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

Kochloukova, Dessislava H. ; Pinto, Aline G. S.

Canadian journal of mathematics, Tome 72 (2020) no. 1, pp. 203-224

Résumé

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 $V=\dim V\otimes _{\mathbb{Z}_{p}}\mathbb{Q}_{p}$.

Détail

DOI : 10.4153/CJM-2018-032-4

Mots-clés : pro-p group, metabelian group, homology

Bibliographie

[1] Baumslag, G., Subgroups of finitely presented metabelian groups . J. Austral. Math. Soc. 16(1973), 98–110. . Google Scholar | DOI

[2] Bieri, R., Homological dimension of discrete groups . Queen Mary College Mathematical Notes, London, 1981. Google Scholar

[3] Bieri, R. and Strebel, R., Valuations and finitely presented metabelian groups . Proc. London Math. Soc. (3) 41(1980), no. 3, 439–464. . Google Scholar | DOI

[4] 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. . Google Scholar | DOI

[5] 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. . Google Scholar | DOI

[6] Corob Cook, G., Bieri-Eckmann criteria for profinite groups . Israel J. Math. 212(2016), no. 2, 857–893. . Google Scholar | DOI

[7] Corob Cook, G., On profinite groups of type FP . Adv. Math. 294(2016), 216–255. . Google Scholar | DOI

[8] 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. . Google Scholar | DOI

[9] Groves, J. R. J., Finitely presented centre-by-metabelian groups . J. London Math. Soc. (2) 18(1978), no. 1, 65–69. . Google Scholar | DOI

[10] 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. . Google Scholar | DOI

[11] King, J., Embedding theorems for pro-p groups . Math. Proc. Cambridge Philos. Soc. 123(1998), no. 2, 217–226. . Google Scholar | DOI

[12] King, J., Homological finiteness conditions for pro-p groups . Comm. Algebra 27(1999), no. 10, 4969–4991. . Google Scholar | DOI

[13] King, J., A geometric invariant for metabelian pro-p groups . J. London Math. Soc. (2) 60(1999), no. 1, 83–94. . Google Scholar | DOI

[14] Kochloukova, D. H., Metabelian pro-p groups of type FP . J. Group Theory 3(2000), no. 4, 419–431. . Google Scholar | DOI

[15] Kochloukova, D. H. and Mokari, F. Y., Virtual rational Betti numbers of abelian-by-polycyclic groups . J. Algebra 443(2015), 75–98. . Google Scholar | DOI

[16] Kochloukova, D. H. and Pinto, A. G. S., Embedding properties of metabelian pro-p groups . J. Group Theory. 9(2006), no. 4, 455–465. . Google Scholar | DOI

[17] Kochloukova, D. H. and Pinto, A. G. S., Centre-by-metabelian pro-p groups of type FP . Math. Proc. Cambridge Philos. Soc. 145(2008), no. 2, 305–309. . Google Scholar | DOI

[18] 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. . Google Scholar | DOI

[19] 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. . Google Scholar | DOI

[20] Pletch, A., Profinite duality groups II . J. Pure Appl. Algebra 16(1980), no. 1, 285–297. . Google Scholar | DOI

[21] Remeslennikov, V. N., Imbedding theorems for profinite groups . Izv. Akad. Nauk SSSR Ser. Mat. 43(1979), no. 2, 399–417, 480. Google Scholar

[22] Ribes, L. and Zalesskii, P., Profinite groups . Ergebnisse der Mathematik und ihrer Grenzgebiete, 40. Springer-Verlag, Berlin, 2000. . Google Scholar | DOI

[23] Weibel, C. A., An introduction to homological algebra . Cambridge Studies in Advanced Mathematics, 38. Cambridge University Press, Cambridge, 1994. . Google Scholar | DOI

[24] Wilson, J. S., Finite presentation of pro-p groups and discrete groups . Invent. Math. 105(1991), no. 1, 177–183. . Google Scholar | DOI

[25] Wilson, J. S., Profinite groups . London Mathematical Society Monographs. New Series, 19. The Clarendon Press, Oxford University Press, New York, 1998. Google Scholar