Pathological and Highly Transitive Representations of Free Groups

Abstract

A well-known result by P. Cameron provides us with a construction of the free group of rank \(2^{\aleph_{0}}\) within the automorphism group of the rationals. We show that the full versatility of doubly transitive automorphism groups is not necessary by extending Cameron’s construction to a larger class of permutation groups and we generalize his result by constructing pathological (permutations of unbounded support) and ω-transitive (highly transitive) representations of free groups. 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 ∈ [ℵ0, \(2^{\aleph_{0}}\)] (resp. of rank \(2^{\aleph_{0}}\)). Assuming the continuum to be a regular cardinal, we show that pathological and ω-transitive representations of uncountable free groups abound within large permutation groups of linear orders. Lastly, we also find a bound on the rank of free subgroups of certain restricted direct products.