cusp

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

Also light direction

One of the lift’s eight ends. They are the sky’s eight points, the directions from which light can arrive.

Plate W.23The cusp torus at

\infty

, the plane modulo the lattice

\mathfrak p

: the seven edges ending at this cusp are the seven observers through the sky point

\infty

, one at each clock, and the corners of clock cubes and line cubes triangulate it as

K_7

A cusp is an end of the lift that runs off to a point at infinity. Near it the manifold looks like a torus times a half-line, and that torus is the cusp torus. The stabilizer of a cusp in the Bianchi group maps onto a Borel subgroup, of order 21 in $\PSL(2,7)$ , and the eight cusps lie one over each point of $\Proj^1(\F_7)$ . On the celestial sphere a cusp is a light direction. Each edge of the lift joins two cusps and is the observer whose pair they are, and the seven edges that end at one cusp cut its torus into the complete graph $K_7$ : the seven observers through that sky point, one at each clock.

Every massless field the lift carries is fixed by its values at the eight cusps, so the light directions decide every coupling: the massless fields scatter between them by finite tables that symmetry fixes up to one constant each, and the observers, the edges joining two light directions, are the channels of that scattering.

As mathematics

A cusp of $M=\Gamma(\mathfrak p)\backslash\mathbb H^3$ is a $\Gamma(\mathfrak p)$ -orbit on $\Proj^1(\Q(\sqrt{-3}))$ . The cusps form the $\PSL(2,7)$ -set $\PSL(2,7)/\bar B=\Proj^1(\F_7)$ , with $\bar B$ the image of the stabilizer of $\infty$ , of order 21. The cross-section at $\infty$ is $\C/\mathfrak p$ , triangulated by the seven edges and fourteen tetrahedra at $\infty$ into $K_7$ ; its triangles $a+\{0,4,5\}$ are the lines $x+\{1,2,4\}$ of the octonion triangle presentation, and $a+\{0,1,5\}$ their mirror image.

The same eight points appear at the prime $\mathfrak p$ itself: the cusps of $M$ are identified, compatibly with $\PSL(2,7)$ , with the eight neighbours of a vertex of the tree of $\PGL(2,\Q_7)$ . And the sphere at infinity of $\mathbb H^3$ is the celestial sphere $\Proj^1(\C)$ , the Bloch sphere of a qubit, so the eight cusps are eight classes of points of the celestial sphere.

Its name in another field	Bridge
a point of $\Proj^1(\F_7)$	built
a Sylow 7-subgroup of $\PSL(2,7)$	built
a neighbour of a vertex of the tree of $\PGL(2,\Q_7)$	built
a class of future null rays on the celestial sphere $\Proj^1(\C)$	classical
a direction from which light can arrive	a reading

Built from: liftsky
In the dictionary: continuumcompletionthe sky
In the volume: XIRulial RelativityXIIIThe Level-Seven ShadowXVSpin from the Double CoverXVIIITwo Parents of the SkyXXThe Commuting SquaresXXIIOne SpeedXXIIILight, Vacuum and Handedness