On Properties of Relatively Hyperbolic Groups

We discuss a number of problems in relatively hyperbolic groups. We show that the word problem and the conjugacy (search) problem are solvable in linear and quadratic time, respectively, for a relatively hyperbolic group, whenever the corresponding problem is solvable in linear and quadratic time in each parabolic subgroup. We also consider the classRof finitely generated toral relatively hyperbolic groups. We show that groups fromRare commutative transitive and generalize a theorem proved by Benjamin Baumslag toR. Moreover, we discuss two definitions of (fully) residually-Cgroups and prove the equivalence of the two definitions forC=R. Let Γ ∈R. We prove that every finitely generated fully residually-Γ group embeds into a group fromR. On the other hand, we give an example of a finitely generated torsion-free fully residually-Hgroup that does not embed into a group fromR;His the class of hyperbolic groups.