Actions of small cancellation groups on hyperbolic spaces - Abbott, C; Hume, D (Springer Verlag, 2020)
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↷X]⪯[G↷Y] in this poset, there is an embeddeding ι:P(ω)→TC such that ι(∅)=[G↷X] and ι(N)=[G↷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.