Title: The Large-Scale Geometry of Locally Compact Abelian Groups
by Nicolò Zava (University of Udine) as part of Topological Groups
Lecture held in Elysium.
Large-scale geometry, also known as coarse geometry, is the branch of mathematics that studies the global, large-scale properties of spaces. This theory is distinguished by its applications which include the Novikov and coarse Baum-Connes conjectures. Since the breakthrough work of Gromov, large-scale geometry has played a prominent role in geometric group theory, in particular, in the study of finitely generated groups and their word metrics. This large-scale approach was successfully extended to all countable groups by Dranishnikov and Smith. A further generalisation introduced by Cornulier and de la Harpe dealt with locally compact σ-compact groups endowed with particular pseudo-metrics.
To study the large-scale geometry of more general groups and topological groups, coarse structures are required. These structures, introduced by Roe, encode global properties of spaces. We also mention the equivalent approach provided by Protasov and Banakh using balleans. Coarse structures compatible with a group structure can be characterised by special ideals of subsets, called group ideals. While the coarse structure induced by the family of all finite subsets is well-suited for abstract groups, the situation is less clear for groups endowed with group topologies, as exemplified by the left coarse structure, introduced by Rosendal, and the compact-group coarse structure, induced by the group ideal of all relatively compact subsets, each suitable in disparate settings.
We present the large-scale geometry of groups via the historically iterative sequence of generalisations, enlisting illustrative examples specific to distinct classes of groups and topological groups. We focus on locally compact abelian groups endowed with compact-group coarse structures. In particular, we discuss the role of Pontryagin duality as a bridge between topological properties and their large-scale counterparts. An overriding theme is an evidence-based tenet that the compact-group coarse structure is the right choice for the category of locally compact abelian groups.