Large free subgroups of automorphism groups of ultrahomogeneous spaces
Volume 140 / 2015
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.
