lift

Part III, Observers of Observers · defined in Chapter XIII, The Level-Seven Shadow

The quotient of velocity space by the integral Lorentz transformations of the report counts that become the identity modulo a prime above seven: Thurston’s congruence link complement. The finite sky is its shadow.

Plate W.24Thurston’s congruence link complement at the complex place: an ideal edge, an observer, in gold and an ideal tetrahedron, half of a clock cube, in blue over the Eisenstein lattice, with the eight cusps as

\Proj^1(\F_7)

and the cusp torus carrying the lines

x+\{1,2,4\}

The report counts of one observer form the Eisenstein lattice $\mathrm{Herm}_2(\Z[\omega])$ , so the Bianchi group $\SL(2,\Z[\omega])$ is their integral Lorentz group. Reduction modulo $\mathfrak p=(3+\omega)$ , a prime of norm seven, maps it onto $\SL(2,\F_7)$ with a torsion-free kernel $\Gamma(\mathfrak p)$ , and $M=\Gamma(\mathfrak p)\backslash\mathbb H^3$ is a hyperbolic three-manifold whose cells are the program’s objects: its eight cusps are the sky points, its twenty-eight edges the observers, and its twenty-eight regular ideal tetrahedra make up the seven clock cubes and the seven line cubes, two to a cube. Its isometry group is $\PGL(2,7)$ and every isometry preserves orientation, so the lift is chiral, and its mirror image is the lift at the conjugate prime.

Its parallel transport, reduced modulo $\mathfrak p$ , is the comparison between observers’ charts: a half-turn on every face, a third-turn on every shortest loop that space forgets, and Thomas precession on spinors. What the lift supplies is kinematics, frames, boosts, holonomy and the massless fields at its ends, every one of them fixed by its values at the eight cusps. It does not supply dynamics.

Theorem(the lift and its cells) computed

Reduction modulo $\mathfrak p=(3+\omega)$ maps $\SL(2,\Z[\omega])$ onto $\SL(2,\F_7)$ , with $\omega\mapsto4$ . Its kernel $\Gamma(\mathfrak p)$ is torsion-free, and the stabilizer of a cusp maps onto a Borel subgroup, of order 21 in $\PSL(2,7)$ . The quotient $M=\Gamma(\mathfrak p)\backslash\mathbb H^3$ is a hyperbolic three-manifold with eight cusps, one over each point of $\Proj^1(\F_7)$ . Its canonical decomposition consists of twenty-eight regular ideal tetrahedra, and its cells are determined by their cusps:

(a) the 8 cusps are the sky points; (b) the 28 edges are the pairs of sky points, each with stabilizer $S_3$ , the anchored observers; (c) the 56 triangles are the triples, each with stabilizer $C_3$ ; (d) the 28 tetrahedra form two disjoint Steiner systems $S(3,4,8)$ , the orbits of the base tetrahedra $(\infty,0,1,1+\omega)$ and $(\infty,0,1,-\omega)$ , the first the fourteen tetrahedra of the seven clock cubes and the second those of the seven line cubes. Every cusp torus is the complete graph $K_7$ with two Fano classes of faces, and its seven vertices are the seven observers through that sky point, one at each clock.

As mathematics

With $\mathcal O=\Z[\zeta]$ , $\zeta=(1+\sqrt{-3})/2$ , and $\mathfrak p=(2+\zeta)$ , the same prime as $(3+\omega)$ since $\zeta=1+\omega$ : $\mathcal O/\mathfrak p\cong\F_7$ , reduction induces $\PSL(2,\mathcal O)/\Gamma(\mathfrak p)\cong\PSL(2,7)$ , $\Gamma(\mathfrak p)$ is torsion-free, and $M$ is a hyperbolic three-manifold of finite volume with $\PSL(2,7)$ as its group of deck transformations over $\PSL(2,\mathcal O)\backslash\mathbb H^3$ . Thurston drew it as the complement of an eight-component link tessellated by 28 regular ideal tetrahedra, and Goerner proved that this complement is the principal congruence manifold of level $2+\zeta$ .

All fifteen classes of subgroups of the group of order 168 are stabilizers of configurations of its cells, among them a cusp ( $7{:}3$ ), an edge ( $S_3$ ), a face ( $C_3$ ), a tetrahedron of either class ( $A_4^a$ , $A_4^b$ ), a complementary pair of tetrahedra ( $S_4^a$ , $S_4^b$ ), and $M$ itself. The tetrahedra of one class are the blocks of a Steiner system $S(3,4,8)$ on the cusps, and a point lies on a line exactly when their tetrahedra share no face, so both lives of the group are seen in one manifold. The bridge from the twenty-eight tetrahedra to the twenty-eight observers is refuted; the edges are the observers.

Its name in another field	Bridge
Thurston’s eight-component congruence link complement	built
the principal congruence Bianchi manifold $\Gamma(\mathfrak p)\backslash\mathbb H^3$ of level $2+\zeta$	built
a link complement in $S^3$ tessellated by 28 regular ideal tetrahedra	classical
the geometry the finite sky is the shadow of	a reading

A second sense

The volume also uses lift in the covering sense, sometimes in one paragraph with the manifold: spin lifts, Clifford lifts, the lift of a rotation to the double cover, the seven clocks’ lifts that generate $\SL(2,7)$ . Seams writes the manifold as $M$ and keeps lift for lifts to a cover, such as the odd lifts of subgroups of $\PSL(2,7)$ to $\SL(2,7)$ .