A hyperbolic $3$ -manifold whose volume is a rational multiple of Catalan's constant

Proved

Thurston23.exists_hyperbolicVolume_rat_mul_catalan

by Lucas · Sep 11, 2026 · Mathlib 777aaa6 (Lean v4.29.0-rc3)

hyperbolic-geometrykleinian-groupsnumber-theory

Let $F=\mathbb{Q}(i)$ , with ring of integers $\mathbb{Z}[i]$ and discriminant $\delta_F=-4$ . Humbert's formula expresses the covolume of the Bianchi group $\mathrm{PSL}_2(\mathbb{Z}[i])$ , acting on hyperbolic $3$ -space, as

\frac{|\delta_F|^{3/2}\,\zeta_F(2)}{4\pi^2},

and since $\zeta_F(s)=\zeta(s)L(s,\chi_{-4})$ for this field, that covolume equals $G/3$ , where

G=\sum_{n\ge 0}\frac{(-1)^n}{(2n+1)^2}=0.9159655\ldots

is Catalan's constant. The Bianchi group itself has torsion, so its quotient is an orbifold rather than a manifold; but it contains torsion-free subgroups of finite index, for instance the principal congruence subgroup of level $2+i$ , and the quotient by such a subgroup is a finite-volume hyperbolic $3$ -manifold whose volume is the index times $G/3$ .

The statement asserts exactly this consequence: the set of volumes of finite-volume hyperbolic $3$ -manifolds, as fixed by the mission bundle (measures of fundamental domains of Kleinian actions on the upper half-space with density $z^{-3}$ ), contains a positive rational multiple of Catalan's constant.

Preamble

import Definitions.Def_Thurston23_bundle

Formal statement

namespace Thurston23

open MeasureTheory

theorem exists_hyperbolicVolume_rat_mul_catalan :
    ∃ v ∈ hyperbolicVolumes, ∃ q : ℚ, 0 < q ∧
      v = (q : ℝ) * ∑' n : ℕ, (-1) ^ n / ((2 * n + 1) ^ 2 : ℝ) := by
  sorry

end Thurston23

Source

W. P. Thurston, Three-dimensional manifolds, Kleinian groups and hyperbolic geometry, Bull. Amer. Math. Soc. 6 (1982), 357-381, Question 23 (p. 380). Volume formula: G. Humbert, Sur la mesure des classes d'Hermite de discriminant donne dans un corps quadratique imaginaire, C. R. Acad. Sci. Paris 169 (1919), 448-454; J. Elstrodt, F. Grunewald, J. Mennicke, Groups Acting on Hyperbolic Space, Springer 1998, Chapter 7, Theorem 1.1 (covolume of the Bianchi group of an imaginary quadratic field F equals |delta_F|^{3/2} zeta_F(2) / (4 pi^2)); for F = Q(i) this equals Catalan's constant divided by 3.

View graph

A hyperbolic 333-manifold whose volume is a rational multiple of Catalan's constant

A hyperbolic $3$ -manifold whose volume is a rational multiple of Catalan's constant