Asymptotically CAT(0) metrics, Z-structures, and the Farrell-Jones Conjecture

TL;DR

This work shows that colorable hierarchically hyperbolic groups admit asymptotically CAT(0) metrics, enabling coarse CAT(0) behavior and the construction of Z-structures. A central technical advance is the Stabler Hull Cubulation framework, which approximates hulls of finite sets by CAT(0) cube complexes in a way that is stable under adding points, yielding invariant metrics $d^p$ (notably $d^2$) that are roughly geodesic and asymptotically CAT(0). Using these metrics, the Vietoris–Rips complexes become contractible at large scales, and a natural boundary theory leads to Z-structures for colorable HHGs; this, in turn, allows a BBF-style Farrell–Jones conjecture verification for many HHGs, including extra-large type Artin groups and mapping class groups. The paper unifies CAT(0) and hyperbolic phenomena within HHGs, providing new tools for geometric group theory and K-/L-theory via a detailed hierarchy of stable trees, cubulations, and boundary analysis. The results yield broad applicability, including establishing Z-structures for mapping class groups and proving FJ for a wide class of HHGs through an inductive, stabilization-based approach.

Abstract

We show that colorable hierarchically hyperbolic groups (HHGs) admit asymptotically CAT(0) metrics, that is, roughly, metrics where the CAT(0) inequality holds up to sublinear error in the size of the triangle. We use the asymptotically CAT(0) metrics to construct contractible simplicial complexes and compactifications that provide $\mathcal{Z}$-structures in the sense of Bestvina and Dranishnikov. It was previously unknown that mapping class groups are asymptotically CAT(0) and admit $\mathcal{Z}$-structures. As an application, we prove that many HHGs satisfy the Farrell--Jones Conjecture, including extra large-type Artin groups. To construct asymptotically CAT(0) metrics, we show that hulls of finitely many points in a colorable HHGs can be approximated by CAT(0) cube complexes in a way that adding a point to the finite set corresponds, up to finitely many hyperplanes deletions, to a convex embedding.

Figures (17)

• Figure 1: A schematic of the two types of fillings we perform at the beginning of the inductive procedure on the skeleta. One is indicated by the dotted line and the other one by the shaded area.
• Figure 2:
• Figure 3: The position of $\rho^Z_Y$ is dictated by the fact that $d_Z(\gamma(t),\gamma(t_0))$ is large.
• Figure 4:
• Figure 5: We obtain a quasi-geodesic by traveling from $c'_n(0)$ to a closest point in $c$, then along $c$ to a closest point to $c'_n(s_n)$, and then to $c'_n(s_n)$.

Theorems & Definitions (267)

• Theorem A
• Theorem B
• Theorem C
• Corollary D
• Corollary E
• Theorem F: Local quasi-cubicality
• Theorem G: Cubical metrics
• Theorem H
• Remark
• Theorem 2.1