Abstract
We define and study a pro-p version of Sidki’s weak commutativity construction. This is the pro-p group \(\mathfrak {X}_p(G)\) generated by two copies G and \(G^{\psi }\) of a pro-p group, subject to the defining relators \([g,g^{\psi }]\) for all \(g \in G\). We show for instance that if G is finitely presented or analytic pro-p, then \(\mathfrak {X}_p(G)\) has the same property. Furthermore we study properties of the non-abelian tensor product and the pro-p version of Rocco’s construction \(\nu (H)\). We also study finiteness properties of subdirect products of pro-p groups. In particular we prove a pro-p version of the \((n-1)-n-(n+1)\) Theorem.
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References
Bestvina, M., Brady, N.: Morse theory and finiteness properties of groups. Invent. Math. 129, 445–470 (1997)
Bridson, M., Howie, J., Miller, C., Short, H.: On the finite presentation of subdirect products and the nature of residually free groups. Am. J. Math. 135(4), 891–933 (2013)
Bridson, M.R., Kochloukova, D.H.: Weak commutativity and finiteness properties of groups. Bull. Lond. Math. Soc. 51, 168–180 (2019)
Bridson, M.R., Kochloukova, D.H.: in preparation
Brown, R., Loday, J.-L.: Van Kampen theorems for diagrams of spaces. Topology 26(3), 311–335 (1987)
Corob Cook, G.: On profinite groups of type \(FP_{\infty }\). Adv. Math. 294, 216–255 (2016)
de Mendonça, L.A.: The weak commutativity construction for Lie algebras. J. Algebra 529, 145–173 (2019)
Dennis, R.K.: In search of new “homology” functors having a close relationship to K-theory. Cornell University, Ithaca, NY, Preprint (1976)
Dixon, J.D., du Sautoy, M.P.F., Mann, A., Segal, D.: Analytic Pro-p-Groups, London Mathematical Society Lecture Note Series, vol. 157. Cambridge University Press, Cambridge (1991)
Ellis, G., Leonard, F.: Computing Schur multipliers and tensor products of finite groups. Proc. R. Irish Acad. Sect. A 95(2), 137–147 (1995)
Gupta, N., Rocco, N., Sidki, S.: Diagonal embeddings of nilpotent groups. Illinois J. Math. 30(2), 274–283 (1986)
King, J.D.: Homological finiteness conditions for pro-p groups. Commun. Algebra 27(10), 4969–4991 (1999)
Kochloukova, D.H., Short, H.: On subdirect products of free pro-p groups and Demushkin groups of infinite depth. J. Algebra 343, 160–172 (2011)
Kochloukova, D.H., Sidki, S.: On weak commutativity in groups. J. Algebra 471, 319–347 (2017)
Kuckuck, B.: Subdirect products of groups and the \(n-(n+1)-(n+2)\) conjecture. Q. J. Math. 65(4), 1293–1318 (2014)
Lazard, M.: Groupes analytiques p-adiques. Inst. Hautes Études Sci. Publ. Math. No. 26, 389–603 (1965)
Lima, B.C.R., Oliveira, R.N.: Weak commutativity between two isomorphic polycyclic groups. J. Group Theory 19(2), 239–248 (2016)
Lubotzky, A., Mann, A.: Powerful p-groups. II. p-adic analytic groups. J. Algebra 105(2), 506–515 (1987)
Miller, C.: The second homology of a group. Proc. Am. Math. Soc. 3, 588–595 (1952)
Moravec, P.: On the Schur multipliers of finite p-groups of given coclass. Israel J. Math. 185, 189–205 (2011)
Moravec, P.: Powerful actions and non-abelian tensor products of powerful p-groups. J. Group Theory 13(3), 417–427 (2010)
Ribes, L., Zalesskii, P.: Profinite Groups, Second Edition, A Series of Modern Surveys in Mathematics, vol. 40. Springer, Berlin (2010)
Rocco, N.R.: On a construction related to the non-abelian tensor square of a group. Bull. Braz. Mat. Soc. (N.S.) 22(1), 63–79 (1991)
Rocco, N.R.: On weak commutativity between finite p-groups, p odd. J. Algebra 76(2), 471–488 (1982)
Sidki, S.: On weak permutability between groups. J. Algebra 63(1), 186–225 (1980)
Symonds, P., Weigel, T.: Cohomology of \(p\)-adic Analytic Groups, New Horizons in Pro-\(p\) Groups, Progress in Mathematics, vol. 184, Birkhäuser Boston, MA, pp. 349–410 (2000)
Wilson, J.: Profinite Groups, London Mathematical Society Monographs, New Series, vol. 19. Clarendon Press, Oxford (1998)
Acknowledgements
During the preparation of this work the Dessislava H. Kochloukova was partially supported by CNPq Grant 301779/2017-1 and by FAPESP Grant 2018/23690-6. The Luís Mendonça was supported by Ph.D. Grant FAPESP 2015/22064-6.
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Kochloukova, D.H., Mendonça, L. Weak commutativity for pro-p groups. Monatsh Math 194, 555–575 (2021). https://doi.org/10.1007/s00605-020-01468-7
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DOI: https://doi.org/10.1007/s00605-020-01468-7