## Self-bumping of deformation spaces of hyperbolic 3-manifolds.(English)Zbl 1030.57028

Let $$N$$ be a hyperbolic 3-manifold and $$B$$ a component of the interior of the deformation space of isometry classes of marked, complete hyperbolic 3-manifolds homotopy equivalent to $$N$$. In the present paper, topological conditions are given on $$N$$ implying the existence of an element $$\rho$$ of the closure of $$B$$ such that the intersection of every sufficiently small neighbourhood of $$\rho$$ with $$B$$ is disconnected (that is, the component “self-bumps”). In particular, the closure of $$B$$ is not a manifold with boundary. In a previous work, J. W. Anderson, R. E. Canary and D. McCullough [Ann. Math. (2) 152, 693-741 (2000; Zbl 0976.57016)] gave necessary and sufficient conditions for two components of the interior of the deformation space to “bump” (that is, to have intersecting closures).

### MSC:

 57M50 General geometric structures on low-dimensional manifolds 30F40 Kleinian groups (aspects of compact Riemann surfaces and uniformization) 57N10 Topology of general $$3$$-manifolds (MSC2010)

### Keywords:

Kleinian group; hyperbolic 3-manifold; deformation space

Zbl 0976.57016
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