Lewis, Wayne; Mader, Adolf Compact connected abelian groups of dimension 1. (English) Zbl 1500.22003 Rend. Semin. Mat. Univ. Padova 146, 163-176 (2021). In the underlying paper, compact connected abelian groups of dimension 1 – the authors call them solenoids – are studied. (This is a continuation of results presented in [E. Hewitt and K. A. Ross, Abstract harmonic analysis. Vol. I: Structure of topological groups. Integration theory. Group representations. Springer, Cham (1963; Zbl 0115.10603)]) An application of the Pontryagin van-Kampen duality theorem yields that these are (up to topological isomorphism) exactly the character groups of discrete torsion-free abelian groups of rank 1, so character groups of non-trivial subgroups of the group \(\mathbb{Q}\) of rational numbers. These groups were classified by R. Baer [Duke Math. J. 3, 68–122 (1937; Zbl 0016.20303)] by means of the type of a height-sequences. The structure theorem presented in the paper makes use of the resolution theorem and the fact, that the exponential mapping in this particular case is canonically given.The structure theorem the authors present (Theorem 1.1) enables them besides other results to classify the torsion subgroups of solenoids. Reviewer: Lydia Außenhofer (Passau) Cited in 1 Document MSC: 22C05 Compact groups 20K15 Torsion-free groups, finite rank Keywords:Compact abelian group; dimension 1; connected groups; Pontryagin duality; resolution theorem; torsion-free abelian group; rank 1 groups; totally disconnected groups Citations:Zbl 0115.10603; Zbl 0016.20303 × Cite Format Result Cite Review PDF Full Text: DOI References: [1] L. F , Abelian groups, Springer, Cham, 2015. · Zbl 1416.20001 [2] E. H -K. A. R , Abstract harmonic analysis, Vol. I, Structure of topological groups. Integration theory, group representations, Die Grundlehren der mathematis-chen Wissenschaften, 115, Academic Press, New York, and Springer-Verlag, Berlin etc., 1963. · Zbl 0115.10603 [3] K. H. H -S. A. M , T , A primer for the student-a handbook for the expert, third edition, revised and augmented, De Gruyter Studies in Mathematics, 25, De Gruyter, Berlin, 2013. · Zbl 1277.22001 [4] M. M , Homological algebra in locally compact abelian groups, Trans. Amer. Math. Soc. 127 (1967), pp. 361-404. · Zbl 0149.26302 [5] Manoscritto pervenuto in redazione il 3 maggio 2018. 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.