Hyperbolic Coxeter groups and minimal growth rates in dimensions four and five

Abstract

For small n, the known compact hyperbolic n-orbifolds of minimal volume are intimately related to Coxeter groups of smallest rank. For n D 2 and 3, these Coxeter groups are given by the triangle group OE7; 3 and the tetrahedral group OE3; 5; 3, and they are also distinguished by the fact that they have minimal growth rate among all cocompact hyperbolic Coxeter groups in Isom H n , respectively. In this work, we consider the cocompact Coxeter simplex group G 4 with Coxeter symbol OE5; 3; 3; 3 in Isom H 4 and the cocompact Coxeter prism group G 5 based on OE5; 3; 3; 3; 3 in Isom H 5 . Both groups are arithmetic and related to the fundamental group of the minimal volume arithmetic compact hyperbolic n-orbifold for n D 4 and 5, respectively. Here, we prove that the group G n is distinguished by having smallest growth rate among all Coxeter groups acting cocompactly on H n for n D 4 and 5, respectively. The proof is based on combinatorial properties of compact hyperbolic Coxeter polyhedra, some partial classification results and certain monotonicity properties of growth rates of the associated Coxeter groups.