Limits of quasi-Fuchsian groups with small bending

Abstract

We study limits of quasi-Fuchsian groups for which the bending measures on the convex hull boundary tend to zero, giving necessary and sufficient conditions for the limit group to exist and be Fuchsian. As an application, we complete the proof of a conjecture made in [24, Conjecture 6.5] that the closures of pleating varieties for quasi-Fuchsian groups meet Fuchsian space exactly in Kerckhoff's lines of minima of length functions. Doubling our examples gives rise to a large class of cone manifolds which degenerate to hyperbolic surfaces as the cone angles approach $2\pi$.