Local connectedness of the space of punctured torus group

Abstract

We will give a necessary condition for local connectedness of the space of Kleinian punctured torus group using Bromgerg’s local coordinate system and provide a sufficient condition for local connectedness on a dense subset of the necessary condition. That is, the collection of the points where the boundary of the space of punctured torus group is not locally connected is a dense subset of the points satisfying the necessary condition.