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

I. V. Protasov

Algebra and discrete mathematics, no. 1 (2009), pp. 83-110.

Résumé

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 $\mathrm{Im}$ and let $\bar{\tau }$ be the set of all ultrafilters on $G$ converging to the unit of $G$ in $\mathrm{Im}$. Then $\bar{\tau }$ is a closed subsemigroup of $\beta G$. We survey the results clarifying the interplays between the algebraic properties of $\bar{\tau }$ and the topological properties of $(G,\mathrm{Im})$ 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 $G^{\ast }$, Ramsey-like ultrafilters, Maximality, Refinements, Resolvability, Potential compactness and ultraranks, Selected open questions.

Détail

Mots-clés : Stone-$\check{C}$ech compactification, product of ultrafilters, idempotents, ideals, maximality, resolvability, extremal disconnectedness.