Examples of relative deformation spaces that are not locally connected (Q2391104)

In this paper, a pared manifold is a pair \((M,P)\) in which \(M\) is a compact orientable 3-manifold that is hyperbolizable and \(P\) a disjoint collection of incompressible tori and annuli in \(\partial M\) containing all tori in \(\partial M\) and satisfying the property that every \(\pi_1\)-injective map \((S^1\times I,S^1\times \partial I)\to (M,P)\) is homotopic, as a map of pairs, into \(P\). The author considers the relative deformation space \(AH(M,P)\), that is, the space of hyperbolic 3-manifolds which are homotopy equivalent to \(M\) with cusps associated to each component of \(P\). (A precise definition of \(AH(M,P)\) is given in the paper in terms of the relative character variety of representations in \(\text{PSL}(2,\mathbb C)\).) The main result of this paper is the existence of an infinite collection of pared manifolds \((M,P)\) whose relative deformation spaces are not locally connected. The author notes that the simplest examples of such manifolds are the following: \(M=S\times I\) where \(S\) is a closed surface of genus \(\geq 2\), and \(P\) is an annulus on \(S\times \{1\}\) such that \(P\) separates \(S\times \{1\}\) into a punctured torus and a once-punctured genus \((g-1)\) surface. The result extends a recent result of Bromberg that shows that the space of Kleinian punctured torus groups is not locally connected.