Free groups in the smallest ideal of \(\beta G\)
The paper under review is devoted to the proof of the following statement (Theorem 1.1): If \(G\) is an infinite group with cardinality \(\kappa\) which is embeddable into a direct sum of countable subgroups, then the structural group of the smallest two-sided ideal of the semigroup \(\beta G\) (the Stone-Čech compactification of \(G\)) contains copies of the free group on \(2^{2^{\kappa}}\) generators. (As the authors recall, the result holds in particular for all infinite Abelian groups.) This generalizes or refines previous results by \textit{N.~Hindman} and \textit{J.~ Pym} [Semigroup Forum 30, 177--193 (1984; Zbl 0536.22005)] and \textit{N.~Hindman} and \textit{D.~Strauss} [Algebra in the Stone-Čech compactification: theory and applications (De Gruyter Expositions in Mathematics 27, Walter de Gruyter, Berlin) (1998; Zbl 0918.22001)].
scientific article; zbMATH DE number 5547498