Surface subgroups of Kleinian groups with torsion

2008/04/08 by Marc Lackenby, Lackenby, Marc
Biochemistry, Genetics and Molecular Biology · Mathematics · #11F06 #57M10 #57N10 #Connective tissue disorders research #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Topology (math.GT) #Geometric and Algebraic Topology #Point processes and geometric inequalities #math.DG #math.GT #msc:11F06 #msc:57M10 #msc:57N10

paper · pdf · doi:10.48550/arxiv.0804.1309

21 pages, 2 figures; final version, to appear in Invent. Math

openalex publication_date 2008/04/08 · arxiv created 2009/07/28 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We prove that every finitely generated Kleinian group that contains a finite, non-cyclic subgroup either is finite or virtually free or contains a surface subgroup. Hence, every arithmetic Kleinian group contains a surface subgroup.

Citations

• The Geometries of 3-Manifolds
• Finite extinction time for the solutions to the Ricci flow on certain three-manifolds

Related

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• Quasi-convex surface subgroups in some one-relator groups with torsion
• Solvable Subgroup Theorem for simplicial nonpositive curvature
• Subgroups of the mapping class group from the geometrical viewpoint
• The Surface Group Conjecture: Cyclically Pinched and Conjugacy Pinched One-Relator Groups