On Properties of Relatively Hyperbolic Groups

Abstract

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 class R of finitely generated toral relatively hyperbolic groups. We show that groups from R are commutative transitive and generalize a theorem proved by Benjamin Baumslag to R. Moreover, we discuss two definitions of (fully) residually-C groups and prove the equivalence of the two definitions for C=R. Let Γ ∈ R . We prove that every finitely generated fully residually-Γ group embeds into a group from R. On the other hand, we give an example of a finitely generated torsion-free fully residually-H group that does not embed into a group from R; H is the class of hyperbolic groups.