Coarse injectivity, hierarchical hyperbolicity and semihyperbolicity

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