Hyperbolic Coxeter groups and minimal growth rate in dimension four

For small $n$, the known compact hyperbolic $n$-orbifolds of minimal volume are intimately related to Coxeter groups of smallest rank. For $n=2$ and $n=3$, these Coxeter groups are intimately related to the triangle group $[7,3]$ and the tetrahedral group $[3,5,3]$, and they are also distinguished by the fact that they have minimal growth rate among {\it all} cocompact hyperbolic Coxeter groups in $\hbox{Isom}\mathbb H^n$, respectively. In this work, we prove that the Coxeter simplex group $[5,3,3,3]$, which is the fundamental group of the minimal volume arithmetic compact hyperbolic $4$-orbifold, has smallest growth rate among all cocompact Coxeter groups in $\hbox{Isom}\mathbb H^4$ as well. The proof is based on certain combinatorial properties of compact hyperbolic Coxeter polyhedra and some monotonicity properties of growth rates of the associated Coxeter groups.