Pro-p groups acting on trees with finitely many maximal vertex stabilizers up to conjugation

DOI10.1007/S11856-022-2287-5zbMath1514.20095arXiv2007.06867MaRDI QIDQ6345043

Publication date: 14 July 2020

Abstract: We prove that a finitely generated pro-$p$ group $G$ acting on a pro-$p$ tree $T$ splits as a free amalgamated pro-$p$ product or a pro-$p$ HNN-extension over an edge stabilizer. If $G$ acts with finitely many vertex stabilizers up to conjugation we show that it is the fundamental pro-$p$ group of a finite graph of pro-$p$ groups $(cal G, Gamma)$ with edge and vertex groups being stabilizers of certain vertices and edges of $T$ respectively. If edge stabilizers are procyclic, we give a bound on $Gamma$ in terms of the minimal number of generators of $G$. We also give a criterion for a pro-$p$ group $G$ to be accessible in terms of the first cohomology $H^1(G, mathbb{F}_pG)$.

Mathematics Subject Classification ID

Free products of groups, free products with amalgamation, Higman-Neumann-Neumann extensions, and generalizations (20E06) Groups acting on trees (20E08) Limits, profinite groups (20E18) Group actions on combinatorial structures (05E18)

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