Relative cohomological dimension of a relatively hyperbolic pair

Abstract

We show that the relative cohomological dimension \(\textrm{cd}_R(G,H)\) 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 \(F_{\infty }\). As a corollary of our methods, we compute \(\textrm{cd}_{\mathbb {Z}}(G,H)\) in several cases.