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• W2964026993 abstract "Let <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper D> <mml:semantics> <mml:mi>D</mml:mi> <mml:annotation encoding=application/x-tex>D</mml:annotation> </mml:semantics> </mml:math> </inline-formula> be the incidence graph of the projective plane over <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=double-struck upper F 3> <mml:semantics> <mml:msub> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi mathvariant=double-struck>F</mml:mi> </mml:mrow> <mml:mn>3</mml:mn> </mml:msub> <mml:annotation encoding=application/x-tex>mathbb {F}_3</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. The Artin group of the graph <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper D> <mml:semantics> <mml:mi>D</mml:mi> <mml:annotation encoding=application/x-tex>D</mml:annotation> </mml:semantics> </mml:math> </inline-formula> maps onto the bimonster and a complex hyperbolic reflection group <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=normal upper Gamma> <mml:semantics> <mml:mi mathvariant=normal>Γ<!-- Γ --></mml:mi> <mml:annotation encoding=application/x-tex>Gamma</mml:annotation> </mml:semantics> </mml:math> </inline-formula> acting on <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=13> <mml:semantics> <mml:mn>13</mml:mn> <mml:annotation encoding=application/x-tex>13</mml:annotation> </mml:semantics> </mml:math> </inline-formula> dimensional complex hyperbolic space <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper Y> <mml:semantics> <mml:mi>Y</mml:mi> <mml:annotation encoding=application/x-tex>Y</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. The generators of the Artin group are mapped to elements of order <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=2> <mml:semantics> <mml:mn>2</mml:mn> <mml:annotation encoding=application/x-tex>2</mml:annotation> </mml:semantics> </mml:math> </inline-formula> (resp. <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=3> <mml:semantics> <mml:mn>3</mml:mn> <mml:annotation encoding=application/x-tex>3</mml:annotation> </mml:semantics> </mml:math> </inline-formula>) in the bimonster (resp. <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=normal upper Gamma> <mml:semantics> <mml:mi mathvariant=normal>Γ<!-- Γ --></mml:mi> <mml:annotation encoding=application/x-tex>Gamma</mml:annotation> </mml:semantics> </mml:math> </inline-formula>). Let <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper Y Superscript ring Baseline subset-of-or-equal-to upper Y> <mml:semantics> <mml:mrow> <mml:msup> <mml:mi>Y</mml:mi> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mo>∘<!-- ∘ --></mml:mo> </mml:mrow> </mml:msup> <mml:mo>⊆<!-- ⊆ --></mml:mo> <mml:mi>Y</mml:mi> </mml:mrow> <mml:annotation encoding=application/x-tex>Y^{circ } subseteq Y</mml:annotation> </mml:semantics> </mml:math> </inline-formula> be the complement of the union of the mirrors of <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=normal upper Gamma> <mml:semantics> <mml:mi mathvariant=normal>Γ<!-- Γ --></mml:mi> <mml:annotation encoding=application/x-tex>Gamma</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. Daniel Allcock has conjectured that the orbifold fundamental group of <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper Y Superscript ring Baseline slash normal upper Gamma> <mml:semantics> <mml:mrow> <mml:msup> <mml:mi>Y</mml:mi> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mo>∘<!-- ∘ --></mml:mo> </mml:mrow> </mml:msup> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mo>/</mml:mo> </mml:mrow> <mml:mi mathvariant=normal>Γ<!-- Γ --></mml:mi> </mml:mrow> <mml:annotation encoding=application/x-tex>Y^{circ }/Gamma</mml:annotation> </mml:semantics> </mml:math> </inline-formula> surjects onto the bimonster. In this article we study the reflection group <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=normal upper Gamma> <mml:semantics> <mml:mi mathvariant=normal>Γ<!-- Γ --></mml:mi> <mml:annotation encoding=application/x-tex>Gamma</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. Our main result shows that there is homomorphism from the Artin group of <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper D> <mml:semantics> <mml:mi>D</mml:mi> <mml:annotation encoding=application/x-tex>D</mml:annotation> </mml:semantics> </mml:math> </inline-formula> to the orbifold fundamental group of <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper Y Superscript ring Baseline slash normal upper Gamma> <mml:semantics> <mml:mrow> <mml:msup> <mml:mi>Y</mml:mi> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mo>∘<!-- ∘ --></mml:mo> </mml:mrow> </mml:msup> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mo>/</mml:mo> </mml:mrow> <mml:mi mathvariant=normal>Γ<!-- Γ --></mml:mi> </mml:mrow> <mml:annotation encoding=application/x-tex>Y^{circ }/Gamma</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, obtained by sending the Artin generators to the generators of monodromy around the mirrors of the generating reflections in <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=normal upper Gamma> <mml:semantics> <mml:mi mathvariant=normal>Γ<!-- Γ --></mml:mi> <mml:annotation encoding=application/x-tex>Gamma</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. 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 <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper P upper G upper L left-parenthesis 3 comma double-struck upper F 3 right-parenthesis subset-of-or-equal-to normal upper A normal u normal t left-parenthesis upper D right-parenthesis> <mml:semantics> <mml:mrow> <mml:mi>PGL</mml:mi> <mml:mo>⁡<!-- ⁡ --></mml:mo> <mml:mo stretchy=false>(</mml:mo> <mml:mn>3</mml:mn> <mml:mo>,</mml:mo> <mml:msub> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi mathvariant=double-struck>F</mml:mi> </mml:mrow> <mml:mn>3</mml:mn> </mml:msub> <mml:mo stretchy=false>)</mml:mo> <mml:mo>⊆<!-- ⊆ --></mml:mo> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi mathvariant=normal>A</mml:mi> <mml:mi mathvariant=normal>u</mml:mi> <mml:mi mathvariant=normal>t</mml:mi> </mml:mrow> <mml:mo stretchy=false>(</mml:mo> <mml:mi>D</mml:mi> <mml:mo stretchy=false>)</mml:mo> </mml:mrow> <mml:annotation encoding=application/x-tex>operatorname {PGL}(3, mathbb {F}_3) subseteq mathrm {Aut}(D)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> acts on <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper Y> <mml:semantics> <mml:mi>Y</mml:mi> <mml:annotation encoding=application/x-tex>Y</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and fixes a complex hyperbolic line pointwise. We show that the restriction of <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=normal upper Gamma> <mml:semantics> <mml:mi mathvariant=normal>Γ<!-- Γ --></mml:mi> <mml:annotation encoding=application/x-tex>Gamma</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-invariant meromorphic automorphic forms on <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper Y> <mml:semantics> <mml:mi>Y</mml:mi> <mml:annotation encoding=application/x-tex>Y</mml:annotation> </mml:semantics> </mml:math> </inline-formula> to the complex hyperbolic line fixed by <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper P upper G upper L left-parenthesis 3 comma double-struck upper F 3 right-parenthesis> <mml:semantics> <mml:mrow> <mml:mi>PGL</mml:mi> <mml:mo>⁡<!-- ⁡ --></mml:mo> <mml:mo stretchy=false>(</mml:mo> <mml:mn>3</mml:mn> <mml:mo>,</mml:mo> <mml:msub> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi mathvariant=double-struck>F</mml:mi> </mml:mrow> <mml:mn>3</mml:mn> </mml:msub> <mml:mo stretchy=false>)</mml:mo> </mml:mrow> <mml:annotation encoding=application/x-tex>operatorname {PGL}(3, mathbb {F}_3)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> gives meromorphic modular forms of level <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=13> <mml:semantics> <mml:mn>13</mml:mn> <mml:annotation encoding=application/x-tex>13</mml:annotation> </mml:semantics> </mml:math> </inline-formula>." @default.
• W2964026993 created "2019-07-30" @default.
• W2964026993 creator A5085174191 @default.
• W2964026993 date "2015-10-05" @default.
• W2964026993 modified "2023-09-27" @default.
• W2964026993 title "The complex Lorentzian Leech lattice and the bimonster (II)" @default.
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• W2964026993 doi "https://doi.org/10.1090/tran/6558" @default.
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