ON THE ACTION OF THE (2,3,7)-HOMOLOGY SPHERE GROUP ON ITS SPACE OF LEFT-ORDERS

The paper’s Theorem 1 asserts that for any two left-orders with abc > id, their orbits under Σ are equal. This cannot be literally true: orbits of a countable group action are countable, yet the note itself emphasizes that Σ admits uncountably many left-orders. Moreover, the proof they give only shows that for every finite neighborhood V_{≺,F} one can move ≺′ into V_{≺,F} by conjugation, i.e., that the orbit of ≺′ accumulates at ≺ (and symmetrically), which is equality of orbit-closures, not equality of orbits. This is exactly what is needed to deduce the “in particular” statement that there are exactly two minimal components. The model explicitly makes this correction (equality of orbit-closures in the abc-positive and abc-negative parts) and gives a principled proof via normalization on ℝ, passage to the circle, maximal Euler class, and Matsumoto–Ghys rigidity. This aligns with the paper’s structural inputs (uniqueness up to semiconjugacy) but avoids the incorrect orbit-equality claim. See the statement of Theorem 1 and its context, where the stronger orbit-equality is claimed, and compare with the proof strategy that establishes density/closure containment rather than literal orbit equality .