Questions on relatively hyperbolic groups and related classes

The notion of a group hyperbolic relative to a collection of subgroups was originally proposed by Gromov [15]. Since then many definitions of relatively hyperbolic groups were suggested by Bowditch [5], Farb [13], Drutu-Osin-Sapir [11, 33], Groves-Manning [17], Osin [29], Yaman [38] and others. All these definitions except for those suggested in [29] are equivalent and require the group to be finitely generated and the collection of subgroup to be finite (while some of them can be easily extended to the general case). The definition from [29] is more general but is equivalent to others when restricted to finitely generated groups. For instance, a free product A∗B is always hyperbolic relative to {A,B} in the sense of [29], but is relatively hyperbolic in the sense of the other definitions iff both A and B are finitely generated. Throughout this list of problems we allow relatively hyperbolic groups to be infinitely generated, unless the converse is explicitly stated. If a group G is hyperbolic relative to a finite collection of finitely presented subgroups {Hλ}λ∈Λ, then G is finitely presented (easy exercise). In the other direction, the only known fact is that if a finitely generated group G is hyperbolic relative to a collection of subgroups {Hλ}λ∈Λ, then the collection is finite and each of the subgroups is finitely generated [29].