Commuting elements with respect to the operator $\wedge $ in infinite groups
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Using the notion of complete nonabelian exterior square $G \widehat{\wedge} G$ of a pro-$p$-group $G$ ($p$ prime), we develop the theory of the exterior degree $\widehat{\mathrm{d}}(G)$ in the infinite case, focusing on its relations with the probability of commuting pairs $\mathrm{d}(G)$. Among the main results of this paper, we describe upper and lower bounds for $\widehat{\mathrm{d}}(G)$ with respect to $\mathrm{d}(G)$. Here the size of the second homology group $H_2(G,\mathbb{Z}_p)$ (over the $p$-adic integers $\mathbb{Z}_p$) plays a fundamental role. A further result of homological nature is placed at the end, in order to emphasize the influence of $H_2(G,\mathbb{Z}_p)$ both on $G$ and $\widehat{\mathrm{d}}(G)$.
Abstract
Using the notion of complete nonabelian exterior square $G \widehat{\wedge} G$ of a pro-$p$-group $G$ ($p$ prime), we develop the theory of the exterior degree $\widehat{\mathrm{d}}(G)$ in the infinite case, focusing on its relations with the probability of commuting pairs $\mathrm{d}(G)$. Among the main results of this paper, we describe upper and lower bounds for $\widehat{\mathrm{d}}(G)$ with respect to $\mathrm{d}(G)$. Here the size of the second homology group $H_2(G,\mathbb{Z}_p)$ (over the $p$-adic integers $\mathbb{Z}_p$) plays a fundamental role. A further result of homological nature is placed at the end, in order to emphasize the influence of $H_2(G,\mathbb{Z}_p)$ both on $G$ and $\widehat{\mathrm{d}}(G)$.
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