continuum

Let $X$ be an object of a finite group $G$. A continuum of $X$ is a homogeneous space $Y$ of a Lie group $L$ together with one of the following. An embedding: an injective homomorphism $\mu\colon G\to L$ and a $G$-equivariant injective map $X\to Y$. An arithmetic realization: a discrete subgroup $\Lambda\le L$, a normal subgroup $\Lambda'$ of $\Lambda$ with $\Lambda/\Lambda'\cong G$, and a $\Lambda$-invariant subset $Z\subseteq Y$ with $\Lambda'\backslash Z\cong X$ as $G$-sets.

The word records that the finite object reappears inside a continuous geometry. Which kind of continuum an object can have is decided by absences.

Let $\zeta=(1+\sqrt{-3})/2$, $\mathcal O=\Z[\zeta]$, $\mathfrak p=(2+\zeta)$, of norm 7, and let $\Gamma(\mathfrak p)$ be the kernel of reduction $\PSL(2,\mathcal O)\to\PSL(2,\mathcal O/\mathfrak p)$.

(1) $\mathcal O/\mathfrak p\cong\F_7$, and $\PSL(2,\mathcal O)/\Gamma(\mathfrak p)\cong\PSL(2,7)$. (2) $\Gamma(\mathfrak p)$ is torsion-free, so $M=\Gamma(\mathfrak p)\backslash\mathbb H^3$ is a hyperbolic 3-manifold of finite volume, on which $\PSL(2,7)$ acts as the group of deck transformations over $\PSL(2,\mathcal O)\backslash\mathbb H^3$. (3) The cusps of $M$ correspond, equivariantly, to the eight points of $\Proj^1(\F_7)$. (4) $M$ is the complement of an eight-component link in the 3-sphere, tessellated by 28 regular ideal tetrahedra (Thurston; Goerner).

The object $\Proj^1(\F_7)$ of $\PSL(2,7)$ has no embedded continuum in $\Proj^1(\C)$ under $\PSL(2,\C)$, and none in Klein’s plane $\Proj^2(\C)$ under Klein’s representation. It has the arithmetic continuum of the cusps of $M$, with $Y=\Proj^1(\C)$, $L=\PSL(2,\C)$, $\Lambda=\PSL(2,\mathcal O)$, $\Lambda'=\Gamma(\mathfrak p)$ and $Z=\Proj^1(K)$.

Each pair of cusps of $M$ is joined by exactly one ideal edge, each triple spans exactly one ideal face, and an ideal tetrahedron is determined by its four cusps: there are 28 edges, 56 faces and 28 tetrahedra. Under $\PSL(2,7)$ the edges form the object of size 28, with stabilizer $S_3$; the faces form the object of size 56, with stabilizer $C_3$; and the tetrahedra fall into two orbits of 14, the rows $A_4^a$ and $A_4^b$ of the seam table, exchanged by $\PGL(2,7)$.

Every conjugacy class of subgroups of $G=\PSL(2,7)$ is the stabilizer class of an orbit of configurations of cells of $M$: for 1, a cusp with a face through it; $C_2$, an edge with a tetrahedron through it (two orbits, one for each class); $C_3$, a face; $C_4$, four cusps that are not the cusps of a tetrahedron; $V_4^a$ and $V_4^b$, a tetrahedron of class $a$ or $b$ with a pair of opposite edges; $S_3$, an edge; $C_7$, a cusp with one of the three classes of parallel edges of its cusp torus; $D_8$, a partition of the cusps into two fours, neither the cusps of a tetrahedron; $A_4^a$ and $A_4^b$, a tetrahedron of class $a$ or $b$; $7{:}3$, a cusp; $S_4^a$ and $S_4^b$, a complementary pair of tetrahedra of class $a$ or $b$; and $G$, the manifold $M$. A tetrahedron is of class $a$ when its cusp set lies in the orbit of $\{0,1,2,5\}$, and the tetrahedron on the complementary cusps has the same class.

Four cusps that are not the cusps of a tetrahedron contain exactly one pair of disjoint edges with harmonic ends, one edge of the Coxeter graph; the cusps of a tetrahedron contain none. The stabilizer of a point of $M$ is trivial or lies in one of the classes $C_2$, $C_3$, $S_3$, $A_4^a$, $A_4^b$.

Let $\mathcal A$ and $\mathcal B$ be Steiner systems $S(3,4,8)$ on a set $X$ of eight points with no block in common. (1) The blocks of $\mathcal A$ with $\emptyset$ and $X$ form a self-dual binary code of dimension 4; the complement of a block is a block, and the nonzero elements of $V_{\mathcal A}$, the code modulo $\{\emptyset,X\}$, are the seven complementary pairs of blocks. The same holds for $\mathcal B$. (2) For complementary pairs $P$ of $\mathcal A$ and $Q$ of $\mathcal B$, either every block of $P$ meets every block of $Q$ in two points, or exactly two of the four pairs of blocks share three points. (3) The pairing $V_{\mathcal A}\times V_{\mathcal B}\to\F_2$, $(A,B)\mapsto|A\cap B|\bmod2$, is non-degenerate, and $P$ and $Q$ are orthogonal exactly in the first case of (2). So, calling the pairs of $\mathcal A$ points and those of $\mathcal B$ lines, the first case is the incidence of the Fano plane $\Proj(V_{\mathcal A})$.

Each face of $M$ lies in exactly one tetrahedron of each class, so the two classes of tetrahedra are two such Steiner systems on the cusps, and a point lies on a line exactly when no tetrahedron of the one shares a face with a tetrahedron of the other. The group $G$ acts linearly and faithfully on the code space $V_a$ of class $a$, so $G\cong\GL(V_a)\cong\GL(3,2)$: both lives of the group are visible in the cells of $M$, the line life on the cusps and the plane life on the code.

$F_4$ is the automorphism group of the Albert algebra $\mathfrak h_3(\Oct)$, and $\Spin(9)$ the stabilizer of a primitive idempotent, a point of the Cayley plane; by the classification of Borel and de Siebenthal $F_4$ also has a maximal subgroup $(\SU(3)\times\SU(3))/\Z_3$ of full rank. Inside $F_4$,

It was checked on Lie algebras in one realization: with $\C=\R1+\R e_1\subset\Oct$, the derivations preserving $\mathfrak h_3(\C)$ form $\mathfrak s\cong\mathfrak{su}(3)\oplus\mathfrak{su}(3)$, with an ideal $\mathfrak c$ of dimension 8 vanishing on $\mathfrak h_3(\C)$, and $\mathfrak s\cap\mathfrak{spin}(9)$ has dimension 12, is isomorphic to $\mathfrak{su}(3)\oplus\mathfrak{su}(2)\oplus\mathfrak u(1)$ and contains $\mathfrak c$.

Let $p=E_{11}$ and let $V=\{X\in\mathfrak h_3(\Oct):p\circ X=\tfrac12X\}\cong\Oct^2$, through the entries $(x_{12},x_{13})$, with the complex structures $J_{\mathrm{row}}(x_{12},x_{13})=(e_1x_{12},e_1x_{13})$ and $J_{\mathrm{col}}(x_{12},x_{13})=(x_{12}e_1,x_{13}e_1)$. (1) $D\mapsto D(p)$ induces an equivariant isomorphism from $\mathfrak f_4/\mathfrak{spin}(9)$ onto $V$. (2) The commutant of $J_{\mathrm{row}}$ in $\mathfrak{spin}(9)$ is $\mathfrak{spin}(9)\cap\mathfrak s\cong\mathfrak{su}(3)\oplus\mathfrak{su}(2)\oplus\mathfrak u(1)$. (3) The commutant of $J_{\mathrm{col}}$ has dimension 18; it is the stabilizer of a splitting $\R^9=\R^3\oplus\R^6$ of the traceless part of the lower block, so it is $\mathfrak{so}(3)\oplus\mathfrak{so}(6)\cong\mathfrak{su}(2)\oplus\mathfrak{su}(4)$, and it contains the algebra of (2).

Let $\omega$ fix $\C$ and act on its complement, a complex 3-space, as the scalar $e^{2\pi e_1/3}$. It generates the centre of the $\SU(3)$ of automorphisms of $\Oct$ fixing $e_1$, the derivations commuting with it form $\mathfrak s$, and

So one point of the Cayley plane and one imaginary unit carry the intersection of Todorov and Dubois-Violette.

In the complexified algebra $\mathfrak h_3(\Oct)\otimes\C$, whose compact symmetry algebra is $\mathfrak e_6=\mathfrak f_4\oplus i\,L(J_0)$, the stabilizer of $E_{11}$ is $\mathfrak{so}(10)$, and $\C^{27}$ splits under it as $\mathbf 1\oplus\mathbf{16}\oplus\mathbf{10}$, the $\mathbf{16}$ being the complexification of $V$. The elements commuting with $\omega$ form $\mathfrak e_6^\omega\cong\mathfrak{su}(3)^{\oplus3}$, and the commutant of $J_{\mathrm{row}}$ in $\mathfrak{so}(10)$ is $\mathfrak{so}(10)\cap\mathfrak e_6^\omega\cong\mathfrak{su}(3)\oplus\mathfrak{su}(2)\oplus\mathfrak{su}(2)\oplus\mathfrak u(1)$, of dimension 15, whose intersection with $\mathfrak f_4$ is $\mathfrak{spin}(9)\cap\mathfrak f_4^\omega$, with its $\mathfrak{su}(2)$ diagonal in the two.

The copy of $\mathfrak{su}(3)\oplus\mathfrak{su}(2)\oplus\mathfrak u(1)$ in $\mathfrak f_4$ acts on $V\otimes\C$ as the complexification of a real representation, so this representation is self-conjugate. A second copy, the stabilizer of a vector killed by $\mathfrak c$ and by one of the two ideals $\mathfrak{su}(2)$, has dimension 12 and is not contained in $\mathfrak f_4$; its centre has eigenvalues on $V\otimes\C$ that are not symmetric under sign change, so its representation is not self-conjugate.

Let $\Lambda=\mathrm{Herm}_2(\Z[\omega])$ with the form $\det$ and its polarization $\langle\,,\rangle$, and let $\Gamma=\SL(2,\Z[\omega])$ act by $X\mapsto gXg^\dagger$. The ideal tetrahedron $T_0$ with cusps $\infty$, 0, 1, $1+\omega$ has reports $N_a=k_ak_a^\dagger$, for $k_\infty=(1,0)$, $k_0=(0,1)$, $k_1=(1,1)$ and $k_{1+\omega}=(1+\omega,1)$, records $T_x=N_a+N_b$, one for each pair $x=\{a,b\}$ of its cusps, with $x^c$ the complementary pair, and centre $D=\sum_aN_a$.

The four reports form a $\Z$-basis of $\Lambda$. The six records have $\det T_x=1$ and span $\Lambda_{\mathrm{even}}=\{\sum n_aN_a:\sum n_a\ \text{even}\}$, of index 2. Moreover $\langle N_a,N_b\rangle=\tfrac12$ for $a\ne b$ and $\det D=6$. In the rest frame of $D$ every record has time component $\sqrt{3/2}$ and spatial part $S_x=T_x-\tfrac12D$, with $\langle S_x,S_x\rangle=-\tfrac12$, $S_{x^c}=-S_x$ and $\langle S_x,S_y\rangle=0$ otherwise: one record per tick, spatial steps of length $1/\sqrt2$ along three orthogonal axes, speed $1/\sqrt3$.

(1) Every nonzero $X\in\Lambda$ has an infinite $\Gamma$-orbit, so no locally finite bond set on $\Lambda$ is invariant under translations and $\Gamma$. (2) The stabilizer in $\Gamma$ of a finite bond set spanning $\Lambda\otimes\R$ is finite and fixes a future timelike vector: every mesh on $\Lambda$ selects a rest frame. (3) For the reports of $T_0$ this stabilizer is the binary tetrahedral group $2T$, the 24 elements of $\Gamma$ that induce the 12 even permutations of the reports and fix $D$.

(1) For every $b\ge2$, $b\Lambda_{\mathrm{even}}\subset\Lambda_{\mathrm{even}}$ with index $b^4$. A coarse bond $bT_x$ is a sum of $b$ records in exactly one way, and a coarse rhombus $b(T_x+T_y)$, for $x,y$ not complementary, is tiled by $b^2$ fine ones; so under the homothety the fluxes of a constant field scale by $b^2$ and the density of sites by $b^{-4}$, and the coarse coupling equals the fine one exactly. (2) The four sublattices $g\Lambda g^\dagger$ with $\det g=1-\omega$ have index 9 and are permuted by $\Gamma$ through $\SL(2,\F_3)$ acting on $\Proj^1(\F_3)$; their bonds have determinant 3 and are not sums of records, so they do not subdivide. (3) $\{X\equiv0\bmod(1-\omega)\}$ has index 27 and is $\Gamma$-invariant, and with the form $\det/3$ it has discriminant $-27$, against $-3$ for $\Lambda$, so it is not similar to $\Lambda$.

The bond graph of $\Lambda_{\mathrm{even}}$ has 33 four-cycles per site up to translation: 15 rhombi, one for each pair of records; 12 matching squares, $T_x$ then $T_{x^c}$ against $T_y$ then $T_{y^c}$; and 6 zig-zag squares $+T_x,-T_y,+T_{x^c},-T_{y^c}$. Together they span the cycle space; the rhombi alone do not.

For a constant field strength $F=(E,B)$ the sum over the smallest loops at a site of the squared fluxes is $\alpha|E|^2+\gamma|B|^2$, isotropic and without an $E\cdot B$ term, with $(\alpha,\gamma)=(15,\tfrac32)$ on all 33. So with one weight on all the smallest loops, waves of the field move at $1/\sqrt{10}$, while records move at $1/\sqrt3$. With separate weights on the four kinds of loop the speed is 1 exactly on one hyperplane of weights, and neither the Minkowski form nor the metric in which the reports are orthonormal gives the smallest loops a circumcentric dual, so no discrete Hodge star fixes the weights.

(1) Every record $T=gg^\dagger$, $g\in\Gamma$, is a sum of two primitive null vectors of $\Lambda$ in exactly one way. (2) Their cusps reduce modulo $\mathfrak p=(3+\omega)$ to two distinct points of $\Proj^1(\F_7)$, so $T$ names a pair $o(T)$; the map satisfies $o(gTg^\dagger)=\bar g\cdot o(T)$, is constant on $\Gamma(\mathfrak p)$-classes and is onto the 28 pairs. (3) The records form $336/12=28$ classes modulo $\Gamma(\mathfrak p)$, and $o$ is a bijection from them onto the pairs. So a pair of points of the finite line is a class of rest frames: an edge of the tessellation, the geodesic of $\mathbb H^3$ joining two cusps. (4) The six records of $T_0$ name six pairs that are pairwise not adjacent in the Coxeter graph, whose antiflags have six distinct points.

The binary tetrahedral group $2T$ of $T_0$ reduces injectively modulo $\mathfrak p$, and its orbits on the 28 pairs have sizes 4, 6, 6 and 12. So no map from $\Lambda_{\mathrm{even}}$ to the pairs is both invariant under translations and equivariant under $2T$: translations do not change rest frames, so such a map is constant, and a constant equivariant map is a fixed pair. And since the Coxeter graph has girth 7, any map of the bonds of $\Lambda_{\mathrm{even}}$ to its edges or to its vertices pulls its comparisons back to a connection that is flat on every smallest loop.