Topological groups and -points

Let κ be an infinite cardinal. A uniform ultrafilter p on κ is a -point if for every decreasing κ-sequence (Aα)α<κ of members of p, there is A∈p such that |A∖Aα|<κ for all α<κ. We show that the existence of a topological group with certain extremal properties implies the existence of a -point for some κ.