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.

中文翻译：

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组上的 DTC 超滤器

如果它确定了半群 $\beta G$ 的拓扑中心，我们就说无限群 $G$ 上的超滤子是 DTC。我们证明了 DTC 超滤器对于虚拟 BFC 组不存在，并且对于实际上不是 FC 的可数组确实存在。特别是，一个无限有限生成群实际上是阿贝尔群，当且仅当它不接受 DTC 超滤器。