refuted

Floor 3, Ce qui est su · introduced in Chapter 2, L’espace en creux

When is it proved that two sets are not one object?

The status of a bridge proved false: no seam exists for the markings in question. A type recurrence proved to be no identification.

Plate 3.3The stabilizer of the point 1 fixes it and no line: the lines through 1 and the lines missing it form orbits of three and four. The polarity matches lines with points only after twisting by an outer automorphism.

Definition(Refuted bridge)

A bridge is refuted when it is proved that its two sides are not incarnations of one object, that is, that no seam exists between them for the markings in question.

Type recurrence is not identification; a refuted bridge is a type recurrence proved to be no identification. A refutation may depend on the marking, or survive every marking.

Proposition(A type bridge that cannot be built)

Let $G=\GL(3,2)$ act on the points $\mathcal P$ and the lines $\mathcal L$ of the Fano plane $\mathrm{PG}(2,2)$ .

(a) Every $g\in G$ fixes as many points as lines, so $\mathcal P$ and $\mathcal L$ have the same permutation character, $1+\chi$ with $\chi$ irreducible of degree 6.

(b) $\mathcal P$ and $\mathcal L$ are not isomorphic $G$ -sets: the stabilizer of a point fixes one point and no line.

(c) The polarity $\pi\colon\mathcal L\to\mathcal P$ , sending the line $\{v: u\cdot v=0\}$ to the point $u$ , satisfies $\pi(gL)=\alpha(g)\pi(L)$ for the automorphism $\alpha(g)=(g^{\mathsf T})^{-1}$ , which is not inner.

Proof

Write $V=\F_2^3$ . (a) The points fixed by $g$ are the nonzero vectors of $\ker(g-1)$ , and the line $\{v:u\cdot v=0\}$ is fixed exactly when $u\in\ker(g^{\mathsf T}-1)$ ; as $g-1$ and its transpose have the same rank, both counts are $2^k-1$ . The character has norm 2, the number of orbits on $\mathcal P\times\mathcal P$ , because $G$ is 2-transitive on points.

(b) The stabilizer of $p$ is transitive on the three lines through $p$ and on the four lines not through $p$ , so it fixes no line; a $G$ -isomorphism $\mathcal P\to\mathcal L$ would carry $p$ to a line fixed by it.

(c) If $\alpha$ were inner, $\mathcal P$ marked through $\alpha$ would be isomorphic to $\mathcal P$ ; but $\pi$ is an isomorphism from $\mathcal L$ onto it, contradicting (b).

Proposition(The Gassmann pairs of PSL(2,7))

Two distinct objects of $\PSL(2,7)$ have the same permutation character exactly for the three pairs $(G/V_4^a,G/V_4^b)$ , $(G/A_4^a,G/A_4^b)$ and $(G/S_4^a,G/S_4^b)$ . Every bridge between the two members of such a pair, for one marking, is refuted.

Proof

The outer automorphism exchanges the two members of each pair and fixes every conjugacy class of elements except $7A$ and $7B$ . The groups $V_4$ , $A_4$ and $S_4$ contain no element of order 7, so the two members meet each conjugacy class equally often, which is the condition for equal permutation characters. Their marks at $V_4^a$ differ, so by Burnside’s theorem they are not isomorphic $G$ -sets, and no seam joins them.

Remark(The coincidence inside

M_{24}

)

The Mathieu group $M_{24}$ acts transitively on the pairs of an octad and a point off it, with stabilizer $A_8$ , and on ordered triples of points, with stabilizer $\PSL(3,4)$ ; both sets have $759\cdot16=12144=24\cdot23\cdot22$ elements. As a bridge between these two objects the coincidence $|A_8|=|\PSL(3,4)|$ has status type, one group and one size, and it is refuted for every choice of markings, because the stabilizers are not even isomorphic as abstract groups.

Example

The 28 ideal tetrahedra of Thurston’s congruence link complement have the size of the object of size 28 and the same group acts on them, but they form two orbits of 14, with stabilizers $A_4$ : the orbits of the cusp sets $\{0,1,2,5\}$ and $\{0,1,2,4\}$ , the rows $A_4^a$ and $A_4^b$ of the seam table. The type bridge between them and the object of size 28 is refuted. The 28 ideal edges are that object.

Remark

A refutation that depends on the marking is undone by a twist. The polarity is a seam over the outer automorphism $g\mapsto(g^{\mathsf T})^{-1}$ from the lines to the points, so the bridge between them is refuted over the identity and built over that automorphism. A seam over $\alpha$ from $X$ to $Y$ exists exactly when $\st(X)=\alpha^{-1}(\st(Y))$ , and inner automorphisms change nothing; so the bridges between the members of the three Gassmann pairs, refuted for a fixed marking, are built over the outer automorphism, while the bridge between the two $M_{24}$ -objects of size 12144 is refuted for every choice of markings.