Sh:729

• Shelah, S., & Strüngmann, L. H. (2001). The failure of the uncountable non-commutative Specker phenomenon. J. Group Theory, 4(4), 417–426. arXiv: math/0009045 DOI: 10.1515/jgth.2001.031 MR: 1859179
• 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)

@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}
}