The complex Lorentzian Leech lattice and the bimonster (II)

Author:

Basak Tathagata

Abstract

Let D D be the incidence graph of the projective plane over F 3 \mathbb {F}_3 . The Artin group of the graph D D maps onto the bimonster and a complex hyperbolic reflection group Γ \Gamma acting on 13 13 dimensional complex hyperbolic space Y Y . The generators of the Artin group are mapped to elements of order 2 2 (resp. 3 3 ) in the bimonster (resp. Γ \Gamma ). Let Y Y Y^{\circ } \subseteq Y be the complement of the union of the mirrors of Γ \Gamma . Daniel Allcock has conjectured that the orbifold fundamental group of Y / Γ Y^{\circ }/\Gamma surjects onto the bimonster.

In this article we study the reflection group Γ \Gamma . Our main result shows that there is homomorphism from the Artin group of D D to the orbifold fundamental group of Y / Γ Y^{\circ }/\Gamma , obtained by sending the Artin generators to the generators of monodromy around the mirrors of the generating reflections in Γ \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 PGL ( 3 , F 3 ) A u t ( D ) \operatorname {PGL}(3, \mathbb {F}_3) \subseteq \mathrm {Aut}(D) acts on Y Y and fixes a complex hyperbolic line pointwise. We show that the restriction of Γ \Gamma -invariant meromorphic automorphic forms on Y Y to the complex hyperbolic line fixed by PGL ( 3 , F 3 ) \operatorname {PGL}(3, \mathbb {F}_3) gives meromorphic modular forms of level 13 13 .

Publisher

American Mathematical Society (AMS)

Subject

Applied Mathematics,General Mathematics

Reference21 articles.

1. The Leech lattice and complex hyperbolic reflections;Allcock, Daniel;Invent. Math.,2000

2. A monstrous proposal;Allcock, Daniel,2009

3. On the 𝑌₅₅₅ complex reflection group;Allcock, Daniel;J. Algebra,2009

4. D. Allcock and T. Basak, Geometric generators for braid-like groups, arXiv:1403:2401, (2014), to appear in Geometry and Topology.

5. The complex Lorentzian Leech lattice and the Bimonster;Basak, Tathagata;J. Algebra,2007

Cited by 3 articles. 订阅此论文施引文献 订阅此论文施引文献,注册后可以免费订阅5篇论文的施引文献,订阅后可以查看论文全部施引文献

1. Generators for a complex hyperbolic braid group;Geometry & Topology;2018-09-23

2. Geometric generators for braid-like groups;Geometry & Topology;2016-04-28

3. The Allcock ball quotient;Pure and Applied Mathematics Quarterly;2015

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