Free subgroups of inverse limits of iterated wreath products of non-abelian finite simple groups in primitive actions

Let W D ¹Gi j 1 i 2 Nº be a set of non-abelian finite simple groups. Set W1 D G1 and choose a faithful transitive primitive W1-set 1. Assume that we have already constructed Wn1 and chosen a transitive faithful primitive Wn1-set n1. The group Wn is then defined as Wn D Gn wrn1 Wn1. If W is the inverse limit W D lim .Wn; n/ with respect to the natural projections nW Wn ! Wn1, we prove that, for each k 2, the set of k-tuples of W that freely generate a free subgroup of rank k is comeagre in W k and its complement has Haar measure zero.