Thurston’s Congruence Link

Klein’s quartic curve may be described as the Riemann surface obtained by taking the quotient of H by the (principal congruence) subgroup Γ(7) = ker {PSL2(Z) → PSL2(Z/7Z)}, and filling points in the cusps (punctures) to get a closed surface (although the punctured surface is sometimes also referred to as Klein’s quartic). It has a cell decomposition by 24 heptagons, centered at each cusp coming from the Epstein-Penner-Ford domain of H/Γ(7). Each heptagon is fixed by a rotation of order 7, which also preserves two other heptagons, giving a grouping of the heptagons into 8 classes which are preserved by the symmetries of the surface. Rotating one heptagon 1/7th of a turn corresponds to rotating one other 2/7ths, and the third 4/7ths. During a lecture at MSRI on Klein’s quartic commemorating the installation of Helaman Ferguson’s sculpture “8-fold way” [2], Thurston noticed that the group of symmetries preserving each class of heptagons is the same as the group of symmetries of the triangulation of the torus whose 1-skeleton is the complete graph on 7 vertices. Thurston wondered if there might be a way of relating these two symmetries, and found a hyperbolic 3-manifold with 8 cusps which gives such a relation. The cusps of S = H/Γ(7) correspond to the orbits of Γ(7) acting on Q̂ = Q ∪∞. These correspond to {±(a, b) ∈ Z| gcd(a, b, 7) = 1}(mod7). Clearly, there are (7 − 1)/2 = 24 such cusps, since they correspond to ±(a, b) ∈ (F7 − {(0, 0)})/{±1}, where F7 = Z/7Z. The matrices fixing ∞ = (1, 0) are the upper triangular matrices in SL2(Z) with trace ±2. These matrices also fix the cusps corresponding to the orbits of (2, 7) and (3, 7) by Γ(7) (remember, these are taken (mod7), so these correspond to (2, 0) and (3, 0)(mod7) respectively, which are clearly fixed by upper triangular matrices (mod7)). The matrices in