Virtually free pro-\(p\) products. (Q285578)

In a recent joint paper with T. Weigel, the author proved that a torsion free finitely generated virtual free pro-\(p\) product is a free pro-\(p\) product. The objective of this paper is to prove a generalization of this theorem and to give a direct simpler proof of it. An infinite free pro-\(p\) product \(H=\coprod_{x\in X}H_x\) is called proper if \(\{H_x\mid H_x\neq 1,\;x\in H\}\) is closed in the subgroup space of \(H\).NEWLINENEWLINE The main result is the following. Let \(G\) be a second countable torsion free pro-\(p\) group having an open subgroup \(H\) that splits as a non-trivial proper free pro-\(p\) product \(H=(\coprod_{x\in X}H_x)\coprod F\) of indecomposable into a free pro-\(p\) product of subgroups \(H_x\not\cong\mathbb Z_p\) of \(H\) indexed by a profinite space \(X\) and a free pro-\(p\) group \(F\); then \(G=(\coprod_{y\in Y}N_G(H_y))\coprod F_1\) is a free pro-\(p\) product of the normalizers, where \(F_1\) is a free pro-\(p\) group and \(Y\) is some closed subset of \(X\). An example is given showing that the second countability hypothesis in the previous statement is essential.