A non-Hopfian relatively hyperbolic group with respect to a Hopfian subgroup (Q6619328)

The notion of a (relatively) hyperbolic group was introduced by \textit{M. Gromov} [Publ., Math. Sci. Res. Inst. 8, 75--263 (1987; Zbl 0634.20015)]. An interesting question around hyperbolic groups was posed by \textit{D. V. Osin} [Mem. Am. Math. Soc. 843, (2006; Zbl 1093.20025), Problem 5.5]: If a group \(G\) is hyperbolic relative to a collection of residually finite subgroups \(\{H_{\lambda} \}_{\lambda \in \Lambda}\), does it follow that \(G\) is residually finite? Later, \textit{D. V. Osin} [Invent. Math. 167, No. 2, 295--326 (2007; Zbl 1116.20031)] proved also that his question is equivalent to Gromov's famous open problem [loc. cit.] to say whether every hyperbolic group is residually finite.\N\NIn the paper under review, the authors produce an example showing that a finitely generated relatively hyperbolic group with respect to a collection of Hopfian subgroups need not be Hopfian, answering Osin's question [loc. cit.] negatively. The construction provided by the authors is relatively simple (see Theorem 1.1) and makes use of successive HNN extensions (in the proof Bass-Serre theory plays a crucial role).