Published online by Cambridge University Press: 11 April 2017
Let $G$ be a group hyperbolic relative to a finite collection of subgroups
${\mathcal{P}}$. Let
${\mathcal{F}}$ be the family of subgroups consisting of all the conjugates of subgroups in
${\mathcal{P}}$, all their subgroups, and all finite subgroups. Then there is a cocompact model for
$E_{{\mathcal{F}}}G$. This result was known in the torsion-free case. In the presence of torsion, a new approach was necessary. Our method is to exploit the notion of dismantlability. A number of sample applications are discussed.