Examples of Non-Busemann Horoballs

Abstract

In the f.g. setting, we construct horoballs which are "not centered around a geodesic" for generalized Heisenberg groups and all wreath products with an infinite acting group. That is, we find are limits of balls -- called horoballs -- which are not increasing unions of balls around points on a geodesic -- called Busemann horoballs. In the case of wreath products this follows from the stronger fact that there exist disconnected horoballs; on the lamplighter group we exhibit limits of Busemann horoballs which are not even coarsely connected. In the case of generalized Heisenberg groups, we use a symmetry argument instead; in fact for the $3$-dimensional Heisenberg group, at least under a particular generating set, we show that all horoballs are connected. This note was written somewhat in isolation from the literature, in response to a recent preprint of Epperlein and Meyerovitch asking for examples of non-Busemann horoballs. It turns out that most of the results mentioned above are known: The results about connectedness follow from known results about almost convexity. The non-coarsely connected horoballs in the lamplighter group, and connected non-Busemann horoballs in the Heisenberg group, may be new statements, but here also we point out some strongly related literature.