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Large free subgroups of automorphism groups of ultrahomogeneous spaces

Volume 140 / 2015

Szymon Głąb, Filip Strobin Colloquium Mathematicum 140 (2015), 279-295 MSC: Primary 20E05; Secondary 20B27, 54H11. DOI: 10.4064/cm140-2-7

Abstract

We consider the following notion of largeness for subgroups of $S_\infty$. A group $G$ is large if it contains a free subgroup on $\mathfrak c$ generators. We give a necessary condition for a countable structure $A$ to have a large group $\mathop{\rm Aut}(A)$ of automorphisms. It turns out that any countable free subgroup of $S_\infty$ can be extended to a large free subgroup of $S_\infty$, and, under Martin's Axiom, any free subgroup of $S_\infty$ of cardinality less than $\mathfrak c$ can also be extended to a large free subgroup of $S_\infty$. Finally, if $G_n$ are countable groups, then either $\prod_{n\in\mathbb N} G_n$ is large, or it does not contain any free subgroup on uncountably many generators.

Authors

  • Szymon GłąbInstitute of Mathematics
    Lodz University of Technology
    Wólczańska 215
    93-005 Łódź, Poland
    e-mail
  • Filip StrobinInstitute of Mathematics
    Lodz University of Technology
    Wólczańska 215
    93-005 Łódź, Poland
    e-mail

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