Groups where free subgroups are abundant

Given an infinite topological group G and a cardinal κ>0, we say that G is almost κ-free if the set of κ-tuples (gi)i∈κ∈Gκ which freely generate free subgroups of G is dense in Gκ. In this note we examine groups having this property and construct examples. For instance, we show that if G is a non-discrete Hausdorff topological group that contains a dense free subgroup of rank κ>0, then G is almost κ-free. A consequence of this is that for any infinite set Ω, the group of all permutations of Ω is almost 2|Ω|-free. We also show that an infinite topological group is almost ℵ0-free if and only if it is almost n-free for each positive integer n. This generalizes the work of Dixon, and Gartside and Knight.