From pure braid groups to hyperbolic groups

TL;DR

This paper proves a rigidity property of homomorphisms from pure surface braid groups to torsion-free hyperbolic groups, showing they are either cyclic or factor through a specific map, extending previous results.

Contribution

It generalizes earlier rigidity results to all torsion-free hyperbolic groups and provides a new proof for the case of free and surface groups.

Findings

Homomorphisms are either cyclic or factor through a forgetful map.

Extends rigidity results to all torsion-free hyperbolic groups.

Provides a new proof for the case of free and surface groups.

Abstract

In this note we show that any homomorphism from a pure surface braid group to a torsion-free hyperbolic group either has a cyclic image or factors through a forgetful map. This extends and gives a new proof of an earlier result of the author which works only when the target is a free group or a surface group. We also prove a similar rigidity result for the pure braid group of the disk.