Lower central series, surface braid groups, surjections and permutations

TL;DR

This paper investigates surjective homomorphisms between surface braid groups, focusing on their lower central series, and classifies certain representations into symmetric groups, extending classical braid group results.

Contribution

It generalizes classical braid group surjection results to surface braid groups and provides a combinatorial approach based on their lower central series.

Findings

Identifies conditions for surjections between surface braid groups

Classifies representations of surface braid groups into symmetric groups

Provides partial results for surfaces with boundary and non-orientable surfaces

Abstract

Generalising previous results on classical braid groups by Artin and Lin, we determine the values of m, n $\in$ N for which there exists a surjection between the n-and m-string braid groups of an orientable surface without boundary. This result is essentially based on specific properties of their lower central series, and the proof is completely combinatorial. We provide similar but partial results in the case of orientable surfaces with boundary components and of non-orientable surfaces without boundary. We give also several results about the classification of different representations of surface braid groups in symmetric groups.