We will discuss what kind of constrains combinatorial covering properties of Menger, Scheepers, and Hurewicz impose on remainders of topological groups. For instance, we show that such a remainder is Hurewicz if and only it it is \(\sigma\)-compact. Also, the existence of a Scheepers non-\(\sigma\)-compact remainder of a topological group follows from CH and yields a \(P\)-point, and hence is independent of ZFC. We also make an attempt to prove a dichotomy for the Menger property of remainders of topological groups in the style of Arhangel'skii.
This is a joint work with Angelo Bella and Secil Tokgoz.