The sky

The object of size 8, stabilizer $7{:}3$ , one class of 8 subgroups · rigid · in the program, the sky

Stabilizer 7:3, rigid: the points of the projective line over $\F_7$ , the Sylow 7-subgroups, the flex triangles and the cyclic orientations of the Fano plane.

The sky: the eight points of

\Proj^1(\F_7)

, each matched with a Singer subgroup of the Fano plane by one of the 336 bijections that carry the Möbius action onto conjugation.

Its incarnations

The seam table names the object in each of the five theories that meet the group. Every entry there is built.

Fano plane: cyclic orientations, for $\{0,1,3\}$ and for $\{0,4,6\}$
Projective line: points
The group: subgroups $C_7$ , $7{:}3$
Klein quartic: flex triangles
Graphs: triples of Coxeter heptagons

The stabilizer is its own normalizer, so between any two incarnations there is exactly one seam, and the seams agree.

The fifteen objects

The object on the line of sizes. In ink, the objects it maps onto; in blue, those that map onto it. Dots count each object’s automorphisms, and dashed brackets join the pairs with one permutation character.

Proposition

$7{:}3$ , the normalizer of a Sylow 7-subgroup, is self-normalizing and the only class of subgroups of order 21, so the object is rigid and its class is fixed by $\Aut(G)$ . Exactly 336 bijections from the eight Sylow 7-subgroups of $\GL(3,2)$ onto $\Proj^1(\F_7)$ carry conjugation onto the Möbius action: the projective line is, in the plane’s own terms, the set of its eight Singer subgroups.

Example

The points of $\Proj^1(\F_7)$ ; the subgroups $C_7$ and $7{:}3$ ; the eight cyclic orientations of the Fano plane, for $\{0,1,3\}$ and for $\{0,4,6\}$ ; the flex triangles of the Klein quartic; the triples of Coxeter heptagons; the eight lattices $\Z^8+\tfrac12C$ whose Fano plane shares no line with the octonion table; the eight cusps of Thurston’s congruence link complement; and the neighbours of a vertex of the tree of $\PGL(2,\Q_7)$ .

Proposition(The antiflags of a triangle presentation)

On $\Z/7$ with lines $x+\{1,2,4\}$ , let $\lambda(x)=x+\{1,2,4\}$ and let $P_7$ be the group of translations. Each $(x,\lambda(x))$ is an antiflag, and the seven form one orbit of $P_7$ . Of the four orbits of $P_7$ on the 28 antiflags, it is the only one stable under the multipliers $x\mapsto2x$ and $x\mapsto4x$ , that is, under $N(P_7)\cong7{:}3$ . So each Sylow 7-subgroup determines a distinguished set of seven antiflags, and sending it to that set is a seam onto the eight $\lambda$ -orbits.

The antiflag of the object of size 28 and the Singer cycle of the object of size 24 meet in the triangle presentation of the octonion table: $\lambda$ chooses one of four orbits for each Sylow 7-subgroup, and the multiplier group $N(P)/P\cong C_3$ , which acts on the object of size 24 by the powers, is exactly what makes the choice canonical.

Proposition(The line life has only an arithmetic continuum in low dimension)

It has no embedded continuum in $\Proj^1(\C)$ , by Klein’s list, nor in Klein’s plane, where the subgroup of order 21 fixes no point. It has the arithmetic continuum of the cusps of Thurston’s link complement.

Proposition(The eight lattices) computed

For each Sylow 7-subgroup $P$ , exactly two of the thirty Fano planes on the octonion units are invariant under $P$ : the plane $\Pi$ of the table and its mirror image $\Pi_P$ , which shares no line with it, and $P\mapsto L_{\Pi_P}$ is the seam onto the eight lattices. Through $A_8\cong\PSL(4,2)$ the thirty lattices are the fifteen points and fifteen planes of $\mathrm{PG}(3,2)$ : up to duality Kirmse’s lattice is a point $P_0$ , the octavian orders are the planes through it, the fourteen lattices sharing one line are the other points, and the eight are the planes not through $P_0$ . The eight lattices are the eight cusps of Thurston’s manifold.

Remark

Over it the double cover adds its smallest new object, the sixteen square classes $\{v,2v,4v\}$ of nonzero vectors, two over each point; in the Weil representation they are the sixteen vectors $\pm v_c$ of its faithful half, whose Gram matrix is $I+C/\sqrt{-7}$ with $C$ a skew conference matrix, after sign changes $\pm$ the Paley matrix. And the sky is where the group’s two arithmetic parents meet: the Bianchi group and Mumford’s group both reduce onto it at a prime over 7, as the link of a vertex of a tree, and the seam between the two links is unique because the object is rigid, while the trees with their symmetries differ.

Examplecomputed

At Klein’s lattice the eight points of the conic modulo $\sqrt{-7}$ are the eight neighbours of the lattice in the tree at 7; the stabilizer of each point of the Fano plane moves them as the rotations of its cube move the vertices. One neighbour is $\mathcal O_L$ , with stabilizer the Frobenius group of order 21, and labelling the points of the plane along the Singer cycle by $\Z/7$ makes the lines the translates $x+\{0,1,3\}$ .

Concepts: objectincarnationnew objectrigid objectbridgestatuskernelreductionlifedouble lifetype lawdictionarycompletionorientationcontinuumspinor systemseam theory
In the Esquisse: 3La table des sutures du groupe d’ordre 1684La monodromie des sutures5Courte marche à travers la théorie de Galois6Quatre groupes à double vie7La trinité de Galois8La famille de Weyl9Immeubles et réseaux10La table en deux, en sept et à l’infini11Le revêtement double et le miroir12Où se rencontrent les deux parents13Orientation et charge14Les continus15La tour assemblée18Exceptionnel veut dire relevable19Une formule du produitÉp.L’horizon : dessins d’enfants
The volume’s word: cuspliftsky
In the volume: IVWorld, Kernel, ObserverVIIISpace as a TallyIXWhat Space ForgetsXThe Finite Celestial SphereXIRulial RelativityXIIThe Coxeter GraphXIIIThe Level-Seven ShadowXIVWhy OctonionsXVSpin from the Double CoverXVIIITwo Parents of the SkyXXThe Commuting SquaresXXIIOne SpeedXXIIILight, Vacuum and HandednessXXIVA Number Nature Could RefuteEp.Forcing, Not Sacred Geometry