Abstract
LetF2 be the free group of rank two, and Φ2 its automorphism group. We consider the problem of describing the representations of Φ2 of degreen for small values ofn. Our main result is the classification (up to equivalence) of all indecomposable representations ρ of Φ2 of degreen≤4 such that ρ(F2) ≠ 1. There are only finitely many such representations, and in all them ρ(F2) is solvable. This is no longer true in higher dimensions. Already forn=6 there exists a 1-parameter family of irreducible nonequivalent representations of Φ2 such that ρ(F2) contains a free non-abelian subgroup. We also obtain some new 4-dimensional representations of the braid groupB4 which are indecomposable and reducible at the same time. It would be interesting to find some applications of these representations.
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