DTC ultrafilters on groups

TL;DR

This paper investigates the existence of DTC ultrafilters on groups, showing they do not exist for virtually BFC groups but do for certain non-virtually FC groups, linking group structure to ultrafilter properties.

Contribution

It characterizes when DTC ultrafilters exist on groups, connecting ultrafilter existence to group-theoretic properties like being virtually abelian.

Findings

DTC ultrafilters do not exist for virtually BFC groups.

DTC ultrafilters exist for countable groups that are not virtually FC.

An infinite finitely generated group is virtually abelian iff it admits no DTC ultrafilter.

Abstract

We say that an ultrafilter on an infinite group $G$ is DTC if it determines the topological centre of the semigroup $\beta G$. We prove that DTC ultrafilters do not exist for virtually BFC groups, and do exist for the countable groups that are not virtually FC. In particular, an infinite finitely generated group is virtually abelian if and only if it does not admit a DTC ultrafilter.