The horofunction boundary is a compactification of metric spaces with roots in works of Busemann and Gromov. It has been widely used in the context of hyperbolic spaces and hyperbolic groups.
Our recent work focuses on studying the structure of these boundaries for "small" finitely generated groups. These have not been studied as extensively, and many basic questions are still open. An important motivation is the so-called "Gap Conjecture" by Grigorchuk.
We prove the following:
A finitely generated group is of linear growth (equivalently, virtually Z) if and only if every Cayley graph has only finitely many Busemann points in its horofunction boundary
(i.e. a finite Busemann boundary).
This is based on joint works with L. Ron-George and M. Tointon.
All notions will be explained, no prior knowledge is assumed.