Actions of small cancellation groups on hyperbolic spaces

We generalize Gruber–Sisto’s construction of the coned–off graph of a small cancellation group to build a partially ordered set $\mathcal{T}\mathcal{C}$ of cobounded actions of a given small cancellation group whose smallest element is the action on the Gruber–Sisto coned–off graph. In almost all cases $\mathcal{T}\mathcal{C}$ is incredibly rich: it has a largest element if and only if it has exactly 1 element, and given any two distinct comparable actions $[G\curvearrowright X]⪯[G\curvearrowright Y]$ in this poset, there is an embeddeding $\iota :P(\omega )\to \mathcal{T}\mathcal{C}$ such that $\iota (\mathrm{\varnothing })=[G\curvearrowright X]$ and $\iota(\mathbb N)=[G\curvearrowright Y]$. We use this poset to prove that there are uncountably many quasi–isometry classes of finitely generated group which admit two cobounded acylindrical actions on hyperbolic spaces such that there is no action on a hyperbolic space which is larger than both.