Translation-finite sets

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 \v{C}ech-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 $Z_R \cup Z_L$.