ESSENTIAL HOLONOMY OF CANTOR ACTIONS

The paper proves that essential holonomy for a minimal equicontinuous Cantor action (under a locally quasi-analytic hypothesis) forces essential holonomy at all depths of the lower central series, via a localization to a small adapted clopen set and a quasi-analytic disjointness argument for commutators; combined with standard structure of equicontinuous actions, this yields elements φ_n ∈ γ_n(Γ) with positive-measure holonomy for all n (Theorem 1.6) . The candidate model solution gives a different, compact-group/averaging proof: it works in the Ellis group closure, uses the identity [a,b] = a^{-1} a^b and stabilizer calculus to show ∂Fix([a,b]) captures intersections of boundaries, and then uses Haar averaging to find h with μ(∂Fix(a) ∩ h^{-1}∂Fix(a)) > 0, iterating up the lower central series. This aligns with the theorem’s conclusion and does not rely on the local quasi-analytic hypothesis; all key steps are standard in the Ellis-group framework (unique ergodicity and G-invariance of μ, X ≅ G/H) as recorded in the paper’s Section 2.3 and the definition of holonomy as boundary points of fixed sets . Minor technical gaps in the model’s continuity claim for the correlation function F(b)=μ(A∩b^{-1}A) can be repaired by uniform clopen approximations. Thus both arguments are substantively correct, with different methods.