Ideal Uniform Polyhedra in and Covolumes of Higher Dimensional Modular Groups

Abstract

Higher dimensional analogues of the modular group $\mathit{PSL}(2,\mathbb{Z})$ are closely related to hyperbolic reflection groups and Coxeter polyhedra with big symmetry groups. In this context, we develop a theory and dissection properties of ideal hyperbolic $k$ -rectified regular polyhedra, which is of independent interest. As an application, we can identify the covolumes of the quaternionic modular groups with certain explicit rational multiples of the Riemann zeta value $\unicode[STIX]{x1D701}(3)$ .

Small‐volume hyperbolic manifolds constructed from Coxeter groups

“…This process yields a finite‐volume hyperbolic Coxeter polyhedron with dihedral angles $$\frac{\pi }{3}$$ and $$\frac{\pi }{2}$$, coinciding with the (hyperbolic) convex hull of all ideal centres of the 2‐faces of $$\hat{S}_{\rm reg}$$, the ideal birectified regular simplex $${\cal B}$$, whose symmetry group is isomorphic to a subgroup of $$S_6$$. For more details, see [15, Section 3.2].…”

“…The covolume of the group Γ 5 is much harder to compute than the covolume of [3,4,3,3,3] and its commensurable groups. In [15], using the proof of [15,Theorem 2] and relations in the crystallographic Napier cycles defined by the group Γ 5 , the second author was able to compute the covolume of Γ 5 = [6,3,3,3,3,6] and that of PS Δ L(2, ℤ[𝜔, 𝑗]). Indeed…”

“…By taking the (hyperbolic) convex hull of all these edge midpoints of $$\hat{S}_{\rm reg}$$, one obtains an ideal hyperbolic 4‐polyhedron $${\cal R}$$ with dihedral angles $$\frac{\pi }{3}$$ and $$\frac{\pi }{2}$$ whose symmetry group is isomorphic to a subgroup of $$S_5$$. More precisely, this process is given by polar truncation of each of the ultra‐ideal vertices of $$\hat{S}_{\rm reg}$$ and is called (simple) rectification of $$\hat{S}_{\rm reg}$$, indicated by $$r_1\hat{S}_{\rm reg}={\cal R}$$; see [15, Section 3.2], [25]. This process also gives rise to a (truncated) finite volume Coxeter polyhedron $$R_4\subset \mathbb {H}^4$$ with symbol $$[\infty ,3,3,3,6]$$, with precisely one ideal vertex, and which barycentrically decomposes $${\cal R}$$ into $$5!$…”

“…The covolume of the group $$\Gamma _5$$ is much harder to compute than the covolume of [3,4,3,3,3] and its commensurable groups. In [15], using the proof of [15, Theorem 2] and relations in the crystallographic Napier cycles defined by the group $$\Gamma _5$$, the second author was able to compute the covolume of $$\Gamma _5=[6,3,3,3,3,6]$$ and that of PS$$_{\Delta }$$L$$(2,\mathbb {Z}[\omega ,j])$$. Indeed $$\begin{equation} \hbox{\rm covol}_5([6,3,3,3,3,6])=\frac{13}{5760}\,\zeta (3)\,\,.$$…”

“…The polyhedron $$\cal B$$ is highly symmetric and can be barycentrically decomposed into $$6! = 720$$ Coxeter pyramids with symbol [6,3,3,3,3,6], with each one having volume $$\frac{13\zeta (3)}{5760}$$; see [15].…”