Corson, Samuel M.; Shelah, Saharon Deeply concatenable subgroups might never be free. (English) Zbl 1458.20047 J. Math. Soc. Japan 71, No. 4, 1123-1136 (2019). Summary: 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. MSC: 20K20 Torsion-free groups, infinite rank 03E25 Axiom of choice and related propositions 03E35 Consistency and independence results 03E75 Applications of set theory Keywords:free group; Baire property; fundamental group; axiom of choice; strongly bounded group; length function PDFBibTeX XMLCite \textit{S. M. Corson} and \textit{S. Shelah}, J. Math. Soc. Japan 71, No. 4, 1123--1136 (2019; Zbl 1458.20047) Full Text: DOI arXiv Euclid References: [1] G. Bergman, Generating infinite symmetric groups, Bull. London Math. Soc., 38 (2006), 429-440. · Zbl 1103.20003 · doi:10.1112/S0024609305018308 [2] A. Blass, Specker’s theorem for Nöbling’s group, Proc. Amer. Math. Soc., 130 (2002), 1581-1587. · Zbl 0998.20047 · doi:10.1090/S0002-9939-01-06222-0 [3] J. Cannon and G. Conner, The combinatorial structure of the Hawaiian earring group, Top. Appl., 106 (2000), 225-271. · Zbl 0955.57002 · doi:10.1016/S0166-8641(99)00103-0 [4] S. Corson, Torsion-free word hyperbolic groups are n-slender, Int. J. Algebra Comput., 26 (2016), 1467-1482. · Zbl 1359.57001 · doi:10.1142/S0218196716500636 [5] G. Conner and S. Corson, A note on automatic continuity, Proc. Amer. Math. Soc., 147 (2019), 1255-1268. · Zbl 1477.20049 · doi:10.1090/proc/14242 [6] Y. de Cornulier, Strongly bounded groups and infinite powers of finite groups, Comm. Algebra, 34 (2006), 2337-2345. · Zbl 1125.20023 · doi:10.1080/00927870600550194 [7] K. Eda, Free \(\sigma \)-products and noncommutatively slender groups, J. Algebra, 148 (1992), 243-263. · Zbl 0779.20012 [8] K. Eda, Free subgroups of the fundamental group of the Hawaiian earring, J. Algebra, 219 (1999), 598-605. · Zbl 0951.20016 · doi:10.1006/jabr.1999.7912 [9] L. Fuchs, Infinite abelian groups, Vols 1, 2, Academic Press, San Diego 1970, 1973. · Zbl 0209.05503 [10] E. Green, Graph products of groups, PhD thesis, University of Leeds, 1990. [11] G. Higman, Unrestricted free products and varieties of topological groups, J. London Math. Soc., 27 (1952), 73-81. · Zbl 0046.02601 · doi:10.1112/jlms/s1-27.1.73 [12] K. Kuratowski and S. Ulam, Quelques propriétés topologiques du produit combinatoire, Fund. Math., 19 (1932), 247-251. · Zbl 0005.18301 · doi:10.4064/fm-19-1-247-251 [13] J. Mycielski, Independent sets in topological algebras, Fund. Math., 55 (1964), 139-147. · Zbl 0124.01301 · doi:10.4064/fm-55-2-139-147 [14] J. Nakamura, Atomic properties of the Hawaiian earring group for HNN extensions, Comm. Algebra, 43 (2015), 4138-4147. · Zbl 1331.20034 · doi:10.1080/00927872.2014.939273 [15] G. Nöbling, Verallgemeinerung eines Satzes von Herrn E. Specker, Invent. Math., 6 (1968), 41-55. · Zbl 0176.29801 [16] R. Nunke, Slender groups, Bull. Amer. Math. Soc., 67 (1961), 274-275. · Zbl 0099.01301 · doi:10.1090/S0002-9904-1961-10582-X [17] E. Specker, Additive Gruppen von folgen Ganzer Zahlen, Portugal. Math., 9 (1950), 131-140. · Zbl 0041.36314 [18] S. Shelah, Can you take Solovay’s inaccessible away?, Israel J. Math., 48 (1984), 1-47. · Zbl 0596.03055 · doi:10.1007/BF02760522 [19] S. Shelah, Can the fundamental (homotopy) group of a space be the rationals?, Proc. Amer. Math. Soc., 103 (1988), 627-632. · Zbl 0661.55012 · doi:10.1090/S0002-9939-1988-0943095-3 [20] R. Solovay, A model of set theory in which every set of reals is Lebesgue measurable, Ann. of Math., 92 (1970), 1-56. · Zbl 0207.00905 · doi:10.2307/1970696 [21] S. Srivastava, A course on Borel sets, Springer, 1998. · Zbl 0903.28001 [22] A. Zastrow, The non-abelian Specker-group is free, J. Algebra, 229 (2000), 55-85. · Zbl 0959.20028 · doi:10.1006/jabr.1999.8261 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.