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

Abstract

It is shown that the growth rate $({\lim }_r|B(r){|}^{1/r})$ of any $k$ faces Dirichlet tiling of the real hyperbolic space ${\mathbb{H}}^d,d>2,$ is at most $k-1-\epsilon$, for an $\epsilon >0$, depending only on $k$ and $d$. We don't 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 $d>2$?