Significance of the horoball shift property

Determine whether the horoball shift property has independent significance beyond abelian groups and groups whose horofunctions are uniformly quasi-subadditive and have no lower bound.

Background

The paper introduces the horoball shift property as a condition ensuring that the set of non-deterministic horofunctions is invariant under the natural group action on the horofunction boundary. The authors prove that this property follows from uniform quasi-subadditivity and the absence of a lower bound for horofunctions, but they do not establish whether the property itself has broader conceptual or geometric significance.

The unresolved issue concerns whether the horoball shift property is useful for groups outside the abelian setting and outside the class of groups already known to satisfy the two sufficient conditions used in the paper.