Subgroups and homology of free products of profinite groups

O. V. Mel'nikov

Izvestiya. Mathematics , Tome 34 (1990) no. 1, pp. 97-119

Résumé

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 $G$. Bibliography: 19 titles.