# 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, $\mathbb{Z} \oplus \mathbb{Z}$. 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.