The complex Lorentzian Leech lattice and the bimonster (II)

Let $D$ be the incidence graph of the projective plane over ${\text{FF}}_3$. The Artin group of the graph $D$ maps onto the bimonster and a complex hyperbolic reflection group $\mathrm{\Gamma }$ 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. $\mathrm{\Gamma }$). Let $Y^{\circ }\subseteq Y$ be the complement of the union of the mirrors of $\mathrm{\Gamma }$. Daniel Allcock has conjectured that the orbifold fundamental group of $Y^{\circ }\mathrm{/}\mathrm{\Gamma }$ surjects onto bimonster. In this article we study the reflection group $\mathrm{\Gamma }$. Our main result shows that there is homomorphism from the Artin group of $D$ to the orbifold fundamental group of $Y^{\circ }\mathrm{/}\mathrm{\Gamma }$, obtained by sending the Artin generators to the generators of monodromy around the mirrors of the generating reflections in $\mathrm{\Gamma }$. This answers a question in Allcock’s article “A monstrous proposal” and takes a step towards the proof of Allcock’s conjecture. The finite group $\text{op}PGL(3,{\text{FF}}_3)\subseteq \text{Aut}(D)$ acts on $Y$ and fixes a complex hyperbolic line pointwise. We show that the restriction of $\mathrm{\Gamma }$-invariant meromorphic automorphic forms on $Y$ to the complex hyperbolic line fixed by $\text{op}PGL(3,{\text{FF}}_3)$ gives meromorphic modular forms of level 13.