Weak commutativity for pro- p groups
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Weak commutativity for pro-p groups Dessislava H. Kochloukova1 · Luís Mendonça2 Received: 9 June 2020 / Accepted: 15 September 2020 © Springer-Verlag GmbH Austria, part of Springer Nature 2020
Abstract We define and study a pro- p version of Sidki’s weak commutativity construction. This is the pro- p group X p (G) generated by two copies G and G ψ of a pro- p group, subject to the defining relators [g, g ψ ] for all g ∈ G. We show for instance that if G is finitely presented or analytic pro- p, then 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 ν(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. Keywords Pro- p groups · Homological finiteness properties · Weak commutativity Mathematics Subject Classification Primary 20J05 · Secondary 20E18
1 Introduction In [25] Sidki defined for an arbitrary discrete group H the group X(H ) and initiated the study of this construction. The case when H is nilpotent was studied by Gupta, Rocco and Sidki in [11,25]. There are links between the weak commutativity construction and homology, in particular Rocco showed in [24] that the Schur multiplier of H is a subquotient of X(H ) isomorphic to W (H )/R(H ), where W (H ) and R(H ) are special normal subgroups of X(H ) defined in [25]. Recently Lima and Oliveira used in [17] the Schur multiplier of H to show that for any virtually polycyclic group H the group X(H ) is virtually polycyclic too. The use of homological methods in the study of X(H ) was further developed by Bridson, Kochloukova and Sidki in [3,14], where finite presentability and the homological finiteness type F Pm of X(H ) were
Communicated by Adrian Constantin.
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Dessislava H. Kochloukova [email protected]
1
Department of Mathematics, State University of Campinas, Campinas, Brazil
2
Mathematisches Institut, Heinrich-Heine-Universität Düsseldorf, Düsseldorf, Germany
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D. K. Kochloukova, L. Mendonça
studied. Recently Kochloukova and Mendonça calculated in low dimensional cases of the Bieri–Strebel–Neumann–Renz -invariants of X(H ). In this paper we study a pro- p version of the construction X(H ) for a fixed prime p. Let G be a pro- p group. We define X p (G) by the pro- p presentation X p (G) = G, G ψ | [g, g ψ ] = 1 for all g ∈ G p , where G ψ is an isomorphic copy of G via g → g ψ and − | − p denotes presentation by generators and relators in the category of pro- p groups. We develop structure theory of X p (G) similar to the discrete case by defining special normal subgroups D p (G), L p (G), W p (G) and R p (G). The structure theory we develop shows that X p (G)/W p (G) is a subdirect product of G × G × G. We need a criterion for homological finiteness properties F Pm of pro- p subdirect products of pro- p groups. The case of a pro- p subdirect product inside a direct product of free pro- p or Demushkin group was considered by Kochloukova and
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