Examples of Non-Busemann Horoballs

TL;DR

This paper constructs and analyzes horoballs in certain groups, showing that some are not centered around geodesics, with implications for understanding the geometric structure of these groups.

Contribution

It provides explicit examples of non-Busemann horoballs in wreath products and generalized Heisenberg groups, highlighting differences from classical Busemann horoballs.

Findings

Existence of disconnected horoballs in wreath products

Presence of non-coarsely connected horoballs in the lamplighter group

All horoballs are connected in the 3D Heisenberg group under certain generators

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.