Joint IAS/PU Groups and Dynamics Seminar

The normal rank (or weight) of a group G is the smallest number of elements that normally generate G. This plays an important role in 3-manifold topology, but it is poorly understood. It is extremely difficult to give lower bounds apart from looking at the abelianization. The Wiegold problem asks if there is a finitely generated perfect group that has normal rank greater than one, namely the group cannot be normally generated by any single element. In a joint work with Yash Lodha, we show free products of nontrivial left-orderable groups all have normal rank greater than one, which solves the Wiegold problem. Our result also has nice implications about Dehn surgeries of 3-manifolds. I will explain the topological and dynamical ingredients in the proof of our theorem.The normal rank (or weight) of a group G is the smallest number of elements that normally generate G. This plays an important role in 3-manifold topology, but it is poorly understood. It is extremely difficult to give lower bounds apart from looking at the abelianization. The Wiegold problem asks if there is a finitely generated perfect group that has normal rank greater than one, namely the group cannot be normally generated by any single element. In a joint work with Yash Lodha, we show free products of nontrivial left-orderable groups all have normal rank greater than one, which solves the Wiegold problem. Our result also has nice implications about Dehn surgeries of 3-manifolds. I will explain the topological and dynamical ingredients in the proof of our theorem.