Actions of small cancellation groups on hyperbolic spaces

Abstract: We generalize Gruber--Sisto's construction of the coned--off graph of a small cancellation group to build a partially ordered set $\mathcal{TC}$ 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 $\mathcal{TC}$ is incredibly rich: it has a largest element if and only if it has exactly 1 element, and given any two distinct comparable actions $[G\curvearrowright X] \preceq [G\curvearrowright Y]$ in this poset, there is an embeddeding $\iota:P(\omega)\to\mathcal{TC}$ such that $\iota(\emptyset)=[G\curvearrowright X]$ 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.