The complex Lorentzian Leech lattice and the bimonster (II)

TL;DR

This paper explores the relationship between the complex Lorentzian Leech lattice, the bimonster, and complex hyperbolic reflection groups, establishing a homomorphism from the Artin group to the orbifold fundamental group, advancing understanding of Allcock's conjecture.

Contribution

It constructs a homomorphism from the Artin group of the incidence graph to the orbifold fundamental group, supporting Allcock's conjecture and linking automorphic forms to modular forms of level 13.

Findings

Established a homomorphism from the Artin group to the orbifold fundamental group.

Connected automorphic forms to modular forms of level 13.

Provided evidence towards Allcock's conjecture.

Abstract

Let $D$ be the incidence graph of the projective plane over $\FF_3$. The Artin group of the graph $D$ maps onto the bimonster and a complex hyperbolic reflection group $\Gamma$ 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. $\Gamma$). Let $Y^{\circ} \subseteq Y$ be the complement of the union of the mirrors of $\Gamma$. Daniel Allcock has conjectured that the orbifold fundamental group of $Y^{\circ}/\Gamma$ surjects onto bimonster. In this article we study the reflection group $\Gamma$. Our main result shows that there is homomorphism from the Artin group of $D$ to the orbifold fundamental group of $Y^{\circ}/\Gamma$, obtained by sending the Artin generators to the generators of monodromy around the mirrors of the generating reflections in $\Gamma$. 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 $\op{PGL}(3, \FF_3) \subseteq \Aut(D)$ acts on $Y$ and fixes a complex hyperbolic line pointwise. We show that the restriction of $\Gamma$-invariant meromorphic automorphic forms on $Y$ to the complex hyperbolic line fixed by $\op{PGL}(3, \FF_3)$ gives meromorphic modular forms of level 13.