Translation-finite sets

TL;DR

This paper introduces translation-finite sets in infinite groups, characterizes their properties as ideals, and explores their relationship with the ch-Stone compactification and the universal WAP compactification.

Contribution

It defines translation-finite sets as ideals in infinite groups and identifies their kernels with specific closures in the ch-Stone compactification, linking algebraic and topological structures.

Findings

Translation-finite sets form ideals in ch-Stone compactification.

Kernels of translation-finite sets are closures of product sets.

The map etarom etainite groups is a homeomorphism outside the kernels.

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 \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 $Z_R \cup Z_L$.