Journal of the Mathematical Society of Japan

Deeply concatenable subgroups might never be free

Samuel M. CORSON and Saharon SHELAH

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We show that certain algebraic structures lack freeness in the absence of the axiom of choice. These include some subgroups of the Baer–Specker group $\mathbb{Z}^{\omega}$ and the Hawaiian earring group. Applications to slenderness, completely metrizable topological groups, length functions and strongly bounded groups are also presented.


The first author was supported by European Research Council grant PCG-336983. The second author was partially supported by European Research Council grant 338821.

Article information

J. Math. Soc. Japan, Advance publication (2019), 14 pages.

Received: 4 May 2018
First available in Project Euclid: 20 March 2019

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Primary: 20K20: Torsion-free groups, infinite rank
Secondary: 03E25: Axiom of choice and related propositions 03E35: Consistency and independence results 03E75: Applications of set theory

free group Baire property fundamental group axiom of choice strongly bounded group length function


CORSON, Samuel M.; SHELAH, Saharon. Deeply concatenable subgroups might never be free. J. Math. Soc. Japan, advance publication, 20 March 2019. doi:10.2969/jmsj/80498049.

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