The Deligne-Mostow 9-ball, and the monster

TL;DR

This paper investigates a complex hyperbolic braid group related to the monster group, proving its quotient is either the bimonster or Z/2, and clarifies the Deligne-Mostow isomorphism with new geometric insights.

Contribution

It establishes the precise nature of the quotient of a 13-dimensional hyperbolic braid group and clarifies the correspondence between braid generators and loops in the Deligne-Mostow moduli space.

Findings

The quotient of the braid group is either the bimonster or Z/2.

Identifies which loops in the 9-ball quotient correspond to standard braid generators.

Provides new geometric understanding of the Deligne-Mostow isomorphism.

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.