Commuting elements with respect to the operator \(\wedge\) in infinite groups (Q2828673)

The exterior degree \(d^{\wedge}(E)\) of a finite group \(E\) is defined as the proportion of pairs \((x, y) \in E^{2}\) such that \(x \wedge y\) is the identity in the nonabelian exterior square of \(E\). This is modelled on the commutative degree \(d(E)\) of \(E\), which is the proportion of commuting pairs. The authors extend the definition of \(d^{\wedge}(E)\) to \(\hat{d}(G)\) that covers pro-\(p\) groups \(G\), and prove various inequalities for \(\hat{d}(G)\) in terms, among others, of \(d(G)\) and the second homology group of \(G\) with \(p\)-adic coefficients.