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- W2952710964 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." @default.
- W2952710964 created "2019-06-27" @default.
- W2952710964 creator A5020412237 @default.
- W2952710964 date "2012-04-25" @default.
- W2952710964 modified "2023-09-27" @default.
- W2952710964 title "Pathological and Omega-transitive Representations of Free Groups" @default.
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