Arzhantseva proved that every infinite-index quasi-convex subgroup $H$ of a torsion-free hyperbolic group $G$ is a free factor in a larger quasi-convex subgroup of $G$. We give a probabilistic generalization of this result. That is, we show that when $R$ is a subgroup generated by independent random walks in $G$, then $⟨H,R⟩\cong H*R$ with probability going to one as the lengths of the random walks go to infinity and this subgroup is quasi-convex in $G$. Moreover, our results hold for a large class of groups acting on hyperbolic metric spaces and subgroups with quasi-convex orbits. In particular, when $G$ is the mapping class group of a surface and $H$ is a convex cocompact subgroup we show that $⟨H,R⟩$ is convex cocompact and isomorphic to $H*R$.

中文翻译：

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圆柱双曲群中的随机游走和拟凸性

Arzhantseva 证明了每一个无穷指数拟凸子群 $H$ 无扭双曲群 $G$ 是更大的拟凸子群中的自由因子 $G$. 我们给出了这个结果的概率概括。也就是说，我们证明当$\mathrm{电阻}$ 是由独立随机游走产生的子群 $G$， 然后 $⟨H,\mathrm{电阻}⟩\cong H*\mathrm{电阻}$ 随着随机游走的长度趋于无穷大，概率趋于 1，并且该子群在 $G$. 此外，我们的结果适用于作用于双曲度量空间的一大类群和具有准凸轨道的子群。特别是，当$G$ 是表面的映射类组，并且 $H$ 是一个凸协紧子群，我们证明 $⟨H,\mathrm{电阻}⟩$ 是凸协紧并同构于 $H*\mathrm{电阻}$.