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 B3 to C3, and for n ≥ 4 we extend this representation under certain restrictions on the parameters of the representation. We determine that the spherical braid group B n (S2) and the mapping class group M(0, n) of an n-punctured sphere are linear for all n ≥ 2. The automorphism group Aut(F n ) is not linear for n ≥ 3, and the group Aut(F2) is linear iff so is the braid group B4. Using the Lawrence—Krammer representation, we construct a faithful linear representation of Aut(F2).