A pro-\(p\) version of Sela's accessibility and Poincaré duality pro-\(p\) groups (Q6619331)

The reviewer would like to quote here excerpts from the introduction of this interesting paper.\N\N``The main notion of the Bass-Serre theory is the notion of graph of groups. The fundamental group of a graph of groups acts naturally on a standard (universal) tree that allows to describe subgroups of these constructions. This theory raised naturally the question of accessibility, namely, whether we can continue to split \(G\) into an amalgamated free product or an HNN-extension forever, or do we reach the situation, after finitely many steps, where we cannot split it any more. In other words, accessibility is the question whether splittings of \(G\) as the fundamental group of a graph of groups have natural bound. Accessibility of splittings over finite groups was studied by \textit{M. J. Dunwoody} [Invent. Math. 81, 449--457 (1985; Zbl 0572.20025); Lond. Math. Soc. Lect. Note Ser. 181, 75--78 (1993; Zbl 0833.20035)], who proved that finitely presented groups are accessible but found an example of an inaccessible finitely generated group. This initiated naturally a search for a kind of accessibility that holds for finitely generated groups. The breakthrough in this direction is due to \textit{Z. Sela} [Invent. Math. 129, No. 3, 527--565 (1997; Zbl 0887.20017)], who proved \(k\)-acylindrical accessibility for any finitely generated group: accessibility provided the stabilizer of any segment of length \(k\) of the group acting on its standard tree is trivial for some \(k\).''\N\NThe pro-\(p\) version of Bass-Serre theory does not give subgroup structure theorems the way it does in the classical Bass-Serre theory: even in the pro-\(p\) case, if \(G\) acts on a pro-\(p\) tree \(T\), then a maximal subtree of the quotient graph \(G \setminus T\) does not always exist and even if it exists, it does not always lift to \(T\). Nevertheless, the pro-\(p\) version of the subgroups structure theorem works for pro-\(p\) groups acting on a pro-\(p\) trees that are accessible with respect to splitting over edge stabilizers [\textit{Z. Chatzidakis} and the second author, Isr. J. Math. 247, No. 2, 593--634 (2022; Zbl 1514.20095)].\N\NThe first main result in the paper under review is the pro-\(p\) version of Sela's celebrated result [loc.cit.].\N\NTheorem 1.1: Let \(G= \Pi_{1}(\mathcal{G}, \Gamma)\) be the fundamental group of a finite reduced \(k\)-acylindrical graph of pro-\(p\) groups. Then\N\[\N\big |E(\Gamma) \big | \leq d(G)(4k-1)-1 \;\; \mbox{and} \;\; \big | V(\Gamma) \big | \leq kd(G).\N\]\N\NThe result below can be considered as the first step towards this theory in the category of pro-\(p\) groups. With the second main result, the authors establish a canonical JSJ-decomposition of Poincaré duality pro-\(p\) groups of dimension \(n\) (i.e., \(PD^{n}\) pro-\(p\) groups).\N\NTheorem 1.2: For every \(PD^{n}\) pro-\(p\) group \(G\), \(n > 2\), there exists a (possibly trivial) \(k\)-acylindrical pro-\(p\) \(G\)-tree \(T\) satisfying the following properties: (i) every edge stabilizer is a maximal polycyclic subgroup of \(G\) of Hirsch length \(n-1\); (ii) every polycyclic subgroup of \(G\) of Hirsch length \(> 1\) stabilizes a vertex; (iii) the underlying graph of groups does not split further \(k\)-acylindrically over polycyclic subgroups of \(G\) of Hirsch length \(n-1\). Moreover, every two pro-\(p\) \(G\)-trees satisfying the properties above are \(G\)-isomorphic.