New examples of groups acting on real trees (Q2797318)

A group \(G\) has the Serre property (FA) if any simplicial action of \(G\) on a simplicial tree, by isometries and without edge inversions, fixes a vertex. Further, \(G\) has the property (F\(\mathbb{R}\)) if it cannot act nontrivially on any \(\mathbb{R}\)-tree. Evidently, (F\(\mathbb{R}\)) implies (FA) and \textit{P. B. Shalen} [in: Essays in group theory, Publ., Math. Sci. Res. Inst. 8, 265--319 (1987; Zbl 0649.20033)] asked whether the converse is true for finitely generated groups. The paper under review is concerned with the proof of the following theorem:NEWLINENEWLINE There exists a finitely generated group \(L\) which has the property (FA) and admits a nontrivial action on some \(\mathbb{R}\)-tree \(T\) such that the stabilizers for this action are finite.NEWLINENEWLINE Furthermore, \(L\) is not a quotient of any finitely presented group with property (FA), thus producing counterexamples to Shalen's question [loc. cit.].