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A small unstable action on a tree. (English) Zbl 1045.20018
This paper deals with groups acting on \(\mathbb{R}\)-trees, a branch of study initiated by J. Tits [in Contrib. to Algebra, Collect. Pap. dedic. E. Kolchin, 377-388 (1977; Zbl 0373.20039)]. A major result was by Rips, who classified the finitely generated groups that have free actions on \(\mathbb{R}\)-trees. Such groups are free products of free Abelian groups and surface groups, a result which had been conjectured by Morgan and Shalen. Let \(T\) be an \(\mathbb{R}\)-tree on which the group \(G\) acts (on the left) by isometries. We say that the action is trivial if for some point \(x\in T\) the stabilizer \(\text{Fix}(x)\) equals \(G\). We say that \(T\) is minimal if it has no proper \(G\)-invariant subtree. A nondegenerate (i.e., containing more than one point) subtree \(S\) of \(T\) is stable if for every nondegenerate subtree \(S'\) of \(S\) we have \(\text{Fix}(S)=\text{Fix}(S')\). A nontrivial action is stable if every nondegenerate subtree of \(T\) contains a stable subtree. M. Bestvina and M. Feighn [Invent. Math. 121, No. 2, 287-321 (1995; Zbl 0837.20047)] have extended the classification of free actions to include all finitely presented groups that have stable minimal actions on \(\mathbb{R}\)-trees.
In the present paper the author gives an example of a finitely generated but not finitely presented group \(H\) that has a nontrivial, unstable action on an \(\mathbb{R}\)-tree \(T\) with finite cyclic arc stabilizers. He then shows that there is no nontrivial action of \(H\) on a simplicial \(\mathbb{R}\)-tree with small edge stabilizers. Small means that they do not contain non-cyclic free subgroups. This example provides a negative answer to Question D of P. B. Shalen [in Essays in group theory, Publ., Math. Sci. Res. Inst. 8, 265-319 (1987; Zbl 0649.20033)]. The construction of the tree \(T\) is by means of a folding sequence of simplicial trees.

MSC:
20E08 Groups acting on trees
20F65 Geometric group theory
57M07 Topological methods in group theory
20F05 Generators, relations, and presentations of groups
20E06 Free products of groups, free products with amalgamation, Higman-Neumann-Neumann extensions, and generalizations
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