The Nonexistence of Expansive Polycyclic Group Actions on the Circle S1

The paper proves Theorem 1.1 (that S^1 admits no expansive polycyclic group actions) by: (i) extracting a minimal open interval (a,b) whose closure [a,b] supports an expansive subaction (Proposition 2.3), (ii) showing every minimal polycyclic action on R is topologically conjugate to an isometric action via Plante’s quasi-invariant measure (Theorem 3.3 and Proposition 3.4), and (iii) ruling out the circle case by a three-way minimal-set analysis (Proposition 4.2) together with the interval contradiction (Proposition 4.1) . The candidate solution largely follows this structure but contains two decisive errors: (1) it asserts there is a minimal expansive subaction on an invariant open interval J, whereas the paper’s Proposition 2.3 gives minimality on (a,b) and expansiveness on the closed interval [a,b], not on (a,b) itself; the model then uses equicontinuity vs expansiveness on a non-compact set, which does not produce the needed contradiction (contrast with the paper’s precise use of [a,b]) . (2) In the Cantor minimal-set case, the model says the stabilizer H has finite index, but the paper explicitly uses that [G:H]=∞ to pass expansiveness to a closed interval and contradict Proposition 4.1 . Aside from these, the model’s Step 2 matches the paper’s conjugacy-to-isometries result on R (Proposition 3.4) based on Plante’s theorem . Given these discrepancies, the paper’s argument is correct and complete, while the model’s argument is materially flawed.