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Martin R. Bridson, Karen Vogtmann, Homomorphisms from Automorphism Groups of Free Groups, Bulletin of the London Mathematical Society, Volume 35, Issue 6, November 2003, Pages 785–792, https://doi.org/10.1112/S0024609303002248
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Abstract
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 Sn +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 Zn −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 )→ Z2 . 2000 Mathematics Subject Classification 20F65, 20F28 (primary).