Homomorphisms from Automorphism Groups of Free Groups |
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Authors: | Bridson Martin R; Vogtmann Karen |
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Institution: | Mathematics Department, Imperial College London 180 Queen's Gate, London SW7 2BZ m.bridson{at}ic.ac.uk
Mathematics Department 555 Malott Hall, Cornell University, Ithaca, NY 14850, USA vogtmann{at}math.cornell.edu |
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Abstract: | The automorphism group of a finitely generated free group isthe normal closure of a single element of order 2. If m <n, then a homomorphism Aut(Fn) Aut(Fm) can have image of cardinalityat most 2. More generally, this is true of homomorphisms fromAut(Fn) to any group that does not contain an isomorphic imageof the symmetric group Sn+1. Strong restrictions are also obtainedon maps to groups that do not contain a copy of Wn = (Z/2)n Sn, or of Zn1. These results place constraints on howAut(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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