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The Deligne-Mostow 9-ball, and the monster

Daniel Allcock, Tathagata Basak

TL;DR

The paper advances the monstrous proposal by showing that the quotient $G/S$ of a complex hyperbolic braid group, after imposing the natural square-relations on meridians, is either the bimonster $(M\times M)\rtimes 2$ or the cyclic group $\mathbb{Z}/2$, ruling out the trivial quotient. It achieves this by embedding the problem in Deligne–Mostow geometry via the 9-dimensional Deligne–Mostow ball $\mathbb{B}^9_{\mathrm{DM}}$, identifying the basepoint and standard braid generators with Deligne–Mostow data, and constructing a neighborhood $U$ around $\mathbb{B}^9_{\mathrm{DM}}$ to extract a Br_5–to–moduli-space extension. A key step is the explicit isomorphism between the moduli space $\mathcal{M}_{12}$ and the DM ball quotient, which clarifies the correspondence between Deligne–Mostow loops and standard braid generators. By analyzing the local group actions and applying the deflation result of Conway–Simons, the authors show that $G/S$ is a quotient of the bimonster; simplicity of the monster forces the nontrivial outcome, and explicit arguments exclude the $\mathbb{Z}/2$ case, yielding the conjectured relation between complex hyperbolic geometry and the monster group. The work also lays groundwork for reinterpretations of Conway–Simons relations in geometric terms, with potential extensions to other moduli spaces such as cubic surfaces.

Abstract

The "monstrous proposal" of the first author is that the quotient of a certain 13-dimensional complex hyperbolic braid group, by the relations that its natural generators have order 2, is the bimonster" (M x M)semidirect Z/2. Here M is the monster simple group. We prove that this quotient is either the bimonster or Z/2. In the process, we give new information about the isomorphism found by Deligne-Mostow, between the moduli space of 12-tuples in CP1 and a quotient of the complex 9-ball. Namely, we identify which loops in the 9-ball quotient correspond to the standard braid generators.

The Deligne-Mostow 9-ball, and the monster

TL;DR

The paper advances the monstrous proposal by showing that the quotient of a complex hyperbolic braid group, after imposing the natural square-relations on meridians, is either the bimonster or the cyclic group , ruling out the trivial quotient. It achieves this by embedding the problem in Deligne–Mostow geometry via the 9-dimensional Deligne–Mostow ball , identifying the basepoint and standard braid generators with Deligne–Mostow data, and constructing a neighborhood around to extract a Br_5–to–moduli-space extension. A key step is the explicit isomorphism between the moduli space and the DM ball quotient, which clarifies the correspondence between Deligne–Mostow loops and standard braid generators. By analyzing the local group actions and applying the deflation result of Conway–Simons, the authors show that is a quotient of the bimonster; simplicity of the monster forces the nontrivial outcome, and explicit arguments exclude the case, yielding the conjectured relation between complex hyperbolic geometry and the monster group. The work also lays groundwork for reinterpretations of Conway–Simons relations in geometric terms, with potential extensions to other moduli spaces such as cubic surfaces.

Abstract

The "monstrous proposal" of the first author is that the quotient of a certain 13-dimensional complex hyperbolic braid group, by the relations that its natural generators have order 2, is the bimonster" (M x M)semidirect Z/2. Here M is the monster simple group. We prove that this quotient is either the bimonster or Z/2. In the process, we give new information about the isomorphism found by Deligne-Mostow, between the moduli space of 12-tuples in CP1 and a quotient of the complex 9-ball. Namely, we identify which loops in the 9-ball quotient correspond to the standard braid generators.
Paper Structure (16 sections, 24 theorems, 55 equations, 1 figure, 1 table)

This paper contains 16 sections, 24 theorems, 55 equations, 1 figure, 1 table.

Table of Contents

  1. Introduction
  2. Background, conventions, notation
  3. Complex hyperbolic space
  4. Eisenstein lattices
  5. The Eisenstein E8 lattice and the hyperbolic cell
  6. The lattice L
  7. Roots, mirrors and the hyperplane arrangement
  8. Special points in B13
  9. The Deligne-Mostow Lattice
  10. Orbifold fundamental groups.
  11. Meridians.
  12. Braid Groups
  13. Change of basepoint in B13
  14. The Deligne-Mostow 9-ball
  15. A neighborhood of the Deligne-Mostow ball, modulo its stabilizer
  16. ...and 1 more sections

Key Result

Theorem 1.1

The complex hyperbolic braid group $G=\pi_1^{\rm orb}((\mathbb{B}^{13}-\EuScript{H})/P\Gamma)$, modulo the subgroup $S$ generated by the squares of all meridians, is isomorphic to either the bimonster or $\mathbb{Z}/2$.

Figures (1)

  • Figure 6.1: The piecewise linear path $t\mapsto\langle\beta(t)\,{|}\,\rho\rangle$ in $\mathbb{C}$, up to scaling. Its reciprocal, scaled, is the path $\alpha$ at the heart of the proof of lemma \ref{['LemSpecialPaths']}\ref{['ItemIncreasingPath']}.

Theorems & Definitions (50)

  • Theorem 1.1
  • Theorem 3.1: Braid Groups
  • proof
  • Theorem 3.2
  • Theorem 4.1
  • proof
  • Lemma 4.2
  • Lemma 4.3
  • proof
  • proof : Proof of lemma \ref{['LemPolygonsMissMirrors']}.
  • ...and 40 more