Translation-finite sets

Abstract

The families of right (left) translation finite subsets of a discrete infinite group $\Gamma$ are defined and shown to be ideals. Their kernels $Z_R$ and $Z_L$ are identified as the closure of the set of products $pq$ ($p\cdot q$) in the Čech-Stone compactification $\beta \Gamma$. Consequently it is shown that the map $\pi :\beta \Gamma \to {\Gamma }^{WAP}$, the canonical semigroup homomorphism from $\beta \Gamma$ onto ${\Gamma }^{WAP}$, the universal semitopological semigroup compactification of $\Gamma$, is a homeomorphism on the complement of $Z_R\cup Z_L$.