Non-positive curvature of fundamental groups and (coarse) topology of manifolds

One way of studying compact manifolds is either by endowing them with a geometry, following the approach used by Thurston for
3-manifolds, or by endowing their fundamental group with a (large scale) geometry. I will explain one particular way of defining
nonpositive curvature for such a fundamental group, using bicombings (an idea introduced by Busemann and pursued by Thurston)
and the impact that this property has on the (coarse) topology of the manifold itself and of its universal cover.
The talk is on joint work with Davide Spriano and Stefanie Zbinden.