Horofunctions and virtual homomorphisms on Groups

In geometric group theory, finitely generated groups are studied from the geometric properties of their Cayley graphs. This point of view sees the group as a metric space, and allows one to apply methods of metric geometry, such as the metric compactification (also called the horofunction compactification). When considering the canonical action of the group on its boundary in such compactifications, it turns out that finite orbits are closely related to virtual homomorphisms. In this talk we will go over definitions, examples, known results and open problems for current and future research.