Relatively hyperbolic groups are a generalization of geometrically finite Kleinian groups, which were introduced by M. Gromov [Publ., Math. Sci. Res. Inst. 8, 75-263 (1987; Zbl 0634.20015)] and have been studied extensively by many authors. In a preprint, Bowditch defined the notion of boundary of a relatively hyperbolic group. The authors study the action of a relatively hyperbolic group on its boundary using the methods of symbolic dynamics. The idea is to extend the work outlined by Gromov and fully developed by M. Coornaert and A. Papadopoulos [Symbolic dynamics and hyperbolic groups. Lect. Notes Math. 1539. Berlin: Springer-Verlag (1993; Zbl 0783.58017)] from hyperbolic groups to the relatively hyperbolic case.
The property of the maximal parabolics that allows the method of Coornaert and Papadopoulos to work is the so-called special symbol property. Thus, the natural question becomes, which groups have the special symbol property?
The main theorems of the paper provide more details about the structure of such groups and provide a large family of examples. Namely, they show that if a relatively hyperbolic group has the special symbol property then it is finitely generated. They also show that poly-hyperbolic and poly-cyclic groups have the special symbol property.