A small unstable action on a tree.

*(English)*Zbl 1045.20018This 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.

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.

Reviewer: S. Andreadakis (MR1739226)