Pathological and Omega-transitive Representations of Free Groups

TL;DR

The paper demonstrates that large permutation groups of linear orders contain highly transitive and pathological free group representations, with results applicable under various set-theoretic assumptions.

Contribution

It establishes the abundance of omega-transitive and pathological free group representations within large automorphism groups of linear orders, including results within ZFC.

Findings

Large subgroups of Aut(Q) contain free groups of any countable rank with omega-transitive representations.

Under GCH, large permutation groups of certain linear orders contain free groups of uncountable rank with omega-transitive representations.

Bound on the rank of free subgroups in certain restricted direct products.

Abstract

Given a linear order $\Omega$ its automorphism group $\Aut(\Omega)$ forms a lattice-ordered group via pointwise order. Assuming the continuum to be a regular cardinal, we show that \emph{pathological} and \emph{$\omega$-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 $\Omega$ for which $|\Omega| = $ cof$(\Omega) = \aleph_i$ ($i \in \N$) then any permutation group that is large in $\Aut(\Omega)$ contains an $\omega$-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 $\omega$-transitive and pathological representation of any free group of rank $\lambda \in [\aleph_0,2^{\aleph_0}]$ (resp. of rank $2^{\aleph_0}$). Lastly, we also find a bound on the rank of free subgroups of certain restricted direct products.