# Automorphisms of pure braid Groups

## TL;DR

This paper characterizes the automorphism groups of pure braid groups for n>3, showing they are generated by specific subgroups and an extra automorphism, and explores automorphism extension problems with negative results.

## Contribution

It provides a detailed description of automorphism groups of pure braid groups and addresses the extension and lifting problems for these automorphisms.

## Key findings

- Automorphism group of P_n for n>3 is generated by specific subgroups and an extra automorphism.
- No non-trivial central automorphism of P_n extends to B_n.
- Extension and lifting problems for automorphisms mostly have negative solutions.

## Abstract

In this paper, we investigate the structure of the automorphism groups of pure braid groups. We prove that, for $n>3$, $\Aut(P_n)$ is generated by the subgroup $\Aut_c(P_n)$ of central automorphisms of $P_n$, the subgroup $\Aut(B_n)$ of restrictions of automorphisms of $B_n$ on $P_n$ and one extra automorphism $w_n$. We also investigate the lifting and extension problem for automorphisms of some well-known exact sequences arising from braid groups, and prove that that answers are negative in most cases. Specifically, we prove that no non-trivial central automorphism of $P_n$ can be extended to an automorphism of $B_n$.