in Advanced Mathematics, 38. Cambridge University Press, Cambridge, 1994. https://doi.org/10.1017/CBO9781139644136.Google Scholar
Wilson, J. S., Finite presentation of pro-p groups and discrete groups . Invent. Math. 105(1991), no. 1, 177–183. https://doi.org/10.1007/BF01232262.Google Scholar
Wilson, J. S., Profinite groups . London Mathematical Society Monographs. New Series, 19. The Clarendon Press, Oxford University Press, New York, 1998.Google Scholar