Submission date: 4 May 2023

Abstract:

We produce a simple group G of cardinality ℵ_1 which is Artinian (every strictly descending chain of subgroups is finite), satisfies a Burnside law and such that for each uncountable subset Y ⊆ G there exists a natural number n_Y for which every element of G may be expressed as a product of length at most n_Y of elements in Y^{± 1}.
In particular this group is Jónsson (every proper subgroup is of strictly smaller cardinality) and strongly bounded (every abstract action on a metric space has bounded orbits); this is the first example of an uncountable group having both of these properties which is constructed without using the continuum hypothesis. The group G can also be made so that all subgroups are simple and all nontrivial subgroups are malnormal in G.

Mathematics Subject Classification: 03E75, 20A15, 20E15

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Full text arXiv 2305.02953: pdf, ps.