On the Topology of the Nonabelian Tensor Product of Profinite Groups

Abstract

Abstract. The properties of the nonabelian tensor products are interesting in different contexts of algebraic topology and group theory. We prove two theorems, dealing with the nonabelian tensor products of projective limits of finite groups. The first describes their topology. Then we show a result of embedding in the second homology group of a pro-pgroup, via the notion of complete exterior centralizer. We end with some open questions, originating from these two results.

Commuting Elements With Respect to the Operator Λ in Infinite Groups

“…where ∧ denotes the operator of nonabelian exterior square in [4,15,16,19] and the last equality is proved in [13,Lemma 2.2]. For instance, d ∧ (E) ≤ d(E) (see [13]) so the commutativity and the exterior degrees are connected.…”

“…This section collects a series of information and some technical results in [4,10,11,16,19]. We begin to recall that a pro-p-group G = lim N ∈P(G) G/N may be written as the projective limit of finite p-groups, where…”

“…The complete nonabelian tensor square G ⊗G of G is the group topologically generated by the symbols x ⊗y, subject to the relations xy ⊗z = (x y ⊗z y )(y ⊗z) and y ⊗tz = (y ⊗z)(y z ⊗t z ) for all x, y, z, t ∈ G, where x y = y −1 xy denotes the conjugate of x with respect to y. This construction can be found in [12,19] and generalizes the nonabelian tensor square in [4], originally formulated by Brown, Johnson and Robertson for abstract groups. A series of interesting properties have been studied with respect to the formation of the nonabelian tensor square (see [16]), in particular if P is a finite p-group, then so is P ⊗ P .…”

“…Corollary 2.2 clarifies the meaning of G ∧G in ( * * ). For the rest of this paper, it is good to recall here that the maps [12,19]). Also…”

“…are epimorphisms of pro-p-groups. The commutative diagram in [4] becomes the following, whose rows are central extensions of pro-p-groups (see also [12,16,19]):…”