Nonmeasurable subgroups of compact groups
- Authors: Hernández, Salvador , Hofmann, Karl , Morris, Sidney
- Date: 2016
- Type: Text , Journal article
- Relation: Journal of Group Theory Vol. 19, no. 1 (2016), p. 179-189
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- Description: In 1985 S. Saeki and K. Stromberg published the following question: Does every infinite compact group have a subgroup which is not Haar measurable? An affirmative answer is given for all compact groups with the exception of some metric profinite groups which are almost perfect and strongly complete. In this spirit it is also shown that every compact group contains a non-Borel subgroup. © 2016 by De Gruyter 2016 Generalitat Valenciana PROMETEO/2014/062 We are grateful for our referee's useful comments. In particular, the suggestion that originally we had overlooked [Pacific J. Math. 116 (1985), 217-241] shortened the proof of Theorem 4.3 considerably.
The separable quotient problem for topological groups
- Authors: Leiderman, Arkady , Morris, Sidney , Tkachenko, Mikhail
- Date: 2019
- Type: Text , Journal article
- Relation: Israel Journal of Mathematics Vol. 234, no. 1 (Oct 2019), p. 331-369
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- Description: The famous Banach-Mazur problem, which asks if every infinite-dimensional Banach space has an infinite-dimensional separable quotient Banach space, has remained unsolved for 85 years, though it has been answered in the affirmative for reflexive Banach spaces and even Banach spaces which are duals. The analogous problem for locally convex spaces has been answered in the negative, but has been shown to be true for large classes of locally convex spaces including all non-normable Frechet spaces. For a topological group G there are four natural analogous problems: Does G have a separable quotient group which is (i) non-trivial; (ii) infinite; (iii) metrizable; (iv) infinite metrizable. Positive answers to all four questions are proved for groups G which belong to the important classes of (a) all compact groups; (b) all locally compact abelian groups; (c) all sigma-compact locally compact groups; (d) all abelian pro-Lie groups; (e) all sigma-compact pro-Lie groups; (f) all pseudocompact groups. However, a surprising example of an uncountable precompact group G is produced which has no non-trivial separable quotient group other than the trivial group. Indeed G(tau) has the same property, for every cardinal number tau >= 1.