The Complex Lorentzian Leech Lattice and the Bimonster (ii)

Let D be the incidence graph of the projective plane over F3. The Artin group of the graph D maps onto the bimonster and a complex hyperbolic reflection group Γ acting on 13 dimensional complex hyperbolic space Y . The generators of the Artin group are mapped to elements of order 2 (resp. 3) in the bimonster (resp. Γ). Let Y ◦ ⊆ Y be the complement of the union of the fixed points of reflections in Γ. Daniel Allcock has conjectured that the orbifold fundamental group of Y ◦/Γ surjects onto bimonster. In this article we study the reflection group Γ. We show that the Artin group of D maps to the orbifold fundamental group of Y ◦/Γ, thus answering a question in Allcock’s article “A monstrous proposal” and taking one step towards the proof of Allcock’s conjecture. The finite group Aut(D) acts on Y . We make a detailed study of the complex hyperbolic line fixed by the subgroup L3(3) ⊆ Aut(D). We define meromorphic automorphic forms on Y , invariant under Γ, with poles along mirrors. We show that the restriction of these forms to the complex hyperbolic line fixed by L3(3) gives meromorphic modular forms of level 13.