What is the “right” large-scale geometry for a locally compact abelian group?

Large-scale geometry, also known as coarse geometry, is the branch of mathematics that studies global, large-scale properties of spaces. Since the breakthrough work of Gromov, large-scale geometry has played a prominent role in geometric group theory and, in particular, in the study of finitely generated groups and their word metrics. This large-scale metric approach was successfully extended up to the class of locally compact -compact groups by Cornulier and de la Harpe. To study more general groups and topological groups, coarse structures, introduced by Roe as the large-scale counterpart of uniformities, are required. Several alternatives for topological groups have appeared in the literature, for example, the left coarse structure, introduced by Rosendal, and the compact-group coarse structure, induced by the family of all relatively compact subsets. 
During the first part of the talk, we present various coarse structures and focus on results concerning the compact-group coarse structure of locally compact abelian groups. In particular, we emphasise the role of Pontryagin duality as a bridge between topological properties and their large-scale counterparts. Then, we discuss the relation between the compact-group coarse structure and the left-coarse structure, showing that they coincide for locally compact abelian groups. To conclude, we provide an application of this result to the theory of Banach spaces.
This talk is based on a joint work with Dmitri Shakhmatov and Takamitsu Yamauchi.