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

Bromberg, K.; Holt, J.

J. Differ. Geom. 57, No. 1, 47-65 (2001).

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).

Reviewer: B.Zimmermann (Trieste)

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

Citations:

Zbl 0976.57016