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.
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.
The cusped hyperbolic n-orbifolds of minimal volume are well known for n ≤ 9. Their fundamental groups are related to the Coxeter n-simplex groups Γn listed in Table 1. In this work, we prove that Γn has minimal growth rate among all non-cocompact Coxeter groups of finite covolume in IsomH n . In this way, we extend previous results of Floyd for n = 2 and of Kellerhals for n = 3 respectively. Our proof is a generalisation of the methods developed in [2] for the cocompact case.
The cusped hyperbolic n-orbifolds of minimal volume are well known for $n\leq 9$ . Their fundamental groups are related to the Coxeter n-simplex groups $\Gamma _{n}$ . In this work, we prove that $\Gamma _{n}$ has minimal growth rate among all non-cocompact Coxeter groups of finite covolume in $\textrm{Isom}\mathbb H^{n}$ . In this way, we extend previous results of Floyd for $n=2$ and of Kellerhals for $n=3$ , respectively. Our proof is a generalization of the methods developed together with Kellerhals for the cocompact case.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.
customersupport@researchsolutions.com
10624 S. Eastern Ave., Ste. A-614
Henderson, NV 89052, USA
Copyright © 2024 scite LLC. All rights reserved.
Made with 💙 for researchers
Part of the Research Solutions Family.