Relatively hyperbolic groups vs 3-manifold groups

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Genevieve Walsh , Tufts University

 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, ZZ.  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 ZZ 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.