# Sh:729

• Shelah, S., & Strüngmann, L. H. (2001). The failure of the uncountable non-commutative Specker phenomenon. J. Group Theory, 4(4), 417–426.
• 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 \bigotimes_\omega{\mathbb Z} to G, then there exists a finite subset F\subseteq\omega and a homomorphism \bar{h}: *_{i\in F} {\mathbb Z}\longrightarrow G such that h=\bar{h}\rho_F, where \rho_F is the natural map from \bigotimes_{i\in\omega}{\mathbb Z} to *_{i\in F}{\mathbb 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.
• Version 2000-12-06_10 (12p) published version (10p)
Bib entry
@article{Sh:729,
author = {Shelah, Saharon and Str{\"u}ngmann, Lutz H.},
title = {{The failure of the uncountable non-commutative Specker phenomenon}},
journal = {J. Group Theory},
fjournal = {Journal of Group Theory},
volume = {4},
number = {4},
year = {2001},
pages = {417--426},
issn = {1433-5883},
mrnumber = {1859179},
mrclass = {20E06 (20E05)},
doi = {10.1515/jgth.2001.031},
note = {\href{https://arxiv.org/abs/math/0009045}{arXiv: math/0009045}},
arxiv_number = {math/0009045}
}