Automorphisms of pure braid groups (Q1673733)

Let $B_n$ be the Artin braid group on $n$-strings, and let $P_n$ be the pure braid group which is the kernel of the natural epimorphism $B_n\to S_n$. In [Am. J. Math. 103, 1151--1169 (1981; Zbl 0476.20026)], \textit{J. L. Dyer} and \textit{E. K. Grossman} completely determined the automorphism group $\Aut(B_n)$ of the braid group $B_n$ to be a semi-direct product, \[ \Aut(B_n) \cong \mathrm{Inn}(B_n) \rtimes \mathbb{Z}_2, \] where $\mathrm{Inn}(B_n) \cong B_n/Z(B_n)$ is the group of inner automorphisms. In [Comment. Math. Helv. 82, No. 4, 725--751 (2007; Zbl 1148.20024)], \textit{R. W. Bell} and \textit{D. Margalit} decomposed the automorphism group $\Aut(P_n)$ of the pure braid group $P_n$ as a semi-direct product, \[ \Aut(P_n) \cong \Aut_c(P_n) \rtimes \Aut(\overline{P}_n), \] where $\Aut_c(P_n)$ is the group of central automorphisms and $\overline{P}_n=P_n/Z(P_n)$, In [Topology Appl. 159, No. 16, 3404--3416 (2012; Zbl 1275.20031)], \textit{D. C. Cohen} gave an explicit presentation for the automorphism group $\mathrm{Aut}(P_n)$. In this paper, the authors investigate the structure of the automorphism group of the pure braid group $\Aut(P_n)$. They 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 $w_n$. For $n=2$ and $3$, $\Aut(P_n)$ is generated by $\Aut_c(P_n)$ and $\Aut(B_n)$. The authors also study the lifting and extension problem for automorphisms of some well-known exact sequences arising from braid groups, and prove that the answers are negative in most cases. More precisely, let \[ 1 \to K \to G \to H \to 1 \] be a short exact sequence of groups. Given $\phi \in \Aut(H)$, does there exists an automorphism of $G$ which induces $\phi?$ Analogously, given $\psi \in \Aut(K)$, does there exists an automorphism of $G$ whose restriction to $K$ is $\psi?$ They prove that the answers to both the questions are negative for the extension $1 \to U_4 \to P_4 \to P_3 \to 1$. Finally, they consider the extension $1 \to P_n \to B_n \to S_n \to 1$ and prove that the non-inner automorphism of $S_6$ cannot be lifted to an automorphism of $B_6$, and that no non-trivial element of $\Aut_c(P_n)$ can be extended to an automorphism of $B_n$.