Detecting Cohomological Dimension Three in Free-by-Demuškin Pro-$p$ Groups

Abstract

Let \[1\longrightarrow N\longrightarrow G\longrightarrow D\longrightarrow1\] be exact, with $N\ne 1$ free pro-$p$ of arbitrary rank and $D$ Demuškin. Writing $W=N/\Phi (N)$, we prove \[H^3(G,\mathbb{F}_p)^\vee\simeq W^D,\qquad \textrm{cd}_p G=3\text{ if and only if }W^D\neq0.\] Thus dimension three is detected on the first Frattini layer of the free kernel. Finite-rank kernels always satisfy the criterion, so a dimension drop requires an infinite-rank kernel, while the criterion remains stable under passage to open subgroups. On the fixed-point-free locus $W^D=0$, the remaining dimension-one/two alternative is governed by $H^1(D,W^{\vee })$ together with the Hochschild--Serre transgression. Dualizing the latter produces a relation-defect class in the topological coinvariants of $W$, which admits an explicit description in terms of a lifted defining relation of the Demuškin quotient. This yields a complete criterion for the lower-dimensional branch and separates the top-degree fixed-point obstruction from the extension data controlling dimensions one and two.

arXiv commentQuestion 10.1 from the previous version has been essentially resolved by an argument privately communicated by Henrique Souza; the resulting statement now appears as Theorem 5.5. The author gratefully acknowledges this contribution