# Relatively hyperbolic groups vs 3-manifold groups

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Genevieve Walsh , Tufts University
Fine Hall 314

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, $\mathbb{Z} \oplus \mathbb{Z}$.  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 $\mathbb{H}^3$, where the maximal parabolic subgroups are the conjugates of $\mathbb{Z} \oplus \mathbb{Z}$ and its Bowditch boundary is $S^2$. We will discuss torsion-free relatively hyperbolic groups whose Bowditch boundaries are $S^2$.  In particular, we show that they are relative $PD(3)$ groups.  This is joint work in progress with Bena Tshishiku.