Suppose $G$ is a 1‐ended finitely generated group that is hyperbolic relative to P, a finite collection of 1‐ended finitely generated proper subgroups of $G$. Our main theorem states that if the boundary $\partial (G,\mathbf{P})$ has no cut point, then $G$ has semistable fundamental group at $\infty$. Under mild conditions on $G$ and the members of P, the 1‐ended hypotheses and the no cut point condition can be eliminated to obtain the same semistability conclusion. We give an example that shows our main result is somewhat optimal. Finally, we improve a ‘double dagger’ result of Dahmani and Groves.

中文翻译：

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相对双曲群，无穷大的半稳定基群

假设 $G$是一个相对于P是双曲的1端有限生成组，它是1端有限生成的适当子组的有限集合。$G$。我们的主要定理指出，如果边界$\partial （G，\mathbf{P}）$ 没有切点，那么 $G$ 拥有半稳定的基本群 $\infty$。在温和条件下$G$并且可以消除P的成员，1端假设和无割点条件，以获得相同的半稳定性结论。我们举一个例子说明我们的主要结果是最佳的。最后，我们改善了Dahmani和Groves的“双匕首”结果。