Geodesic
Algebra in the Stone-$\check C$ech compactification: applications to topologies on groups
Algebra and discrete mathematics, no. 1 (2009), pp. 83-110.

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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.
Keywords: Stone-$\check{C}$ech compactification, product of ultrafilters, idempotents, ideals, maximality, resolvability, extremal disconnectedness. 
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     author = {I. V. Protasov},
     title = {Algebra in the {Stone-}$\check C$ech compactification: applications to topologies on groups},
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     number = {1},
     year = {2009},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/ADM_2009_1_a8/}
}
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I. V. Protasov. Algebra in the Stone-$\check C$ech compactification: applications to topologies on groups. Algebra and discrete mathematics, no. 1 (2009), pp. 83-110. http://geodesic.mathdoc.fr/item/ADM_2009_1_a8/