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Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

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The complex Lorentzian Leech lattice and the bimonster (II)
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by Tathagata Basak PDF
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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Additional Information
  • Tathagata Basak
  • Affiliation: Department of Mathematics, Iowa State University, Ames, Iowa 50011
  • Email: tathagat@iastate.edu
  • Received by editor(s): December 9, 2013
  • Received by editor(s) in revised form: April 15, 2014
  • Published electronically: October 5, 2015
  • © Copyright 2015 American Mathematical Society
  • 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
  • MathSciNet review: 3453368