AN ALTERNATIVE FOR MINIMAL GROUP ACTIONS ON TOTALLY REGULAR CURVES

The paper’s Theorem 1.3/7.3 states exactly the alternative the candidate solves: for a minimal action on a totally regular curve, either the system is conjugate to isometries on S1 (hence virtually abelian) or the group contains a quasi-Schottky (free) subgroup. The proof splits into equicontinuous vs sensitive cases (Theorem 2.10). In the equicontinuous branch, the paper uses the standard structure theorem (Theorem 2.11) plus homogeneity and classical characterizations to conclude the space is a circle and the action is conjugate to isometries; the candidate follows the same path with Auslander and Whyburn–Ayres, adding a standard invariant-metric argument. In the sensitive branch, the paper proves contractible neighborhoods (Theorem 3.4), builds an atomless measure with the ‘μ→0 ⇒ diam→0’ property (Theorem 5.7), uses strong ε-proximality (Proposition 2.14) to approximate by finitely supported measures, invokes the escaping lemma (Lemma 6.4), and then constructs ping–pong domains to obtain a quasi-Schottky subgroup (Section 2.6, and the final construction in Section 7). The candidate mirrors these steps closely, citing the same ingredients and arriving at the same ping–pong conclusion. Small wording differences (e.g., presenting strong proximality “on some open subset” rather than globally) do not affect the logic. Overall, the candidate’s solution faithfully tracks the paper’s argument and is correct. See Theorem 1.3 and the proof in Section 7 for the paper’s statement and structure, including the dichotomy (Theorem 2.10), the equicontinuous structure (Theorem 2.11), contractible neighborhoods (Theorem 3.4), the measure characterization (Theorem 5.7), the escaping lemma (Lemma 6.4), and the quasi-Schottky definition and ping–pong (Section 2.6) .