Algebra in the Stone-\(\check{C}\)ech compactification: applications to topologies on groups

Abstract

For every discrete group \(G\), the Stone-\(\check{C}\)ech compactification \(\beta G\) of \(G\) has a natural structure of compact right topological semigroup. Assume that \(G\) is endowed with some left invariant topology \(\Im\) and let \(\overline{\tau}\) be the set of all ultrafilters on \(G\) converging to the unit of \(G\) in \(\Im\). Then \(\overline{\tau}\) is a closed subsemigroup of \(\beta G\). We survey the results clarifying the interplays between the algebraic properties of \(\overline{\tau}\) and the topological properties of \((G,\Im)\) and apply these results to solve some open problems in the topological group theory.

The paper consists of 13 sections: Filters on groups, Semigroup of ultrafilters, Ideals, Idempotents, Equations, Continuity in \(\beta G\) and \(G^*\), Ramsey-like ultrafilters, Maximality, Refinements, Resolvability, Potential compactness and ultraranks, Selected open questions.