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Translation-finite sets

The families of right (left) translation finite subsets of a discrete infinite group $Γ$ 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 $βΓ$. Consequently it is shown that the map $π: βΓ\to Γ^{WAP}$, the canonical semigroup homomorphism from $βΓ$ onto $Γ^{WAP}$, the universal semitopological semigroup compactification of $Γ$, is a homeomorphism on the complement of $Z_R \cup Z_L$.