## 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

### Citations:

Zbl 1018.20020; Zbl 0927.20013; Zbl 1140.20305
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