Quasirationality and Aspherical (Pro-$p$-) Presentations

A. Mikhovich

Matematičeskie zametki, Tome 105 (2019) no. 4, pp. 553-563.

Résumé

It is shown that the property of quasirationality of a (pro-$p$-) presentation of a (pro-$p$-) group $G$ is a property of the (pro-$p$-) group itself and does not depend on the choice of a presentation. It is proved that the class of quasirational presentations is wider than the class of aspherical pro-$p$-presentations (and of combinatorially aspherical presentations in the discrete case). For quasirational presentations, the notion of generalized permutationality of the module of relations is introduced, which turns out to be equivalent to the permutationality of the $\operatorname{mod}(p)$ quotient of the module.

Détail

Keywords: quasirationality, asphericity, generalized permutationality.

Bibliographie

[1] A. Mikhovich, “Quasirational relation modules and $p$-adic Malcev completions”, Topology Appl., 201 (2016), 86–91 | MR | Zbl

[2] I. Chiswell, D. J. Collins, J. Huebschmann, “Aspherical group presentations”, Math. Z., 178 (1981), 1–36 | MR | Zbl

[3] J. Huebschmann, “Cohomology theory of aspherical groups and of small cancellation groups”, J. Pure Appl. Algebra, 14:2 (1979), 137–143 | MR | Zbl

[4] J. Huebschmann, “The homotopy type of a combinatorially aspherical presentation”, Math. Z., 173:2 (1980), 163–169 | MR | Zbl

[5] O. V. Melnikov, “Asfericheskie pro-$p$-gruppy”, Matem. sb., 193:11 (2002), 71–104 | DOI | MR | Zbl

[6] A. Mikhovich, Proalgebraic Crossed Modules of Quasirational Presentations, 2016, arXiv: 1507.03155v4

[7] A. Mikhovich, “Quasirationality and prounipotent crossed modules”, J. Knot Theory Ramifications (to appear); 2017, arXiv: 1701.04793

[8] A. Mikhovich, “Identity theorem for pro-$p$-groups”, Springer Proc. Math. Stat. (to appear); 2017, arXiv: 1703.02996

[9] O. V. Melnikov, A. A. Shishkevich, “Pro-$p$-gruppy s virtualnoi dvoistvennostyu Puankare razmernosti 2”, Dokl. NAN Belarusi, 46:1 (2002), 13–15 | MR | Zbl

[10] A. A. Korenev, “Gruppy kogomologii pro-$p$-grupp s koeffitsientami v gruppovom koltse i virtualnaya dvoistvennost Puankare”, Matem. zametki, 78:6 (2005), 853–863 | DOI | MR | Zbl

[11] Kh. Kokh, Teoriya Galua $p$-rasshirenii, Mir, M., 1973 | MR

[12] L. Ribes, P. Zalesskii, Profinite Groups, Springer-Verlag, Berlin, 2000 | MR | Zbl

[13] D. Passman, The Algebraic Structure of Group Rings, Wiley-Interscience, New York, 1977 | MR | Zbl

[14] K. S. Brown, Cohomology of Groups, Springer-Verlag, Berlin, 1982 | MR | Zbl

[15] R. Brown, J. Huebschmann, “Identities among relations”, Low-Dimensional Topology, London Math. Soc. Lecture Note Ser., 48, Cambridge Univ. Press, Cambridge, 1982, 153–202 | MR

[16] Zh. P. Serr, “Struktura nekotorykh pro-$p$-grupp”, Sobranie sochinenii, T. 3, MTsNMO, M., 2007

[17] Y. Ihara, “On Galois representations arising from towers of coverings of $\mathbb{P}^1-\{0,1,\infty\}$”, Invent. Math., 86:3 (1986), 427–459 | MR | Zbl

[18] G. Karpilovsky, The Schur Multiplier, Clarendon Press, New York, 1987 | MR | Zbl

[19] F. E. A. Johnson, Stable Modules and the D(2)-Problem, London Math. Soc. Lecture Note Ser., 301, Cambridge Univ. Press, Cambridge, 2003 | MR | Zbl

[20] J. Neukirch, Algebraic Number Theory, Springer-Verlag, Berlin, 1999 | MR | Zbl

[21] A. Weiss, “Rigidity of $p$-adic $p$-torsion”, Ann. of Math. (2), 127:3 (1988), 317–332 | DOI | MR | Zbl

[22] D. Benson, Representations and Cohomology. I. Basic Representation Theory of Finite Groups and Associative Algebras, Cambridge Univ. Press, Cambridge, 1998 | MR | Zbl