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UNCOUNTABLE HOMOMORPHIC IMAGES OF POLISH GROUPS ARE NOT $\aleph _{1}$-FREE GROUPS

Published online by Cambridge University Press:  08 February 2005

ANATOLE KHELIF
ANATOLE KHELIF
Affiliation:
Equipe de Logique Mathématique, Université Paris 7, 2 Place Jussieu, 75251 Paris Cedex 05, France e-mail: [email protected]
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Abstract

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 $\aleph _{1}$-free group be a Polish group? A negative answer is given here; indeed, it is proved that an $\aleph _{1}$-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.

Keywords

20E05
Type
Papers
Information
Bulletin of the London Mathematical Society , Volume 37 , Issue 1 , January 2005 , pp. 54 - 60
Copyright
© The London Mathematical Society 2005

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