Limits of quasi-Fuchsian groups with small bending

Abstract

We study limits of quasifuchsian 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 [22], that the closure of pleating varieties for quasifuchsian 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 2π. AMS classification numbers: 30F40, 20H10, 32G15.

Lengths are coordinates for convex structures

“…It is not hard to show that when restricted to P(N, α), the map L : R(N) → C d is real-valued and coincides with L. We remark that the map L is not globally non-singular; in fact we showed in [32] that if G is quasifuchsian, then dL is singular at Fuchsian groups on the boundary of P(N, α).…”

“…The results should have various applications. For example, combining Theorem A with [32], when the holonomy of N is quasifuchsian, one should be able to exactly locate P(N, α) in R(N).…”

The Maskit embedding of the twice punctured torus

“…We restrict to pleating rays for which OE is rational, that is, supported on closed curves, and for simplicity write P in place of P OE , although noting that P depends only on OE . From general results of Bonahon and Otal [5] (see Theorem 3.1 in Section 3), for any pants decomposition 1 ; 2 such that 1 ; 2 ; 1 ; 2 are mutually nonhomotopic and fill up †, and any pair of angles  i 2 .0; /, there is a unique group in M for which the bending measure of @C C =G is In contrast to the quasifuchsian situation studied by the author in [28], there is no algebraic limit along P as  ! 0; see Corollary 6.5.…”

“…Proof Join P to a variable point X on between P and P 0 (see Figure 3 in [28]). If PX has length x , the distance from P to X along is t , and the acute angle between PX and at X is , then at every non-bend point of , one has the usual variational formula dx=dt D cos (see Lemma 4.2.12 of [6]).…”

Lines of minima are uniformly quasigeodesic

“…Let S be a hyperbolizable surface of finite type and -(S) be the Teichmüller space of S. Let ν + and ν − be two measured laminations that fill up S. The associated line of minima is the path t → ᏸ t ∈ -(S), where ᏸ t = ᏸ t (ν + , ν − ) is the unique hyperbolic surface that minimizes the length function e t l(ν + ) + e −t l(ν − ) on -(S); see [Kerckhoff 1992] and Section 2 below. Lines of minima have significance for hyperbolic 3-manifolds: infinitesimally bending ᏸ t along the lamination ν + results in a quasifuchsian group whose convex core boundary has bending measures in the projective classes ν + and ν − and in the ratio e 2t : 1; see [Series 2005]. In this paper we prove: Theorem A.…”