拟凸虚拟并的结构
Structure of quasiconvex virtual joins
AI总结：
本文研究了相对双曲群中相对拟凸子群的虚拟并结构，证明了此类并与极大抛物子群的交集可分解为因子子群与极大抛物子群交集的并，并为抛物子群几乎相容的拟凸子群提供了组合定理。
摘要：
设$G$为相对双曲群，$Q$和$R$为相对拟凸子群。已知在$G$的profinite拓扑适当假设下，存在多对有限指数子群$Q' \leqslant_f Q$和$R' \leqslant_f R$，使得子群并$\langle Q', R' \rangle$也是相对拟凸的。我们证明，此类并与$G$的极大抛物子群的交集，本身就是因子子群$Q'$和$R'$与$G$的极大抛物子群的交集的并。作为推论，我们证明抛物子群几乎相容的拟凸子群具有抛物子群相容的有限指数子群，并为这类子群提供了一个组合定理。
英文摘要：
Let $G$ be a relatively hyperbolic group and let $Q$ and $R$ be relatively quasiconvex subgroups. It is known that there are many pairs of finite index subgroups $Q' \leqslant_f Q$ and $R' \leqslant_f R$ such that the subgroup join $\langle Q', R' \rangle$ is also relatively quasiconvex, given suitable assumptions on the profinite topology of $G$. We show that the intersections of such joins with maximal parabolic subgroups of $G$ are themselves joins of intersections of the factor subgroups $Q'$ and $R'$ with maximal parabolic subgroups of $G$. As a consequence, we show that quasiconvex subgroups whose parabolic subgroups are almost compatible have finite index subgroups whose parabolic subgroups are compatible, and provide a combination theorem for such subgroups.
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