Coarse injectivity, hierarchical hyperbolicity and semihyperbolicity

We relate three classes of nonpositively curved metric spaces: hierarchically hy-perbolic spaces, coarsely injective spaces and strongly shortcut spaces. We show that every hierarchically hyperbolic space admits a new metric that is coarsely in-jective. 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 hyper-bolic groups, including that they are semihyperbolic, they have solvable conjugacy problem and finitely many conjugacy classes of finite subgroups, and their finitely generated abelian subgroups are undistorted. Along the way we show that hierar-chically quasiconvex subgroups of hierarchically hyperbolic groups have bounded packing.