seam system

A seam system for an object $X$ is a family of incarnations of $X$ together with a set $E$ of seams between members of the family; a seam from an incarnation to itself is allowed. A cycle is a closed walk $\gamma=(s_1^{\epsilon_1},\dots,s_m^{\epsilon_m})$ with $s_k\in E$ and $\epsilon_k=\pm1$, starting and ending at one incarnation.

For a non-rigid object any automorphism is the monodromy of some cycle, so the notion has content only for seams given by the theories rather than chosen. Such seams are called natural, without formalizing the word, and for each one the construction that defines it is recorded.

On the object of size 24 the following maps are seams.

(1) Inversion $7A\to7B$, $y\mapsto y^{-1}$.

(2) Transvection: a vector $\pm v$ of $\F_7^2$ goes to the transvection $w\mapsto w+\det(v,w)\,v$. It maps onto $7A$, sending $\pm(1,0)$ to $g\colon z\mapsto z+1$.

(3) Rotation: a flex $P$ of the Klein quartic goes to the element of its stabilizer that acts on the tangent line $T_PX$ by $\zeta=e^{2\pi i/7}$. It maps onto $7A$.

(4) Tangent: a flex goes to its tangent line. Residual point: a flex tangent meets the quartic at its flex with multiplicity 3 and at exactly one other point, again a flex, and the tangent goes to that point.

(5) Singer: a cyclic labelling $\ell$ of the Fano plane goes to the collineation $\ell^{-1}\circ(x\mapsto x+1)\circ\ell$. For the marking fixed in the book the labellings for $\{0,1,3\}$ go onto $7B$ and those for $\{0,4,6\}$ onto $7A$. Reversal, $\ell\mapsto-\ell$, goes from the first kind to the second.

(6) Roles: for $d\in\{0,1,3\}$, a labelling $\ell$ for $\{0,1,3\}$ goes to the perfect matching of the Heawood graph that joins each point $v$ to the line in which $v$ plays the role $d$, namely $\ell^{-1}(\ell(v)-d+\{0,1,3\})$.

(7) One step: a heptagon of the Coxeter graph goes to the element of $7A$ that rotates it by one step.

On each of the two trees at 7, the Bianchi group’s and Mumford’s, the link $\mathrm{Lk}(x)$ of a vertex is an incarnation of the projective line over $\F_7$. Let $K_x$ be the elements of the vertex group of $x$ that act trivially on $\mathrm{Lk}(x)$. For a step $x\to y$ the mixed-square group $M(x\to y)$ is the image of $K_x$ on $\mathrm{Lk}(y)$: the holonomy of a square made of a loop at $x$ that $x$ cannot see, followed by the step.

(1) On Mumford’s tree, the kernel of $\Gamma_1$ on the link of $\Lambda_0$, of order 98, acts on the link of each neighbour through a dihedral group of order 14: seven translations and seven involutions outside $\PSL(2,7)$, one of them the central element of $\mathrm{Sp}(N)$. (2) On the Bianchi group’s tree, the kernel of $\PSL(2,\Z/49)$ on the link acts on the new neighbours of each neighbour through a cyclic group of order 7, by even permutations.

Let $X_7$ be the object of size 7 with stabilizer $S_4^a$, the points of the Fano plane together with its lines, and $X_7'$ the object with stabilizer $S_4^b$; both are rigid, and the outer automorphism exchanges them. Place at each vertex of a tree an incarnation of $X_7$, and carry it across each step.

(1) On the Bianchi group’s tree the seam system is forced: every mixed square twists the incarnation by an element of $\PSL(2,7)$, the twisted incarnation is again one of $X_7$, and since $X_7$ is rigid it is joined to the untwisted one by exactly one seam. So the seam system exists and is unique; its gauge group $N(S_4^a)/S_4^a$ is trivial, and the squares act by automorphisms of $G$ that are inner. (2) On Mumford’s tree there is none: the central element of $\mathrm{Sp}(N)$ twists the incarnation at the next vertex by an involution outside $\PSL(2,7)$, which carries the seven groups $S_4^a$ onto the seven groups $S_4^b$ and the octonion table $x+\{0,1,3\}$ onto its mirror $x+\{0,4,6\}$; the twisted incarnation is one of $X_7'$, and no seam joins it to the untwisted one. (3) Along Mumford’s building, at a vertex of the type of $\Lambda_0$, every element acts on the link through $\PSL(2,7)$, so the incarnation is carried; along a slice through Klein vertices every move from one Klein vertex to another at distance two acts improperly: the groups $S_4^a$ of the second are the groups $S_4^b$ of the first, and labelled by the Singer cycle one carries the table and the other its mirror.

A choice of one parent for each vertex is the remaining datum of a seam system along a tree. Let every vertex of a locally finite infinite tree choose one neighbour, its parent, so that every edge is chosen by at least one of its endpoints. Then either exactly one edge is chosen by both of its endpoints, and every path of parents ends by oscillating on that edge, or no edge is, and all paths of parents run to one common end. The symmetries of the choice fix that edge, or that end; so no such choice is invariant under a group that fixes no vertex, edge or end, and none is invariant under either parent.