Coarse injectivity, hierarchical hyperbolicity, and semihyperbolicity

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.

35 pages

References in corpus (5)

• Poc Sets, Median Algebras and Group Actions
• A combinatorial take on hierarchical hyperbolicity and applications to quotients of mapping class groups
• Stable cubulations, bicombings, and barycenters
• Hierarchical hyperbolicity of hyperbolic-2-decomposable groups
• Strongly Shortcut Spaces

Cited by in corpus (7)

• Stable cubulations, bicombings, and barycenters
• Lattices in a product of trees, hierarchically hyperbolic groups, and virtual torsion-freeness
• Extending and improving conical bicombings
• Strongly Shortcut Spaces
• Extensions of Veech groups II: Hierarchical hyperbolicity and quasi-isometric rigidity
• Relatively hyperbolic groups with strongly shortcut parabolics are strongly shortcut
• Visual maps between coarsely convex spaces