Hyperbolic groups and local connectivity
In the 1980s Gromov introduced the family of hyperbolic groups, a family of finitely generated groups that are “negatively curved” in a certain sense. Hyperbolic groups share many features with the classical hyperbolic plane. The Gromov boundary of a hyperbolic group is a certain compact space “at infinity” on which the group naturally acts. Many features of this compact fractal boundary reflect algebraic and geometric properties of the group itself.
I will discuss hyperbolic groups and their boundaries and the remarkable result that the boundary of a hyperbolic group is locally connected whenever it is connected. This deep theorem combines work of Bestvina–Mess, Bowditch, Levitt, and Swarup. Perhaps surprisingly, the key to the proof is to show that the boundary does not contain a cut point!