Relatively hyperbolic groups vs 3-manifold groups
Relatively hyperbolic groups vs 3-manifold groups
An illustrative example of a relatively hyperbolic group is the fundamental group of a hyperbolic knot complement. In this case, the peripheral subgroup corresponds to the group of the cusp cross-section, Z⊕Z. Bowditch described the boundary of a relatively hyperbolic group pair (G,P) as the boundary of any hyperbolic space that G acts geometrically finitely upon, where the maximal parabolic subgroups are conjugates of the peripheral group P. For example, the fundamental group of a hyperbolic knot complement acts as a geometrically finitely on H3, where the maximal parabolic subgroups are the conjugates of Z⊕Z and its Bowditch boundary is S2. We will discuss torsion-free relatively hyperbolic groups whose Bowditch boundaries are S2. In particular, we show that they are relative PD(3) groups. This is joint work in progress with Bena Tshishiku.