BIRS Workshop Lecture Videos
Generalized torsion in 3-manifold groups and normal closures of slope elements Ito, Tetsuya
If a 3-manifold group does not admit a bi-ordering, then we may expect that it has a generalized torsion element. As a particular case, the fundamental group of any 3-manifold obtained by non zero surgery on a knot in the 3-sphere may have such an element. Then there are two situations: (1) a generalized torsion element in a knot group becomes a generalized torsion element in the surgered 3-manifold, or (2) a generalized torsion element arises via the Dehn filling. The first situation leads us to study of normal closures of slope elements in a knot group. In the first talk we investigate relationships among such normal subgroups. In particular, we establish the peripheral Magnus property. In the second talk we focus on generalized torsion elements in Dehn surgered manifolds arisen from the first or the second situations. We also take a closer look at some explicit examples. In the third talk we prove that any generalized torsion element in a free product of torsion-free groups is conjugate to a generalized torsion element in some factor group. This implies that the fundamental group of a compact orientable 3-manifold M has a generalized torsion element if and only if the fundamental group of some prime factor of M has a generalized torsion element. On the other hand, we demonstrate that there are infinitely many toroidal 3-manifolds whose fundamental group has a generalized torsion element, while the fundamental group of each decomposing piece has no such elements. Additionally, in the course of the proof of the former result, we give an upper bound for the stable commutator length of generalized torsion elements.
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