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Homomorphisms from Automorphism Groups of Free Groups
Author(s) -
Bridson Martin R.,
Vogtmann Karen
Publication year - 2003
Publication title -
bulletin of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 2.396
H-Index - 48
eISSN - 1469-2120
pISSN - 0024-6093
DOI - 10.1112/s0024609303002248
Subject(s) - mathematics , homomorphism , automorphism , image (mathematics) , group (periodic table) , combinatorics , free group , mathematics subject classification , cardinality (data modeling) , order (exchange) , outer automorphism group , automorphism group , pure mathematics , artificial intelligence , chemistry , organic chemistry , finance , economics , computer science , data mining
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).