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arXiv · 2305.02953

Steep uncountable groups

Samuel M. Corson·Alexander Olshanskii·Olga Varghese
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Abstract

We produce a simple group $G$ of cardinality $\aleph_1$ which is Artinian (every strictly descending chain of subgroups is finite), satisfies a Burnside law and such that for each uncountable subset $Y \subseteq G$ 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 $Y^{\pm 1}$. 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 $G$.

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Samuel M. Corson, Alexander Olshanskii, Olga Varghese. 2023-05-04. Steep uncountable groups. https://doi.org/10.1112/blms.12959

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