In this paper we develop a geometric framework to address asymptoticity and nonexpansivity in topological dynamics when the acting group is second countable and locally compact. As an application, we show extensions of Schwartzman’s theorem in this context. Also, we get new results when the acting group is Z d {\mathbb Z}^d : any half-space of R d \mathbb {R}^d contains a vector defining a (oriented) nonexpansive direction in the sense of Boyle and Lind. Finally, we deduce rigidity properties of distal Cantor systems.