[Paper Review] Coarse injectivity, hierarchical hyperbolicity, and semihyperbolicity

Thomas Haettel, Nima Hoda+1 more|arXiv (Cornell University)|Sep 29, 2020

Geometric and Algebraic Topology55 references17 citations

TL;DR

This paper establishes a new metric on hierarchically hyperbolic spaces that is coarsely injective and quasi-isometric to the original metric, proving that all hierarchically hyperbolic groups are coarsely injective and hence strongly shortcut. It further shows that coarsely injective spaces of uniformly bounded geometry are strongly shortcut, leading to strong structural and algorithmic properties for these groups, including semihyperbolicity, solvable conjugacy problem, and bounded packing for hierarchically quasiconvex subgroups.

ABSTRACT

We relate three classes of nonpositively curved metric spaces: hierarchically hyperbolic spaces, coarsely injective spaces, and strongly shortcut spaces. We show that every hierarchically hyperbolic space admits a new metric that is coarsely injective. The new metric is quasi-isometric to the original metric and is preserved under automorphisms of the hierarchically hyperbolic space. We show that every coarsely injective metric space of uniformly bounded geometry is strongly shortcut. Consequently, hierarchically hyperbolic groups -- including mapping class groups of surfaces -- are coarsely injective and coarsely injective groups are strongly shortcut. Using these results, we deduce several important properties of hierarchically hyperbolic groups, including that they are semihyperbolic, have solvable conjugacy problem, have finitely many conjugacy classes of finite subgroups, and that their finitely generated abelian subgroups are undistorted. Along the way we show that hierarchically quasiconvex subgroups of hierarchically hyperbolic groups have bounded packing.

Motivation & Objective

• To establish a new quasi-isometric metric on hierarchically hyperbolic spaces that is coarsely injective and preserved under automorphisms.
• To prove that every coarsely injective space of uniformly bounded geometry is strongly shortcut.
• To deduce strong algorithmic and geometric properties—such as semihyperbolicity and solvable conjugacy problem—for hierarchically hyperbolic groups.
• To show that hierarchically quasiconvex subgroups of hierarchically hyperbolic groups have bounded packing.
• To unify three classes of nonpositively curved spaces: hierarchically hyperbolic, coarsely injective, and strongly shortcut spaces.

Proposed method

• Construct a new metric σ on a hierarchically hyperbolic space (X, S) using a generalization of Bowditch’s injective metric construction on finite-rank median spaces.
• Prove that the new metric σ is coarsely injective and quasi-isometric to the original metric d via a coarse Helly property for balls, derived from a hierarchical generalization of Chepoi, Dragan, and Vaxès’ result on hyperbolic spaces.
• Use the fact that a metric space is coarsely injective if and only if it is coarsely dense in its injective hull, and construct a minimal radius function in the injective hull to define the new metric.
• Apply a coarse Helly property for balls in hierarchically hyperbolic spaces by showing that pairwise close balls have uniformly bounded intersection under the new metric.
• Prove that coarsely injective spaces of uniformly bounded geometry satisfy the strong shortcut property by bounding the number of points in a ball of fixed radius under a (K,C)-quasi-isometric embedding of a Riemannian circle.
• Use the construction of a sequence of minimal radius functions in the injective hull to derive a contradiction when assuming unbounded geometry under such embeddings, proving strong shortcutness.

Experimental results

Research questions

• RQ1 Can every hierarchically hyperbolic space be equipped with a coarsely injective metric quasi-isometric to its original metric, preserved under automorphisms?
• RQ2 Is every coarsely injective metric space of uniformly bounded geometry strongly shortcut?
• RQ3 Do hierarchically hyperbolic groups inherit strong algorithmic properties such as semihyperbolicity and solvable conjugacy problem from their coarsely injective structure?
• RQ4 Do hierarchically quasiconvex subgroups of hierarchically hyperbolic groups have bounded packing?
• RQ5 How do the classes of hierarchically hyperbolic, coarsely injective, and strongly shortcut spaces relate in terms of inclusion and geometric properties?

Key findings

• Every hierarchically hyperbolic space admits a new metric σ that is coarsely injective and quasi-isometric to the original metric, and σ is invariant under the automorphism group of the space.
• Every coarsely injective metric space of uniformly bounded geometry is strongly shortcut, establishing a direct link between coarse injectivity and strong shortcutness.
• As a consequence, all hierarchically hyperbolic groups—including mapping class groups—are coarsely injective and hence strongly shortcut.
• Hierarchically hyperbolic groups are semihyperbolic, have solvable conjugacy problem, and finitely many conjugacy classes of finite subgroups.
• Finitely generated abelian subgroups of hierarchically hyperbolic groups are undistorted, and hierarchically quasiconvex subgroups have bounded packing.
• The construction yields a canonical, equivariant, and roughly geodesic metric on hierarchically hyperbolic groups that satisfies the strong shortcut property.