×

zbMATH — the first resource for mathematics

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
PDF BibTeX XML Cite
Full Text: DOI
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
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.