A survey on classifying hyperbolic actions of groups

This paper is a survey of results proved in recent years that pertain to classifying cobounded hyperbolic actions of any group $G$. In other words, we discuss results that allow us to describe the partially ordered set $\mathcal{H}(G)$, first introduced in by Abbott-Balasubramanya-Osin. In certain cases, a complete classification of the poset is possible. In cases where this seems out of reach, we provide descriptions of large subsets of the poset. This covers a wide range of groups, including acylindrically hyperbolic groups, nilpotent groups, many solvable groups and ``Thompson-like" groups. We also cover the range of strategies utilized to obtain these classification results. As a result, we produce some new examples of groups with interesting $\mathcal{H}(G)$ structure.