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Let  =   {   G  i   ∣   1  ≤  i  ∈  ℕ   }{\mathcal{W}=\{G_{i}\mid 1\leq i\in\mathbb{N}\}}    be a set of non-abelian finite simple groups. SetW  1   =   G  1{W_{1}=G_{1}}    and choose a faithful transitive primitiveW  1W_{1}   -setΔ  1\varDelta_{1}   . Assume that we have already constructedW   n  -  1W_{n-1}    and chosen a transitive faithful primitiveW   n  -  1W_{n-1}   -setΔ   n  -  1\varDelta_{n-1}   . The groupW  nW_{n}    is then defined asW  n   =    G  n   ⁢    wr   Δ   n  -  1⁡   W   n  -  1{W_{n}=G_{n}\operatorname{wr}_{\varDelta_{n-1}}W_{n-1}}   . If  W  is the inverse limitW  =    lim  ←   ⁡   (   W  n   ,   ρ  n   ){W=}{\varprojlim(W_{n},\rho_{n})}    with respect to the natural projectionsρ  n   :    W  n   →   W   n  -  1{\rho_{n}\colon W_{n}\to W_{n-1}}   , we prove that, for eachk  ≥  2k\geq 2   , the set of  k -tuples of  W  that freely generate a free subgroup of rank  k  is comeagre inW  kW^{k}    and its complement has Haar measure zero.

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