The complex Lorentzian Leech lattice and the bimonster (II)

Basak, Tathagata

doi:10.1090/tran/6558

The complex Lorentzian Leech lattice and the bimonster (II)
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Trans. Amer. Math. Soc. 368 (2016), 4171-4195 Request permission

Abstract:

Let $D$ be the incidence graph of the projective plane over $\mathbb {F}_3$. 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 the 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 $\operatorname {PGL}(3, \mathbb {F}_3) \subseteq \mathrm {Aut}(D)$ 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 $\operatorname {PGL}(3, \mathbb {F}_3)$ gives meromorphic modular forms of level $13$.

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