University of Arkansas Topology Seminar: 11/19/2020

Speaker: Nima Hoda

Title: Hierarchical hyperbolicity, coarse Helly spaces and the strong shortcut property

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.