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Bulletin of the London Mathematical Society 2005 37(1):54-60; doi:10.1112/S0024609304003960
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© London Mathematical Society

Uncountable Homomorphic Images of Polish Groups are not N1-Free Groups

Anatole Khelif

Equipe de Logique Mathématique, Université Paris 7 2 Place Jussieu, 75251 Paris Cedex 05, France; khelif{at}logique.jussieu.fr

Received 4 November 2002. Revision received 18 March 2004.

Shelah has recently proved that an uncountable free group cannot be the automorphism group of a countable structure. In fact, he proved a more general result: an uncountable free group cannot be a Polish group. A natural question is: can an uncountable N1-free group be a Polish group? A negative answer is given here; indeed, it is proved that an N1-free group cannot be a homomorphic image of a Polish group. In fact, a stronger result is proved, involving a non-commutative analogue of the notion of separable group. 2000 Mathematics Subject Classification 20E05.


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