The three smallest compact arithmetic hyperbolic 5–orbifolds

Abstract

We determine the three hyperbolic 5-orbifolds of smallest volume among compact arithmetic orbifolds, and we identify their fundamental groups with hyperbolic Coxeter groups.

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

“…The smallest known Salem number α L ≈ 1.176281 with minimal polynomial L(t) = t 10 + t 9 − t 7 − t 6 − t 5 − t 4 − t 3 + t + 1 equals the growth rate τ [7,3] of the cocompact Coxeter triangle group G = [7,3] with Coxeter graph •-7 ---•---• which in turn is the smallest growth rate among all cocompact planar hyperbolic Coxeter groups (see [16,20]). The second smallest growth rate is realised by the Coxeter triangle group [8,3] with Coxeter graph •- The denominator polynomial q(t) of f S (t) is palindromic and of degree 60. By means of the software PARI/GP [26], one checks that q(t) is irreducible and has -beside non-real roots some of them being of absolute value oneexactly two inversive pairs α ±1 , β ±1 of real roots such that α > β > 1.…”

“…Hyperbolic Coxeter groups are not only characterised by a simple presentation but they are also distinguished in other ways. For example, for small n, they appear as fundamental groups of smallest volume orbifolds O n = H n /Γ where Γ ⊂ IsomH n is a discrete subgroup (see [1,2], [15], [8] and [19], for example). In particular, for n = 2, 3 and 4, the compact hyperbolic n-orbifold of minimal volume is the quotient of H n by a Coxeter group of smallest rank and given by the triangle group [7,3], the Z 2 -extension of the tetrahedral group [3,5,3] and, when restricted to the arithmetic context, by the simplex group [5,3,3,3], respectively.…”

“…In [4], and by an extension of our methods, an analogous result for growth rates of hyperbolic Coxeter groups acting cocompactly on hyperbolic space of higher dimension is shown. For n = 5, the group of reference is the Coxeter prism group [5,3,3,3,3] found by Makarov whose quotient space, by [8], has minimal volume among all arithmetic compact hyperbolic 5-orbifolds.…”

The optimal hyperball packings related to the smallest compact arithmetic 5-orbifolds

“…In [4] the second and third values in the volume spectrum of compact orientable arithmetic hyperbolic 5-orbifolds are determined and has proved the following Theorem 1.1 (Emery-Kellerhals) The lattices Γ ′ 0 , Γ ′ 1 , Γ ′ 2 (ordered by increasing covolume) are the three cocompact arithmetic lattices in Isom + (H 5 ) of minimal covolume. They are unique, in the sense that any cocompact arithmetic lattice in Isom + (H 5 ) of covolume smaller than or equal to Γ ′ 2 is conjugate in Isom(H 5 ) to one of the Γ ′ i (i = 0, 1, 2).…”

The least dense hyperball covering of regular prism tilings in hyperbolic $$n$$ n -space

“…The formulas for the hyperbolic covolumes of the considered n-dimensional Coxeter tilings are determined in [7], [8] and [4], therefore, it is possible to compute the covolumes of the regular prisms and the densities of the corresponding ball, horoball and hyperball packings.…”