论文标题
虚拟连接的准清理和相对双曲组中产品的可分离性
Quasiconvexity of virtual joins and separability of products in relatively hyperbolic groups
论文作者
论文摘要
如果所有有限生成的相对Quasiconvex亚组在Profinite拓扑中关闭了$ G $，则相对双曲线$ G $ ne QCERF。假设$ g $是一个相对双曲线的QCERF，具有双固定（例如，实际上是多环形）外围亚组。给定两个有限生成的相对Quasiconvex子组$ q，r \ leqslant g $，我们证明存在有限索引子组的存在$ q'\ leqslant_f q $和$ r'\ leqslant_f r $ join $ \ langle q'，r'，r'\ rangle $ rang $ s y s y yes $ s in然后，我们表明，在周围亚组的最小必要假设下，有限生成的相对Quasiconvex子组的产物在$ g $上的profinite拓扑中封闭。由此，我们获得了有限生成的亚组的乘积，包括限制组，kleinian组和具有环状边缘组的自由组有限图的平衡基本组。
A relatively hyperbolic group $G$ is said to be QCERF if all finitely generated relatively quasiconvex subgroups are closed in the profinite topology on $G$. Assume that $G$ is a QCERF relatively hyperbolic group with double coset separable (e.g., virtually polycyclic) peripheral subgroups. Given any two finitely generated relatively quasiconvex subgroups $Q,R \leqslant G$ we prove the existence of finite index subgroups $Q'\leqslant_f Q$ and $R' \leqslant_f R$ such that the join $\langle Q',R'\rangle$ is again relatively quasiconvex in $G$. We then show that, under the minimal necessary hypotheses on the peripheral subgroups, products of finitely generated relatively quasiconvex subgroups are closed in the profinite topology on $G$. From this we obtain the separability of products of finitely generated subgroups for several classes of groups, including limit groups, Kleinian groups and balanced fundamental groups of finite graphs of free groups with cyclic edge groups.
