Annales Academiæ Scientiarum Fennicæ Series A. I. Mathematica Volumen 20, 1995, 245–250

QUASICONFORMAL GROUPS ACTING ON B^{3} THAT ARE NOT QUASICONFORMALLY CONJUGATE TO MÖBIUS GROUPS

Manouchehr Ghamsari

University of Cincinnati, Department of Mathematical Sciences

Old Chemistry Building (ML 0025), Cincinnati OH 45221-0025, U.S.A.; Manouchehr.Ghamsari@uc.edu

Abstract. We construct a quasiconformal group acting on the unit ball in 3-space which is not quasiconformally conjugate to any Möbius group. F. Gehring and P. Palka first asked if each quasiconformal group acting on Euclidean n-space is quasiconformally conjugate to a Möbius group. The answer to the question is positive for n = 2 (D. Sullivan and P. Tukia) and is negative for higher dimensions (P. Tukia). A similar question for groups leaving the unit ball in n-space invariant was answered negatively for n > 3; but the question has remained open for n = 3.

1. Introduction

The purpose of this paper is to prove the following result.

1.1 Theorem. There exists a quasiconformal group acting on B^{3} that is not quasiconformally conjugate to any Möbius group.

The above theorem answers a question raised by G. Martin in [M] about the existence of such groups. F. Gehring and B. Palka first asked whether each quasiconformal group acting in \overline{R}^{n} is conjugate to a Möbius group via a quasiconformal map [GP]. For n = 2, the question was answered affirmatively by D. Sullivan [S] and by P. Tukia [T1]. For n ≥ 3, the answer to the question is negative. P. Tukia [T2] constructed a quasiconformal group not quasiconformally conjugate to a Möbius group. Later, G. Martin modified Tukia’s construction to provide a discrete group with the above property [M]. Moreover, for n ≥ 4 there are quasiconformal Fuchsian groups (discrete groups that leave a ball invariant) which are not quasiconformally conjugate to any Möbius group [M, Theorem 2.4]. But the problem of existence of a quasiconformal group acting on B^{3} which is not quasiconformally conjugate to any Möbius group remained open. The basic idea in Tukia’s construction is to use the rigidity of quasiconformal maps in higher dimensions to construct a quasiconformal group of the form G = fΓf^{−1} where Γ is a Möbius group and f is not a quasiconformal map and show that G is not quasiconformally conjugate to any Möbius group. We follow the same idea. Our construction can be best described as the “cylindrical version” of Tukia’s construction.

1991 Mathematics Subject Classification: Primary 30C62; Secondary30C65.

After some preparations in Section 2, we prove the main result in Section 3 and in Section 4 we present an example of a finite quasiconformal reflection group acting in B^{3} which is not conjugate to any Möbius group via a quasiconformal map. The group constructed in Section 3 is not discrete and the group presented in Section 4 is finite. I do not know if there is a non-elementary discrete group G for Theorem 1.1.

2. Preliminaries

2.1. Notation. As usual, R^{n} denotes the Euclidean n-space and \overline{R}^{n} = R^{n} ∪ \{∞\}. For a set A ⊂ \overline{R}^{n} we write ∂A, Ā for the boundary and the closure of A, respectively. The ball centered at x or radius r is denoted by B^{n}(x, r) and

R^{1} = R,  B^{n} = B^{n}(0, 1),  S^{n−1} = ∂B^{n},  H^{n} = {x ∈ R^{n} : x_{n} > 0}.

When working in R^{2} we sometimes use the complex notation z = x_{1} + ix_{2} .

2.2. Möbius and quasiconformal groups. Let X be a non-empty set. If G is any group of permutations of X, then the G-orbit of x is

G(x) = {g(x) : g ∈ G}.

The fixed points of G is denoted by Fix(G). Two subgroups G_{1} and G_{2} of G are conjugate in G if for some h ∈ G, G_{2} = hG_{1}h^{−1} . Clearly G_{2}(hx) = h(G_{1}(x)).

A Möbius transformation acting in \overline{R}^{n} is a finite composition of reflections in spheres and planes. The group of Möbius transformations acting on \overline{R}^{n} is called general Möbius group and is denoted by GM(\overline{R}^{n}). The Möbius group M(\overline{R}^{n}) is the subgroup of GM(\overline{R}^{n}) consisting of all orientation preserving Möbius transformations. Finally, we call G a quasiconformal group if it is a K -quasiconformal group for some K.

We need the next two lemmas for the proof of Theorem 1.1.

2.3. Lemma. Suppose that Γ_{1} is a subgroup of M(R̄^{3}). If Γ_{1} fixes more than two points in \overline{R}^{3} , then Γ_{1} is conjugate to a group of rotations about a line L.

Proof. After conjugating with a Möbius transformation, we may assume that Γ_{1} fixes the points 0, ∞ and x_{0} . Let g ∈ Γ_{1} and let L be the line passing through 0 and x_{0} . Then g fixes L pointwise because g(L) = L and g fixes three points in L. Since g fixes 0 and ∞, g is a similarity by [A, II, Lemma 1] and since g fixes L pointwise g is an isometry.

If P is a hyperplane perpendicular to L, then g(P) is a plane perpendicular to L again containing P ∩ L. Hence g(P) = P and g in P is an isometry fixing P ∩ L. Therefore g in P is either a rotation of P about the point P ∩ L, or it is a reflection in a line L_{1} through P ∩ L in P.

Quasiconformal groups acting on B^{3} 247

If g in P is a rotation, we choose a rotation g_{0} of R^{3} about L which agrees with g on P. Then g_{0}^{−1}g = id in P and hence g_{0}^{−1}g = id in R^{3} or g_{0}^{−1}g is a reflection in P by [B; Theorem 3.2.4]. But both g_{0} and g are sense preserving and g is a rotation about L as desired.

It remains to see what happens if g is a reflection about L_{1} in P. In that case, g fixes the plane P_{1} containing the lines L and L_{1} because it fixes both L_{1} and L pointwise. Therefore g is a reflection in P_{1} which is impossible.

2.4. Lemma. Let Γ_{1} and L be as in Lemma 2.3 and let Γ_{2} be a subgroup of M(\overline{R}^{3}). If there is x_{0} ∈ \overline{R}^{3} so that Fix(g) = \{x_{0}\} for each g ∈ Γ_{2} , g ≠ id and if Γ = ⟨Γ_{1}, Γ_{2}⟩ is abelian, then Γ_{2} | L is conjugate to a group of translations along L and each element of Γ_{2} is a combination of a translation and a rotation along L.

Proof. Let g_{2} ∈ Γ_{2} , g_{2} ≠ id and let g_{1} ∈ Γ_{1} . Then g_{1}(x_{0}) = g_{1}g_{2}(x_{0}) = g_{2}g_{1}(x_{0}) and hence g_{1}(x_{0}) ∈ Fix(Γ_{2}) = {x_{0}} and thus x_{0} ∈ Fix(Γ_{1}). After conjugation, we may assume that x_{0} = ∞ and Fix(Γ_{1}) = L is a line passing through ∞. Next if x ∈ L and g_{2} ∈ Γ_{2} , then

g_{1}g_{2}(x) = g_{2}g_{1}(x) = g_{2}(x)

for each g_{1} ∈ Γ_{1} . Hence g_{2}(x) ∈ Fix(Γ_{1}) = L and g_{2} | L is a translation, or g_{2} | L = ax + b. Now if a ≠ 1, then g_{2} | L has a finite fixed point contrary to our assumption. Therefore g_{2} | L = x + b and g_{2} − b fixes L_{2} pointwise. Hence g_{2} − b is a rotation along L.

2.5. Maps. We consider K -quasiconformal, η -quasisymmetric and Lbilipschitz maps. For definitions of quasiconformal and quasisymmetric maps see [V1], [V2; 1.3 and 3.1].

2.6. Quasihyperbolic metric. Suppose that D ⊂ R^{n}, D ≠ R^{n} is a domain. The quasihyperbolic metric K_{D} of D is defined by the element of length |dx|/d(x, ∂D) where d(x, ∂D) is the distance from x to ∂D, see [GP]. If D ⊂ R^{2} is simply connected then

h_{D}/2 =^{<} K_{D} =^{<} 2h_{D}

where h_{D} is the hyperbolic metric of D.

The following result is contained in the proof of [G2; 2.11] for a special case, the general case needs only minor modifications.

2.7. Lemma. Suppose that D and D′ are domains in R^{n} with connected boundaries and that f: \overline{D} → \overline{D}′ is a homeomorphism such that

1. f(D) = D′,

2. f|∂D is L-bilipschitz,

3. f|D is M -bilipschitz in the quasihyperbolic metrics of D and D′. Then f is L_{1} -bilipschitz with L_{1} = L_{1}(L, M, n).

We close this section with an extension result that is essential for the proof of Theorem 1.1.

2.8. Lemma. Suppose that C is a bounded Jordan curve in R^{2} and that f_{0}: S^{1} → C is η -quasisymmetric. Then f_{0} has an extension to a quasiconformal self map f of R^{2} so that

1. f|S^{1} = f_{0} ,

2. f is bilipschitz with respect to quasihyperbolic metrics in both components of R^{2} \ S^{1} ,

3. f(z) = z for |z| ≥ r, r = r(η, diam C).

Proof. Performing similarity transformations, we may assume that C is contained in an annulus A = {1 < |z| < r_{1}} where r_{1} is a positive constant depending only on η. Let D and D^{∗} be the interior and exterior of C_{1} respectively. We first consider extension to interiors.

Let φ: \overline{D} → \overline{B}^{2} be a homeomorphism which φ is conformal in D and φ(0) = 0. Since C is a quasicircle, φ is η′ -quasisymmetric, where η′ depends only on the constant of C and dist(0, ∂D) both of which depend only on η and so does η′. Now h = f_{0} ∘ φ^{−1} is a quasisymmetric map of S^{1} and can be extended to a self-homeomorphism h_{1} of \overline{B}^{2} so that h_{1} is quasisymmetric in B^{2} , h_{1}(0) = 0 and h_{1} is bilipschitz with respect to quasihyperbolic metrics of B^{2} and D, by [GNV; Lemma 2.10]. Furthermore f_{0} is bilipschitz in {|z| < 1/2} and we may assume that f_{1}(z) = z for |z| < r_{2} by a version of annulus theorem [M; Theorem 2.6], where r_{2} = r_{2}(η).

We now turn to extension of f_{0} to exteriors. Let T (z) = 1/z. Then g = T f_{0}T is a quasisymmetric map that carries S^{1} onto T C, because T is bilipschitz and hence quasisymmetric in A. The map g has an extension to a quasisymmetric map g_{1}: B^{2} → T D^{∗} so that g_{1}(z) = z in B^{2}(r_{3}) and g_{1} is bilipschitz with respect to quasihyperbolic metric, as is shown for f_{1} above.

Next f_{2} = T g_{1}T maps {|z| > 1} onto D^{∗} and f_{2}(z) = z for |z| ≥ 1/r_{3} . Because T is bilipschitz in {r_{3} < |z| < 1/r_{3}} and g_{1} is bilipschitz with respect to quasihyperbolic metric, f_{2} is bilipschitz with respect to quasihyperbolic metrics of {|z| > 1} and D^{∗} whenever z and f_{2}(z) are both in {r_{3} < |z| < 1/r^{3}}. But this implies that f_{2} is bilipschitz with respect to quasihyperbolic metrics of {|z| > 1} and D^{∗}, because f_{2}(z) = z for |z| ≥ 1/r_{3} . Finally

f(z) = \{ \begin{matrix}f_{1}(z) & for |z| ≤ 1, \\ f_{2}(z) & for |z| > 1 \end{matrix}

is the desired map.

3. Proof of Theorem 1.1

We now turn to the proof of Theorem 1.1. For this we begin with the familiar bounded snowflake curve S and construct a K -quasiconformal group G acting on a round cylinder so that (1) G(a) = S × R for each a ∈ S × R. (2) If Γ is any Möbius group conjugate to G with ∞ ∈ Fix(Γ), then Γ(b) is contained in a round cylinder. Appealing to a well established result of Tukia and Väisälä, that S × R cannot be imbedded in R^{2} or S^{1} × R by a locally quasisymmetric map, we see that G is not quasiconformally conjugate to a Möbius group.

3.1. The Jordan curve S. Let S_{n} as shown below. Then {S_{n}} converges to a Jordan curve S, where dim_{H}(S) = α > 1. We may assume that H_{α}(S) = 2π and 1 ∈ S, where H_{α}(S) is the Hausdorff measure of S.

250 Manouchehr Ghamsari

S_{1}  S_{2}  S_{3}

Choose an orientation on S and define θ(x) = H_{α}(S(1, x)) where x ∈ S and S(1, x) is the oriented arc in S from 1 to x. Now f_{0}^{−1}(x) = e^{iθ(x)} is a bihölder map from S onto S^{1} , see [FM; Theorem]. Hence f_{0}^{−1} and f_{0} are quasisymmetric. By Lemma 2.8 there exists a quasiconformal self map f of R^{2} so that f | S = f_{0} , f is bilipschitz with respect to quasihyperbolic metrics in both components of R^{2} \ S^{1} and f(z) = z for |z| > r.

3.2. The group G. Let R be the group of all rotations R_{θ}: z → ze^{iθ} in R^{2} . Then R_{S} = fRf^{−1} is a group of self maps of R^{2} acting transitively on S. Since R_{S} leaves B^{2}(r) invariant, we regard R_{S} as a group of transformations of B^{2}(r). Next fR_{θ}f^{−1} is bilipschitz with respect to hyperbolic metrics in both components of R^{2} \ S, being a composition of such maps. Moreover, fR_{θ}f^{−1} | S = f_{0}R_{θ}f_{0}^{−1} is bilipschitz by [FM; Corollary] and hence fR_{θ}f^{−1} is bilipschitz throughout by Lemma 2.7. Let G_{1} = R_{S} × id and let G_{2} be the group of translations along the x_{3} -axis. Then G = ⟨G_{1}, G_{2}⟩ is a bilipschitz group of B^{2}(r) × R and we have the following result.

3.3. Theorem. The group G above is a quasiconformal group of B^{2}(r) × R that is not quasiconformally conjugate to any Möbius group.

Proof. Suppose that there exists a quasiconformal map h so that Γ = hGh^{−1} is a Möbius group. Let Γ_{j} = hG_{j}h^{−1} where G_{1} and G_{2} are as in 3.2. By Lemmas 2.3 and 2.4, after conjugation, Γ_{1} is a group of rotations about the x_{3} - axis and Γ_{2} is a group of translations along that axis and Γ = ⟨Γ_{1}, Γ_{2}⟩. Let h^{−1}(b) = a ∈ S × R. Then G(a) = h(Γ(b)) = S × R, or

h^{−1}(S × R) = Γ(b) ⊂ S^{1}(r) × R,

which cannot be, by [T2; Theorem 5]. Hence G is not quasiconformally conjugate to any Möbius group.

3.4. Completion of the proof of Theorem 1.1. Let g_{0} map B^{2}(r) × R onto B^{3} quasiconformally. Then

G′ = g_{0}Gg_{0}^{−1}

is a quasiconformal group of B^{3} that is not quasiconformally conjugate to any Möbius group.

252 Manouchehr Ghamsari

4. Remarks

Let C be a quasicircle through ∞ that contains a nonrectifiable bounded subarc and let σ be a bilipschitz reflection in σ. Then ⟨σx × id⟩ is a (sensereversing) QC group leaving the upper half space in R^{3} invariant. This observation is due to Väisälä, and Martio [Remark 4.29, MV].

I would like to thank the referee for his comments that simplified and reduced Section 4 to a very short example.

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Received 18 July 1992