Wojciech J. Gajda

A finitely generated group is called "hyperbolic" (in the sense of Gromov) if all geodesic triangles in a Cayley graph of the group are uniformly thin. Basic examples are free groups and fundamental groups of closed Riemannian manifolds of negative sectional curvature, e.g. fundamental groups of closed surfaces of genus at least two. For a given hyperbolic group there exists a topological space associated to it - its Gromov boundary. It is the collection of classes of all geodesic rays in a Cayley graph up to a finite Hausdorff distance. The Cannon Conjecture asserts that if the Gromov boundary of a hyperbolic group is homeomorphic to the two-sphere then the group is virtually (i.e. up to a finite index subgroup) the fundamental group of a closed 3-manifold of constant curvature -1. I will explain all the notions involved in the formulation of the Cannon Conjecture and say a word or two about our recent work on the subject, joint with Ganiel Groves, Peter Haïssinsky, Jason Manning, Alessandro Sisto, and Genevieve Walsh.