An upper bound on the growth of Dirichlet tilings of hyperbolic spaces

It is shown that the growth rate (lim(r) |B(r)|(1/r)) of any k faces Dirichlet tiling of H-d, d > 2, is at most k - 1 - epsilon, for an epsilon > 0, depending only on k and d. We do not know if there is a universal epsilon(u) > 0, such that k - 1 - epsilon(u) upperbounds the growth rate for any k- regular tiling, when d >2>