On groups with few Busemann points

The horofunction boundary of groups is a way to compactify any metric space. It was introduced by Gromov, and primarily studied for hyperbolic groups. I’ll explain why studying the action of more general groups on their boundary -- in particular finding finite orbits and small invariant subsets -- can be interesting from an algebraic point of view. A natural invariant subset to look at consists of Busemann points, i.e. limits of geodesic rays. In joint work with Liran Ron-Geva, Ariel Yadin, and Kenshiro Tashiro, we characterise when the set of Busemann points is finite and explore when it can be countable.