Relevance of the RCD(0,∞) phenomenon to hyperbolicity and CAT(0) in the 1‑Grushin hyperbolic plane

Ascertain whether the RCD(0,∞) property of the sub-Riemannian 1‑Grushin hyperbolic half-plane H^+_1 (defined on the hyperbolic plane H^2 by the generating fields X = ∂_x and Y = (sinh(x)/cosh(x))∂_y and restricted to {x > 0}) is relevant to the validity of Gromov-hyperbolicity or the CAT(0) property for the full 1‑Grushin hyperbolic plane H_1.

Background

Within the class of sub-Riemannian α‑Grushin hyperbolic half-planes H+_α equipped with β‑weighted measures, the authors show that the curvature-dimension condition RCD(K,N) is only achievable with negative K, except for an exceptional case at α=1 where RCD(0,∞) holds.

This unexpected behavior at α=1 raises questions about how the measured metric properties (RCD(0,∞)) of the half-space may relate to classical geometric properties (Gromov-hyperbolicity and CAT(0)) of the full 1‑Grushin hyperbolic plane, prompting investigation of any implications or connections.

References

An odd phenomenon occurs at $\alpha=1$, where the $1$-Grushin hyperbolic half-plane satisfies $RCD(0,+\infty)$. It is unclear if this phenomenon is relevant to the validity of hyperbolicity or $\mathsf{CAT}(0)$ property in the $1$-Grushin hyperbolic plane.

— Curvature-dimension condition of sub-Riemannian $α$-Grushin half-spaces (2409.11177 - Borza et al., 17 Sep 2024) in Section 1 (Introduction), after Theorem 1.2