标题： 渐近CAT(0)度量，Z-结构，以及Farrell-Jones猜想 显示英文标题

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

摘要： 我们证明了可着色的分层双曲群（HHG）可以具有渐近CAT(0)度量，也就是说，大致上，三角形大小的子线性误差内满足CAT(0)不等式。 我们利用渐近CAT(0)度量构造了可收缩的单纯复形和紧化，这些提供了Bestvina和Dranishnikov意义上的$\mathcal{Z}$-结构。 之前未知映射类群是渐近CAT(0)且具有$\mathcal{Z}$-结构。 作为应用，我们证明了许多HHG满足Farrell-Jones猜想，包括额外大型类型的Artin群。 为了构造渐近CAT(0)度量，我们证明了可着色HHG中有限多个点的包络可以通过CAT(0)立方复形来逼近，这种方式下，向有限集添加一个点对应于凸嵌入，最多经过有限多个超平面删除。 显示英文摘要

摘要： 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.