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 = 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 all cocompact hyperbolic Coxeter groups in IsomH 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 IsomH 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.

Hyperbolic Coxeter groups of minimal growth rates in higher dimensions

“…Their related growth rates will be useful when comparing with the one of . This approach is similar to the one developed in [2].…”

“…Recall that R i is the smallest positive pole of f i , and that τ W i = 1 R i . We establish the growth functions f i according to Steinberg's formula (2). They are given as follows:…”

Hyperbolic Coxeter groups of minimal growth rates in higher dimensions

“…Their related growth rates will be useful when comparing with the one of Γ. This approach is similar to the one developed in [2]. Let W 0 = [∞, 3, 3] be the abstract Coxeter group depicted in Figure 5.…”