Groups Acting on the Line with at Most 2 Fixed Points: An Extension of Solodov’s Theorem

The paper’s Theorem A states exactly the dichotomy the candidate proves, and the proof strategies line up step-for-step. With a global fixed point, the paper shows G+ is abelian and any fixed point of a nontrivial element is global (Lemma 3.1), via restriction to the two invariant half-lines and an application of Solodov’s theorem; the candidate reproduces the same argument, including the “≥3 fixed points” contradiction if both half-line actions were nonabelian affine . Without a global fixed point, both argue: (i) if every nontrivial element has ≤1 fixed point, then Solodov gives a semi-conjugacy to an affine action; (ii) otherwise, an element with exactly two fixed points yields a G-wandering open interval by a case analysis (Lemma 3.4), and collapsing all such components via a monotone proper map reduces the action to the ≤1-fixed-point case, hence affine by Solodov; the candidate explicitly cites this and mirrors the paper’s construction and conclusion . The small commentary about exclusivity (not claimed by the paper) is correct: translation actions are both abelian and affine, so the cases are not strictly exclusive unless “affine” is read as “nonabelian affine.” Overall, the model’s solution faithfully implements the paper’s reasoning.