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

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arxiv 2504.17048 v1 pith:ALQSIAFZ submitted 2025-04-23 math.GT math.GRmath.MG

Matthew Gentry Durham , Yair Minsky , Alessandro Sisto This is my paper · ORCID
classification math.GTmath.GRmath.MG
keywords asymptoticallymetricsgroupshhgsmanyadmitcolorablecomplexes
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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.

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Cited by 4 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. Periodic quasiflats in hierarchically hyperbolic spaces

    math.GR 2026-08 accept novelty 8.0 of 10

    Every hierarchically hyperbolic group that is not hyperbolic contains a Z^2 subgroup, and every virtually Z^n subgroup lies in an A-invariant uniform quasi-flat whose points are joined by hierarchy paths.

  2. Bounded cohomology, quotient extensions, and hierarchical hyperbolicity

    math.GR 2025-05 conditional novelty 6.0 of 10

    A central extension of a hierarchically hyperbolic group is hierarchically hyperbolic if and only if its Euler class is bounded.

  3. An EZ-structure for the mapping class group

    math.GT 2025-05 conditional novelty 6.0 of 10

    Hamenstadt defines an explicit geometric boundary X(S) for Mod(S) and compactifies the thick part of Teichmuller space by it, producing an EZ-structure whose boundary has strong dynamical properties.

  4. New tools in hierarchical hyperbolicity: A survey

    math.GR 2025-07 accept

    A survey of tools for hierarchical hyperbolicity, including combinatorial HHSs, injective metrics, asymptotically CAT(0) metrics, curtains, R-cubings, and higher-rank JSJ decompositions.

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