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

TL;DR

This paper establishes an upper bound on the growth rate of Dirichlet tilings in hyperbolic spaces, showing it is strictly less than a linear function of the number of faces, with implications for understanding tiling complexity.

Contribution

It provides a new upper bound on the growth rate of Dirichlet tilings in hyperbolic spaces, depending only on the number of faces and the dimension.

Findings

Growth rate of Dirichlet tilings is at most $k-1- ext{epsilon}$

The bound depends only on the number of faces and the space dimension

Open question about universal epsilon for all $k$-regular tilings

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$?