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.

中文翻译：

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双曲空间上小抵消群的作用

我们概括 Gruber--Sisto 对一个小取消群的锥度图的构造，以构建一个给定小取消群的共界动作的偏序集 $\mathcal{TC}$，其最小元素是 Gruber 上的动作--Sisto 锥形--关闭图。在几乎所有情况下，$\mathcal{TC}$ 都非常丰富：当且仅当它恰好有 1 个元素，并且给定任何两个不同的可比操作 $[G\curvearrowright X] \preceq [G\curvearrowright Y]$ 在这个偏序组中，有一个嵌入的 $\iota:P(\omega)\to\mathcal{TC}$ 使得 $\iota(\emptyset)=[G\curvearrowright X]$ 和 $\iota( \mathbb N)=[G\curvearrowright Y]$。