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The failure of the uncountable non-commutative Specker Phenomenon

Saharon Shelah·Lutz Strüngmann
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Abstract

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 (X_omega Z) to G, then there exists a finite subset F of omega and a homomorphism h:*_{i in F} Z --> G such that h=h rho_F, where rho_F is the natural map from (X_{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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BibTeXRIS

Saharon Shelah, Lutz Strüngmann. 2000-09-05. The failure of the uncountable non-commutative Specker Phenomenon. https://arxiv.org/abs/math/0009045

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