In this paper, we study strongly quasiconvex subgroups in a finitely generated 3-manifold group $$\pi _1(M)$$ π 1 ( M ) . We prove that if M is a compact, orientable 3-manifold that does not have a summand supporting the Sol geometry in its sphere-disc decomposition then a finitely generated subgroup $$H \le \pi _1(M)$$ H ≤ π 1 ( M ) has finite height if and only if H is strongly quasiconvex. On the other hand, if M has a summand supporting the Sol geometry in its sphere-disc decomposition then $$\pi _1(M)$$ π 1 ( M ) contains finitely generated, finite height subgroups which are not strongly quasiconvex. We also characterize strongly quasiconvex subgroups of graph manifold groups by using their finite height, their Morse elements, and their actions on the Bass-Serre tree of $$\pi _1(M)$$ π 1 ( M ) . This result strengthens analogous results in right-angled Artin groups and mapping class groups. Finally, we characterize hyperbolic strongly quasiconvex subgroups of a finitely generated 3-manifold group $$\pi _1(M)$$ π 1 ( M ) by using their undistortedness property and their Morse elements.

中文翻译：

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三流形群中的拟凸性

在本文中，我们研究了有限生成的 3 流形群 $$\pi _1(M)$$ π 1 ( M ) 中的强拟凸子群。我们证明，如果 M 是一个紧凑的、可定向的 3-流形，在其球盘分解中没有支持 Sol 几何的被加数，那么一个有限生成子群 $$H \le \pi _1(M)$$ H ≤ π 1 ( M ) 具有有限高度当且仅当 H 是强拟凸。另一方面，如果 M 有一个在其球盘分解中支持 Sol 几何的被加数，则 $$\pi _1(M)$$ π 1 ( M ) 包含不是强拟凸的有限生成的有限高度子群。我们还通过使用它们的有限高度、它们的莫尔斯元素以及它们在 $$\pi _1(M)$$ π 1 ( M ) 的 Bass-Serre 树上的作用来刻画图流形群的强拟凸子群。该结果加强了直角 Artin 组和映射类组中的类似结果。最后，我们通过使用它们的不失真性质和它们的莫尔斯元素来刻画有限生成的 3 流形群 $$\pi_1(M)$$ π 1 ( M ) 的双曲强拟凸子群。