Bounded subgroups of relatively finitely presented groups
The notion of a group \(G\) that is hyperbolic relative to a finite set \(H_{\Lambda}\) of its subgroups was introduced by \textit{M. Gromov}, in [Publ., Math. Sci. Res. Inst. 8, 75--263 (1987; Zbl 0634.20015)], as a generalization of a word hyperbolic group. In that definition, the groups \(H\in H_{\Lambda}\) are stabilizers of points at infinity of a certain hyperbolic space \(X\) the group \(G\) acts on.\N\NLet \(G\) be a finitely generated group that is relatively finitely presented with respect to a collection \(H_{\Lambda}\) of peripheral subgroups such that the corresponding relative Dehn function is well-defined. In the paper under review, the author proves that every infinite subgroup \(H\) of \(G\) that is bounded in the relative Cayley graph of \(G\) with respect to \(H_{\Lambda}\) is conjugate into a peripheral subgroup. As an application, he obtains a trichotomy for subgroups of relatively hyperbolic groups. Furthermore, he proves the existence of the relative exponential growth rate for all subgroups of limit groups.
scientific article; zbMATH DE number 7901546
scientific article