I.V. Protasov
Topologies on groups determined by right cancellable ultrafilters

Comment.Math.Univ.Carolin. 50,1 (2009) 135-139.

Abstract:For every discrete group $G$, the Stone-\v {C}ech compactification $\beta G$ of $G$ has a natural structure of a compact right topological semigroup. An ultrafilter $p\in G^*$, where $G^*=\beta G\setminus G$, is called right cancellable if, given any $q,r\in G^*$, $qp=rp$ implies $q=r$. For every right cancellable ultrafilter $p\in G^*$, we denote by $G(p)$ the group $G$ endowed with the strongest left invariant topology in which $p$ converges to the identity of $G$. For any countable group $G$ and any right cancellable ultrafilters $p,q\in G^*$, we show that $G(p)$ is homeomorphic to $G(q)$ if and only if $p$ and $q$ are of the same type.

Keywords: Stone-\v {C}ech compactification, right cancellable ultrafilters, left invariant topologies
AMS Subject Classification: Primary 54H11; Secondary 54C05, 54G15