Lattices like the Leech lattice: classification of Lorentzian lattices having a vector with bounded inner product against all simple roots, and the uniqueness of $\mathrm{II}_{25,1}$ in dimension at least 26

In the closing section of 'Lattices like the Leech lattice' (Journal of Algebra 130 (1990) 219-234), Borcherds posed the problem of finding all Lorentzian lattices that possess a nonzero vector whose inner product with every simple root of the reflection group is bounded - the property enjoyed by the even unimodular lattice $\mathrm{II}_{25,1}$ of signature (25,1), whose simple roots are indexed by the Leech lattice and all have inner product $-1$ with the norm-0 Weyl vector. The cluster asks: (i) for a complete classification, which subsumes the classification of reflective Lorentzian lattices (those whose reflection group has finite index in the automorphism group); (ii) whether there are essentially only finitely many such lattices of dimension greater than 2; (iii) whether $\mathrm{II}_{25,1}$ is the only such lattice of dimension at least 26; (iv) whether $\mathrm{I}_{9,1}$ is the only lattice of dimension at least 10 whose reflection group generated by strong roots has infinitely many simple roots satisfying the analogous boundedness condition; plus an item on presenting non-reflection groups of 'type 2' lattices as direct limits of finite groups. The cluster is open: the classification is complete only in rank 3, and the finiteness, maximal-dimension, strong-root and direct-limit questions remain open or only partially resolved.

Background

A lattice is a free abelian group $L$ of finite rank together with an integer-valued symmetric bilinear form $(\,,\,)$; it is called Lorentzian if the form is nondegenerate over $\mathbb{R}$ and has signature $(n,1)$, that is, exactly one negative direction. Its automorphism group $\mathrm{Aut}(L)$ is, following the source's convention, the group of integral isometries that fix each of the two cones of negative-norm vectors (an index-2 subgroup of the full isometry group), and it embeds as a discrete arithmetic group of isometries of hyperbolic $n$-space $H^n$. For a vector $r$ of nonzero norm, the reflection $s_r$ maps $v$ to $v-2(v,r)/(r,r)\,r$; it preserves $L$ exactly when $2(v,r)/(r,r)$ is an integer for every $v\in L$. A root is a primitive vector $r$ of positive norm (primitive meaning not a nontrivial multiple of another lattice vector) whose reflection preserves $L$; in an even lattice every vector of norm $2$ is a root, and in an odd lattice so is every vector of norm $1$, but roots of higher norm also occur. The subgroup $W(L)$ of $\mathrm{Aut}(L)$ generated by the reflections in all roots is the reflection group of $L$. By the standard theory of arithmetic reflection groups, $H^n$ is tiled by the $W(L)$-translates of a fundamental chamber $D$ whose walls correspond to the simple roots of $L$; these can be enumerated by Vinberg's algorithm, the standard iterative procedure for computing simple roots of integral reflection groups. A lattice is called reflective if $W(L)$ has finite index in $\mathrm{Aut}(L)$; the quotient $\mathrm{Aut}(L)/W(L)$ is called the non-reflection group of $L$, and it is either finite, or virtually free abelian (the parabolic or quasi-reflective case), or non-amenable (by the Tits alternative, which states that a finitely generated subgroup of a real linear group either contains a nonabelian free subgroup or is virtually solvable). A Weyl vector is a vector $\rho$ with $(\rho,\alpha)=-(\alpha,\alpha)/2$ for every simple root $\alpha$.

The archetypes are as follows. The Leech lattice $\Lambda$ is the unique even unimodular ('even' meaning all vector norms even, 'unimodular' meaning determinant $\pm 1$) positive-definite lattice of rank $24$ with no vectors of norm $2$, and $\mathrm{II}_{25,1}$ is the unique even unimodular Lorentzian lattice of signature $(25,1)$. Conway showed that the simple roots of $\mathrm{II}_{25,1}$ are in natural bijection with the points of $\Lambda$, that the Weyl vector $\rho$ has norm $0$ with $(r,\rho)=-1$ for every simple root $r$, and that $\mathrm{Aut}(\mathrm{II}_{25,1})$ is a semidirect product of its reflection subgroup - which has infinite index - with the affine automorphism group of $\Lambda$ (the isometries of $\Lambda\otimes\mathbb{R}$ preserving $\Lambda$, generated by the isometries of $\Lambda$ and its translations). Thus $\mathrm{II}_{25,1}$ has infinitely many simple roots, yet all of them have bounded inner product with the single vector $\rho$: it is the prototype of what Borcherds called lattices like the Leech lattice. Similarly, $\mathrm{I}_{9,1}$ denotes the odd unimodular Lorentzian lattice of signature $(9,1)$; a strong root is a root $r$ such that $(r,r)$ divides $(r,v)$ for every $v\in L$ (in an odd lattice this includes all vectors of norm $1$), and Borcherds showed that the simple roots of the group generated by the norm-$1$ reflections of $\mathrm{I}_{9,1}$ are isometric to the $E_8$ lattice, mirroring the $\mathrm{II}_{25,1}$/$\Lambda$ picture (both statements reflect the covering radii of $\Lambda$ and $E_8$, i.e. the facts that balls of radius $\sqrt2$ about the points of $\Lambda$ and balls of radius $1$ about the points of $E_8$ cover their ambient Euclidean spaces).

The 1990 paper proves a dichotomy (its Theorem 2.4) for a nonzero vector $c$ in the closed fundamental chamber whose inner products with all simple roots are bounded: either the existence of such a vector of negative norm is equivalent to there being only finitely many simple roots, in which case the reflection group is essentially of finite index (the elliptic case), or $c$ has norm $0$, the reflection group has infinite index, and the simple roots are organized like finitely many cosets of a positive-definite lattice together with finitely many exceptional roots (the parabolic case, the type 2 of that theorem). The same lattices govern a chapter of Lie theory: the fake monster Lie algebra (called the monster Lie algebra in the 1990 paper) is a generalized Kac-Moody algebra - defined like a Kac-Moody algebra but allowing simple roots of nonpositive norm - with root lattice $\mathrm{II}_{25,1}$, whose real simple roots are the simple roots of the lattice; its subalgebras fixed by finite groups of diagram automorphisms (the baby monster and $Fi_{24}$ Lie algebras) are again generalized Kac-Moody algebras, but not Kac-Moody algebras, having simple roots of norm $0$ that are multiples of the vector $c$. Borcherds' 1998 notes settle the negative-norm simple-root question for these algebras: the fake monster Lie algebra has no simple roots of negative norm, while the baby monster and $Fi_{24}$ Lie algebras do.

The classification landscape surrounding the problem: elliptic reflective Lorentzian lattices have rank at most $22$ (improving an earlier general bound of $30$), the bound being attained essentially only by the even sublattice of the odd unimodular lattice $\mathbb{Z}^{21,1}$ and its dual, first recognized as reflective by Borcherds in 1987, so there are no elliptic reflective examples in dimensions $23$ through $25$; parabolic examples in high dimension begin with Conway's $\mathrm{II}_{25,1}$ (dimension $26$) and include further lattices produced by Borcherds' Theorem 3.3 from fixed-point sublattices of the Leech lattice; the rank-$3$ classification is complete (all $8595$ reflective Lorentzian lattices of rank $3$, in $374$ reflection groups and $39$ commensurability classes (two groups being commensurable when their intersection has finite index in both), due to Allcock), with partial classifications of specific families in ranks $4$ and $5$; and the high-dimensional parabolic picture is governed by the standing conjecture that hyperbolic dimension $25$ is the largest admitting a quasi-reflective group. Nikulin (1996) proved that for each rank at least $3$ there are only finitely many reflective Lorentzian lattices up to similarity and isomorphism, settling the finiteness question for such lattices whose vector $c$ has norm at most $0$. Barnard (2003) proved the dimension-$25$ bound for the parabolic case, for reflection groups with a Weyl vector, conditionally on the Burger-Li-Sarnak conjecture on the automorphic spectra of orthogonal groups.

Problem Statement

Original formulation (Borcherds, 'Lattices like the Leech lattice', J. Algebra 130 (1990), final section). Find all Lorentzian lattices $L$ (free $\mathbb{Z}$-modules of finite rank with an integral symmetric bilinear form of signature $(n,1)$) which possess a nonzero vector $c$ whose inner product with every simple root of the reflection group $W(L)$ is bounded; the problem includes as a special case the classification of all Lorentzian lattices whose reflection group has finite index in $\mathrm{Aut}(L)$ (reflective lattices). Throughout, lattices are compared up to isomorphism and scaling: multiplying the form by a nonzero scalar leaves the reflection group and the simple roots unchanged, so every nonzero multiple of a lattice with the bounded-vector property again has the property. Explicit sub-questions: (1) Finiteness: are there, up to isomorphism and scaling, only essentially finitely many such lattices of dimension greater than 2? (2) Maximal dimension: is $\mathrm{II}_{25,1}$ the only such lattice of dimension at least $26$? (3) Strong-root variant: is $\mathrm{I}_{9,1}$ the only lattice of dimension at least 10 such that the reflection group generated by strong roots (roots $r$ with $(r,r)$ dividing $(r,v)$ for all $v\in L$) has infinitely many simple roots, all of bounded inner product with some vector $c$? (4) Non-reflection group presentations: find more lattices of type 2 in the source's Theorem 2.4 (the parabolic case) whose non-reflection group $\mathrm{Aut}(L)/W(L)$ can be presented as a direct limit of a finite number of finite groups, extending the computations for a few unimodular lattices in Borcherds' 1987 paper.

References

1. R. E. Borcherds, 'Lattices like the Leech lattice', Journal of Algebra 130 (1990), no. 1, 219-234. DOI 10.1016/0021-8693(90)90110-A, https://doi.org/10.1016/0021-8693(90)90110-A; TeX source, https://math.berkeley.edu/~reb/papers/llll/llll.tex.
2. R. E. Borcherds, 'Automorphism groups of Lorentzian lattices', Journal of Algebra 111 (1987), no. 1, 133-153. DOI 10.1016/0021-8693(87)90245-6, https://doi.org/10.1016/0021-8693(87)90245-6.
3. J. H. Conway, 'The automorphism group of the 26-dimensional even unimodular Lorentzian lattice', Journal of Algebra 80 (1983), no. 1, 159-163. DOI 10.1016/0021-8693(83)90025-X, https://doi.org/10.1016/0021-8693(83)90025-X.
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10. G. Höhn and N. R. Scheithauer, 'A natural construction of Borcherds' fake baby monster Lie algebra', American Journal of Mathematics 125 (2003), no. 3, 655-667. DOI 10.1353/AJM.2003.0016, https://doi.org/10.1353/AJM.2003.0016; preprint arXiv:math/0312106, https://arxiv.org/abs/math/0312106.
11. A. Barnard, 'The Singular Theta Correspondence, Lorentzian Lattices and Borcherds-Kac-Moody Algebras', Ph.D. thesis, University of California, Berkeley, 2003. arXiv:math/0307102, https://arxiv.org/abs/math/0307102.
12. D. Allcock, 'The reflective Lorentzian lattices of rank 3', Memoirs of the American Mathematical Society 220 (2012), no. 1033. DOI 10.1090/S0065-9266-2012-00648-4, https://doi.org/10.1090/S0065-9266-2012-00648-4; preprint arXiv:1010.0486, https://arxiv.org/abs/1010.0486.
13. M. Belolipetsky, 'Arithmetic hyperbolic reflection groups', Bulletin of the American Mathematical Society 53 (2016), no. 3, 437-475. DOI 10.1090/bull/1530, https://doi.org/10.1090/bull/1530; preprint arXiv:1506.03111, https://arxiv.org/abs/1506.03111.