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(Fn)→Aut(Fm) can have image of cardinality at most 2. More generally, this is true of homomorphisms from Aut(Fn) 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 Wn = (Z/2)n ⋊ Sn, or of Zn−1. These results place constraints on how Aut(Fn) can act. For example, if n ⩾ 3, any action of Aut(Fn) on the circle (by homeomorphisms) factors through det : Aut(Fn)→Z2. 2000 Mathematics Subject Classification 20F65, 20F28 (primary).
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