Discontinuity of multiplication and left translations in $\beta G$

Abstract:

The operation of a discrete group $G$ naturally extends to the Stone-Čech compactification $\beta G$ of $G$ so that for each $a\in G$, the left translation $\beta G\ni x\mapsto ax\in \beta G$ is continuous, and for each $q\in \beta G$, the right translation $\beta G\ni x\mapsto xq\in \beta G$ is continuous. We show that for every Abelian group $G$ with finitely many elements of order 2 such that $|G|$ is not Ulam-measurable and for every $p,q\in G^*=\beta G\setminus G$, the multiplication $\beta G\times \beta G\ni (x,y)\mapsto xy\in \beta G$ is discontinuous at $(p,q)$. We also show that it is consistent with ZFC, the system of usual axioms of set theory, that for every Abelian group $G$ and for every $p,q\in G^*$, the left translation $G^*\ni x\mapsto px\in G^*$ is discontinuous at $q$.