Annales Academiæ Scientiarum Fennicæ Mathematica Volumen 25, 2000, 131–150

JØRGENSEN’S INEQUALITY FOR DISCRETE CONVERGENCE GROUPS

Petra Bonfert-Taylor

Wesleyan University, Department of Mathematics 265 Church Street, Middletown, CT 06459, U.S.A.; pbonfert@mail.wesleyan.edu

Abstract. We explore in this paper whether certain fundamental properties of the action of Kleinian groups on the Riemann sphere extend to the action of discrete convergence groups on \overline{R^{2}} . A Jørgensen inequality for discrete K -quasiconformal groups is developed, and it is shown that such an inequality depends naturally on the quasiconformal dilatation K. Furthermore, it is established that no such inequality can hold for general discrete convergence groups. In the discontinuous case a universal constraint on discreteness is formulated for both quasiconformal and general convergence groups.

1. Basic definitions and notation

The group of all orientation-preserving Möbius transformations in \overline{R^{2}} is denoted by Möb. All maps in this article are assumed to be orientation-preserving.

A group G of homeomorphisms of \overline{R^{2}} is a K -quasiconformal group if each of its elements is K -quasiconformal, and we call the group simply quasiconformal if it is K -quasiconformal for some K. Recall that every Möbius transformation is 1-quasiconformal, and that the converse also holds; i.e. every 1-quasiconformal homeomorphism of \overline{R^{2}} is a Möbius transformation (see [TV2] for a nice geometric proof). One natural way to construct a quasiconformal group is to conjugate a conformal group by a quasiconformal mapping. In \overline{R^{2}} one obtains every quasiconformal group in this way ([Sul], [Tuk2]), whereas in higher dimensions there exist quasiconformal groups which are not quasiconformally conjugate to Möbius groups ([Tuk2], [Mar], [McK], [FrSk]).

A group G of homeomorphisms of \overline{R^{2}} is discrete if no sequence \{f_{n}\} ⊂ G of distinct elements converges to the identity uniformly in \overline{R^{2}} . A discrete subgroup of Möb is called a Kleinian group.

A (not necessarily discrete) group G of homeomorphisms of \overline{R^{2}} is said to be a convergence group if each infinite subfamily of G contains a sequence {f_{n}}, such that one of the following is true:

(i) There exists a homeomorphism f of \overline{R^{2}} such that

lim_{n→∞} f_{n} = f  and  lim_{n→∞} f_{n}^{−1} = f^{−1}

1991 Mathematics Subject Classification: Primary 30F40, 57S30, 30C62, 20H10.

This work was done while the author was at the University of Michigan.

uniformly in \overline{R^{2}} .

(ii) There exist points x_{0}, y_{0} ∈ \overline{R^{2}} such that

lim_{n→∞} f_{n} = x_{0}  and  lim_{n→∞} f_{n}^{−1} = y_{0}

locally uniformly in \overline{R^{2}}∖\{y_{0}\} and \overline{R^{2}}∖\{x_{0}\}, respectively. Here we allow x_{0} = y_{0} .

In (ii) we call x_{0} and y_{0} the attracting and repelling limit point of the sequence {f_{n}}, respectively, if these two points are distinct. Möbius groups and quasiconformal groups are examples of convergence groups, see [GM1]. Homeomorphic conjugates of quasiconformal groups are also convergence groups, so that the class of convergence groups of \overline{R^{2}} is strictly larger than the class of quasiconformal groups.

Convergence groups in many essential ways resemble their conformal counterparts. As with Möbius groups, we define the limit set L(G) of the convergence group G to be the set of all limit points of those sequences {f_{n}} converging in the sense of (ii). Likewise, we define the regular set Ω(G) to be the set of points where G acts discontinuously; i.e. the set of all x that have a neighborhood U satisfying g(U) ∩ U = ∅ for all but finitely many g ∈ G. The regular set is an open set, and the limit set is closed; both sets are G-invariant. If L(G) contains more than two points then L(G) is an infinite perfect set. If Ω(G) ≠ ∅, then G is necessarily discrete. For discrete G, the limit set L(G) is the complement of the regular set Ω(G). (See e.g. [GM1], [Tuk3] for proofs.)

There exists a classification of the elements of a convergence group that is topologically analogous to the classification of Möbius maps. If G is a convergence group and g ∈ G, then we say that g is elliptic if ⟨g⟩, the group generated by g, is pre-compact, i.e. if every sequence in ⟨g⟩ contains a subsequence converging uniformly to a homeomorphism. If g is not elliptic, then g is loxodromic if it has exactly two fixed points, or g is parabolic if g fixes exactly one point. It is not hard to see that every element in a convergence group is either elliptic, parabolic or loxodromic ([Tuk3]). In a discrete convergence group the elliptic elements are those g ∈ G that satisfy g^{n} = id for some n ∈ N. The sequence {g^{n}} of iterates of a loxodromic element of a discrete convergence group converges to a fixed point a of g locally uniformly in the exterior of the other fixed point b; we call a the attracting and b the repelling fixed point of g. For parabolic g, the sequence of iterates converges to the fixed point of g locally uniformly in the exterior of that fixed point (see [GM1]).

As is customary, we define a discrete convergence group G to be non-elemen-tary if L(G) contains more than two points. We can extend this definition to non-discrete G but then in addition we must require that no x ∈ L(G) is fixed by the entire group G. In both cases one can show that a convergence group G is elementary if and only if either L(G) = ∅ or there is a one or two-point set which is fixed setwise by G. Furthermore, G is non-elementary if and only if there are two loxodromic g, h ∈ G without common fixed points (see [Tuk3]).

Acknowledgments. I thank Fred Gehring for introducing the problem to me and for communicating Theorem 3.5 to me. I am grateful to Matti Vuorinen for an improvement in Lemma 4.1, and to Gaven Martin for pointing out to me that Corollary 3.4 is a consequence of Theorem 3.3. I enjoyed productive conversations on this subject with Juha Heinonen and Edward Taylor. Finally I would like to thank the referee for valuable comments and suggestions.

2. Measuring discreteness

Let q(x, y) denote the chordal distance of the points x, y ∈ \overline{R^{2}} ; it is the Euclidean distance of their stereographic projections onto S^{2} ⊂ R^{3} and is given by

q(x, y) = \frac{2|x − y|}{\sqrt{1 + |x|^{2}} \sqrt{1 + |y|^{2}}} .

For two homeomorphisms f, g of \overline{R^{2}} , define their chordal distance to be

d(f, g) := sup_{x∈\overline{R^{2}}} q(f(x), g(x)).

Likewise, let d(f) denote the chordal distance of f to the identity:

(1)  d(f) := d(f, id) = sup_{x∈\overline{R^{2}}} q(f(x), x).

Suppose that G is a fixed convergence group acting on \overline{R^{2}} . If G is discrete, then all f ∈ G∖{id} are uniformly bounded away from the identity in the metric given by (1), i.e. there is a constant c > 0 (depending on G) such that

d(f) ≥ c

for all f ∈ G ∖ {id}.

The following theorems of Gehring and Martin [GM2, Theorems 4.19, 4.26, 6.14] make this observation more precise in the case of Kleinian groups; the authors find a uniform estimate in the chordal metric, independent of the group G. The first result is a consequence of Jørgensen’s inequality [Jør] and we shall refer to it as chordal Jørgensen inequality.

Theorem 2.1 (Chordal Jørgensen inequality). There is a constant c_{1} > 0 so that if f and g generate a discrete non-elementary subgroup of Möb then f and g satisfy:

max{d(f), d(g)} ≥ c_{1}  and  d(f) + d(g) ≥ 2c_{1}.

Furthermore,

2(\sqrt{2} − 1) = 0.828 … ≤ c_{1} ≤ 0.911 … = 2\sqrt{\frac{cos(2π/7) + cos(π/7) − 1}{cos(2π/7) + cos(π/7) + 1}} .

Recall that Jørgensen’s inequality [Jør] in its original form was stated as follows:

Theorem 2.2 (Jørgensen’s inequality). If the two Möbius transformations f, g generate a discrete non-elementary group then

| tr^{2}(f) − 4| + | tr(f ∘ g ∘ f^{−1} ∘ g^{−1}) − 2| ≥ 1,

where tr denotes the trace function.

The next result by Gehring and Martin says that we can conjugate any Kleinian group, so that the resulting group has the property that its non-identity elements are bounded away from zero in the chordal distance given in (1).

Theorem 2.3. If G is a discrete subgroup of Möb, then there exists an h ∈ Möb such that

d(f) ≥ c_{1}

for all f ∈ h ∘ G ∘ h^{−1} ∖ {id}. Here c_{1} is the same constant as in Theorem 2.1.

3. Statement of results

Our first results show that there are analogs to Theorem 2.1 and Theorem 2.3 for discrete quasiconformal groups. In particular, there is a chordal Jørgensen inequality for discrete K -quasiconformal groups:

Theorem 3.1. For each K ≥ 1, there is a constant c_{K} > 0 so that if f and g generate a discrete non-elementary K -quasiconformal group on \overline{R^{2}} , then f and g satisfy:

max{d(f), d(g)} ≥ c_{K}  and  (d(f))^{1/K} + (d(g))^{1/K} ≥ 2(c_{K})^{1/K}.

One (non-sharp) choice for the constant c_{K} is

c_{K} = \frac{1}{\sqrt{2}}(\frac{\sqrt{2} c_{1}}{128} )^{K},

where c_{1} is the constant from Theorem 2.1.

A relative version of Theorem 2.3 holds for discrete K -quasiconformal groups:

Theorem 3.2. If G is a discrete, torsion-free K -quasiconformal group on \overline{R^{2}} , then there exists an h ∈ Möb such that

d(f) ≥ c_{K}

for all f ∈ h ∘ G ∘ h^{−1} ∖ {id}. Here c_{K} is the same constant as in Theorem 3.1.

Our main result is that Theorem 3.1 does not hold for general discrete convergence groups:

Theorem 3.3 (Main theorem). There are sequences {f_{n}}, {g_{n}} of homeomorphisms on \overline{R^{2}} so that the group generated by f_{n} and g_{n} is a free, discrete non-elementary convergence group with non-empty regular set for each n ∈ N, but

max{d(f_{n}), d(g_{n})} → 0 as n → ∞.

We give a brief outline of the proof of Theorem 3.3. First we construct a sequence {G_{n}} of topological Schottky groups, where G_{n} is freely generated by f_{n} and g_{n}. We shall denote this by G_{n} = ⟨f_{n}, g_{n}⟩. Suppressing the index n in the notation, the idea of the construction of the groups G is as follows:

We shall find four mutually disjoint, simply connected regions S_{0} , S_{1} , S_{2} , S_{3} . The homeomorphism f will map the exterior of S_{2} onto the interior of S_{0} and S_{2} onto the exterior of S_{0} . The boundary of S_{2} will be mapped onto the boundary of S_{0} . The homeomorphism g will be constructed in a similar way, with S_{1} and S_{3} replacing S_{0} and S_{2} . By appropriately shrinking S_{0} , S_{1} and S_{3} under the map f, shrinking S_{1} , S_{2} , S_{3} under f^{−1} , shrinking S_{0} , S_{1} , S_{2} under g and shrinking S_{0} , S_{2} , S_{3} under g^{−1} , the group G = ⟨f, g⟩ will have the convergence property. Since G is the topological version of a Schottky group, it is discontinuous by construction.

The main difficulty consists in making d(f) and d(g) as small as desired. To this end it is necessary to have any point of the exterior of S_{0} be close to S_{2} and any point of the exterior of S_{2} be close to S_{0} . The same must be true with S_{1} and S_{3} replacing S_{0} and S_{2} . One way to satisfy these assumptions is to give the regions S_{0}, …, S_{3} the shape of spirals, see Figure 1. The hardest part now is to tune the action of f and g in such a way that d(f) and d(g) are small, while ensuring on the other hand that ⟨f, g⟩ has the convergence property.

Once the main theorem has been proved, using the density of diffeomorphisms in the homeomorphisms of \overline{R^{2}} it is not hard to see the following:

Corollary 3.4. There are sequences {f̂_{n}}, {ĝ_{n}} of quasiconformal mappings such that f̂_{n} and ĝ_{n} generate a discrete, non-elementary, K_{n} -quasiconformal group where K_{n} → ∞ and d(f̂_{n}) → 0 and d(ĝ_{n}) → 0 as n → ∞.

Note that by Theorem 3.1 K_{n} must necessarily become unbounded as n → ∞.

Even though there is no chordal Jørgensen inequality on the full class of discrete convergence groups, there is an analog of Theorem 3.2 for convergence groups with non-empty regular set [Geh1]:

Theorem 3.5. If G is a convergence group with non-empty regular set, then for each constant c with 0 < c < 2 there exists an h ∈ Möb such that

d(f) ≥ c

for all f ∈ h ∘ G ∘ h^{−1} ∖ {id}.

4. Proofs

We first show that normalized K -quasiconformal maps satisfy a chordal Hölder inequality. The result is probably known, though for the reader’s convenience we include a proof.

Lemma 4.1. Any K -quasiconformal homeomorphism φ of \overline{R^{2}} fixing 0, 1 and ∞ satisfies

q(φ(x), φ(y)) ≤ 128 · 2^{(1−K)/(2K)}q(x, y)^{1/K}

for all x, y ∈ \overline{R^{2}} .

Proof. The proof uses the following fact [Geh3, Theorem 4.1]: If φ: D → D′ is a K -quasiconformal map of the domain D ⊂ \overline{R^{2}} onto D′ ⊂ \overline{R^{2}} where \overline{R^{2}} ∖ D ≠ ∅, then

(2)  q(φ(x), φ(y)) · q(\overline{R^{2}} ∖ D′) ≤ 128 · (\frac{q(x, y)}{q(x, ∂D)})^{1/K}

for all x, y ∈ \overline{R^{2}} with x ≠ y. Here for a set E the expression q(E) denotes the chordal diameter of E and q(x, ∂E) is the chordal distance of x to the boundary of E.

Note that q(0, 1) = q(1, ∞) = \sqrt{2} and q(0, ∞) = 2. Let x, y ∈ \overline{R^{2}} .
Case 1. x, y ∈ {0, 1, ∞}. Then

\begin{aligned}q(φ(x), φ(y)) & = q(x, y) ≤ 2 ≤ 128 · 2^{(1−K)/(2K)} · \sqrt{2}^{1/K} \\ & ≤ 128 · 2^{(1−K)/(2K)} · q(x, y)^{1/K}. \end{aligned}

Case 2. y ∉ {0, 1, ∞}. If we are not in case 1 then we can always assume we are in case 2 by relabeling x and y.

(a) Assume first that q(x, 0) < \sqrt{2} /2. In this case we have q(x, 1) ≥ \sqrt{2} /2 and q(x, ∞) ≥ \sqrt{2} /2. Choosing D = D′ = \overline{R^{2}} ∖ \{1, ∞\} we obtain q(\overline{R^{2}} ∖ D′) = q(1, ∞) = \sqrt{2} and q(x, ∂D) ≥ 1/\sqrt{2} . Hence by (2) we have

q(φ(x), φ(y)) · \sqrt{2} ≤ 128 · q(x, y)^{1/K} · \sqrt{2}^{1/K},

and from this the claim follows.

(b) Assume next that q(x, 1) < \sqrt{2} /2. Choosing D = D′ = \overline{R^{2}} ∖ \{0, ∞\} and again applying (2) we obtain the desired result in this case.

(c) Assume finally that q(x, 0) ≥ \sqrt{2} /2 and q(x, 1) ≥ \sqrt{2} /2. Choose D = D′ = \overline{R^{2}} ∖ \{0, 1\} and again use (2) to complete the proof of the lemma.

We can now prove Theorem 3.1.

Proof of Theorem 3.1. Let G be a K -quasiconformal, discrete, non-elementa-ry convergence group, generated by the quasiconformal mappings f, g: \overline{R^{2}} → \overline{R^{2}} . Then G is K -quasiconformally conjugate to a Möbius group [Sul], [Tuk1], i.e. there is a K -quasiconformal map φ and a Möbius group Γ such that G = φ^{−1} ∘ Γ ∘ φ. By conjugating Γ with a Möbius map we may assume that φ fixes 0, 1, and ∞. Setting f̃ = φ ∘ f ∘ φ^{−1} and g̃ = φ ∘ g ∘ φ^{−1} we obtain that Γ is generated by f̃ and g̃, and furthermore Γ is a discrete, non-elementary group of Möbius transformations. Hence, by the chordal Jørgensen inequality (Theorem 2.1) we have

(3)  max{d(f̃), d(g̃)} ≥ c_{1},

where c_{1} > 0 is as in Theorem 2.1. Note that φ satisfies the inequality of Lemma 4.1. Let

c_{K} := \frac{1}{\sqrt{2}} (\frac{\sqrt{2} c_{1}}{128} )^{K}.

We argue by contradiction: Assume that max{d(f), d(g)} < c_{K}. Observe that

d(f̃) = d(φ ∘ f ∘ φ^{−1}) = d(φ ∘ f, φ) = sup_{x∈\overline{R^{2}}} q(φ(f(x)), φ(x)).

Since \overline{R^{2}} is compact, the supremum is obtained at some point x_{0} ∈ \overline{R^{2}} . For this x_{0} we have

\begin{aligned}q(φ(f(x_{0})), φ(x_{0})) & ≤ 128 · 2^{(1−K)/(2K)}q(f(x_{0}), x_{0})^{1/K} \\ & ≤ 128 · 2^{(1−K)/(2K)}d(f)^{1/K} < 128 · 2^{(1−K)/(2K)}(c_{K})^{1/K} = c_{1}, \end{aligned}

hence we have shown that d(f̃) < c_{1} . In the same way we obtain d(g̃) < c_{1} , contradicting (3).

For the proof of the second part of the theorem, we again argue by contradiction and assume that (d(f))^{1/K} + (d(g))^{1/K} < 2(c_{K})^{1/K}. As before, we obtain

\begin{aligned}d(f̃) + d(g̃) & ≤ 128 · 2^{(1−K)/(2K)} · (d(f))^{1/K} + 128 · 2^{(1−K)/(2K)} · (d(g))^{1/K} \\ & < 2 · 128 · 2^{(1−K)/(2K)} · (c_{K})^{1/K} = 2c_{1}. \end{aligned}

This contradicts Theorem 2.1.

We can now prove Theorem 3.2, following an argument given by Waterman [Wat].

Proof of Theorem 3.2. By conjugation with a Möbius transformation we may assume that 0 and ∞ are not fixed by any g ∈ G ∖ {id}. Define h_{t}(x) := t · x for 0 < t ≤ 1. Suppose that there is no t ∈ (0, 1] such that each element f ∈ h_{t} ∘ G ∘ h_{t}^{−1} ∖ {id} satisfies d(f) ≥ c_{K}. We seek a contradiction.

Observe that for g ∈ G ∖ {id} we have

(4)  d(h_{t} ∘ g ∘ h_{t}^{−1}) ≥ q(t · g(∞), ∞) → 2  as t → 0.

Observe also that, for fixed g ∈ G, the distance of h_{t} ∘ g ∘ h_{t}^{−1} to the identity (i.e. d(h_{t} ∘ g ∘ h_{t}^{−1})) varies continuously in t.

Let t̃_{0} ≤ 1. Then by assumption there exists g_{0} ∈ G ∖ {id} so that d(ht̃_{0} ∘ g_{0} ∘ h−1 ; t̃_{0}) < c_{K}. By continuity and using (4) we can find a largest t_{0}^{∗} < t̃_{0} so that d(h_{t} ∘ g_{0} ∘ h_{t}^{−1}) ≥ c_{K} for all t ≤ t_{0}^{∗}. Hence, by assumption, we find another element g_{1} ∈ G ∖ {id}, g_{1} ≠ g_{0} , so that d(h_{t_{0}^{∗}} ∘ g_{1} ∘ h_{t_{0}^{∗}}^{−1}) < c_{K}.

By continuity we can now find t_{0} ∈ (t_{0}^{∗}, t̃_{0}), so that

d(h_{t_{0}} ∘ g_{0} ∘ h_{t_{0}}^{−1}) < c_{K}  and  d(h_{t_{0}} ∘ g_{1} ∘ h_{t_{0}}^{−1}) < c_{K}.

Defining t̃_{1} := t_{0}^{∗} we have d(ht̃_{1} ∘ g_{1} ∘ h−1 ; t̃_{1}) < c_{K} and can restart our construction with t̃_{0} being replaced by t̃_{1} .

In general we find t_{n+1} < t_{n} and mutually distinct elements g_{n} ∈ G ∖ {id} so that

d(h_{t_{n}} ∘ g_{n} ∘ h_{t_{n}}^{−1}) < c_{K}  and  d(h_{t_{n}} ∘ g_{n+1} ∘ h_{t_{n}}^{−1}) < c_{K}

for all n ∈ N. By Theorem 3.1 we conclude that the groups ⟨g_{n}, g_{n+1}⟩ are elementary for each n ∈ N. The discrete elementary torsion-free convergence groups have been studied in [GM1, Theorems 5.7, 5.10, 5.11]. We conclude that either all g_{n} are loxodromic and have common fixed points a, b; or all g_{n} are parabolic and fix a common point a. By choosing a subsequence (and relabeling a, b if necessary) and using the convergence property, we may assume that

g_{n} → a  locally uniformly in \overline{R^{2}} ∖ \{b\} as n → ∞  and
g_{n}^{−1} → b  locally uniformly in \overline{R^{2}} ∖ \{a\} as n → ∞;

where a = b in the parabolic case. Note that {a, b} ∩ {0, ∞} = ∅ by assumption. Since each h_{t} fixes 0 and ∞, we obtain

(h_{t_{n}} ∘ g_{n} ∘ h_{t_{n}}^{−1})(∞) = t_{n} · g_{n}(∞)  and  (h_{t_{n}} ∘ g_{n} ∘ h_{t_{n}}^{−1})(0) = t_{n} · g_{n}(0),

where

g_{n}(0) → a  and  g_{n}(∞) → a as n → ∞.

Note that the decreasing sequence {t_{n}} converges to some t^{∗} ≥ 0 as n → ∞, hence

(h_{t_{n}} ∘ g_{n} ∘ h_{t_{n}}^{−1})(∞) → t^{∗}a  and  (h_{t_{n}} ∘ g_{n} ∘ h_{t_{n}}^{−1})(0) → t^{∗}a as n → ∞.

But q(0, t^{∗}a) + q(∞, t^{∗}a) ≥ q(0, ∞) = 2, so that we conclude

lim inf_{n→∞} d(h_{t_{n}} ∘ g_{n} ∘ h_{t_{n}}^{−1}) ≥ 1,

contradicting the fact that d(h_{t_{n}} ∘ g_{n} ∘ h_{t_{n}}^{−1}) < c_{K} ≤ c_{1} < 1 for all n.

In the proof of the fact that there is no chordal Jørgensen inequality for general discrete convergence groups we construct a sequence of free two generator discrete convergence groups, where both generators are arbitrarily close to the identity in the chordal distance.

Proof of Theorem 3.3. We construct a sequence of discrete non-elementary convergence groups ⟨f_{n}, g_{n}⟩, where f_{n}, g_{n} are homeomorphisms of \overline{R^{2}} , and max{d(f_{n}), d(g_{n})} → 0 as n → ∞. In the following we will suppress the index n in the notation; we write f, g instead of f_{n}, g_{n}.

Part I: A topological analog of Schottky groups. The group generated by f and g that we construct is a topological analog of a two generator Schottky group. That is, there are four disjoint regions S_{0} , S_{1} , S_{2} , S_{3} ; the homeomorphism f will map the exterior of S_{2} onto the interior of S_{0} , its inverse will map the exterior of S_{0} onto the interior of S_{2} , boundary will be mapped onto boundary. The homeomorphism g shall be constructed in the same way, using S_{1} and S_{3} instead of S_{0} and S_{2} .

The special shape of these regions will make it possible for d(f) and d(g) to be small and at the same time ensure the group generated by f and g has the convergence property. As already mentioned in the outline of the proof, in order to make d(f) as small as desired, it is necessary to have any point of the exterior of S_{0} be close to S_{2} and any point of the exterior of S_{2} be close to S_{0} . The same must hold for the regions S_{1} and S_{3} in order to make d(g) as small as desired. We can satisfy these assumptions by giving the regions S_{0}, …, S_{3} the shapes of spirals: Define

γ_{k}(t) := e^{2πit}e^{t/n+k/(8n)},  k = 0, …, 7,  −n^{2} ≤ t ≤ n^{2}.

Let S_{0} be the region bounded by γ_{0} , γ_{1} , the line segment

{e^{−2πin^{2}}e^{−n+s/(8n)} | 0 ≤ s ≤ 1}

(which connects γ_{0}(−n^{2}) and γ_{1}(−n^{2})) and the line segment

{e^{2πin^{2}}e^{n+s/(8n)} | 0 ≤ s ≤ 1}

(which connects γ_{0}(n^{2}) and γ_{1}(n^{2})). Define S_{1} to be the region bounded by γ_{2} , γ_{3} and similar radial line pieces. Define S_{2} and S_{3} analogously (see Figure 1), that is

S_{j} = {e^{2πit}e^{t/n+(2j+s)/(8n)} | −n^{2} ≤ t ≤ n^{2}, 0 ≤ s ≤ 1}, j = 0, 1, 2, 3.

Figure 1. The spiral regions

A point z ∈ S_{j}, j = 0, 1, 2, 3, can be described by its argument parameter t and its radius parameter s:

(5)  z = z(j, s, t) = e^{2πit}e^{t/n+(2j+s)/(8n)},  −n^{2} ≤ t ≤ n^{2}, 0 ≤ s ≤ 1.

Note that for example on the boundary of S_{0} we have

γ_{0}(t) = z(0, 0, t), γ_{1}(t) = z(0, 1, t).

Part II: The construction of the maps f and g. The following seven steps describe the construction of f. In steps 1–3 the map f is constructed on S_{0} ∪S_{1} ∪ S_{3} as a composition h_{3} ∘ h_{2} ∘ h_{1} of shrinking (via h_{1} , h_{2} ) and shifting (via h_{3} ) processes. In step 4, f is defined on ∂S_{2} , step 5 extends f to all of the exterior of S_{2} , so that f becomes a homeomorphism between the exterior of S_{2} and the interior of S_{0} ; boundary gets mapped onto boundary. In step 6 we define f^{−1} as a homeomorphism between the exterior of S_{0} and the interior of S_{2} in the same way as f has been defined before. Finally, step 7 combines both definitions and we obtain a homeomorphism f: \overline{R^{2}} → \overline{R^{2}} .

Step 1. The map h_{1} shrinks in t-direction. The map h_{1} shrinks the spirals S_{0} , S_{1} , and S_{3} by a factor (1 − 2/n^{2}) in “length” (t-direction), keeping points with t = 0 fixed, i.e. for

z = z(j, s, t) = e^{2πit}e^{t/n+(2j+s)/(8n)} ∈ S_{j}, j = 0, 1, 3, 0 ≤ s ≤ 1, −n^{2} ≤ t ≤ n^{2},

define

h_{1}(z) := z(j, s, t(1 − \frac{2}{n^{2}} )) = e^{2πit(1−2/n^{2})}e^{(t/n)(1−2/n^{2})+(2j+s)/(8n)}.

Then for j = 0, 1, 3 h_{1}(S_{j}) is a subspiral of S_{j}, which is as “thick” as S_{j}, but “shorter”:

h_{1}(S_{j}) = {e^{2πit}e^{t/n+(2j+s)/(8n)} | −n^{2} + 2 ≤ t ≤ n^{2} − 2, 0 ≤ s ≤ 1} ⊂ S_{j}.

Note that for t ≠ 0 the point z(j, s, t) travels towards the point e^{(2j+s)/(8n)} on the median of S_{j}, but chordally no points get moved far as we shall show in the following:

Let z = z(j, s, t) ∈ S_{j} as in (5) for j ∈ {0, 1, 3}. We consider three cases:

(i) −n\sqrt{n} ≤ t ≤ n\sqrt{n} . In this case
q(z, h_{1}(z)) = q(e^{2πit}|z|, e^{2πit(1−2/n^{2})}|h_{1}(z)|)

≤ q(e^{2πit}|z|, e^{2πit(1−2/n^{2})}|z|) + q(|z|, |h_{1}(z)|)

≤ q(e^{2πit}, e^{2πit(1−2/n^{2})}) + \frac{2|1 − |h_{1}(z)|/|z||}{\sqrt{|z|^{−2} + 1} \sqrt{1 + |h_{1}(z)|^{2}}}

≤ q(1, e^{−4πit/n^{2}}) + 2|1 − \frac{|h_{1}(z)|}{|z|}| ≤ q(1, e^{4πi/\sqrt{n}}) + 2|1 − e^{−2t/n^{3}}|

≤ q(1, e^{4πi/\sqrt{n}}) + 2|1 − e^{2/(n\sqrt{n})}|.

Hence q(z, h_{1}(z)) is arbitrarily small for large n, uniformly in −n\sqrt{n} ≤ t ≤ n\sqrt{n} . (ii) t > n\sqrt{n} . In this case

\begin{aligned}|z| & = e^{t/n+(2j+s)/(8n)} ≥ e^{t/n} ≥ e^{\sqrt{n}} and \\ |h_{1}(z)| & = e^{(t/n)(1−2/n^{2})+(2j+s)/(8n)} ≥ e^{(t/n)(1−2/n^{2})} ≥ e^{\sqrt{n} −2/(n\sqrt{n})}. \end{aligned}

Hence q(z, ∞) and q(h_{1}(z), ∞) are arbitrarily small for large n, and the same holds for q(z, h_{1}(z)) by the triangle inequality.

(iii) t < −n\sqrt{n} . In this case

|z| ≤ e^{−\sqrt{n} +7/(8n)}  and  |h_{1}(z)| ≤ e^{−\sqrt{n} +2/(n\sqrt{n})+7/(8n)},

so that q(z, 0), q(h_{1}(z), 0) and hence q(z, h_{1}(z)) are arbitrarily small for large n.

Summing up, we have shown that for given ε > 0 we can find a large enough n ∈ N such that any z ∈ S_{0} ∪ S_{1} ∪ S_{3} satisfies q(z, h_{1}(z)) < ε.

Step 2. The map h_{2} shrinks in s-direction. On the new shorter spirals h_{1}(S_{0} ∪ S_{1} ∪ S_{3}) we define a map h_{2} which shrinks in “width” (s-direction) by a factor 8, keeping each of the longitudinal center spiral lines given by s = \frac{1}{2} fixed. That is for −n^{2} + 2 ≤ t ≤ n^{2} − 2, 0 ≤ s ≤ 1, and

w = w(j, s, t) = e^{2πit}e^{t/n+(2j+s)/(8n)} ∈ h_{1}(S_{j}), j = 0, 1, 3

define

h_{2}(w) := w(j, \frac{s}{8} + \frac{7}{16}, t) = e^{2πit}e^{t/n+2j/(8n)+(1/8n)(s/8+7/16)}.

Thus for j = 0, 1, 3

(h_{2} ∘h_{1})(S_{j}) = \{e^{2πit}e^{t/n+(2j+s)/(8n)} | −n^{2} +2 ≤ t ≤ n^{2} −2, \frac{7}{16} ≤ s ≤ \frac{9}{16}\} ⊂ S_{j}.

As before we see that q(w, h_{2}(w)) is arbitrarily and uniformly small for all w ∈ h_{1}(S_{0} ∪ S_{1} ∪ S_{3}) given large enough n:

q(w, h_{2}(w)) = q(|w|, |h_{2}(w)|) ≤ q(e^{t/n+2j/(8n)}, e^{t/n+(2j+1)/(8n)})

= 2\frac{|1 − e^{1/(8n)}|}{\sqrt{e^{−2t/n−4j/(8n)} + 1} \sqrt{1 + e^{2t/n+(4j+2)/(8n)}}}

≤ 2|1 − e^{1/(8n)}| → 0 as n → ∞.

Step 3. Shifting (h_{2} ∘ h_{1})(S_{0} ∪ S_{1} ∪ S_{3}) into S_{0} . Next with a map h_{3} we move v ∈ (h_{2} ∘ h_{1})(S_{0} ∪ S_{1} ∪ S_{3}) along a radial line into S_{0} without meeting S_{2} , i.e. we keep v ’s argument e^{2πit} fixed. The map keeps points in (h_{2} ∘h_{1})(S_{0}) fixed, decreases the radius of points in (h_{2} ∘h_{1})(S_{1}), and increases the radius of points in (h_{2} ∘ h_{1})(S_{3}) so that these three shrunk spirals come to be placed equally spaced in S_{0} . To be precise: For −n^{2} + 2 ≤ t ≤ n^{2} − 2, \frac{7}{16} ≤ s ≤ \frac{9}{16} , and j = 0, 1, 3 the point

v = v(j, s, t) = e^{2πit}e^{t/n+(2j+s)/(8n)}

gets mapped to

h_{3}(v) := \{ \begin{matrix}v, & for j = 0, \\ v(0, s + \frac{9}{32}, t) = e^{2πit}e^{t/n+(s+(9/32))/(8n)}, & for j = 1, \\ v(0, s − \frac{9}{32}, t + 1) = e^{2πit}e^{(t+1)/n+(s−(9/32))/(8n)}, & for j = 3. \end{matrix}

Again, it is easy to verify that h_{3} moves points arbitrarily little, uniformly for all v ∈ (h_{2} ∘ h_{1})(S_{0} ∪ S_{1} ∪ S_{3}).

Together, the maps constructed above define f := h_{3} ∘h_{2} ∘h_{1} on S_{0} ∪S_{1} ∪S_{3} . Note that the only fixed point of f on these three spirals is the point

z = z(0, \frac{1}{2}, 0) = e^{2πi·0}e^{(1/n)·0+(1/2)/(8n)} = e^{1/(16n)}

which lies in S_{0} .

Step 4. The map f on ∂S_{2} . The map f maps ∂S_{2} onto ∂S_{0} as follows:

\begin{aligned}f(γ_{4}(t)) & := γ_{1}(t),  for − n^{2} ≤ t ≤ n^{2}, \\ f(γ_{5}(t)) & := γ_{0}(t + 1),  for − n^{2} ≤ t ≤ n^{2} − 1. \end{aligned}

Furthermore, define f so that it maps the radial segment

{e^{−2πin^{2}}e^{−n+(4+s)/(8n)} | 0 ≤ s ≤ 1} ⊂ ∂S_{2}

homeomorphically onto the set

{γ_{0}(t) | −n^{2} ≤ t ≤ −n^{2} + 1} ∪ {e^{−2πin^{2}}e^{−n+s/(8n)} | 0 ≤ s ≤ 1} ⊂ ∂S_{0}.

Finally, let f map the set

{γ_{5}(t) | n^{2} − 1 ≤ t ≤ n^{2}} ∪ {e^{2πin^{2}}e^{n+(4+s)/(8n)} | 0 ≤ s ≤ 1} ⊂ ∂S_{2}

homeomorphically onto the radial segment

{e^{2πin^{2}}e^{n+s/(8n)} | 0 ≤ s ≤ 1} ⊂ ∂S_{0}.

As before, no point gets moved far, given large enough n.

Step 5. Extending f to all of the exterior of S_{2} . With steps 1–4, f is defined on S_{0} ∪ S_{1} ∪ S_{3} ⊂ ext(S_{2}) and on the boundary of S_{2} . We now extend f to the remaining part of the exterior of S_{2} to become a homeomorphism mapping the exterior of S_{2} onto the interior of S_{0} , so that no point gets moved far. This can be done as follows: We first extend f to the four regions between the spirals, and then extend f to the two simply connected domains that are left around the origin and around the point infinity, respectively.

In order to extend f to the regions between the spirals, note that points in these regions can be described as in (5), but with j = 0.5, 1.5, 2.5, 3.5, i.e.

z(j, s, t) = e^{2πit}e^{t/n+(2j+s)/(8n)}, 0 < s < 1,

and −n^{2} ≤ t ≤ n^{2} for j = 0.5, 1.5, 2.5 and −n^{2} ≤ t ≤ (n^{2} − 1) for j = 3.5.

For the region between the spirals S_{0} and S_{1} , i.e. points of the form z(0.5, s, t), we define f by “interpolating” between the images of the points γ_{1}(t) and γ_{2}(t). Note that γ_{1}(t) and γ_{2}(t) are points on the adjacent boundaries of S_{0} and S_{1} , respectively, that have the same argument t as the point z. Since f(γ_{1}(t)) ∈ S_{0} and f(γ_{2}(t)) ∈ S_{0} , there are unique parameters s_{1}, s_{2} ∈ [0, 1] and t_{1}, t_{2} ∈ [−n^{2}, n^{2}] such that

f(γ_{1}(t)) = z(0, s_{1}, t_{1})  and  f(γ_{2}(t)) = z(0, s_{2}, t_{2}).

We now set

f(z(0.5, s, t)) := z(0, (1 − s) · s_{1} + s · s_{2}, (1 − s) · t_{1} + s · t_{2}),

i.e. we interpolate linearly between the “width” and “length” parameters of the images of γ_{1}(t) and γ_{2}(t). Since z(0.5, s, t) is arbitrarily close to both γ_{1}(t) and γ_{2}(t) for large enough n, and since f moves the points γ_{1}(t) and γ_{2}(t) an arbitrarily small distance, we conclude that the extension of f to the region between S_{0} and S_{1} moves points an arbitrarily small distance, as well.

For points z(1.5, s, t) between the spirals S_{1} and S_{2} we define f in exactly the same way as above by using the boundary curves γ_{3} and γ_{4} instead of γ_{1} and γ_{2} . For points z(2.5, s, t) between the spirals S_{2} and S_{3} we use the boundary curves γ_{5} and γ_{6} instead. As before, we see that f moves points as little as desired, given large enough n.

For points z(3.5, s, t) (−n^{2} ≤ t ≤ (n^{2} − 1)) we define f by “interpolating” between the images of the points γ_{7}(t) and γ_{0}(t + 1), which are on the adjacent boundaries of S_{3} and S_{0} , respectively. Again, f does not move points far.

The only parts in the exterior of S_{2} , where f has not been defined yet are two simply connected domains. One of these domains, denoted V_{0} , is bounded by

{γ_{0}(t) | −n^{2} ≤ t ≤ −n^{2} + 1} ∪ {e^{−2πin^{2}}e^{−n+s/(8n)} | 0 ≤ s ≤ 8},

i.e. the first spiral part of γ_{0} and the line segment joining γ_{0}(−n^{2}) and γ_{0}(−n^{2}+1). The other domain, denoted V_{1} , is bounded by

{γ_{7}(t) | n^{2} − 1 ≤ t ≤ n^{2}} ∪ {e^{2πin^{2}}e^{n+s/(8n)} | −1 ≤ s ≤ 7},

i.e. the last spiral part of γ_{7} and the line segment joining γ_{7}(n^{2} − 1) and γ_{7}(n^{2}). Note that f has already been defined on the boundaries of the domains V_{0} and V_{1} . Note furthermore, that any two points in V_{0} are chordally close to each other for large enough n, and the same holds for any two points in V_{1} . Hence we can extend f to all of V_{0} and V_{1} using the topological Schoenflies theorem, so that f becomes a homeomorphism between V_{0} and its image, and between V_{1} and its image, where no point gets moved far.

With the above definitions, f becomes a homeomorphism of the exterior of S_{2} onto the interior of S_{0} .

Step 6. Defining f^{−1} on the exterior of S_{0} . In the same way as f was defined on S_{0}∪S_{1}∪S_{3} in steps 1–3, we now define its inverse f^{−1} on S_{1}∪S_{2}∪S_{3} . I.e. f^{−1} shrinks S_{1} , S_{2} , and S_{3} in length and width, then moves these spirals into S_{2} . On ∂S_{0} , we define f^{−1}: ∂S_{0} → ∂S_{2} as the inverse of the already defined map f: ∂S_{2} → ∂S_{0} (compare step 4). Note that f^{−1} fixes z(2, \frac{1}{2}, 0) = e^{9/(16n)} ∈ S_{2} and no other point on S_{1} ∪S_{2} ∪S_{3} . In the same way as it was done for f in step 5, we now extend f^{−1} to all of the exterior of S_{0} , so that f^{−1} maps the exterior of S_{0} homeomorphically onto the interior of S_{2} .

Step 7. The map f. Combining the definitions of f and f^{−1} , we obtain a homeomorphism f: \overline{R^{2}} → \overline{R^{2}} , which has exactly two fixed points, and which is as close to the identity as desired.

With these steps the construction of the map f is complete. In the same way as above, we now construct the map g. That is: g maps the exterior of S_{3} onto the interior of S_{1} and g^{−1} maps the exterior of S_{1} onto the interior of S_{3} , where boundary gets mapped onto boundary.

Next we show that the free group generated by f and g is a convergence group.

Part III: ⟨f, g⟩ is a convergence group. By construction, ⟨f, g⟩ is a free group, i.e. every element h ∈ ⟨f, g⟩ has a unique (shortest) representation as a word

h = h^{(k)} ∘ h^{(k−1)} ∘ ⋯ ∘ h^{(1)}, where h^{(l)} ∈ {f, f^{−1}, g, g^{−1}}.

In the following we shall call the letter h^{(1)} the “first letter” or “beginning” and the letter h^{(k)} the “last letter” or “end” of the word h. Denote by ℓ_{l}(h) the lth letter of the unique representation of h, i.e.

ℓ_{l}(h) = h^{(l)}, l = 1, …, k.

We shall call the regions S_{0} , S_{1} , S_{2} , and S_{3} “spirals of generation 0”, whereas images of a spiral S_{j} under a k -letter word h which does not start with the letter that maps the exterior of S_{j} onto some other spiral will be called “k th generation spirals”. For example, f(S_{0}), f(S_{1}) and f(S_{3}) are spirals of generation 1 (they are all small subspirals of S_{0} ), but f(S_{2}) is not a spiral anymore. Note that the chordal diameter of all k th generation spirals tends uniformly to 0 as k tends to ∞.

Let {h_{m}} be an infinite sequence of distinct elements in ⟨f, g⟩. Then the word length of h_{m} is unbounded as m → ∞. Choose a subsequence of {h_{m}} so that the word length of the mth element of the new sequence is ≥ 2m. Denote this new sequence by {h_{m}}. We can choose a subsequence {h_{m}^{1}} of {h_{m}} so that ℓ_{1}(h_{m}^{1}) is constant for all m, and also ℓ_{1}((h_{m}^{1})^{−1}) is constant for all m. That is, all words in this subsequence start with the same letter w_{1} ∈ {f, f^{−1}, g, g^{−1}} and end with the same letter v_{1} . From this sequence {h_{m}^{1}} we can choose another subsequence {h_{m}^{2}} so that ℓ_{2}(h_{m}^{2}) is equal to some w_{2} ∈ {f, f^{−1}, g, g^{−1}} for all m and ℓ_{2}((h_{m}^{2})^{−1}) = v_{2}^{−1} for all m. Proceed like this, and denote the diagonal sequence {h_{m}^{m}} by {H_{m}}. Then

H_{m} = v_{1} ∘ v_{2} ∘ ⋯ ∘ v_{m} ∘ r_{m} ∘ w_{m} ∘ w_{m−1} ∘ ⋯ ∘ w_{1},

where r_{m} is some word of unknown length, which does not start with w_{m}^{−1} and does not end with v_{m}^{−1} .

We now show that there are a, b ∈ S_{0}∪S_{1}∪S_{2}∪S_{3} , so that H_{m} → a uniformly on U = \overline{R^{2}} ∖ (S_{0} ∪ S_{1} ∪ S_{2} ∪ S_{3}), and H_{m}^{−1} → b uniformly on U.

By definition, the map v_{m} (which is one of the maps f, f^{−1}, g, g^{−1} ) maps the exterior of some 0th generation spiral onto the interior of some other 0th generation spiral, and v_{m}^{−1} reverses this process. Denote by S_{E(m)} the spiral whose exterior is mapped by v_{m} onto another spiral, called S_{I(m)} . Since the last letter of r_{m} is not v_{m}^{−1} , we know that (r_{m} ∘w_{m} ∘w_{m−1} ∘⋯∘w_{1})(U) is contained in a 0th generation spiral different from S_{E(m)} and hence is in the exterior of S_{E(m)} . Thus (v_{m} ∘ r_{m} ∘ w_{m} ∘ w_{m−1} ∘ ⋯ ∘ w_{1})(U) is contained in S_{I(m)} . Furthermore, (v_{1} ∘ v_{2} ∘ ⋯ ∘ v_{m−1})(S_{I(m)}) is an (m − 1)st generation spiral, whose “length” has been shrunk by a factor (1 − 2/n^{2})^{m−1} , where n is the variable but now fixed parameter of the construction of f and g. Hence, H_{m}(U) is contained in this (m − 1)st generation spiral. Observe now that H_{m+1}(U) is contained in an mth generation spiral, being a subspiral of the previous (m − 1)st generation spiral. By construction, the chordal diameter of all mth generation spirals converges uniformly to 0 as m → ∞. Hence there exists a unique point a so that H_{m}(x) → a uniformly in x ∈ U.

A similar argument shows H_{m}^{−1} → b uniformly in U for some b ∈ S_{0} ∪ S_{1} ∪ S_{2} ∪ S_{3} . Both points a and b are limit points of descending “Cantor-type” spiral sequences.

Finally, we show that

H_{m} → a locally uniformly in \overline{R^{2}} ∖ \{b\},

and

H_{m}^{−1} → b locally uniformly in \overline{R^{2}} ∖ \{a\}.

Let K be a compact connected set in \overline{R^{2}} ∖ \{b\}. Then there is an open neighborhood of b, which does not intersect K. Thus K intersects only finitely many elements of the Cantor spiral sequence converging to b, say only spirals of generation ≤ k. Then (w_{k+2} ∘ w_{k+1} ∘ ⋯ ∘ w_{1})(K) is entirely contained in one of the spirals S_{0} , S_{1} , S_{2} or S_{3} : Otherwise there would be x ∈ U, so that

(w_{1}^{−1} ∘ w_{2}^{−1} ∘ ⋯ ∘ w_{k+2}^{−1})(x) ∈ K.

But this contradicts the fact that (w_{1}^{−1} ∘ w_{2}^{−1} ∘ ⋯ ∘ w_{k+2}^{−1})(U) is contained in a (k + 1)st generation spiral, which is part of the sequence converging to b.

Again, since there are no cancellations between the letters of H_{m}, we see that (v_{m} ∘ r_{m} ∘ w_{m} ∘ ⋯ ∘ w_{1})(K) is contained in S_{I(m)} for m ≥ k + 2, so that H_{m}(K) is contained in a (m − 1)st generation spiral for m ≥ k + 2. This shows that H_{m} → a uniformly in K. Similarly we obtain H_{m}^{−1} → b locally uniformly in \overline{R^{2}} ∖ \{a\}.

Hence we have shown that ⟨f, g⟩ is a convergence group. Furthermore it is obvious that ⟨f, g⟩ acts discontinuously on U, so that ⟨f, g⟩ is discrete. Finally ⟨f, g⟩ is non-elementary, since both f and g are loxodromic and have disjoint fixed point sets. This completes the proof.

Remark 1. Note that the limit set L(⟨f, g⟩) of ⟨f, g⟩ is a Cantor set with

L(⟨f, g⟩) ⊂ {e^{s/(8n)} | 0 ≤ s ≤ 7}.

Hence the chordal diameter of the limit sets converges to 0 as n → ∞.

Remark 2. We can modify step 1 of the above construction, so that for each ε > 0 there exists n ∈ N such that L(⟨f, g⟩) is ε-dense in \overline{R^{2}} , i.e. for each x ∈ \overline{R^{2}} the chordal ball B_{ε}(x) meets L(⟨f, g⟩). We can do this by only modifying one of the maps, say f. In step 1, instead of giving f an attracting fixed point at z(0, \frac{1}{2}, 0) in S_{0} and a repelling fixed point at z(2, \frac{1}{2}, 0) in S_{2} , we let the attracting fixed point be z(0, \frac{1}{2}, −n\sqrt{n} ) in S_{0} , and the repelling fixed point z(2, \frac{1}{2}, n\sqrt{n} ) in S_{2} . This can be done by redefining h_{1} on S_{0} ∪ S_{1} ∪ S_{3} : Let h_{1} map the point

z = z(j, s, t) = e^{2πit}e^{t/n+(2j+s)/(8n)} ∈ S_{j}

onto the point

\begin{aligned}h_{1}(z) & = z(j, s, t(1 − \frac{2}{n^{2}} ) − \frac{2}{\sqrt{n}} ) \\ & = e^{2πi[t(1−2/n^{2})−2/\sqrt{n}]}e^{(1/n)[t(1−(2/n^{2}))−(2/\sqrt{n})]+(2j+s)/(8n)}. \end{aligned}

With the same modification for the definition of f^{−1} on S_{1} ∪ S_{2} ∪ S_{3} the map f fixes

e^{−2πin\sqrt{n}} e^{−\sqrt{n}+(1/2)/(8n)} ∈ S_{0}

and

e^{2πin\sqrt{n}} e^{\sqrt{n}+(4+1/2)/(8n)} ∈ S_{2}.

Let g be unchanged. If ε > 0 is given then we can find n ∈ N such that d(f) < \frac{1}{2}ε, d(g) < \frac{1}{2}ε, and the attracting fixed point of f is in an \frac{1}{2}ε-neighborhood of 0, the repelling fixed point of f is in an \frac{1}{2}ε-neighborhood of ∞, and the distance from

z = z(j, s, t) = e^{2πit}e^{t/n+(2j+s)/(8n)}

to

z′ = z′(j, s, t + 1) = e^{2πi(t+1)}e^{(t+1)/n+(2j+s)/(8n)}

is less than \frac{1}{2}ε for all j, s, t.

Note that the images of all fixed points of f and g under maps in ⟨f, g⟩ are contained in the limit set L(⟨f, g⟩). Observe now that g moves the fixed points of f in steps of length ≤ \frac{1}{2}ε towards its attracting fixed point, i.e. these images will be \frac{1}{2}ε-dense in g ’s attracting spiral S_{3} . Similar for g^{−1} . The map f maps this picture into S_{0} , whereas f^{−1} maps it into S_{2} . Since by construction the spirals S_{j} and S_{(j+1)mod 4} are within distance \frac{1}{2}ε of each other, the limit set L(⟨f, g⟩) is ε-dense in \overline{R^{2}} .

Proof of Corollary 3.4. Let G_{n} = ⟨f_{n}, g_{n}⟩ be the discrete, non-elementary convergence groups constructed in Theorem 3.3. Then by construction G_{n} does not contain parabolics and its limit set L(G_{n}) is a Cantor set. Thus by [MS] the group G_{n} is topologically conjugate to a Möbius group, i.e. there exists a homeomorphism Φ_{n}: \overline{R^{2}} → \overline{R^{2}} and a Möbius group Γ_{n} = ⟨f̃_{n}, g̃_{n}⟩ such that f_{n} = Φ_{n} ∘ f̃_{n} ∘ Φ_{n}^{−1} , g_{n} = Φ_{n} ∘ g̃_{n} ∘ Φ_{n}^{−1} , and hence

G_{n} = Φ_{n} ∘ Γ_{n} ∘ Φ_{n}^{−1}.

Because of the topological conjugacy, Γ_{n} is a discrete, non-elementary Möbius group. Since the diffeomorphisms of \overline{R^{2}} are dense in the homeomorphisms or \overline{R^{2}} , we can choose a quasiconformal map Ψ_{n}: \overline{R^{2}} → \overline{R^{2}} with d(Ψ_{n}, Φ_{n}) < 1/n. Define f̂_{n} = Ψ_{n} ∘ f̃_{n} ∘ Ψ_{n}^{−1} and ĝ_{n} = Ψ_{n} ∘ g̃_{n} ∘ Ψ_{n}^{−1} . Then

d(f̂_{n}) = d(Ψ_{n} ∘ f̃_{n}, Ψ_{n})
≤ d(Ψ_{n} ∘ f̃_{n}, Φ_{n} ∘ f̃_{n}) + d(Φ_{n} ∘ f̃_{n}, Φ_{n}) + d(Φ_{n}, Ψ_{n})
= 2d(Φ_{n}, Ψ_{n}) + d(f_{n}) → 0 as n → ∞,

and similarly

d(ĝ_{n}) → 0 as n → ∞.

For the proof of Theorem 3.5 we first show a lemma:

Lemma 4.2. Suppose that G is a convergence group with non-empty regular set. Then there exists an open ball U such that f(U)∩U = ∅ for all f ∈ G∖{id}.

Proof. Since by hypothesis the regular set of G is non-empty, there exists a point y_{0} ∈ Ω(G) and an open neighborhood V of y_{0} such that f(V) ∩ V = ∅ for all but a finite number of elements f ∈ G, let f_{1}, …, f_{k} ∈ G ∖ {id} be these elements. If E_{j} denotes the set of fixed points of f_{j}, then E_{j} is closed by continuity of f_{j}. Furthermore, E_{j} contains no interior points: If E_{j} is finite then this is clear, otherwise f_{j} is elliptic and E_{j} has no interior points by a theorem of Newman’s [New] on periodic homeomorphisms of spaces. Thus we find x_{0} ∈ V which is not fixed by any f_{j}, and hence

s := min_{j=1,...,k} q(f_{j}(x_{0}), x_{0}) > 0.

Let U be a chordal ball about x_{0} with radius r > 0, where r is chosen so that r < \frac{1}{2}s, U ⊂ V , and so that x ∈ U implies q(f_{j}(x), f_{j}(x_{0})) < \frac{1}{2}s for j = 1, …, k. Then for x ∈ U we have

q(f_{j}(x), x_{0}) ≥ q(f_{j}(x_{0}), x_{0}) − q(f_{j}(x), f_{j}(x_{0})) > s − \frac{1}{2}s > r,

hence f_{j}(x) ∉ U. Thus U satisfies

f(U) ∩ U = ∅ for all f ∈ G ∖ {id}.

This lemma enables us to prove our final theorem.

Proof of Theorem 3.5. Let 0 < c < 2. Choose U as in Lemma 4.2 and let h be a Möbius transformation which maps U onto the chordal ball V with center ∞ and radius c. Then if

f̃ = h ∘ f ∘ h^{−1}, f ∈ G ∖ {id},

we have

f̃(V) ∩ V = h(f(U) ∩ U) = ∅,

and hence

d(f̃, id) ≥ q(f̃(∞), ∞) ≥ c.

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Received 21 April 1998