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

Abstract

Abstract Let 𝒲 = { 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. Set W 1 = G 1 {W_{1}=G_{1}} and choose a faithful transitive primitive W 1 W_{1} -set Δ 1 \varDelta_{1} . Assume that we have already constructed W n - 1 W_{n-1} and chosen a transitive faithful primitive W n - 1 W_{n-1} -set Δ n - 1 \varDelta_{n-1} . The group W n W_{n} is then defined as W 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 limit W = lim ← ⁡ ( W n , ρ n ) {W=}{\varprojlim(W_{n},\rho_{n})} with respect to the natural projections ρ n : W n → W n - 1 {\rho_{n}\colon W_{n}\to W_{n-1}} , we prove that, for each k ≥ 2 k\geq 2 , the set of k-tuples of W that freely generate a free subgroup of rank k is comeagre in W k W^{k} and its complement has Haar measure zero.