On the geometry of left-orders

Given a finitely generated group with a left invariant order we investigate the geometry of the subset of the Cayley graph consisting of elements greater than the identity (the positive cone of the order). We will give examples where all the positive cones are coarsely connected (e.g. Braid groups) and examples where no positive cone is coarsely connected (e. g. hyperbolic surface groups and free groups). This is based on a joint work with J. Alonso, J. Brum and C. Rivas.