Abstract:
The author defines a new construction of free product
$G=\mathop{\text{\LARGE{$*$}}}^{\mathfrak K}_TG_t$ in the variety $\mathfrak K$ of profinite groups of the family $\{G_t\mid t\in T\}$ of groups in $\mathfrak K$, continuously indexed by points of the profinite space $T$. In the case where $\mathfrak K$ is closed relative to extensions with Abelian kernels, a number of assertions about the homology groups of $G$ are obtained. Using homological methods, a theorem of Kurosh type on decomposition of an arbitrary pro-$p$-subgroup in $G$ into a free pro-$p$-product is proved, under a certain separability condition on $G$.
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