Hyperbolic structures on Houghton groups

Given a group G, its poset of hyperbolic structures H(G) encodes all the possible cobounded actions of G on hyperbolic spaces. In this article, we describe the poset H(Hn) for every Houghton group Hn, n ≥ 2. In particular, we show that Hn admits exactly n focal hyperbolic structures. As an application, we construct the first example of a group admitting exactly one focal hyperbolic structure, answering a question of Abbott, Balasubramanya, and Osin.