We show that a group that is hyperbolic relative to strongly shortcut groups is itself strongly shortcut, thus obtaining new examples of strongly shortcut groups. The proof relies on a result of independent interest: we show that every relatively hyperbolic group acts properly and cocompactly on a graph in which the parabolic subgroups act properly and cocompactly on convex subgraphs.

中文翻译：

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具有强捷径抛物线的相对双曲群是强捷径

我们证明，相对于强快捷群是双曲线的群本身就是强快捷群，从而获得了强快捷群的新例子。证明依赖于独立兴趣的结果：我们证明每个相对双曲群在图上正确且协紧地起作用，其中抛物线子群在凸子图上正确且协紧地起作用。