Bohr almost periodic maps into $K(\pi ,1)$ spaces

Abstract:

Let $X$ be a locally finite simplicial complex of finite topological dimension. Assume further that $X$ is a $K(\pi ,1)$ space where $\pi$ is a group whose only abelian subgroups are infinite cyclic. We prove that a Bohr almost periodic map of the real line into $X$ is uniformly homotopic to a periodic map. As a consequence we show that a Bohr almost periodic geodesic on a compact Riemannian manifold of everywhere negative curvature is necessarily periodic.