Cyclic splittings of pro-\(p\) groups (Q6612138)

Let \(G\) be a pro-\(p\) group. A splitting of \(G\) as an amalgamated free pro-\(p\) product or HNN-extension over \(\mathbb{Z}_{p}\) is called a \(\mathbb{Z}_{p}\)-splitting (in this paper). An element of \(G\) is called elliptic with respect to a splitting \(G = G_{1} \coprod_{\mathbb{Z}_{p}} G_{2}\) as an amalgamated free pro-\(p\) product (pro-\(p\) HNN-extension \(G=\mathrm{HNN}(G_{1},\mathbb{Z}_{p},t)\)) if it is conjugate into \(G_{1} \cup G_{2}\) (into \(G_{1}\)) and is called hyperbolic otherwise. A pair of given \(\mathbb{Z}_{p}\)-splittings \(A_{1} \coprod_{C_{1}} B_{1}\) (or \(\mathrm{HNN}(A_{1}, C_{1}, t)\)) and \(A_{2} \coprod_{C_{2}} B_{2}\) (or \(\mathrm{HNN}(A_{2}, C_{2},t)\)) over \(C_{1}=\langle c_{1} \rangle\), \(C_{2}=\langle c_{2} \rangle\) is called\N\begin{itemize} \N\item[(1)] elliptic-elliptic if \(c_{1}\) is elliptic in \(A_{2} \coprod_{C_{2}} B_{2}\) and \(c_{2}\) is elliptic in \(A_{1} \coprod_{C_{1}} B_{1}\); \N\item[(2)] hyperbolic-hyperbolic if \(c_{1}\) is hyperbolic in \(A_{2} \coprod_{C_{2}} B_{2}\) and \(c_{2}\) is hyperbolic in \(A_{1} \coprod_{C_{1}} B_{1}\); \N\item[(3)] hyperbolic-elliptic if \(c_{1}\) is hyperbolic in \(A_{2} \coprod_{C_{2}} B_{2}\) and \(c_{2}\) is elliptic in \(A_{1} \coprod_{C_{1}} B_{1}\).\N\end{itemize}\N\NThe first result in the paper under review is the pro-\(p\) analog of a result by \textit{E. Rips} and \textit{Z. Sela} [Ann. Math. (2) 146, No. 1, 55--109 (1997; Zbl 0910.57002), Theorem 2.1].\N\NTheorem 1.1: Let \(G\) be a finitely generated pro-\(p\) group that does not split as a free pro-\(p\) product. Then any two \(\mathbb{Z}_{p}\)-splittings of \(G\) are either elliptic-elliptic or hyperbolic-hyperbolic.\N\NA second result is Theorem 1.3: Let \(p >2\) and let \(G\) be a finitely generated pro-\(p\) group that does not split as a free pro-\(p\) product. Let \(G=A_{1} \coprod_{C_{1}} B_{1}\) (or \(G=\mathrm{HNN}(A_{1}, C_{1}, t)\)) and \(G=A_{2} \coprod_{C_{2}} B_{2}\) (or \(G=\mathrm{HNN}(A_{2}, C_{2},t)\)), be two hyperbolic-hyperbolic \(\mathbb{Z}_{p}\)-splittings of \(G\). Suppose that \(N_{G}(C_{1})\) is not cyclic. Then \(G \simeq \mathbb{Z}_{p} \times \mathbb{Z}_{p}\).\N\NA third result is the counterpart of Theorem 1.3 in the case where \(p=2\) (see Proposition 1.6 and Theorem 1.7).