Asymptotically CAT(0) metrics, Z-structures, and the Farrell-Jones Conjecture
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arXiv
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| Autores principales: | , , |
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| Formato: | Preprint |
| Publicado: |
2025
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| _version_ | 1866913806502854656 |
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| author | Durham, Matthew Gentry Minsky, Yair Sisto, Alessandro |
| author_facet | Durham, Matthew Gentry Minsky, Yair Sisto, Alessandro |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2504_17048 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Asymptotically CAT(0) metrics, Z-structures, and the Farrell-Jones Conjecture Durham, Matthew Gentry Minsky, Yair Sisto, Alessandro Geometric Topology Group Theory Metric Geometry 20F67, 20F65, 20F69, 57K20 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. |
| title | Asymptotically CAT(0) metrics, Z-structures, and the Farrell-Jones Conjecture |
| topic | Geometric Topology Group Theory Metric Geometry 20F67, 20F65, 20F69, 57K20 |
| url | https://arxiv.org/abs/2504.17048 |