Free minimal actions of solvable Lie groups which are not affable

The paper constructs a Sol(a,b)-tiling T̂ from the hyperbolic Penrose tiling, proves that when Δ^{-1}=2^{b/a} is an integer, one can refine to a face-to-face, repetitive tiling, and then uses a two-color decoration (via Morse or Oxtoby sequences) to break the residual symmetry, yielding an aperiodic tiling T̂(ω). It then shows the continuous hull M(a,b,ω) is compact, with a free minimal Sol(a,b)-action, giving a transversely Cantor lamination; see the explicit construction of R̂,Ŝ,Δ and T̂ (Proposition 1) and the integer condition for face-to-face/repetitivity plus decoration (Proposition 2), culminating in Theorem 1 (compact, free, minimal, transversely Cantor) . The candidate solution largely mirrors this blueprint for compactness, minimality, and the lamination structure, but its freeness argument is flawed: it claims freeness holds regardless of whether the decorating sequence ω is periodic or aperiodic, and uses an invalid “collared-tile implies g^m fixes a collar” step. In contrast, the paper explicitly chooses repetitive aperiodic decorations (Morse/Oxtoby) to ensure aperiodicity, which is the standard route to freeness of the hull action (see also the hyperbolic discussion of repetitive aperiodic tilings leading to a minimal free action) . The model also omits the paper’s second sign case (±b/a = log n/log 2) noted in the statement of Theorem 1 .