Relative cohomological dimension of a relatively hyperbolic pair

We show that the relative cohomological dimension cdR(G,H) cd R ( G , H ) of a relatively hyperbolic pair ( G , H ) is always finite when G does not contain R -torsion. We also show that this dimension is preserved under quasi-isometries, provided that G is torsion-free and the peripheral subgroup H is unconstricted and of type F∞ F ∞ . As a corollary of our methods, we compute cdZ(G,H) cd Z ( G , H ) in several cases.