Here is a new paper of mine. The story is this. A couple of years ago I was reading the paper (the first version). I was surprised finding there a statement answering a 30-years old question of van den Dries and Wilkie (if one asymptotic cone of a group is locally compact, should the group be virtually nilpotent?). The question appears in the very first paper about asymptotic cones. At that time it was not even known that a group can have several asymptotic cones (up to homeomorphism). I became less surprised when I discovered that the proof was wrong. My then student Curt Kent actually found a mistake, But that made me think about this question. I thought that the positive answer may follow from a then recent paper by Shalom and Tao. So I asked Yehuda Shalom about it. He asked Terry Tao, and Terry responded that although he does not see how the answer can follow from their paper, it most probably follows from the paper by Hrushovsky. And indeed, the result easily follows from that paper. So now that my paper was needed for a special issue of IJAC dedicated to Stuart Margolis’ birthday, i decided to write a short note about it. Although the proof is very easy modulo Hrushovsky’s paper, it does solve a 30 years old problem. In fact I prove that even if one ball in one asymptotic cone of a group is compact, then the group is virtually nilpotent (and then all asymptotic cones are locally compact and bi-Lipschitz equivalent).
A paper about locally compact asymptotic cones