DTC ultrafilters on groups (Q2663127)

Let \(G\) be an infinite group and let \(\beta G\) be its Čech-Stone compactification, viewed as the set of ultrafilters on \(G\) with the topology which admits as a basis the family \(\{\hat A:A\subseteq G\}\) where \(\hat{A}=\{u\in \beta G: A\in u\}\). It is known that there are two ways of extending the internal operation on \(G\) to \(\beta G\) in such a way that \(\beta G\) becomes a semigroup: \(u\square v=\{ A\subseteq G : \{x \in G : x^{-1}A\in v\}\in u\} \) \(u\lozenge v=\{A\subseteq G : \{x \in G : Ax^{-1}\in u\}\in v\}\) The authors of this highly interesting paper call DTC ultrafilters those elements \(v\in \beta G\) such that \(u\square v \ne u \lozenge v\) for every \(u\in \beta G \setminus G.\) The name DTC is an abbreviation of ``determining for the [left] topological center''; this notion was studied in [\textit{H. G. Dales} et al., Banach algebras on semigroups and on their compactifications. Providence, RI: American Mathematical Society (AMS) (2010; Zbl 1192.43001)] in the context of Banach algebras on semigroups. Recall that \(\textsf{FC}(G)\) is the subgroup of \(G\) formed by all elements with finitely many conjugates. The main results of the paper are the following: Theorem 3.1: Let \(G\) be a group and \(H\) a subgroup of finite index in \(G\). Then \(G\) admits a DTC ultrafilter if and only if \(H\) does. Theorem 3.3: Let \(G\) be an infinite group such that \(\textsf{FC}(G)=G\) and there exists \(n\in \omega,\) \(n\ge 1,\) such that \(x^ny=yx^n\) for all \(x,\,y\in G.\) Then \(G\) does not admit any DTC ultrafilter. Theorem 4.3: Let \(G\) be a countable group such that \(\textsf{FC}(G)\) has countable index in \(G\). Then \(G\) has a DTC ultrafilter. Corollary 4.6: An infinite, finitely generated group admits no DTC ultrafilters if and only if it is virtually abelian. In the final sections the authors give several illuminating examples and open problems.