The impact of Pontryagin and Bohr functors on large-scale properties of locally compact abelian groups

Following the recent introduction of functorial coarse structures on groups, we consider functorial coarse structures on topological groups. In particular, we focus on the compact-group coarse structure and we study the impact of Pontryagin and Bohr functor in small-scale and large-scale properties of locally compact abelian groups. We provide results concerning both the covering dimension and the asymptotic dimension.