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Acylindrical accessibility for groups acting on \(\mathbb{R}\)-trees. (English) Zbl 1080.20035
Summary: We prove an acylindrical accessibility theorem for finitely generated groups acting on \(\mathbb{R}\)-trees. Namely, we show that if \(G\) is a freely indecomposable non-cyclic \(k\)-generated group acting minimally and \(D\)-acylindrically on an \(\mathbb{R}\)-tree \(X\) then there is a finite subtree \(T_\varepsilon\subseteq X\) of measure at most \(2D(k-1)+\varepsilon\) such that \(GT_\varepsilon=X\). This generalizes theorems of Z. Sela and T. Delzant about actions on simplicial trees.

MSC:
20F67 Hyperbolic groups and nonpositively curved groups
20E08 Groups acting on trees
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