Linear representations of the group of conjugating automorphisms and the braid groups of some manifolds

We extend the Burau representation to the group $C_n$ of conjugating automorphisms. We extend the Lawrence–Krammer faithful linear representation of the braid group $B_3$ to $C_3$, and for $n\geqslant4$ we extend this representation under certain restrictions on the parameters of the representation. We determine that the spherical braid group $B_n(S^2)$ and the mapping class group $M(0,n)$ of an $n$-punctured sphere are linear for all $n\geqslant2$. The automorphism group $\operatorname{Aut}(F_n)$ is not linear for $n\geqslant3$, and the group $\operatorname{Aut}(F_2)$ is linear iff so is the braid group $B_4$. Using the Lawrence–Krammer representation, we construct a faithful linear representation of $\operatorname{Aut}(F_2)$.