Abstract
We generalize Gruber-Sisto's construction of the coned-off graph of a small cancellation group to build a partially ordered set 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 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 curved right arrow X] <= [ G curved right arrow Y] in this poset, there is an embeddeding iota : P(omega) -> T C such that iota(empty set) = [G curved right arrow X] and iota(N) = [ G curved right arrow 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.