2019/11/18 by Hoang Thanh Nguyen, Nguyen, Hoang Thanh, Hung Cong Tran +3
Mathematics · #Advanced Operator Algebra Research #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometric Topology (math.GT) #Geometric and Algebraic Topology #Group Theory (math.GR) #math.GR #math.GT
paper · pdf · doi:10.48550/arxiv.1911.07807
36 pages. Version 2 incorporates the referee's comments. To appear in Mathematische Annalen
openalex publication_date 2019/11/18 · arxiv created 2020/07/05 · arxiv updated 2020/07/07 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
In this paper, we study strongly quasiconvex subgroups in a finitely generated 3--manifold group π1(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 ≤ π1(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 π1(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 π1(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 π1(M) by using their undistortedness property and their Morse elements.