Homomorphisms from Automorphism Groups of Free Groups

The automorphism group of a finitely generated free group is the normal closure of a single element of order 2. If m < n , then a homomorphism Aut( F n )→Aut( F m ) can have image of cardinality at most 2. More generally, this is true of homomorphisms from Aut( F n ) to any group that does not contain an isomorphic image of the symmetric group S n +1 . Strong restrictions are also obtained on maps to groups that do not contain a copy of W n = ( Z /2) n ⋊ S n , or of Z n −1 . These results place constraints on how Aut( F n ) can act. For example, if n ⩾ 3, any action of Aut( F n ) on the circle (by homeomorphisms) factors through det : Aut( F n )→ Z 2 . 2000 Mathematics Subject Classification 20F65, 20F28 (primary).