Automorphism groups of Lorentzian lattices

Journal of Algebra, Vol. 111, No. 1, Nov 1987, 133–153. Richard E. Borcherds, D.P.M.M.S., University of Cambridge, 16 Mill Lane, Cambridge, CB2 1SB, England. The study of automorphism groups of unimodular Lorentzian lattices In,1 was started by Vinberg. These lattices have an infinite reflection group (if n ≥ 2) and Vinberg showed that the quotient of the automorphism group by the reflection groups was finite if and only if n ≤ 19. Conway and Sloane rewrote Vinberg’s result in terms of the Leech lattice Λ, showing that this quotient (for n ≤ 19) was a subgroup of ·0 = Aut(Λ). In this paper we continue Conway and Sloane’s work and describe Aut(In,1) for n ≤ 23. In these cases there is a natural complex U associated to In,1, whose dimension is the virtual cohomological dimension of the “non-reflection part” Gn of Aut(In,1), and which is a point if and only if n ≤ 19. For n = 20, 21, and 22 the group Gn is an amalgamated product of 2 subgroups of ·0, while G23 is a direct limit of 6 subgroups of ·0. The group G24 seems to be much more complicated (although it would probably be just about possible to describe it). We also have a few results about Gn for large n; for example, if n is at least 18 and congruent to 2, 3, 4, 5, or 6 mod 8 then Aut(In,1) is a nontrivial amalgamated product. We find a few new lattices whose reflection group has finite index in the automorphism group; for example, the even sublattice of I21,1 of determinant 4.