Limits of quasi-Fuchsian groups with small bending

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 cl...

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Bibliographic Details
Published in:Duke mathematical journal 2005-06, Vol.128 (2), p.285-329
Main Author: Series, Caroline
Format: Article
Language:English
Subjects:
20H10
22E40
30F35
30F40
30Fxx
32G15
32Nxx
Fuchsian groups and their generalizations [See also 11F06
Kleinian groups [See also 20H10]
Moduli of Riemann surfaces
Teichmüller theory [See also 14H15
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Summary: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 π .
ISSN:0012-7094
1547-7398
DOI:10.1215/S0012-7094-04-12823-4