For relatively hyperbolic groups, we investigate conditions under which the subgroup generated by two quasiconvex subgroups Q1 and Q2 is quasiconvex and isomorphic to Q1 ∗Q1∩Q2 Q2. Our results generalized known combination theorems for quasiconvex subgroups of word-hyperbolic groups. Some applications are presented. In addition, it is proved that the… (More)

Suppose that all hyperbolic groups are residually finite. The following statements follow: In relatively hyperbolic groups with peripheral structures consisting of finitely generated nilpotent subgroups, quasiconvex subgroups are separable; Geometrically finite subgroups of non-uniform lattices in rank one symmetric spaces are separable; Kleinian groups are… (More)

The firefighter game problem on locally finite connected graphs was introduced by Bert Hartnell [6]. The game on a graph G can be described as follows: let fn be a sequence of positive integers; an initial fire starts at a finite set of vertices; at each (integer) time n ≥ 1, fn vertices which are not on fire become protected, and then the fire spreads to… (More)

Let G be a group hyperbolic relative to a finite collection of subgroups P. Let F be the family of subgroups consisting of all the conjugates of subgroups in P, all their subgroups, and all finite subgroups. Then there is a cocompact model for EFG. This result was known in the torsion-free case. In the presence of torsion, a new approach was necessary. Our… (More)

Let G be a group which is hyperbolic relative to a collection of subgroups H1, and it is also hyperbolic relative to a collection of subgroups H2. Suppose that H2 ⊂ H1. We characterize, for subgroups of G, when quasiconvexity relative to H1 implies quasiconvexity relative to H2. We also show that quasiconvexity relative toH2 implies quasiconvexity relative… (More)