Abstract:We investigate groups whose Cayley graphs have poorly connected subgraphs. We prove that a finitely generated group has bounded separation in the sense of Benjamini-Schramm-Timár if and only if it is virtually free. We then prove a gap theorem for connectivity of finitely presented groups, and prove that there is no comparable theorem for all finitely generated groups. Finally, we formulate a connectivity version of the conjecture that every group of type with no Baumslag-Solitar subgroup is hyperbolic, and prove it for groups with at most quadratic Dehn function.

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中文翻译：

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连接不良的群体

摘要：我们调查了Cayley图的子图连接不良的组。我们证明，当且仅当它实际上是自由的时，从Benjamini-Schramm-Timár的意义上说，一个有限生成的群已经限制了分离。然后，我们证明了有限表示的组的连通性的间隙定理，并证明了所有有限生成的组都没有可比的定理。最后，我们建立了猜想的连通性版本，即没有Baumslag-Solitar子组的每个类型的组都是双曲线的，并针对具有最多二次Dehn函数的组证明了这一关系。

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