completion

Let $K$ be a field complete for a discrete valuation, with valuation ring $\mathcal O$, uniformizer $\pi$ and finite residue field $k$. The building of $\PGL(n,K)$ has as vertices the homothety classes $[L]$ of lattices in $K^n$, two being adjacent when they have representatives with $\pi L\subsetneq L'\subsetneq L$; for $n=2$ it is a tree in which every vertex has $|k|+1$ neighbours. The link of a vertex is the flag complex of $\mathrm{PG}(n-1,k)$, and the stabilizer of the vertex acts on it through $\PGL(n,k)$.

A completion of a finite projective geometry $\Pi$ over $k$ is a building of type $\tilde A_{n-1}$ together with a vertex whose link is isomorphic to the flag complex of $\Pi$; for $n=2$, a tree with a vertex whose neighbours are identified with the points of $\Proj^1(k)$. The name records that the building lives over $K$, a completion of a global field, while the finite geometry lives over the residue field. Completions are not unique: $\Q_2$ and $\F_2(\!(t)\!)$ both have residue field $\F_2$.

(1) In the building of $\PGL(3,\Q_2)$ the link of a vertex is the Heawood graph, the incidence graph of the Fano plane, and the vertex stabilizer acts on it through $\GL(3,2)$, the automorphisms of the link that preserve the two types of vertices. The full automorphism group of the link has order 336.

(2) In the tree of $\PGL(2,\Q_7)$ the link of a vertex is $\Proj^1(\F_7)$, and the vertex stabilizer acts on it through $\PGL(2,7)$.

(3) A bijection from the eight Sylow 7-subgroups of the type-preserving automorphisms of the 2-adic link onto $\Proj^1(\F_7)$ carries the full automorphism group of the link onto $\PGL(2,7)$, the type-preserving part onto $\PSL(2,7)$, and the 168 dualities, which exchange points and lines, onto $\PGL(2,7)\setminus\PSL(2,7)$.

(4) The 28 pairs of vertices at distance 3 in the 2-adic link, a point and a line not through it, and the 28 pairs of neighbours in the 7-adic link are incarnations of one object, and there is exactly one seam between them.

The group $\Gamma_{\Oct}=\langle a_x\ (x\in\Z/7)\mid a_xa_{x+1}a_{x+3}=1\rangle$, presented by the oriented triples of the octonion table $e_xe_{x+1}=e_{x+3}$, acts simply transitively on the vertices of a building of type $\tilde A_2$ whose vertex links are Fano incidence graphs, by the theorem of Cartwright, Mantero, Steger and Zappa. That building is the building of $\PGL(3)$ over $\F_2(\!(t)\!)$: with $\F_8=\F_2[w]/(w^3+w+1)$, $M_x(v)=w^xv^2$ and $A_x=I+(1+t)M_x$,

and $a_x\mapsto[A_x]$ is injective, with image acting simply transitively on the vertices of the building of $\PGL(3,\F_2(\!(t)\!))$.

No subgroup of finite index in $\Gamma_{\Oct}$ is isomorphic to a subgroup of finite index in Mumford’s lattice, which acts on the building of $\PGL(3,\Q_2)$. In particular $\Gamma_{\Oct}$ is not Mumford’s lattice.

At the prime 2 the group of order 168 acts on the link of a vertex of the building of $\PGL(3,\Q_2)$, the incidence graph of the Fano plane, and the octonion completion gives the same link over $\F_2(\!(t)\!)$. At the prime 7 it acts on the eight neighbours of a vertex of the tree of $\PGL(2,\Q_7)$. At the archimedean place it is the deck group of Thurston’s congruence link complement, with its eight cusps. The plane life is seen at 2, and the line life at 7 and at infinity.

Let $E=\Q(\sqrt{-7})$, $\zeta=e^{2\pi i/7}$, $L=\Q(\zeta)$ with the hermitian form $h(x,y)=\operatorname{tr}_{L/E}(x\bar y)$, and following Kato let $\Gamma_1$ be the image in $\PGL(3,\Q_2)$ of the similitudes of $h$ that preserve $\mathcal O_L\otimes\Z_\ell$ for every prime $\ell\neq2$. Modulo $\sqrt{-7}$ the form has a null plane $N$ over $\F_7$, and Mumford’s lattice $\Gamma_M$ is the image of those similitudes whose action on $N$ lies in a fixed Sylow 2-subgroup of the elements of $\GL(2,7)$ of determinant $\pm1$.

(1) The stabilizer of the vertex $v_0=[\Z_2^3]$ in $\Gamma_1$ is the Frobenius group of order 21 generated by multiplication by $\zeta$ and by $\zeta\mapsto\zeta^2$. Reduction is a homomorphism from $\Gamma_1$ onto $\PSL(2,7)$, and $\Gamma_M$ is the preimage of a Sylow 2-subgroup, dihedral of order 8; its stabilizer of $v_0$ is trivial. (2) The triples $(U,V,W)$ of planes of $\F_2^3$ with $a_Ua_Va_W=1$, where $a_V\in\Gamma_M$ moves $v_0$ to the neighbour of $V$, form a triangle presentation $T_M$: with a suitable numbering, the rotations of $(0,0,1)$, $(0,2,3)$, $(1,3,4)$, $(1,5,2)$, $(2,4,6)$, $(3,6,5)$, $(4,5,6)$. (3) $\Gamma_{T_M}\cong\Gamma_M$, acting simply transitively on the vertices of the building of $\PGL(3,\Q_2)$. (4) Up to relabelling, $T_M$ depends neither on the Sylow subgroup nor on the conventions.

Two finite invariants separate it from the octonion completion: the abelianization of $\Gamma_M$ is $\Z/2\times\Z/6$ against $\Z/2\times\Z/2\times\Z/6$ for $\Gamma_{\Oct}$, and only the identity relabelling preserves $T_M$, while 21 preserve the octonion presentation. Both lattices glue Fano planes into a building over a field with residue field $\F_2$; the octonion table glues them with the symmetry $x\mapsto2x$, Mumford’s gluing has no symmetry at all, and the symmetry of order 21 sits instead in the vertex stabilizer of $\Gamma_1$, which meets $\Gamma_M$ trivially.

Let $T$ be a triangle presentation for the Fano plane whose group of relabellings contains a subgroup of order 21. Then $T$ is equivalent, by a relabelling, to the octonion presentation or to its reversal, so $\Gamma_T\cong\Gamma_{\Oct}$ and the building of $\Gamma_T$ is that of $\PGL(3,\F_2(\!(t)\!))$.

Consequently, if a group acts on a building of type $\tilde A_2$ with Fano vertex links, by type-rotating automorphisms, transitively on the vertices, with vertex stabilizers of order 21 acting faithfully on the links, and has a normal subgroup acting simply transitively on the vertices, then the building is that of $\PGL(3,\F_2(\!(t)\!))$ and not that of $\PGL(3,\Q_2)$. So $\Gamma_1$ has no normal subgroup acting simply transitively, while the octonion lattice is normal in its overgroup by the Frobenius group of order 21: a gluing of Fano planes that this group respects is the octonion gluing, and it lives in characteristic 2.

Over $E_7=\Q_7(\sqrt{-7})$, the self-dual lattices of $h$ and the lattices $\Lambda$ of type 2, with $\Lambda\subset\Lambda^\vee\subset\sqrt{-7}^{\,-1}\Lambda$ and $\Lambda^\vee/\Lambda$ of length 2, form the Bruhat–Tits tree $T_7$ of the unitary group of $h$, regular of valence 8. The lattice $\Lambda_0=\mathcal O_L\otimes\mathcal O_7$ is of type 2, and $\sqrt{-7}$ maps $\Lambda_0^\vee/\Lambda_0$ onto the null plane $N$, with a nondegenerate alternating form. $\Gamma_1$ fixes $\Lambda_0$ and permutes its eight neighbours as its reduction permutes the eight lines of $N$: the link of $\Lambda_0$ is the sky, with the group $\PSL(2,7)$.

The unitary group over $\mathcal O_E[1/7]$, modulo $\pm1$, acts on $T_7$ with quotient a path $[S_4]$ — $[F_{21}]$ — $[\PSL(2,7)]$, whose vertex groups are the automorphism groups of the standard lattice, of $\mathcal O_L$ and of Klein’s lattice $L_\infty$, with $\tfrac1{21}=\tfrac1{24}+\tfrac1{168}$; so it is the amalgam $S_4*_{C_3}\PSL(2,7)$. Over $\Z[1/14]$ it acts cocompactly on the product of the building at 2 and $T_7$ with three orbits of vertices, with stabilizers $F_{21}$, $S_4$ and $\PSL(2,7)$.

Let $\mathcal O=\Z[\omega]$, $\omega$ a primitive cube root of unity, and $\pi=3+\omega$, a prime of norm 7. Give the tree $T_{\mathfrak p}$ of $\PGL(2,\Q_7)$ at $\mathfrak p=(\pi)$ the action of $\PSL(2,\mathcal O[1/\pi])$, and $T_7$ that of Mumford’s unitary group over $\Z[1/14]$, with base vertices whose stabilizers $\PSL(2,\mathcal O)$ and $\Gamma_1$ act on the two links as $\PSL(2,7)$ on the sky.

(1) The two links are incarnations of the object of size 8, which is rigid, so there is exactly one seam between them. (2) Both trees are regular of valence 8, so isomorphisms extending that seam exist, and none is distinguished. (3) No such isomorphism carries the local symmetry of one parent to that of the other: on the ball of radius 2 about the base vertex, $\PSL(2,\mathcal O)$ induces a group of order $168\cdot7^3$ and $\Gamma_1$ one of order $168\cdot2\cdot7^2$; and every vertex stabilizer of the first acts on its link by even permutations, while the stabilizer of $L_\infty$ in the second acts with odd permutations as well.

So the bridge “the two parents complete the sky to one tree” is built on the link, a type on the bare trees, and refuted on the trees with their symmetry. The orders differ for a structural reason: $168\cdot7^3=|\PSL(2,7)|\cdot|\mathfrak{sl}(2,\F_7)|$, while $168\cdot2\cdot7^2=|\SL(2,7)|\cdot|\F_7^2|$.

Call a free $\mathcal O_E$-module of rank 3 with a positive definite hermitian form and a faithful action of $G=\PSL(2,7)$ by isometries a hermitian lattice for $G$. (1) Any two become isometric after the form of one is multiplied by a positive rational number, by an isometry that is equivariant up to an automorphism of $G$; up to isometry exactly one is unimodular. (2) For the lattice $L_\infty$ above, the automorphism group of $(L_\infty,h)$ is $\{\pm1\}\times G_0$, with $G_0\cong G$ having the character values $3,-1,0,1,\alpha,\bar\alpha$ of Klein’s representation: $L_\infty$ is the unimodular hermitian lattice for $G$, Klein’s lattice. (3) $L_\infty$ is the fractional ideal $(1-\zeta)^{-1}\Z[\zeta]$ of $L$ with the form $\operatorname{tr}_{L/E}(x\bar y)$; it is isometric to Elkies’ lattice, and it has no vectors of norm 1, 42 of norm 2 and 56 of norm 3.

The unitary group of $L_\infty$ over $\mathcal O_E[1/2]$ contains a sibling of Mumford’s lattice: the preimage of $\{1,\tau\}$, for an involution $\tau$ of $\PGL(2,7)$ outside $\PSL(2,7)$, under the action on the link at 7, is torsion-free of index 168 and acts simply transitively on the vertices of Klein’s type in the building at 2; its quotient has 8 vertices, 56 edges and 56 triangles. Every move from a Klein vertex to another at distance two acts on the link at 7 by an odd permutation, so the Klein vertices fall into two classes, and Klein vertices at distance two lie in different classes.

The lattice is classical. Uniqueness of the stable lattice follows from Gross, as Elkies records; Allcock and Kato give it, with its isometry group $\PSL(2,7)\times\{\pm1\}$, as the lattice whose group is one of the two densest lattices in $\PGL_3(\Q_2)$; and Nebe identifies it as the Hermitian Barnes lattice, whose trace form is the Barnes lattice $P_6$. The vertex orbits, stabilizers and covolume of its unitary group over $\mathcal O_E[1/2]$, and the surjection onto $\PGL(2,7)$ with torsion-free kernel, are due to Allcock and Kato; the sibling was not found in their papers.

Two Klein vertices $v_0$ and $xv_0$ lie in the same class exactly when the action of $x$ on the eight neighbours at 7 is proper, that is, lies in $\PSL(2,7)$. When they differ, an element of order seven acting on the link by a given Möbius map has trace $\alpha$ on one Klein lattice and $\bar\alpha$ on the other: the two vertices carry Klein’s representation and its conjugate.

Let $N=\Lambda_0^\vee/\Lambda_0$ with its alternating form, let $R$ be the group induced by $\Gamma_1$ on the ball of radius two about $\Lambda_0$ in Mumford’s tree, and let $O$ be the Sylow 7-subgroup of the kernel of $R$ on the link. (1) $R/O\cong\mathrm{Sp}(N)=\SL(2,7)$, and $O\cong N$ as modules for it: the intertwiners $O\to N$ form one line, and they are invertible. The centre $-1$ acts trivially on the link of $\Lambda_0$, by $-1$ on $O$, and on the link of each neighbour of $\Lambda_0$ as an involution of $\PGL(2,7)$ outside $\PSL(2,7)$, fixing $\Lambda_0$ and one other point. (2) The kernel of the action of the Bianchi group on the ball of radius two of its tree is $\mathfrak{sl}_2(\F_7)$ with the adjoint action, and $\mathfrak{sl}_2(\F_7)\cong\mathrm{Sym}^2(V)$ for the natural module $V$; identifying the two copies of $\SL(2,7)$ so that the unique seam between the two links is equivariant, $V\cong N$. So the second layer of the one tree is the symmetric square of the second layer of the other. (3) The Weil representation of $\mathrm{Sp}(N)$ on the functions on a line of $N$ is the even quartet, on which $-1$ acts as $-1$, plus the odd triplet, on which $-1$ acts trivially and whose character is that of Klein’s representation on $L_\infty\otimes\C$. (4) $(L_\infty/\sqrt{-7}L_\infty,h)$ is isometric to $(\mathfrak{sl}_2(\F_7),c\det)$ for some $c\in\F_7^\times$, by a map equivariant up to an automorphism of $\PSL(2,7)$ and unique up to scalars; its isotropic, interior and exterior points are the nilpotent, non-split and split lines.

So the layer of Klein’s lattice read at $\sqrt{-7}$, which is the first layer at Klein’s vertex, is the second layer of the other parent’s tree.

Every subgroup of finite index in $\Gamma_S$ is dense in $\SL(2,\Q_7)$, so at every vertex $t$ of $T_{\mathfrak p}$ its stabilizer acts on $\mathrm{Lk}(t)$ through all of $\SL(2,7)$.

Let $\Gamma$ be a subgroup of finite index in $\Gamma_1\times\Gamma_S$. For every vertex $t$ of $T_{\mathfrak p}$, the elements $(1,\delta)\in\Gamma$ with $\delta(t)=t$ fix every vertex $(v,t)$, act trivially on the object $X_7$ of the points through $\Gamma_1$, and induce all of $\SL(2,7)$ on $\mathrm{Lk}(t)$. Consequently no vertex $(v,t)$ admits a $\Gamma_{(v,t)}$-equivariant bijection between the incarnation of $X_7$ through $\tilde\varpi$ and the incarnation of $X_7$ on $\mathrm{Lk}(t)$, nor a nonzero equivariant map between the even Weil representations through $\tilde\varpi$ and through $\rho_t$. The same holds for every nontrivial irreducible representation, and for any group in place of $\Gamma_1$.

Fix a vertex $t$ and an isomorphism $\iota$ of the two copies of $\SL(2,7)$ carrying the class of $S_4^a$ to the class of $S_4^a$, and let $\Gamma^{(t)}=\{(\gamma,\delta)\in\Gamma_1\times\Gamma_S(t):\iota\tilde\varpi(\gamma)=\rho_t(\delta)\}$.

(1) $\Gamma^{(t)}$ has index 336 in $\Gamma_1\times\Gamma_S(t)$; modulo scalars it is a non-cocompact reducible lattice in $\PGL_3(\Q_2)\times\SL(2,\C)$, acting on $\Delta_2\times\mathbb H^3$. (2) The two incarnations of $X_7$ are joined by exactly one $\Gamma^{(t)}$-equivariant seam, and they form a seam system over $\Delta_2\times\mathbb H^3$ with trivial gauge group. (3) The even Weil representation through $\tilde\varpi$ is isomorphic, as a representation of $\Gamma^{(t)}$, to one of the two even Weil representations through $\rho_t$; the equivariant isomorphisms form $\C^\times$, the unitary ones $U(1)$. (4) The other class of $\iota$ gives the subgroup conjugate under $(1,D_t)$, with $D_t$ a conjugate of $D=\mathrm{diag}(-1,1)$ fixing $t$; it exchanges the classes of $S_4$ and the two even Weil representations. (5) A subgroup of $\Gamma_1\times\Gamma_S(t)$ that maps onto $\SL(2,7)$ under $\tilde\varpi$ and admits an equivariant isomorphism between the even Weil representations through the two reductions lies in some $\Gamma^{(t)}$; by the theorem above, $\Gamma^{(t)}$ is not the stabilizer of $t$ in any subgroup of finite index in $\Gamma_1\times\Gamma_S$.

Let $\Gamma_J=\Gamma_1\times_{\SL(2,7)}\Gamma_S$, the fiber product of $\tilde\varpi$ and reduction modulo $\bar{\mathfrak p}$. (1) Reduction modulo $\mathfrak p$ and $\bar{\mathfrak p}$ maps $\SL(2,\mathcal O)$ onto $\SL(2,7)\times\SL(2,7)$. (2) The stabilizer of $(v_0,t_0)$ acts on the pair (incarnation through $\tilde\varpi$, points of $\mathrm{Lk}(t_0)$) through $F_{21}\times\PSL(2,7)$, a direct product of order 3528. (3) Its elements acting trivially on $\mathrm{Lk}(t_0)$ act on the link at $v_0$ and on the seven points one step deeper below a neighbour of $t_0$ through $F_{21}\times C_7$, a direct product of order 147; its elements acting trivially through $\tilde\varpi$ act on $\mathrm{Lk}(t_0)$ through all of $\PSL(2,7)$.

So the only lattice that joins the two parents across scales attaches the finite line of Mumford’s arithmetic to the line at $\bar{\mathfrak p}$, which does not move with the scale, and at the scale-carrying prime $\mathfrak p$ no attachment survives.