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On accessibility of finitely generated groups. (English) Zbl 1280.20032

From the introduction: In this note, we prove an accessibility result that is to Linnell’s theorem what Sela’s result is to Grushko’s theorem.
Let \(G\) be a group acting on a simplicial tree \(T\) and \(\pi\colon T\to G\backslash T\) be the canonical projection. Let \([v,w]\) be a simplicial segment of \(T\). We say that \([v,w]\subset T\) has projective length \(k\) and write \(\text{pl}([v,w])=k\) if \(\pi([v,w])\) meets \(k\) edges (edge pairs) of \(G\backslash T\).
Let \(\mathbb A\) be a graph of groups, \(G=\pi_1(\mathbb A)\) and \(T\) be the associated Bass-Serre tree. We say that \(T\) or \(\mathbb A\) is \((k,C)\)-acylindrical if the stabilizer of any segment \([v,w]\subset T\) with \(\text{pl}([v,w])>k\) is of order at most \(C\). We further call a graph of groups weakly reduced if there exists no valence 2 vertex \(v\) such that both boundary monomorphisms into the vertex group \(A_v\) are isomorphisms.
We prove the following theorem; here, \(\#EA\) is the number of edges of the graph \(A\) underlying the graph of groups \(\mathbb A\).
Theorem 1. Let \(\mathbb A\) be a weakly reduced minimal \((k,C)\)-acylindrical graph of groups with \(k\geq 1\). Then \[ \#EA\leq (2k+1)\cdot C\cdot(\text{rank\,}\pi_1(\mathbb A)-1). \] Note that the theorem does not hold for \(k=0\). To see this note that the free group of rank \(n\) is the fundamental group of a weakly reduced minimal graph of trivial groups with \(3n-3\) edges, this splitting is clearly \((0,1)\)-acylindrical. However, as any \((k,C)\)-acylindrical splitting is also a \((k+1,C)\)-splitting, our theorem does still give a bound for the case \(k=0\).
The proof of the theorem combines ideas of the author’s proof of acylindrical accessibility [Proc. Lond. Math. Soc., III. Ser. 85, No. 1, 93-118 (2002; Zbl 1018.20020)] and M. J. Dunwoody’s proof of Linnell accessibility [Geom. Topol. Monogr. 1, 139-158 (1998; Zbl 0927.20013)] in the language of R. Weidmann [Contemp. Math. 372, 99-108 (2005; Zbl 1140.20305)]. It applies Stallings’ folding arguments and relies on defining a complexity of decorated splittings that does not increase under foldings and related operations.

MSC:

20E08 Groups acting on trees
20E06 Free products of groups, free products with amalgamation, Higman-Neumann-Neumann extensions, and generalizations
20F65 Geometric group theory
57M07 Topological methods in group theory
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