The automorphism group of a free group is not linear

Abstract

A linear group has ascending chain condition on centralizers. (Malcev [S, p. 511). Finitely generated linear groups are residually finite. (Malcev [S, p. 453). A solvable linear group is nilpotent-by-(abelian-byfinite). (Tits [5, pp. 14551461). A linear group either contains a free group of rank two, or is solvable-by-(locally finite). Lubotzky [3] has characterized finitely generated linear groups over a field of characteristic zero by purely group theoretic conditions. These conditions appear difficult to check, although they have been used to show linearity for certain groups. However, we do not use his theorem. The main result of this paper (Theorem 5) is that for n 2 3, the automorphism group of a free group of rank n is not a linear group. The proof uses the representation theory of algebraic groups to show that a kind of “diophantine equation” between the irreducible representations of