Spaces with distinguished geodesics

Contact: Christian Kaiser (kaiser@mpim-bonn.mpg.de)

A bicombing on a metric space distinguishes for each pair of points a geodesic connecting them. Recently, bicombings have become a useful tool in geometric group theory, especially in the context of Helly groups. An important result of Descombes and Lang states that every Gromov hyperbolic group acts geometrically on a proper, finite-dimensional metric space with a unique bicombing satisfying a strong convexity condition. This convexity condition can be seen as a notion of non-positive curvature in metric spaces with not necessarily unique geodesics. In this talk I will give an overview of recent results on the geometry of metric spaces with such bicombings and the groups acting on them. For example, we will prove that any group acting geometrically on a proper metric space with a conical bicombing has a Z-boundary in the sense of Bestvina.