Hyperbolic Orbifolds of Minimal Volume

Abstract

We provide a survey of hyperbolic orbifolds of minimal volume, starting with the results of Siegel in two dimensions and with the contributions of Gehring, Martin and others in three dimensions. For higher dimensions, we summarise some of the most important results, due to Belolipetsky, Emery and Hild, by discussing related features such as hyperbolic Coxeter groups, arithmeticity and consequences of Prasad's volume, as well as canonical cusps, crystallography and packing densities. We also present some new results about volume minimisers in dimensions six and eight related to Bugaenko's cocompact arithmetic Coxeter groups.

All hyperbolic Coxeter n-cubes

“…Identifying the facets of a hyperbolic Coxeter polyhedron by using the reflections in their supporting hyperplanes is a simple way to construct hyperbolic n-orbifolds and n-manifolds. In the known cases, such polyhedra are responsible for minimal volume hyperbolic orbifolds and manifolds [10].…”

Counting problems for geodesics on arithmetic hyperbolic surfaces

“…Note that such a prime exists because we have already shown that there are infinitely many primes of k which are inert in all of the extensions L i /k. Before continuing we note that because the compact (respectively non-compact) hyperbolic 2-orbifold of minimal area has area π/42 (respectively, π/6), the fact that N(p 0 ) > 13 ensures that V 0 · (N(p 0 ) − 1) > 1 (see [6]). We will now construct our quaternion algebras B by choosing primes p of k which are unramified in B 0 and inert in all of the extensions L i /k, and then defining B to be the quaternion algebra for which Ram(B) = Ram(B 0 ) ∪ {p 0 , p}.…”

On commensurable hyperbolic Coxeter groups

“…3, (3.1)) of the groups 2,3 and 2,4 is a discrete subgroup of Isom(H 3 ) containing both groups as subgroups of finite index. Observe that C is a non-cocompact but cofinite (nonarithmetic) group so that its covolume is universally bounded from below by the minimal covolume K(π/3)/8 in this class which is realised by the tetrahedral group [3,3,6] (see [25] and [19,Table 2]). This allows us to rewrite (4.4) according to Table 2 Commensurability classes N n in the non-arithmetic case…”