This talk concerns three classes of metric spaces defined by
nonpositive curvature-like properties: hierarchically hyperbolic
spaces, coarse Helly spaces and strongly shortcut spaces. Roughly
speaking, hierarchically hyperbolic spaces are metric spaces whose
geometry can be understood through a structured family of projections
onto Gromov-hyperbolic spaces; coarse Helly spaces are metric spaces
whose balls satisfy a coarse version of the classical 1-dimensional
Helly property; and strongly shortcut spaces are metric spaces whose
subspaces cannot approximate long circles with arbitrary precision.
I will discuss recent joint work with Thomas Haettel and Harry Petyt in which we prove that hierarchically hyperbolic spaces are coarsely Helly and that coarse Helly spaces of uniformly bounded geometry are strongly shortcut. This work has important consequences for hierarchically hyperbolic groups: it shows that they are semihyperbolic, have solvable conjugacy problem, are of type FP∞, have finitely many conjugacy classes of finite subgroups and that their finitely generated abelian subgroups are undistorted.