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Compact connected abelian groups of dimension 1. (English) Zbl 1500.22003

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.

MSC:

22C05 Compact groups
20K15 Torsion-free groups, finite rank
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.
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