ESSENTIAL FREEDOM, ALLOSTERY AND Z-STABILITY OF CROSSED PRODUCTS

Eusebio Gardella, Shirly Geffen, Rafaela Gesing, Grigoris Kopsacheilis, Petr Naryshkin

correctmedium confidence

Category: math.DS
Journal tier: Strong Field
Processed: Sep 28, 2025, 12:56 AM
arXiv Links: Abstract ↗PDF ↗

Audit review

The paper proves that for actions of countable amenable groups on compact zero-dimensional spaces, almost finiteness in measure (AFIM) is equivalent to essential freeness: AFIM ⇒ essentially free (Lemma 2.7) and, conversely, essentially free ⇒ AFIM (Theorem 3.6) . The candidate solution reproduces the AFIM ⇒ essential freeness direction via the standard 'paired levels' argument and gives a different, quasitiling/marker-based route for the converse. Aside from minor gaps (e.g., explicitly enlarging the marker set to a difference set and citing the density estimate), the model’s approach aligns with the paper’s conclusions.

Referee report (LaTeX)

\textbf{Recommendation:} minor revisions

\textbf{Journal Tier:} strong field

\textbf{Justification:}

The paper rigorously extends almost-finiteness techniques to essentially free actions, proving AFIM ⇔ essential freeness on zero-dimensional spaces and enabling further structural results. The arguments are careful and technically solid, with clear handling of fixed-point sets via Banach density. Minor clarifications could aid readability, particularly around the marker modifications and the role of minimality.