ar X iv : d g - ga / 9 60 30 10 v 1 1 9 M ar 1 99 6 QUASI - RIGIDITY OF HYPERBOLIC 3 - MANIFOLDS AND SCATTERING THEORY

Suppose Γ1 and Γ2 are two convex co-compact co-infinite volume discrete subgroups of PSL(2,C), so that there exists a C∞ diffeomorphism ψ : Ω(Γ1)/Γ1 → Ω(Γ2)/Γ2 that induces an isomorphism φ : Γ1 → Γ2. For fixed s ∈ C, let Si(s) be the scattering operator on Ω(Γi)/Γi (i = 1, 2). Define Srel(s) = S1(s)− ψ ∗S2(s), where ψ ∗S2(s) is the pull-back of S2(s) to an operator acting on the appropriate complex line bundle over Ω(Γ1)/Γ1. Our main result is: if the operator norm of Srel(s) is ǫ-small, then the Γi are K(ǫ)-quasi-conformally conjugate and the dilatation K(ǫ) decreases to 1 as ǫ decreases to 0. 1. Statement of Results Geometrically finite Kleinian groups uniformizing infinite volume hyperbolic 3-manifolds exhibit a rich deformation theory due to work of Ahlfors, Bers, Kra, Marden, Maskit, Thurston and others. The purpose of this note is to introduce scattering theory as an analytic tool in the study of the deformations of complete geometrically finite hyperbolic structures. The results in this paper are restricted to a sub-class of geometrically finite Kleinian groups called convex co-compact groups, i.e. those containing no parabolic subgroups. Assume that Γ is a convex co-compact, torsion-free Kleinian group with non-empty regular set Ω(Γ) (see Section 2 for definitions). The compact (possibly disconnected) quotient surface Ω(Γ)/Γ is the conformal boundary at infinity of the hyperbolic 3-manifold M(Γ) = H3/Γ. To the Laplacian ∆ on M(Γ) we can associate a scattering operator acting on sections of certain complex line bundles over the conformal boundary Ω(Γ)/Γ. These sections are most conveniently described as automorphic forms on Ω(Γ). For a complex parameter s, let Fs(Γ) be the space of automorphic forms of weight s on Ω(Γ) (see Section 3 for the definition). The scattering operator S(s) is a pseudodifferential operator [10] with known singularity mapping F2−s(Γ) → Fs(Γ). For Re s = 1 we have a natural L 2 inner product on Fs(Γ), so we can complete these spaces to form Hilbert spaces. Date: February 7, 2008. First author supported in part by NSF grant DMS-9401807. Third author partially supported by a University of Michigan Rackham Fellowship. 1 2 DAVID BORTHWICK, ALAN MCRAE, AND EDWARD C. TAYLOR Now take two convex co-compact, torsion-free Kleinian groups Γi=1,2, with Ω(Γi=1,2) 6= ∅. Assume there exists an orientation-preserving diffeomorphism ψ : Ω(Γ1) → Ω(Γ2) that induces an isomorphism φ : Γ1 → Γ2. Denote by S2(s) the scattering operator acting on sections over Ω(Γ2)/Γ2. Recall the diffeomorphism ψ descends to a diffeomorphism ψ : Ω(Γ1)/Γ1 → Ω(Γ2)/Γ2. Thus we can pull S2(s) back, via the diffeomorphism ψ, to an operator taking acting on sections over Ω(Γ1)/Γ1. Denote this pull-back of S2 by ψ S2(s). Perry [17] shows that if for some Re s = 1, s 6= 1, the operator Srel(s) = S1(s)− ψ S2(s) is trace-class with respect to the Hilbert space completions of F2−s(Γ1) and Fs(Γ1), then Srel(s) = 0 and the diffeomorphism is actually a Möbius transformation, i.e. the manifolds M(Γi) = H /Γi are isometric. Our results show that the size of the operator Srel(s) detects how close to being isometric the quotient manifolds are. Recall that the deformation theory of Kleinian groups is based on the notion of quasi-conformal conjugacy (see section 2). Our main result is: Main Theorem: Suppose Γi=1,2 are convex co-compact, torsion-free Kleinian groups so that M(Γi=1,2) has infinite hyperbolic volume. Let ψ : Ω(Γ1) → Ω(Γ2) be an orientation-preserving C∞-diffeomorphism conjugating Γ1 to Γ2. Fix s ∈ C : Re(s) = 1, s 6= 1 and let ǫ > 0. There is K(ǫ) > 1 so that ‖Srel(s)‖ < ǫ implies that Γ2 is a K(ǫ)-quasi-conformal deformation of Γ1, where K(ǫ) → 1 as ǫ → 0. The norm ‖ · ‖ in the Theorem is the operator norm for the L2 space of sections. Independently, Douady-Earle [8], Reimann [18] and Thurston [19] have shown that each K-quasi-conformal deformation of a Kleinian group can be extended to an equivariant K̃-quasi-isometry of H3, where K̃ → 1 as K → 1. Thus the Main Theorem says that if Srel(s) is small in the operator norm, then M(Γ1) is “nearly isometric” to M(Γ2). In particular, Srel(s) = 0 implies that M(Γ1) is isometric to M(Γ2) (a fact which was contained in Perry’s result [17]). The plan for this paper is as follows: Section 2 and and Section 3 discuss respectively the basics of Kleinian group theory and scattering theory we will be using. Section 4 contains the proof of the Main Theorem, as well as various remarks and conjectures. Acknowledgements: The authors would like to thank Peter Perry for explaining the contents of [17] to the third author while Professor Perry was visiting the SUNY-Stony Brook during the Spring of 1993. We are also indebted to Richard Canary of the University of Michigan for helpful and enjoyable conversations on the contents of this paper.