Endomorphisms of automorphism groups of free groups

It is proved that any non-trivial endomorphism of an automorphism group AutFn of a free group Fn, for n ≥ 3, either is an automorphism or factorization over a proper automorphism subgroup. An endomorphism of AutF2 is an automorphism, or else a homomorphism onto one of the groups S3, D8, Z2 × Z 2, Z2, or S3*Z2 (Z2 × Z2). A non-trivial homomorphism of AutFn into AutFm, for n ≥ 3, m ≥ 2, and n > m, is a homomorphism onto Z2 with kernel SAutFn. As a consequence, we obtain that AutFn is co-Hopfian. © 2005 Springer Science+Business Media, Inc.