# Surjective homomorphisms between surface braid groups

## TL;DR

This paper investigates the structure of surjective homomorphisms and automorphisms of pure surface braid groups, showing they are mostly governed by geometric forgetful maps and extending known automorphism results to punctured surfaces.

## Contribution

It proves that all surjective homomorphisms between pure surface braid groups factor through forgetful maps and characterizes the automorphism group for punctured cases, revealing geometric automorphisms for n>1.

## Key findings

- Surjective homomorphisms factor through forgetful maps.
- Automorphism group is explicitly computed for punctured surface braid groups.
- Automorphisms for n>1 are geometric, contrasting with the n=1 case.

## Abstract

Let $PB_n(S_{g,p})$ be the pure braid group of a genus $g>1$ surface with $p$ punctures. In this paper we prove that any surjective homomorphism $PB_n(S_{g,p})\to PB_m(S_{g,p})$ factors through one of the forgetful homomorphisms. We then compute the automorphism group of $PB_n(S_{g,p})$, extending Irmak, Ivanov and McCarthy's result to the punctured case. Surprisingly, in contrast to the $n=1$ case, any automorphism of $PB_n(S_{g,p})$, $n>1$ is geometric.