SHADOWING OF ACTIONS OF HYPERBOLIC GROUPS ON THEIR BOUNDARIES

Michal Doucha

correcthigh confidence

Category: Not specified
Journal tier: Specialist/Solid
Processed: Sep 28, 2025, 12:56 AM
arXiv Links: Abstract ↗PDF ↗

Audit review

The paper proves the shadowing property with a carefully constructed auxiliary cover (V_j) and canonical representatives m_g (via Q: D → ∂G) that guarantee quantitative fellow-traveling needed to glue rays and propagate asymptoticity (Definition 2.4; Lemmas 3.18–3.19; Proposition 3.9; Section 4 construction; Proposition 4.3) . The candidate outline reverses a crucial implication about the auxiliary cover W: it assumes that membership in a common W-piece implies long fellow-travel for chosen geodesic representatives, but only states/uses the converse (“fellow-travel ⇒ same W-piece”). This gap prevents deducing the edge-by-edge fellow-travel needed to glue a global quasi-geodesic. The paper’s proof supplies the missing direction by building V so that x, x′ in the same piece force representative rays to be close on long initial segments (Lemma 3.19 and the N^ℓ_x machinery), thereby fixing the gap .

Referee report (LaTeX)

\textbf{Recommendation:} minor revisions

\textbf{Journal Tier:} specialist/solid

\textbf{Justification:}

This paper gives a precise and technically robust proof that the canonical boundary action of any hyperbolic group has shadowing. The approach is well-motivated, uses standard but carefully quantified tools from hyperbolic geometry, and connects neatly to recent work on topological stability. With slight enhancements to exposition (clarifying the role of Q: D→∂G and the N\^ℓ\_x neighborhoods when first introduced, and a brief roadmap of the constants), the paper will be accessible to a broad segment of geometric group theorists and dynamical systems researchers.