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

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.