On the topology of the nonabelian tensor product of profinite groups (Q2812472)

In the paper under review, the author first shows that the (topological) nonabelian tensor product commutes with profinite closures. The author then considers the complete nonabelian exterior square \(G \widehat\wedge G\) of a pro-\(p\) group \(G\), and the exterior centralizer of \(x \in G\), defined as \(\widehat{C_{G}}(x) = \{ a \in G : a \widehat\wedge x = 1 \}\). This is shown to be a closed subgroup, and a normal subgroup of the usual centralizer \(C_{G}(x)\). Moreover \(C_{G}(x) / \widehat{C_{G}}(x)\) is abelian, and isomorphic to a closed subgroup of the Schur multiplier of \(G\) over the \(p\)-adic integers.