[Geometry, Topology and Dynamics Seminar] On torsion in discrete isometry groups of negatively curved manifolds by Dr. Michael Kapovich (UC Davis)
Abstract: It is a classical theorem of Selberg that every finitely generated isometry group of a symmetric space (i.e. a finitely generated group of matrices with real coefficients) contains a finite index torsion-free subgroup. In his 1974 ICM talk Gregory Margulis asked if the same property holds for isometry groups of Hadamard manifolds (complete, simply connected manifolds of nonpositive curvature). I will prove that Selberg's property fails already for discrete isometry groups of negatively curved Hadamard manifolds of dimension 4 and what does it have to do with hyperbolic 3-manifolds fibered over the circle. I will also explain why Selberg's property holds for 3-dimensional Hadamard manifolds.