Hyperbolic structures on Houghton groups

TL;DR

This paper characterizes the hyperbolic structures on Houghton groups, revealing they have exactly n focal hyperbolic structures, and constructs a group with a unique focal hyperbolic structure, answering an open question.

Contribution

It provides a complete description of the poset of hyperbolic structures for Houghton groups and constructs the first example of a group with exactly one focal hyperbolic structure.

Findings

Houghton group H_n has exactly n focal hyperbolic structures

The poset of hyperbolic structures for H_n is explicitly described

Constructed the first example of a group with a unique focal hyperbolic structure

Abstract

Given a group $G$, its poset of hyperbolic structures $\mathcal{H}(G)$ encodes all the possible cobounded actions of $G$ on hyperbolic spaces. In this article, we describe the poset $\mathcal{H}(H_n)$ for every Houghton group $H_n$, $n \geq 2$. In particular, we show that $H_n$ 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.