Pathological and Omega-transitive Representations of Free Groups

Abstract

Given a linear order $Ω$ its automorphism group $\Aut(Ω)$ forms a lattice-ordered group via pointwise order. Assuming the continuum to be a regular cardinal, we show that \emph{pathological} and \emph{$ω$-transitive} (i.e. highly transitive) representations of free groups abound within \emph{large} permutation groups of linear orders. Consequently, under the Generalized Continuum Hypothesis it is then true that given any linear order $Ω$ for which $|Ω| = $ cof$(Ω) = \aleph_i$ ($i \in \N$) then any permutation group that is large in $\Aut(Ω)$ contains an $ω$-transitive representation of $G_{\aleph_{i}^+}$ (i.e. the free group of rank $2^{\aleph_i}$). In particular, and working solely within ZFC, we show that any large subgroup of $\Aut(\Q)$ (resp. $\Aut(\R)$) contains an $ω$-transitive and pathological representation of any free group of rank $λ\in [\aleph_0,2^{\aleph_0}]$ (resp. of rank $2^{\aleph_0}$). Lastly, we also find a bound on the rank of free subgroups of certain restricted direct products.