Surfaces, braids, Stokes matrices, and points on spheres

TL;DR

This paper explores the symmetries of moduli spaces of points on spheres, revealing their connections to braid and mapping class groups, and investigates the finiteness of integral braid group orbits in certain Stokes matrix spaces.

Contribution

It establishes new isomorphisms between braid group actions on spheres and mapping class group actions, and analyzes the finiteness of integral orbits in Stokes matrices.

Findings

Braid group actions extend to mapping class groups for spheres in dimensions 0, 1, and 3.

Exceptional isomorphisms relate moduli spaces of points on spheres to local systems.

Finitely many integral braid group orbits exist in certain Stokes matrix spaces.

Abstract

Moduli spaces of points on $n$-spheres carry natural actions of braid groups. For $n=0$, $1$, and $3$, we prove that these symmetries extend to actions of mapping class groups of positive genus surfaces, by establishing exceptional isomorphisms with certain moduli of local systems. This relies on the existence of group structure for spheres in these dimensions. We also use the connection to demonstrate that the space of rank 4 Stokes matrices with fixed Coxeter invariant of nonzero discriminant contains only finitely many integral braid group orbits.