Giuliano Basso (MPIM Bonn) - Spaces with distinguished geodesics

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.