On commensurable hyperbolic Coxeter groups

Abstract

For Coxeter groups acting non-cocompactly but with finite covolume on real hyperbolic space H n , new methods are presented to distinguish them up to (wide) commensurability. We exploit these ideas and determine the commensurability classes of all hyperbolic Coxeter groups whose fundamental polyhedra are pyramids over a product of two simplices of positive dimensions.

New hyperbolic 4–manifolds of low volume

“…The arithmeticity of the 24-cell is proved in [18,Section 4], while arithmeticity of the polytope P is observed in [7] (it can be easily verified from the Coxeter diagram in Figure 1-right by applying Vinberg's algorithm [22] as explained in [7,Section 13.3]). The rectified 5-cell R is clearly commensurable with the pyramid R/ S 5 of Lemma 2.2, which is shown to be arithmetic in [6].…”

On Minimal Covolume Hyperbolic Lattices

“…According to the corresponding commensurability classification performed in [7], one has five Coxeter pyramid groups in Isom H 11 falling into two commensurability classes, three Coxeter pyramid groups in Isom H 13 forming one commensurability class, and finally the single Coxeter pyramid group Γ * ⊂ Isom H 17 that is closely related to the automorphism group of the even unimodular group PO(II 17,1 ) (see Example 2). Among the five arithmetic Coxeter pyramid groups Isom H 11 , which fall into two commensurability classes, the group Γ 11 given by the graph in Figure 9 has smallest covolume, and among the three commensurable Coxeter pyramid groups in Isom H 13 , the group Γ 13 given by Figure 11 has smallest covolume (see [7] and [32]). In order to identify explicitly-if possible-the minimal volume orientable cusped arithmetic hyperbolic n-orbifolds for n ≥ 11 odd, we provide details of the corresponding result of Belolipetsky and Emery (see Section 3.1.1).…”

“…In the sequel of our commensurability classification of Coxeter pyramid groups with n + 2 generators existing up to dimension 17 (see [7]), Guglielmetti [8] developed the software program CoxIter testing various properties such as arithmeticity and providing invariants such as the Euler characteristic of a hyperbolic Coxeter group. In Section 2.6, we give several instructive examples.…”

Problem on Mutant Pairs of Hyperbolic Polyhedra

“…Study of these polyhedral reflection groups Γ(P ) is a classical topic in hyperbolic geometry [19]. They are also of considerable recent interest from several different perspectives; as a sample, we refer the reader to [1,4,9,11,15]. Definition 1.1.…”