The failure of the uncountable non-commutative Specker Phenomenon

by Shelah and Struengmann. [ShSm:729]
J Group Theory, 2001
Higman proved in 1952 that every free group is non-commutatively slender, this is to say that if G is a free group and h is a homomorphism from the countable complete free product bigotimes_omega Z to G, then there exists a finite subset F subseteq omega and a homomorphism bar {h}: *_{i in F} Z ---> G such that h= bar {h} rho_F, where rho_F is the natural map from bigotimes_{i in omega} Z to *_{i in F} Z . Corresponding to the abelian case this phenomenon was called the non-commutative Specker Phenomenon. In this paper we show that Higman's result fails if one passes from countable to uncountable. In particular, we show that for non-trivial groups G_alpha (alpha in lambda) and uncountable cardinal lambda there are 2^{2^lambda} homomorphisms from the complete free product of the G_alpha 's to the ring of integers.


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