Annales Academiæ Scientiarum Fennicæ Mathematica Volumen 23, 1998, 83–100

ON THE DYNAMICS OF PSEUDO-ANOSOV HOMEOMORPHISMS ON REPRESENTATION VARIETIES OF SURFACE GROUPS

Michael Kapovich

The University of Utah, Department of Mathematics Salt Lake City, Utah 84112, U.S.A.; kapovich@math.utah.edu

Abstract. We study the action of pseudo-Anosov homeomorphisms f: R → R on the character varieties of SL(2, C)-representations of the fundamental groups π_{1}(R) of closed orientable hyperbolic surfaces R. We prove that the representation π_{1}(R) ↪ SL(2, C) corresponding to the holonomy representation of the hyperbolic structure on the mapping torus of f is a hyperbolic fixed point for the action of f on the character variety X(π_{1}(R)).

1. Introduction

Suppose that R is a closed oriented hyperbolic surface and f: R → R an orientation-preserving pseudo-Anosov homeomorphism inducing an automorphism ϕ of the fundamental group π_{1}(R). According to Thurston’s hyperbolization theorem the mapping torus M_{f} of f admits a hyperbolic structure (see [O]). The fundamental group π_{1}(M_{f}) is the semidirect product G = π_{1}(R) ⋊ Z, where a generator t of Z acts on π_{1}(R) by the conjugation x ↦ t^{−1}xt, this action is the same as the automorphism ϕ. We use the hyperbolic structure on M_{f} to realize the group G as a discrete group of isometries of the hyperbolic 3-space H^{3} . Let X(π_{1}(R)) = Hom(π_{1}(R), SL(2, C))// SL(2, C) be the character variety. The isomorphism ϕ: π_{1}(R) → π_{1}(R) induces a holomorphic automorphism

Φ: X(π_{1}(R)) → X(π_{1}(R)), Φ : [ρ] ↦ [ρ ∘ ϕ]

where [ρ] denotes the SL(2, C)-equivalence class of a representation ρ: π_{1}(R) → SL(2, C). The equivalence class of the identity embedding ι: π_{1}(R) ↪ G ⊂ SL(2, C) is a fixed point of Φ.

The main goal of this paper is to prove the following theorem that was conjectured by Curt McMullen in [Mc]:

Theorem 1.1. The point [ι] is a hyperbolic fixed point of Φ, i.e. the derivative

dΦ: T_{[ι]}X(π_{1}(R)) ⟶ T_{[ι]}X(π_{1}(R))

has no eigenvalues λ such that |λ| = 1.

1991 Mathematics Subject Classification: Primary 57M50, 22E40; Secondary 30C65, 30F40.

Remark 1.2. McMullen proved in [Mc] that there are at least two eigenvalues of dΦ which do not belong to the unit circle.

Unlike the approach of McMullen, our proof is mostly cohomological, we study the action of ϕ on the cohomology group H^{1}(π_{1}(R), sl(2, C)) of π_{1}(R), where sl(2, C) is the Lie algebra of SL(2, C) and π_{1}(R) acts on sl(2, C) via the adjoint representation ad ∘ ι. We prove the following theorem which implies Theorem 1.1:

Theorem 1.3. The action ϕ^{∗}: H^{1}(π_{1}(R), sl(2, C)) → H^{1}(π_{1}(R), sl(2, C)) has no eigenvalues λ such that |λ| = 1.

Note however that not every fixed point of Φ is hyperbolic. For instance, we can choose ϕ that acts trivially on H_{1}(R, Z), thus the trivial representation of π_{1}(R) to SL(2, C) is not a hyperbolic fixed point of the corresponding mapping Φ. More generally, for each m ≥ 1 there are hyperbolic 3-manifolds M fibered over circles so that the character variety X(π_{1}(M)) contains a smooth complex mdimensional submanifold Y^{m}. Let π_{1}(R) denote a normal surface subgroup in π_{1}(M) and Φ: X(π_{1}(R)) → X(π_{1}(R)) the automorphism corresponding to the fibration of M over S^{1} with the fiber R. Thus, for each [ρ] ∈ Y^{m} the point [ρ|_{π_{1}(R)}] is a fixed point of Φ and the derivative of Φ at [ρ|_{π_{1}(R)}] has at least m-dimensional fixed subspace.

As a warm-up to the proof of Theorem 1.3 we prove that ϕ^{∗} has no roots of unity as eigenvalues. Indeed, if λ is an eigenvalue of ϕ^{∗} such that λ^{n} = 1, then (ϕ^{n})^{∗} has a nonzero invariant vector in H^{1}(π_{1}(R), sl(2, C)). Note that for every n ≥ 1 the group G_{n} := π_{1}(R) ⋊ ⟨t^{n}⟩ acts on the hyperbolic 3-space as a uniform lattice. Now the Serre–Hochschild exact sequence (see [Br, Corollary 6.4]) implies that H^{1}(G_{n}, sl(2, C)) ≠ 0 since it is isomorphic to the space of (ϕ^{n})^{∗} - invariants on H^{1}(π_{1}(R), sl(2, C)); this contradicts the Calabi–Weil infinitesimal rigidity theorem [Ra].

Below is an outline of our proof in the general case: Suppose that we have a cocycle σ ∈ Z^{1}(π_{1}(R), sl(2, C)) such that ϕ^{#}(σ) = λσ, λ ∈ C^{∗}. Then there is a smooth vector-field ξ on H^{3} and its lift ξ̃ to the group SL(2, C) such that:

(a) ad(γ)ξ − ξ = σ_{γ} for all γ ∈ π_{1}(R).

(b) ad(t)ξ̃ = λ^{−1}ξ.

If |λ| = 1 it follows that ξ is a quasiconformal vector-field on H^{3} in the sense of Ahlfors. Then we construct a tangential extension ξ_{∞} of ξ to the ideal boundary S^{2} of H^{3} so that ξ_{∞} is a quasiconformal vector-field on S^{2} ≅ C∪{∞} which still satisfies the property (a) with respect to the action of π_{1}(R) on S^{2} . Then μ = ¯∂ξ_{∞} is a π_{1}(R)-invariant Beltrami differential, hence Sullivan’s rigidity theorem [Su] implies that μ = 0 almost everywhere.^{1} Therefore ξ_{∞} is actually a Moebius vector-field and the property (a) implies that the cocycle σ is a coboundary. This proves that the only solution τ of the equation

1 Note that we have to use Sullivan’s theorem since it is the only rigidity theorem which deals with discrete groups of infinite covolume such as π_{1}(R) ⊂ SL(2, C).

ϕ^{∗}(τ) = λτ, |λ| = 1, τ ∈ H^{1}(π_{1}(R), sl(2, C))

is the trivial cohomology class which concludes the proof.

The reader will notice that our arguments in the proof of Theorem 1.3 provide an alternative proof of the Calabi–Weil infinitesimal rigidity theorem à la Mostow. Namely, suppose that Γ ⊂ Isom(H^{n}) is a uniform lattice (n ≥ 3). Then the Calabi–Weil infinitesimal rigidity theorem states that H^{1}(Γ, ad) = 0. The usual way to prove this is to take a harmonic representative α of a class [α] ∈ H^{1}(Γ, ad) and then verify that α = 0 via Bochner’s technique. Instead we take a quasiconformal vector field ξ on H^{n} representing [α], extend ξ (tangentially) to the sphere at infinity and then check (using for instance Sullivan’s rigidity theorem) that this extension is actually a Moebius vector field. This proves that α is a coboundary.

The main technical difficulty in our proof is to establish existence of a “continuous extension” of quasiconformal vector fields from the open ball in R^{n} to its boundary. We prove that under some extra condition^{2} there is a tangential continuous extension. This extension will suffice for our purposes. The nontrivial analytical ingredient of our construction of the tangential extension is Semenov’s stability theorem (see Theorem 3.5), which is a part of the general stability theory for the spatial quasiconformal mappings. This theorem proves that k -quasiconformal vector fields in the unit open ball in R^{n}, n ≥ 3, are bounded. We discuss this and some basic facts about quasiconformal vector fields in Section 3. In Section 4 we establish existence of tangential extensions of quasiconformal vector fields. Theorem 1.3 is proven in Section 5. In Section 2 we show how to deduce Theorem 1.1 from Theorem 1.3.

The proof of Theorem 1.3 given here is not entirely satisfactory: it does not tell how eigenvalues of ϕ^{∗} are related to combinatorial invariants of the pseudo-Anosov homeomorphism f: R → R (like its Perron–Frobenius matrix, see [FLP]). Another interesting question is to describe stable and unstable manifolds of the fixed point [ι] of the mapping Φ. Note that Φ preserves the natural symplectic structure on X(π_{1}(R)) (see [Go]). Thus ϕ^{∗} has 3g −3 eigenvalues whose absolute value is less than 1, and the same number of eigenvalues outside the unit disc (where g is the genus of R). In particular, the (complex) dimension of stable and unstable manifolds E^{s}, E^{u} of Φ at [ι] is 3g − 3. McMullen in [Mc] proved the following theorem about E^{s} and E^{u}:

Theorem 1.4. Take a pair of singly degenerate Kleinian groups F^{+}, F^{−} ⊂ SL(2, C), whose ending laminations L^{+} , L^{−} are those of the discrete doubly degenerate group π_{1}(R) ⊂ G ⊂ SL(2, C). Then the Teichmuller spaces T (F^{+}), T (F^{−}) are open subsets in E^{s} and E^{u}.

2 The vector field must be automorphic under the action of a discrete group of Moebius transformations whose limit set is the boundary of the ball, see Definition 4.4.

However it is unclear how E^{s}, E^{u} behave outside of the domains T (F^{+}), T (F^{−}).

Acknowledgements. This research was partially supported by the NSF grant DMS-96-26633 at University of Utah. I am grateful to Curt McMullen for asking an interesting question and to the referee for several suggestions.

2. Local dynamics on the representation variety

Most of the material of this section is fairly well known, we present it here for the sake of completeness. Let π_{1}(R), ϕ be as in the introduction. Let ad(g)ξ = g^{−1}ξg denote the adjoint action of the group SL(2, C) on its Lie algebra sl(2, C). We consider the action of the automorphism ϕ on the variety Hom(π_{1}(R), SL(2, C)) given by:

ϕ : ρ ↦ ρ ∘ ϕ, ρ ∈ Hom(π_{1}(R), SL(2, C)).

This action of ϕ projects to a holomorphic automorphism Φ of the character variety X(π_{1}(R)) = Hom(π_{1}(R), SL(2, C))// SL(2, C).

Remark 2.1. The character variety is the quotient of Hom(π_{1}(R), SL(2, C)) in the sense of the geometric invariants theory and in general is different from the “naive” set-theoretic quotient Hom(π_{1}(R), SL(2, C))/ SL(2, C). However if we restrict to the open subvariety Hom(π_{1}(R), SL(2, C))^{−} consisting of Zariski dense representations then the projection X(π_{1}(R))^{−} of Hom(π_{1}(R), SL(2, C))^{−} to X(π_{1}(R)) is naturally isomorphic to the set-theoretic quotient Hom(π_{1}(R), SL(2, C))^{−}/ SL(2, C), see [JM, Theorem 1.1]. Moreover X(π_{1}(R))^{−} is a smooth complex manifold of the dimension 6g − 6, see [W], [Go].

Suppose that [ρ_{0}] ∈ X(π_{1}(R)) is a fixed point for Φ and the image of ρ_{0} is Zariski dense in SL(2, C). Then there is an element t ∈ SL(2, C) such that

tρ_{0}(ϕ(γ))t^{−1} = ρ_{0}(γ), for all γ ∈ π_{1}(R).

We consider the induced action T_{0}(Φ) of Φ on the tangent space T_{[ρ_{0}]} to X(π_{1}(R)) at [ρ_{0}]. Our goal is to identify this action with the natural action ϕ^{∗} of ϕ on the first cohomology group

H^{1}(π_{1}(R), sl(2, C)) = Z^{1}(π_{1}(R), sl(2, C))/B^{1}(π_{1}(R), sl(2, C)).

Recall that the action ϕ^{∗} is defined by

ϕ^{#}(c)_{γ} = ad(t^{−1})c_{ϕ(γ)}, ϕ^{#} : Z^{1}(π_{1}(R), sl(2, C)) ⟶ Z^{1}(π_{1}(R), sl(2, C))

ϕ^{∗}[c] = [ϕ^{#}(c)], [c] ∈ H^{1}(π_{1}(R), sl(2, C))

where c: γ ↦ c_{γ} ∈ sl(2, C) are cocycles in Z^{1}(π_{1}(R), sl(2, C)).

Remark 2.2. Here and in what follows we use the notation c_{γ} for the value of the cocycle c on the element γ ∈ π_{1}(R). Note that c_{γ} belongs to sl(2, C) and is a vector field on SL(2, C) and H^{3} . Thus given g ∈ SL(2, C) or x ∈ H^{3} the evaluations c_{γ}(g) and c_{γ}(x) are tangent vectors in T_{g}(SL(2, C)) and T_{x}H^{3} respectively.

It was first noticed by Andre Weil in [W] (see also [Go]) that the (Zariski) tangent space T_{[ρ_{0}]} is naturally isomorphic to H^{1}(π_{1}(R), sl(2, C)).

Lemma 2.3. Under the natural isomorphism between T_{[ρ_{0}]} and H^{1}(π_{1}(R), sl(2, C)) we have: T_{0}(Φ) = ϕ^{∗}.

Proof. Recall (see [Go]) that curves of representations ρ_{ε} through the point ρ_{0} can be described as

ρ_{ε}(g) = exp(εc_{γ} + O(ε^{2}))ρ_{0}(γ), ε → 0

where c belongs to Z^{1}(π_{1}(R), sl(2, C)). The isomorphism

T_{ρ_{0}} Hom(π_{1}(R), SL(2, C)) ≅ Z^{1}(π_{1}(R), sl(2, C))

is given by:

\frac{d}{dε}ρ_{ε} |_{ε=0}↦ c.

Thus

ρ_{ε}(ϕ(γ)) ≈ exp(ε c_{ϕ(γ)})ρ_{0}(ϕ(γ)) = t^{−1} exp(ε ad(t^{−1})c_{ϕ(γ)})ρ_{0}(γ)t.

The last has the same projection to X(π_{1}(R)) as exp(εad(t^{−1})c_{ϕ(γ)})ρ_{0}(γ), which has the tangent vector ϕ^{#}(c). Thus

T_{0}(Φ)(c) − ϕ^{#}(c) ∈ B^{1}(π_{1}(R), sl(2, C))

which concludes the proof.

Corollary 2.4. Theorem 1.3 is equivalent to Theorem 1.1.

3. Quasiconformal vector fields and the S -operator

For the proof of Theorem 1.3 we will need definitions and properties of quasiconformal vector fields that were introduced by Lars Ahlfors in [Ah1] under the name of quasiconformal deformations. Our discussion here will follow [Ah1] and [Ah2].

Let U denote the upper half-space Rn_{+} = {(x_{1}, …, x_{n}) : x_{n} > 0}, we will identify it with the hyperbolic n-space H^{n} with the hyperbolic metric |dx|/x_{n}. Let Mob_{n} denote the group of Moebius transformations of \overline{R^{n}} and Mob_{n}(U) denote the stabilizer of U in Mob_{n}. The group Mob_{n}(U) acts as the group of isometries Isom(H^{n}) of H^{n}. We shall use the notations mob_{n} and isom(H^{n}) for the Lie algebras of the groups Mob_{n} and Isom(H^{n}). We will realize mob_{n} and isom(H^{n}) as subalgebras in the space of all vector fields on R^{n} and U.

If X is a smooth manifold and A: X → X is a diffeomorphism of X then we shall denote by A_{∗}: ξ ↦ A_{∗}(ξ) the action of A on vector fields on X. In particular, the action of Moebius transformations A on vector fields ξ is given by the formula:

A_{∗}(ξ(x)) = (DA_{x})^{−1}ξ(Ax)

where DA_{x} is the Jacobian matrix of A at x. Then A_{∗}(ξ) = ad(A)(ξ) for all ξ ∈ mob_{n}. Similarly, if we identify mob_{n} with the Lie algebra of left-invariant vector fields on Mob_{n} then A_{∗}(ξ) = ad(A)(ξ). Elements ξ of mob_{n} have the form of “second degree polynomials”:

ξ(x⃗) = u⃗ + (Q + aI)x⃗ + |x⃗|^{2}b⃗ − 2⟨x⃗, b⃗ ⟩x⃗.

Here ⟨ · , · ⟩ is the scalar product in R^{n}, u⃗, b⃗ are vectors in R^{n}, Q = −Q^{T} , a ∈ R and I is the identity matrix. If ξ ∈ isom(H^{n}), then:

⟨u⃗, e⃗_{n}⟩ = ⟨b⃗, e⃗_{n}⟩ = 0, Qe⃗_{n} = 0.

We define the norm ∥A∥ for n×n-matrices as ∥A∥ = \sqrt{Tr(AA^{T})}. Define the supremum-norm ∥h∥_{∞} for h: Ω ⊂ R^{n} → R^{n} by ∥h∥_{∞} = sup_{x∈R^{n}} |f(x)| using the Euclidean metric on R^{n} (the issue here is that we have to consider hyperbolic metrics on domains in R^{n} as well). We shall denote by B_{x}(r) the metric ball with the radius r and center x in R^{n}, let B(1) = B_{0}(1) and S(1) = ∂B(1). We shall identify S(1) with the sphere S^{n−1} .

Consider a domain Ω in R^{n} (n ≥ 2) and a continuous vector field f: Ω → R^{n} on Ω. Suppose that f has locally integrable distributional partial derivatives. Define the matrix S = S_{n}[f](x) by the formula:

S = \frac{1}{2}(Df + Df^{T}) − \frac{1}{n} Tr(Df).

The operator S[f] is called the Ahlfors S -operator, it was introduced by Lars Ahlfors in [Ah1]. If n = 2 then the S -operator becomes the ¯∂-operator:

S[f] = [ \begin{matrix}Re(∂f/∂z̄) & Im(∂f/∂z̄) \\ Im(∂f/∂z̄) & − Re(∂f/∂z̄) \end{matrix} ]

under the standard matrix realization of C.

Definition 3.1. A vector-field f above is called a k -quasiconformal vectorfield if ∥S[f]∥ ∈ L_{∞}(Ω) and ∥S[f]∥_{∞} ≤ k\sqrt{n}.

Quasiconformal vector-fields are infinitesimal analogues of quasiconformal mappings in R^{n}. More precisely, suppose that h_{t}, t ∈ [0, 1], is a smooth family of quasiconformal homeomorphisms such that h_{0} = id. Then (d/dt)h_{t}|_{t=0} = f is a quasiconformal vector field. If M_{t} = M(h_{t}) is the Beltrami differential of h_{t} then (d/dt)M_{t}|_{t=0} = S[f], see [Ah1], [Ah2] for more details. Similarly to quasiconformal mappings quasiconformal vector fields are differentiable a.e. in their domains.

Assume that n = 2, f(z) transforms as a vector-field under Moebius mappings α. Then μ = S[f] = ¯∂(f) transforms as a Beltrami differential under α:

α_{∗}: μ(z) ↦ μ(γz)\overline{γ′(z)}/γ′(z).

More generally, if n ≥ 2 then

(1)  A_{∗}S[f] := S[A_{∗}(f)] = (DA)^{−1}(S[f] ∘ A)DA

(see [Ah1]). This implies:

(2)  ∥S[A_{∗}(f)]∥ = ∥S[f] ∘ A∥.

Therefore Moebius transformations send k -quasiconformal vector-fields to k -qua-siconformal vector-fields. This allows us to define quasiconformal vector fields on the extended Euclidean space \overline{R^{n}} = R^{n} ∪ \{∞\}.

Note that the matrix S[f] is symmetric and traceless. Suppose that S(z) = S[f](z) ≠ 0, let φ_{z} be the corresponding quadratic form. Then S(z) determines a nontrivial splitting of R^{n} (which we identify with the tangent space at z):

(3)  R^{n} = P_{S,z} ⊕ N_{S,z}

where the restriction of φ_{z} to P_{z} is a positive-definite quadratic form and the restriction of φ_{z} to N_{z} is a non-positive quadratic form. Thus the formula (1) implies

Lemma 3.2. Let Ω ⊂ R^{n} be a domain where S ≠ 0 almost everywhere. Suppose that A: Ω → Ω is a Moebius transformation and A_{∗}S = S. Then a.e. defined splitting P_{S,z} ⊕ N_{S,z} of the tangent bundle T Ω is invariant under A.

The following lemma is the usual Weyl lemma if n = 2, for n ≥ 3 it was proven by Ahlfors in [Ah2]:

Lemma 3.3. If f is a quasiconformal vector field on a connected domain Ω ⊂ \overline{R^{n}} and S[f] = 0 a.e. in Ω, then f is a conformal vector field. If n ≥ 3 or Ω = \overline{R^{n}} this implies that f is a Moebius vector field , i.e. f ∈ mob_{n} |_{Ω} .

J. Sarvas proved in [Sar] that k -quasiconformal vector-fields have the following “convergence property” similar to one for quasiconformal mappings:

Lemma 3.4 (J. Sarvas, [Sar]). Suppose that f_{j} is a sequence of k -quasi-conformal vector fields in a domain Ω ⊂ R^{n} such that ∥f_{j}∥ ≤ C for all j. Then {f_{j}} contains a subsequence which is convergent uniformly on compacts in Ω to a k -quasiconformal vector field on Ω.

The following theorem was proven by V. Semenov in [Se]:

Theorem 3.5 (Semenov’s stability theorem, [Se, Lemma 1]). Suppose that n ≥ 3. Then there exists a universal constant c_{n} depending only on the dimension n of R^{n} so that the following holds:

Let f be a k -quasiconformal vector field on the open unit ball B(1) ⊂ R^{n}. Then there exists λ ∈ mob_{n} such that ∥f − λ∥_{∞} ≤ c_{n}k.

In particular, every quasiconformal vector field in B(1) is bounded (this obviously fails if n = 2).

In what follows we will need to establish a relation between the operators S_{n} and S_{n−1} in R^{n} and R^{n−1} . Let n ≥ 2 and ξ(x) = (ξ_{1}, …, ξ_{n}) be a differentiable vector field in a domain Ω ⊂ R^{n}, take the hyperplane L = {x_{n} = 0} which intersects Ω along a domain Ω′ ≠ ∅. Define a vector field ξ̂ on the domain Ω′ by

ξ̂(x′) = (ξ_{1}, …, ξ_{n−1}), x′ = (x_{1}, …, x_{n−1}).

Lemma 3.6. For every vector field ξ we have: ∥S_{n}[ξ]∥ ≥ ∥S_{n−1}[ξ̂]∥.

Proof. Let diag(Dξ) denote the diagonal matrix whose diagonal entries are the diagonal entries of Dξ. Then

∥S_{n}[ξ]∥^{2} = ∥\frac{1}{2}(Dξ + Dξ^{T}) − diag(Dξ)∥^{2} + ∥diag(Dξ) − \frac{1}{n} Tr(diag(Dξ))∥^{2}.

Let q_{n} := ∥ diag(Dξ) − (1/n) Tr(diag(Dξ))∥^{2} . It is obvious that

∥\frac{1}{2}(Dξ + Dξ^{T}) − diag(Dξ)∥^{2} ≥ ∥\frac{1}{2}(Dξ̂+ Dξ̂^{T}) − diag(Dξ̂)∥^{2}.

Thus what we need to prove is: q_{n} ≥ q_{n−1} . We introduce the notations: z_{i} = ∂ξ_{i}/∂x_{i}, τ_{n} = z_{1} + ⋯ + z_{n}, then τ_{n−1} = τ_{n} − z_{n}. Clearly

q_{n} = ∑_{j=1}^{n} z_{j}^{2} − \frac{1}{n}τ_{n}^{2}

q_{n−1} = ∑_{j=1}^{n−1} z_{j}^{2} − \frac{1}{n − 1}τ_{n−1}^{2}

and

q_{n} − q_{n−1} = z_{n}^{2} − \frac{1}{n}τ_{n}^{2} + \frac{1}{n − 1}τ_{n−1}^{2} = \frac{1}{n − 1}(nz_{n}^{2} + \frac{1}{n}τ_{n}^{2} − 2τ_{n}z_{n}).

The determinant of the matrix of the quadratic form Q(z_{n}, τ_{n}) = nz_{n}^{2} + (1/n)τ_{n}^{2} − 2τ_{n}z_{n} is zero and diagonal entries are positive, thus q_{n} − q_{n−1} = Q(z_{n}, τ_{n}) ≥ 0.

4. Tangential extension of quasiconformal vector fields

It will be crucial for us to construct continuous extensions of quasiconformal vector fields from the open unit ball B(1) to its boundary S(1). The extension that we will construct is not a vector field on R^{n} but rather a tangent vector field on the sphere S(1) ≅ S^{n−1} . The only extension results for quasiconformal vector fields that I know are theorems of L. Ahlfors [A3] and J. Sarvas [Sar] below:

Theorem 4.1 (L. Ahlfors, [A3]). Suppose that f is a quasiconformal vector field on the open unit ball B(1) ⊂ R^{n}, such that:

(a) |f(x)| is uniformly bounded in B(1),

(b) lim_{|x|→1}⟨f(x), x⟩ = 0, i.e. f is asymptotically tangential. Then f admits a continuous extension to the unit sphere S(1) and, by reflection in S(1), an extension to a quasiconformal vector field on \overline{R^{n}}.

Theorem 4.2 (J. Sarvas, [Sar]). Isolated singularities of quasiconformal vector fields in \overline{R^{n}} (n ≥ 3) are removable.

Remark 4.3. To understand the difficulty note that for any holomorphic function f(z) defined in the upper half-plane U := {Im(z) > 0} ⊂ C the vector field f(z)∂/∂z is 0-quasiconformal in U, but usually it cannot be continuously extended to the boundary.

As far as I can see, Ahlfors’ theorem is not strong enough to suffice for our purposes. Namely, quasiconformal vector fields that we consider are not apriori asymptotically tangential. On the other hand, if ξ is a quasiconformal vector field on B(1), then its projection 𝒫_{p}(ξ) (defined below) is not in general a quasiconformal vector field on B(1).

We will prove existence of a continuous tangential extension under the following technical assumption:

Definition 4.4. We say that a continuous vector field f in B(1) is automorphic if the following holds:

There exists a discrete group F of Moebius transformations of B(1) whose limit set equals S(1) and a 1-cocycle σ: F → isom(H^{n}) so that

A_{∗}f − f = σ_{A}

for all A ∈ F.

To describe the extension we will need several preliminary constructions. Pick a point p ∈ S(1) and consider the family 𝒰_{p} of horospheres U_{p} in H^{n} = B(1) centered at p. Let ν_{x,p} denote the unit (with respect to the Euclidean metric) normal vector to the sphere U_{p} at the point x ∈ U_{p}. We assume that ν_{x,p} is directed outside U_{p}. Let ξ be a vector field on B(1). We define another vector field 𝒫_{p}(ξ) in B(1) by the formula

𝒫_{p}(ξ)(x) = f(x) − ⟨f(x), ν_{x,p}⟩ν_{x,p}

where ⟨ · , · ⟩ is the standard scalar product in R^{n}. Clearly 𝒫_{p}(f)(x) is tangent to the sphere U_{p} at the point x and is obtained via the orthogonal projections of f(x) to T_{x}(U_{p}).

Proposition 4.5. Suppose that f is a smooth automorphic k -quasicon-formal vector field on the open unit ball B(1) in R^{n}, n ≥ 3. Then for any p ∈ S(1) the field 𝒫_{p}(f) admits a continuous extension to [B(1) ∪ S(1)] − {p} whose restriction to the unit sphere S(1) ≅ S^{n−1} is a vector field f_{p,∞} which is tangent to S(1) and is k -quasiconformal on S^{n−1} − {p} ≅ R^{n}.

Proof. Let F denote the discrete group and σ the cocycle so that f is automorphic with respect to F and σ.

Step 1. We first prove that the field f has well-defined limits (with respect to the conical topology) on a dense countable subset in S(1), which consists of fixed points of loxodromic elements of the group F.

Lemma 4.6. Let L ⊂ S(1) be the collection of fixed points of loxodromic elements of the group F. Then the vector field f admits a continuous (with respect to the conical topology) extension to L.

Proof. It is enough to consider the upper half-space model U = Rn_{+} of the hyperbolic n-space H^{n}, assume that the origin O is a fixed point of a loxodromic element A: x⃗ ↦ λW x⃗ of the group F. We assume that 0 < λ < 1, the orthogonal matrix W belongs to O(n − 1) (the stabilizer of ∂U in SO(n)).

Pick a sequence of points z_{k} ∈ U that is convergent to the origin O in the conical topology, i.e. there is r > 0 so that z_{k} belong to the hyperbolic rneighborhood of the geodesic {(0, 0, …, 0, x_{n}), x_{n} > 0} ⊂ H^{n}. Let K denote the metric ball in H^{n} with the center at e⃗_{n} and hyperbolic radius r + exp(λ). Then each z_{k} belongs to the ⟨A⟩-orbit of K:

A^{−k}(z_{k}) = y_{k} ∈ K, for some k ∈ Z.

Let σ = σ_{A} ∈ isom(H^{n}) be the value of the cocycle σ at A, then

(A_{∗}^{k}f)(y) = ∑_{j=0}^{k−1}(A_{∗}^{j}σ)(y) + f(y)

for each y ∈ K. This means:

f(A^{k}y) = λ^{k}W^{k}f(y) + λ^{k}W^{k} ∑_{j=0}^{k−1}(A_{∗}^{j}σ)(y).

Since 0 < λ < 1 it is clear that lim_{k→∞} λ^{k}U^{k}f(y) = 0 uniformly for y ∈ K. Note that

|(A_{∗}^{j}σ)(y)| = λ^{−j}|σ(A^{j}y)|.

To estimate |σ(A^{j}y)| we let

σ(x) = v⃗ + (T + sI)x⃗ + (|x|^{2}b⃗ − 2⟨x⃗, b⃗⟩x⃗) = σ_{(−1)}(x) + σ_{(0)}(x) + σ_{(1)}(x).

Then

σ_{(0)}(A^{j}y⃗) = (T + sI)A^{j}y⃗ = A^{j}(T_{j} + sI)y⃗ = λ^{j}W^{j}(T_{j} + sI)y⃗

where T_{j} is a sequence of skew-symmetric matrices which have uniformly bounded norm (T_{j} = A^{−j}T A^{j}). Thus

|(T + sI)A^{j}y⃗| ≤ Const ·λ^{j}

and

|λ^{k}W^{k} ∑_{j=0}^{k−1} A_{∗}^{j}σ_{(0)}(y)| = |λ^{k}W^{k} ∑_{j=0}^{k−1} λ^{−j}(T + sI)A^{j}y⃗| ≤ Const ·k · λ^{k}

which tends to zero as k → ∞. The estimate for |σ_{(1)}(A^{j}y)| is:

|σ_{(1)}(A^{j}y)| ≤ 3λ^{2j}|b⃗| · |y⃗|^{2}.

Thus

|λ^{k}W^{k} ∑_{j=0}^{k−1} λ^{−j}σ_{(1)}[A^{j}y⃗]| ≤ 3|b⃗| · |y⃗|^{2}λ^{k} \frac{λ^{2k} − 1}{λ^{2} − 1}

which again tends to zero as k → ∞ uniformly for y ∈ K. We conclude that

lim_{k→∞} f(A^{k}y) = lim_{k→∞} ∑_{j=0}^{n−1} λ^{k}W^{k}A_{∗}^{j}σ_{(−1)}(y)

= lim_{k→∞} ∑_{j=0}^{k−1} λ^{k−j}W^{k−j}v⃗ = −v⃗ + ∑_{j=0}^{∞} λ^{j}W^{j}v⃗ = [1 − A]^{−1}v⃗ − v⃗.

Therefore

lim_{k→∞} f(A^{k}y) = [1 − A]^{−1}v⃗ − v⃗ = ([1 − A]^{−1} − 1)σ(0)

uniformly for y ∈ K. Hence, for the sequence z_{k} convergent to O in the conical topology we get:

lim_{k→∞} f(z_{k}) = lim_{k→∞} f(A^{k}y_{k}) = ([1 − A]^{−1} − 1)σ(0)

where A^{−k}z_{k} = y_{k} ∈ K.

Step 2. Now we change the coordinates by an isometry of H^{n} = U so that the point p becomes the point ∞ ∈ \overline{R^{n}}. Then f: U → R^{n} is a k -quasiconformal deformation whose sup-norm is bounded in U ∩ B(R), 0 < R < ∞.

We will prove that 𝒫_{p}(f) has a continuous extension f_{p,∞} to cl(U)∩B(R/2), so that f_{p,∞}|_{B(R/2)∩{x_{n}=0}} is a k -quasiconformal vector field in R^{n−1} .

For each 0 < t < 1 we consider the restriction f_{t} of 𝒫_{p}(f) to the unit (n − 1)-disc B_{C_{t}}(1) ⊂ {x_{n} = t} with the center at C_{t} = (0, …, 0, t). We regard f_{t} as a vector field on B^{n−1}(R/2) ⊂ R^{n−1} by “forgetting” the last coordinate x_{n}. Lemma 3.6 implies that f_{t} is a k -quasiconformal vector field.

Note that sup-norms of the k -quasiconformal vector fields f_{t} are uniformly bounded (Theorem 3.5). Hence, according to Lemma 3.4, up to a subsequence t_{j}, f_{t} are convergent to a k -quasiconformal vector field f_{p,∞} on B(R/2) ∩ R^{n−1} . The limit f_{p,∞} is supposed to be the boundary value of the extension of 𝒫_{p}(f) to the hyperplane {x_{n} = 0}. The problem is that for different sequences t_{j} → 0 we may get distinct limits. However we know that different limits must coincide on a dense set of points (Step 1). Thus the vector field 𝒫_{p}f admits a continuous extension to B(R/2) ∩ {x_{n} = 0}, which is k -quasiconformal in R^{n−1} .

Note however that f_{p,∞} is not defined at the point p = ∞. Thus we have to consider extensions corresponding to another family of horospheres. Let q ∈ S^{n−1} − {p}. From Theorem 4.5 we know that both vector fields 𝒫_{p}(f), 𝒫_{q}(f) admit continuous extensions f_{p,∞}, f_{q,∞} to the sphere S^{n−1} .

Proposition 4.7. For every pair of points p, g ∈ S^{n−1} we have: f_{p,∞} = f_{q,∞} on S^{n−1} − {p, q}.

Proof. We go back to the unit ball model of the hyperbolic space. Pick a pair of horospheres U_{p} ∈ 𝒰_{p}, U_{q} ∈ 𝒰_{q}, let ρ denote the minimum of their Euclidean radii. Take a point x ∈ U_{p} ∩ U_{q}. Then obviously

lim_{ρ→1} |ν_{x,p} − ν_{x,q}| = 0

where | · | denotes the Euclidean norm.

lim_{ρ→1} |𝒫_{p}(f)(x) − 𝒫_{q}(f)(x)| ≤ lim_{ρ→1} Const |ν_{x,p} − ν_{x,q}| = 0.

Thus for each automorphic vector field f there is a k -quasiconformal vector field f_{∞} tangent to S^{n−1} which coincides with each f_{p,∞} on S^{n−1}−{p}. Therefore we have proved

Theorem 4.8. Suppose that f is a smooth automorphic k -quasiconformal vector field on the open unit ball B(1) in R^{n}, n ≥ 3. Then f admits a continuous tangential extension f_{∞} to S(1). The vector field f_{∞} is again a k -quasiconformal vector field on the sphere S(1) = S^{n−1} .

Proposition 4.9. Let f be a quasiconformal vector field on B = B(1) which is automorphic with respect to a group F ⊂ Mob_{n} and a cocycle σ: A_{∗}f −f = σ_{A} for all A ∈ F. Assume that F is torsion-free. Then

A_{∗}f_{∞} − f_{∞} = σ_{A}|_{S^{n−1}} for all A ∈ F.

Proof. Let A ∈ F −{1}, then A has a fixed point p ∈ S^{n−1} . Thus the family of horospheres 𝒰_{p} is invariant under A. Therefore

A_{∗}𝒫_{p}(f) = 𝒫_{p}(A_{∗}f).

Since 𝒫_{p}(f) has continuous extension f_{∞} to S^{n−1} we conclude that

A_{∗}f_{∞} − f_{∞} = (σ_{A})_{∞} = σ_{A}|_{S^{n−1}}.

Remark 4.10. It is easy to check that the above proposition is also valid for groups with torsion.

5. Proof of the main theorem

In what follows we will identify the hyperbolic 3-space with the quotient space SU(2)\ SL(2, C); let 𝒫 : SL(2, C) → H^{3} denote the projection. Let Vec be the space of C^{∞} -vector fields on SL(2, C) which are invariant under the left SU(2)-action. Clearly we have the well-defined projection P_{∗}: Vec → H^{0}(T H^{3}) to the space of vector-fields on the hyperbolic 3-space. This projection commutes with the (right) action of the group SL(2, C) on vector-fields. Note that Vec is invariant under the multiplication by complex numbers (since the left action of SU(2) on T (SL(2, C)) is C-linear). We shall identify the Lie algebra sl(2, C) of SL(2, C) with the algebra of left-invariant vector-fields on SL(2, C).

The following simple lemma is critical for proving quasiconformality of the vector field ξ that we will construct in Theorem 5.2(c). Note that Lemma 5.1 clearly fails without the assumption |λ| = 1.

Lemma 5.1. Let ξ̃ ∈ Vec , U ⊂ H^{3} is a metric ball, Ũ = P^{−1}(U). Then there exists a constant C = C(ξ, U) such that:

∥S[P_{∗}(λ · ξ̃|_{U})]∥ ≤ C

for all λ ∈ C such that |λ| = 1.

Proof. The family of vector fields λ · ξ̃|_{U}, |λ| = 1, is compact (with respect to the C^{1} -topology of uniform convergence). Thus the assertion follows from continuity of the S -operator.

We consider the action of the group G = π_{1}(R) ⋊ ⟨t⟩ on the hyperbolic 3space as in the introduction. This action extends to the (right) action of G on SL(2, C). Our main goal is to prove the following

Theorem 5.2. Suppose that σ ∈ Z^{1}(π_{1}(R), sl(2, C)) is such that ϕ^{#}(σ) = λσ, λ ∈ C^{∗}. Then there is a smooth vector-field ξ on H^{3} and its lift ξ̃ ∈ Vec to the group SL(2, C) such that:

(a) ad(γ)ξ − ξ = σ(γ) for all γ ∈ π_{1}(R).

(b) ad(t)ξ̃ = λ^{−1}ξ̃.

(c) The vector field ξ is quasiconformal.

Proof. To construct the vector field ξ we will need a partition of unity η̃ on the space SL(2, C) corresponding to the subgroup π_{1}(R) and invariant under the action of t:

Lemma 5.3. There exists a positive bounded C^{∞} -function η̃ on SL(2, C) which satisfies the following properties:

(1) η̃ is invariant under the right action of t and the left action of SU(2).

(2) ∑_{γ∈π_{1}(R)} η̃(gγ) = 1 for all g ∈ SL(2, C).

Proof. Our proof follows [Kr, Chapter V, Lemma 3.1]. Take a finite covering {D_{j}} of the manifold SU(2)\ SL(2, C)/G by open metric balls D_{j} which satisfy the condition:

The radii of the balls D_{j} are smaller than the injectivity radius of H^{3}/G.

Let m denote the multiplicity of the covering {D_{j}} (i.e. the maximal number of balls which have nonempty intersection). The group G acts on the lift of this covering to H^{3} with trivial stabilizers. Let {η̄_{j}} denote the partition of unity on H^{3}/G corresponding to the covering {D_{j}}. For each D_{j} choose a connected component V_{j} of its lift to H^{3} and let η_{j} be the lift of η̄_{j} to V_{j}. Extend the function η_{j} to H^{3} by:

η_{j}(x) = \{ \begin{matrix}η_{j}(t^{n}x) & if x ∈ t^{−n}(V_{j}), \\ 0 & otherwise. \end{matrix}

Finally let

η(x) = ∑_{j} η_{j}(x).

It is clear that η is invariant under the action of t, ∑_{γ∈π_{1}(R)} η(γ(x)) = 1 for all x and 0 ≤ η ≤ 1. Then we lift the function η to η̃ = η ∘ P: SL(2, C) → R.

Suppose that σ ∈ Z^{1}(π_{1}(R), sl(2, C)) is such that ϕ^{#}(σ) = λσ, λ ∈ C^{∗} (at this stage we do not need the assumption |λ| = 1).

Lemma 5.4. Under the assumptions above there exists a vector field ξ̃ ∈ Vec which satisfies the following properties:

(i) t_{∗}ξ̃ = λ^{−1}ξ̃;

(ii) α_{∗}ξ̃− ξ̃ = σ_{α}, for all α ∈ π_{1}(R).

Proof. Our proof again follows [Kr, Chapter V, Theorem 3.2] (see [KM] for more general constructions). Let

ξ̃(g) = − ∑_{γ∈π_{1}(R)} η̃(gγ) · σ_{γ}(g).

(Here σ_{γ}(g) is a tangent vector to SL(2, C) at g, see Remark 2.2.) Note that for every g ∈ SL(2, C) not more than m terms of this series are different from zero (where m is the multiplicity of the covering {D_{j}}). Hence the infinite series which we use to define ξ̃ is convergent to a smooth vector field on SL(2, C). To show that ξ̃ is invariant under the (left) SU(2)-action we note that the elements σ_{γ} of the Lie algebra are left-invariant under SL(2, C) and the function η̃ on SL(2, C) is invariant under the left action of SU(2).

Now we verify (i). We have:

\begin{aligned}ξ̃(g) & = − ∑_{ϕ(γ)∈π_{1}(R)} η̃(gϕ(γ)) · σ_{ϕ(γ)}(g), \\ t_{∗}^{−1}ξ̃(g) & = − ∑_{γ∈π_{1}(R)} η̃(gγ) · t_{∗}^{−1}σ_{ϕ(γ)}(g) \end{aligned}

since η̃ is t-invariant.

− ∑_{γ∈π_{1}(R)} η̃(gγ) · t_{∗}^{−1}σ_{ϕ(γ)}(g) = − ∑_{γ∈π_{1}(R)} η̃(gγ) · (ϕ^{#}σ)_{γ}(g).

The latter equals

−λ ∑_{γ∈π_{1}(R)} η̃(gγ) · σ_{γ}(g) = λξ̃(g)

which proves (i). The second assertion of Lemma 5.4 again follows from the direct computation:

(α_{∗}ξ̃)(g) = − ∑_{γ∈π_{1}(R)} η̃(gαγ)(α_{∗}σ_{γ})(g) =

and hence, by the cocycle condition σ_{γ∘α} − σ_{α} = α_{∗}σ_{γ},

= ∑_{γ∈π_{1}(R)} η̃(gαγ)σ_{α}(g) − ∑_{γ∈π_{1}(R)} η̃(gαγ)σ_{γ∘α}(g) = σ_{α}(g) + ξ̃(g).

Thus ξ := P_{∗}(ξ̃) is a smooth vector-field on H^{3} , where ξ̃ is the vector field constructed in the previous lemma.

Lemma 5.5. The field ξ satisfies the following properties:

(i) α_{∗}ξ − ξ = σ_{α} for all α ∈ π_{1}(R). (ii) ∥S[ξ(x)]∥ ≤ Const for all x ∈ H^{3} .

Proof. The first assertion follows from the statement (ii) of Lemma 5.4. Let us prove the second assertion. Recall that the quotient H^{3}/G is compact, pick a compact K ⊂ H^{3} which is a fundamental domain for the action of G. Let C_{1} = max ∥S[ξ](x)∥, x ∈ K. The equality γ_{∗}ξ − ξ = σ_{γ} ∈ sl(2, C) implies:

S[γ_{∗}ξ] = S[ξ]

for all γ ∈ π_{1}(R). Hence it follows from the equality (2) in Section 3 that

∥S[ξ]∥ = ∥S[ξ] ∘ γ∥

and ∥S[ξ]∥ is bounded by C_{1} along the π_{1}(R)-orbit of K. Similarly,

∥S[ξ](t^{n}x)∥ = ∥S[P_{∗}(λ^{−n}ξ̃)](x)∥, n ∈ Z,

thus Lemma 5.1 implies that ∥S[ξ](t^{n}x)∥ ≤ C_{2} = C(ξ, K) for all n ∈ Z. Therefore ∥S[ξ(x)]∥ ≤ Const = max(C_{1}, C_{2}) for all x ∈ H^{3} .

This concludes the proof of Theorem 5.2.

Now we can start proving Theorem 1.3. We assume that ϕ^{∗} has an eigenvalue λ such that |λ| = 1, let τ ∈ H^{1}(π_{1}(R), sl(2, C)) − {0} be an eigenvector: ϕ^{∗}(τ) = λτ. As we proved in the introduction, λ ≠ 1.

Lemma 5.6. Under the conditions above there exists a cocycle σ ∈ Z^{1}(π_{1}(R), sl(2, C)) such that τ = [σ] and ϕ^{#}(σ) = λσ.

Proof. Choose bases in H^{1}(π_{1}(R), sl(2, C)) and B^{1}(π_{1}(R), sl(2, C)); the automorphism ϕ^{∗}: H^{1}(π_{1}(R), sl(2, C)) → H^{1}(π_{1}(R), sl(2, C)) is represented by a matrix D. Hence, ϕ^{#}: Z^{1}(π_{1}(R), sl(2, C)) → Z^{1}(π_{1}(R), sl(2, C)) is represented by a matrix:

[ \begin{matrix}A & B \\ 0 & D \end{matrix} ] .

Note that A is the matrix of the action of ϕ^{#} on the complex 3-dimensional space B^{1}(π_{1}(R), sl(2, C)). The element t ∈ SL(2, C) is loxodromic, the eigenvalues of its adjoint representation are: 1, μ^{2} , μ^{−2} , where |μ| ≠ 1, hence none of these numbers equals λ. The spectrum of the action of ϕ on B^{1}(π_{1}(R), sl(2, C)) is the same as the spectrum of the adjoint representation of t^{−1} (since the subgroup π_{1}(R) ⊂ SL(2, C) is Zariski dense). Now existence of the eigenvector σ follows from the linear algebra.

We apply Theorem 5.2 to the cocycle σ (given by Lemma 5.6) and construct a quasiconformal vector field ξ on H^{3} such that

ad(γ)ξ − ξ = σ_{γ} for all γ ∈ π_{1}(R).

Hence by Theorem 4.8 the vector field ξ admits a tangential extension ζ = ξ_{∞} to S^{2} = ∂B, which is a quasiconformal tangent vector field on S^{2} such that

(4)  γ_{∗}ζ − ζ = σ_{γ}  for all γ ∈ π_{1}(R)

(see Proposition 4.9). Recall that S[χ] = 0 for any conformal vector field χ. Thus, by applying the operator S to both sides of the equation (4), we get:

γ_{∗}S[ζ] − S[ζ] = 0 for all γ ∈ π_{1}(R).

The matrix-valued function S[ζ] is measurable and belongs to L_{∞}(\overline{C}). Let L_{S} be the subset of \overline{C} where S[ζ] ≠ 0. This subset is measurable, π_{1}(R)-invariant and the field S[ζ] determines a nontrivial π_{1}(R)-invariant measurable splitting of the tangent bundle T L_{S} (see Lemma 3.2). Now we can apply

Theorem 5.7 (Sullivan’s rigidity theorem, [Su], see also [O]). Suppose that F ⊂ PSL(2, C) is a discrete finitely-generated group whose limit set is \overline{C} and L ⊂ \overline{C} is an F -invariant measurable subset of nonzero measure. Then there are no F -invariant measurable line-fields defined on L.

We let F = π_{1}(R), then Sullivan’s theorem implies that the set L_{S} has zero measure. This means that S[ζ] = 0 almost everywhere and, according to Lemma 3.3, ζ is a Moebius vector field : ζ ∈ mob_{2} = sl(2, C). We conclude that the cocycle σ is a coboundary:

δ_{ζ}(γ) = ad(γ)ζ − ζ = σ_{γ}

which contradicts our assumption that the class [σ] = τ ∈ H^{1}(π_{1}(R), sl(2, C)) is nontrivial. This proves Theorem 1.3.

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Received 6 June 1996