double life

Floor 4, Les doubles vies · introduced in Chapter 6, Quatre groupes à double vie

Which groups carry the geometries of two families at once?

Two lives of one group in different members of the families; by Artin’s absence exactly four groups have one: $A_5$ , $\PSL(2,7)$ , $A_6$ and $A_8$ .

Plate 4.2The four double lives, at orders 60, 168, 360 and 20160. In three of them an outer automorphism exchanges a dual pair in one life and is unremarkable in the other.

Definition(Life, double life, dictionary)

A double life of $G$ is a pair of lives $\mu_1\colon G\to\Gamma_1$ and $\mu_2\colon G\to\Gamma_2$ in different members of the families. It is built when the isomorphism $\mu_2\mu_1^{-1}\colon\Gamma_1\to\Gamma_2$ is given explicitly.

The word is meant plainly: the same group lives as the symmetry group of two different geometries.

Proposition

The groups with a double life are, up to isomorphism, exactly four: $A_5$ , with $A_5$ on five letters, $\PSL(2,4)$ on $\Proj^1(\F_4)$ and $\PSL(2,5)$ on $\Proj^1(\F_5)$ ; $\PSL(2,7)$ , with $\PSL(2,7)$ on $\Proj^1(\F_7)$ and $\PSL(3,2)$ on $\mathrm{PG}(2,2)$ ; $A_6$ , with $A_6$ on six letters and $\PSL(2,9)$ on $\Proj^1(\F_9)$ ; and $A_8$ , with $A_8$ on eight letters and $\PSL(4,2)$ on $\mathrm{PG}(3,2)$ .

So the double lives are the terminal imprint of Artin’s absence, read as seams.

Proof

Isomorphic groups have equal orders, so by Artin’s theorem a double life can only occur among the groups of orders 60, 168, 360 and 20160 that he lists, and $\PSL(3,4)$ is excluded because it is isomorphic to neither $\PSL(4,2)$ nor $A_8$ . Each remaining pair is shown to be a double life by an explicit isomorphism.

Remark(The pattern)

In three of the four double lives an outer automorphism of the group exchanges two objects of the same size that one life sees as dual: for $\PSL(2,7)$ the points and the lines of the Fano plane, for $A_8$ the points and the planes of $\mathrm{PG}(3,2)$ , and for $A_6$ the six letters and the second six-point action of $\PSL(2,9)$ . The other life sees the same automorphism as unremarkable: a Möbius map of non-square determinant on $\Proj^1(\F_7)$ or $\Proj^1(\F_9)$ , an odd permutation of the eight letters. A bridge refuted for one marking becomes a seam after twisting by the outer automorphism. The double life of $A_5$ has no such pair: every conjugacy class of its subgroups is invariant under $S_5$ .

Propositionproved

The type law explains which double lives carry a dual pair exchanged by the outer automorphism. For $\PSL(2,7)$ and $A_6$ that automorphism is the complex conjugation of the CM field of Klein’s, respectively the Valentiner, lattice: the prime of the life that sees two dual objects splits in $K/K^+$ , and the prime of the life that sees a Möbius map of non-square determinant ramifies. For $A_5$ the character field is totally real, so no Galois element acts as a duality, and this is why $A_5$ has no dual pair. The dual pair of $A_8$ , the points and planes of $\mathrm{PG}(3,2)$ , lies outside the law.

Example

At 168 the double life is built twice: from the line, because Galois’s seven-point action carries a Fano plane, and from the plane, through its eight Singer subgroups. Its best-known dictionary entry is the object of size 28: the antiflags of the plane are the pairs of points of the line.

Remark(Both lives in one place)

The double life at 168 can be assembled in single structures. In Thurston’s congruence link complement the line life is on the eight cusps and the plane life on the code of the tetrahedra of one class, on which the group acts as $\GL(3,2)$ . And in the unitary group of Mumford’s hermitian form over $\Z[1/14]$ , acting on the product of the building of $\PGL(3,\Q_2)$ and a tree at 7, the stabilizer of the vertex of Klein’s lattice is a group $\PSL(2,7)$ that acts on the link at 2, the incidence graph of the Fano plane, through $\GL(3,2)$ , and on the link at 7 as $\PSL(2,7)$ on $\Proj^1(\F_7)$ .

Corollary(One finite group, two arithmetic parents)

The group of order 168 is a congruence quotient, at primes over 7, of two arithmetic groups: of $\PSL(2,\mathcal O)$ , $\mathcal O=\Z[\zeta]$ , $\zeta=(1+\sqrt{-3})/2$ , by $\Gamma(\mathfrak p)$ , $\mathfrak p=(2+\zeta)$ , a lattice in $\PSL(2,\C)$ ; and of Mumford’s group $\Gamma_1$ , by reduction modulo $\sqrt{-7}$ , a lattice in $\PGL(3,\Q_2)$ made of rational points of a group whose adjoint group over $\R$ is compact. Both reductions give the line life of the group, and in $\Gamma_1$ the plane life is present as well, as the vertex link of its building at 2.

This is not a double life, since both parents reduce to the same life. It is one finite life with two arithmetic parents of opposite real type, and at the prime 7 the two parents share the sky but not its completion.

Remark(The shape of a double life)

Comparing the seam table of the icosahedral group $A_5$ , whose lives are $\Proj^1(\F_5)$ , $\Proj^1(\F_4)$ and the five letters, with that of the group of order 168 separates what a double life has from what is specific to 7. In each life the cyclic rows are tori or unipotent groups, and a prime that is the characteristic of one life gives a unipotent row in that life and a toral row in the other. A non-split torus row is incarnated by imaginary points, with the Frobenius as automorphism. The Weil representation of the double cover holds a geometric representation of dimension 3 of the group, Klein’s for 168 and the icosahedral rotations for 60, and a faithful half. And the new objects of the double cover lie exactly over the rows of odd order.

Specific to 7: the outer automorphism moves classes, so the labels $a$ and $b$ , the three Gassmann pairs and the refuted bridge between points and lines have no counterpart for $A_5$ ; since $-1$ is not a square modulo 7, the square classes over the points form a new object; and there is no spinor of dimension 2, so the faithful half has dimension 4, monomial on sixteen vectors and carrying a lattice $E_8$ .