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The complex Lorentzian Leech lattice and the bimonster (II)


Author: Tathagata Basak
Journal: Trans. Amer. Math. Soc. 368 (2016), 4171-4195
MSC (2010): Primary 11H56, 20F05, 20F55; Secondary 20D08, 20F36, 51M10
DOI: https://doi.org/10.1090/tran/6558
Published electronically: October 5, 2015
MathSciNet review: 3453368
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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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Additional Information

Tathagata Basak
Affiliation: Department of Mathematics, Iowa State University, Ames, Iowa 50011
Email: tathagat@iastate.edu

DOI: https://doi.org/10.1090/tran/6558
Keywords: Complex hyperbolic reflection group, Leech lattice, bimonster, Artin group, orbifold fundamental group, generator and relation
Received by editor(s): December 9, 2013
Received by editor(s) in revised form: April 15, 2014
Published electronically: October 5, 2015
Article copyright: © Copyright 2015 American Mathematical Society