# Generators for a complex hyperbolic braid group

## TL;DR

This paper identifies natural generators for a complex hyperbolic braid group derived from hyperplane arrangements in 13-dimensional space, supporting a conjecture linking it to the monster finite simple group.

## Contribution

It provides explicit generators for a complex hyperbolic braid group associated with a hyperplane arrangement in 13-space, supporting the 'monstrous proposal' connecting it to the monster group.

## Key findings

- Generators are loops around hyperplanes nearest the basepoint.
- Supports the conjecture relating the braid group to the monster group.
- Provides a geometric understanding of the orbifold fundamental group.

## Abstract

We give generators for a certain complex hyperbolic braid group. That is, we remove a hyperplane arrangement from complex hyperbolic $13$-space, take the quotient of the remaining space by a discrete group, and find generators for the orbifold fundamental group of the quotient. These generators have the most natural form: loops corresponding to the hyperplanes which come nearest the basepoint. Our results support the conjecture that motivated this study, the "monstrous proposal", which posits a relationship between this braid group and the monster finite simple group.