  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.
