Polish Algebras shy from freedom

Abstract

Our first motivation was the question: can a countable structure have an automorphism group, which a free uncountable group? This is answered negatively in [Sh:744]. Lecturing in a conference in Rutgers, February 2001, I was asked whether I am really speaking on Polish groups. We can prove this using a more restrictive condition on the set of equations. Parallel theorems, hold for semi groups and for metric algebras, e.g. with non-isolated unit. Here we do the general case. For instance we show that there is no Polish group which as a group is free and uncountable.