Large free subgroups of automorphism groups of ultrahomogeneous spaces

Szymon Głąb ¹   ; Filip Strobin ¹

¹ Institute of Mathematics Lodz University of Technology Wólczańska 215 93-005 Łódź, Poland

Colloquium Mathematicum, Tome 140 (2015) no. 2, pp. 279-295

Résumé

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.

Détail

DOI : 10.4064/cm140-2-7

Keywords: consider following notion largeness subgroups infty group large contains subgroup mathfrak generators necessary condition countable structure have large group mathop aut automorphisms turns out countable subgroup infty extended large subgroup infty under martins axiom subgroup infty cardinality mathfrak extended large subgroup infty finally countable groups either prod mathbb large does contain subgroup uncountably many generators