LOW-DIMENSIONAL REPRESENTATIONS OF AUT (F2)

Let F-2 be the free group of rank two, and Phi 2 its automorphism grou p. We consider the problem;of describing the representations of Phi(2) of degree n for small values of n. Our main result is the classificat ion (up to equivalence) of all indecomposable representations rho of P hi(2) of degree n less than or equal to 4 such that rho(F-2) not equal 1. There are only finitely many such representations, and in all them rho(F-2) is solvable. This is no longer true in higher dimensions. Al ready for n = 6 there exists a 1-parameter family of irreducible noneq uivalent representations of Phi(2) such that rho(F-2) contains a free non-abelian subgroup. We also obtain some new 4-dimensional representa tions of the braid group B-4 which are indecomposable and reducible at the same time. It would be interesting to find some applications of t hese representations.