Pathological and Highly Transitive Representations of Free Groups

A well-known result by P. Cameron provides us with a construction of the free group of rank 2א0 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 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, 2א0 ] (resp. of rank 2א0). 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.