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On $$\omega$$-categorical groups and their completions. (English) Zbl 1202.20035
The article is an investigation of a certain topological completion of an $$\omega$$-categorical group.
The starting point is a construction from V. V. Belyaev [Sib. Math. J. 34, No. 2, 218-232 (1993); translation from Sib. Mat. Zh. 34, No. 2, 23-41 (1993 ; Zbl 0836.20051)] which allows one to embed a group $$G$$ which possesses an inert, residually finite subgroup $$H$$ (this is the case for $$\omega$$-categorical $$G$$) into a locally compact group $$\overline G$$, by letting $$G$$ act on cosets of all subgroups commensurable with $$H$$. This construction coincides with the profinite completion if $$G$$ is residually finite.
Using the work of A. B. Apps [J. Algebra 81, 320-339 (1983; Zbl 0512.20013)] on the structure of $$\omega$$-categorical groups, the author studies properties of the completion $$\overline G$$: among other things, its cofinality and the possibility for it to be $$\omega$$-categorical.
MSC:
 20E18 Limits, profinite groups 03C60 Model-theoretic algebra 03C15 Model theory of denumerable and separable structures 22C05 Compact groups
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References:
 [1] DOI: 10.1017/S0305004100059442 · Zbl 0497.20017 [2] DOI: 10.1016/0021-8693(83)90192-8 · Zbl 0512.20013 [3] DOI: 10.1017/S0305004196001387 · Zbl 0883.03019 [4] DOI: 10.2307/2586577 · Zbl 0966.03037 [5] DOI: 10.1007/BF00975160 [6] DOI: 10.2307/2271901 · Zbl 0309.02059 [7] DOI: 10.1016/0021-8693(79)90230-8 · Zbl 0401.03012 [8] DOI: 10.2307/2586568 · Zbl 0965.03050 [9] DOI: 10.1006/jabr.1997.7264 · Zbl 0896.20029 [10] Koppelberg S., C. R. Math. Acad. Sci. Paris Ser. A 279 pp 583– (1974) [11] Macintyre A. J., J. Algebra 43 pp 483– (1998) [12] DOI: 10.1093/qmath/39.4.483 · Zbl 0667.03027 [13] DOI: 10.1007/BF01109904 · Zbl 0122.02901 [14] DOI: 10.1112/blms/21.5.456 · Zbl 0695.20018 [15] DOI: 10.2307/2695049 · Zbl 0993.03049 [16] DOI: 10.1016/0021-8693(73)90092-6 · Zbl 0256.20003 [17] DOI: 10.1090/S0002-9947-96-01746-1 · Zbl 0867.20026
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