The complex Lorentzian Leech lattice and the bimonster (II)

Abstract

LetDDbe the incidence graph of the projective plane overF3\mathbb {F}_3. The Artin group of the graphDDmaps onto the bimonster and a complex hyperbolic reflection groupΓ\Gammaacting on1313dimensional complex hyperbolic spaceYY. The generators of the Artin group are mapped to elements of order22(resp.33) in the bimonster (resp.Γ\Gamma). LetY∘⊆YY^{\circ } \subseteq Ybe the complement of the union of the mirrors ofΓ\Gamma. Daniel Allcock has conjectured that the orbifold fundamental group ofY∘/ΓY^{\circ }/\Gammasurjects onto the bimonster.In this article we study the reflection groupΓ\Gamma. Our main result shows that there is homomorphism from the Artin group ofDDto the orbifold fundamental group ofY∘/ΓY^{\circ }/\Gamma, obtained by sending the Artin generators to the generators of monodromy around the mirrors of the generating reflections inΓ\Gamma. This answers a question in Allcock’s article “A monstrous proposal” and takes a step towards the proof of Allcock’s conjecture. The finite groupPGL⁡(3,F3)⊆Aut(D)\operatorname {PGL}(3, \mathbb {F}_3) \subseteq \mathrm {Aut}(D)acts onYYand fixes a complex hyperbolic line pointwise. We show that the restriction ofΓ\Gamma-invariant meromorphic automorphic forms onYYto the complex hyperbolic line fixed byPGL⁡(3,F3)\operatorname {PGL}(3, \mathbb {F}_3)gives meromorphic modular forms of level1313.