# 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)

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} }