Fundamental groups of finite volume, bounded negatively curved 4-manifolds are not 3-manifold groups

2013/08/30 by Grigori Avramidi, Avramidi, Grigori, T. Tâm Nguyễn Phan +4
Mathematics · #57R99 #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometric Topology (math.GT) #Geometric and Algebraic Topology #Geometry and complex manifolds #math.DG #math.GT #msc:57R99

paper · pdf · doi:10.48550/arxiv.1309.0043

9 pages

arxiv created 2013/08/30 · openalex publication_date 2013/08/30 · arxiv updated 2013/09/03 · openalex created_date 2016/06/24 · openalex updated_date 2026/07/28

Abstract

We study noncompact, complete, finite volume, Riemannian 4-manifolds M with sectional curvature -1<K<0. We prove that π₁ M cannot be a 3-manifold group. A classical theorem of Gromov says that M is homeomorphic to the interior of a compact manifold \M with boundary ∂\barM. We show that for each π₁-injective boundary component C of \M, the map i_* induced by inclusion i\colon C→ \M has infinite index image i_*(π₁ C) in π₁ \M. We also prove that M cannot be homotoped to be contained in ∂\M.

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