  In this paper we consider convex co-compact subgroups of the projective
linear group. We prove that such a group is relatively hyperbolic with respect
to a collection of virtually Abelian subgroups of rank two if and only if each
open face in the ideal boundary has dimension at most one. We also introduce
the "coarse Hilbert dimension" of a subset of a convex set and use it to
characterize when a naive convex co-compact subgroup is word hyperbolic or
relatively hyperbolic with respect to a collection of virtually Abelian
subgroups of rank two.
