Annales Academiæ Scientiarum Fennicæ Mathematica Volumen 22, 1997, 387–394

CONSERVATIVE ACTION AND THE HOROSPHERIC LIMIT SET

Pekka Tukia

University of Helsinki, Department of Mathematics P.O. Box 4 (Yliopistonkatu 5), FIN-00014 Helsinki, Finland; pekka.tukia@helsinki.fi

Abstract. D. Sullivan [4] showed that a discrete Möbius group G of S^{n} acts conservatively on the horospheric limit set ℋ (G) of G, that is if A ⊂ ℋ (G) has positive measure, then so has A ∩ gA for infinitely many g in G. On the other hand, G has a measurable fundamental set outside ℋ (G). We show that if the definition of the horospheric limit set slightly changed, these results are also valid for any conformal measure μ of G.

Let X be a topological space with a finite Borel measure μ and let G be a countable group of measurable, absolutely continuous bijections of X. If A ⊂ X is setwise G-invariant, then the action of G is conservative on A (sometimes the word “recurrent” is also used) if, whenever U ⊂ A is measurable and μ(U) > 0, then μ(U ∩gU) > 0 for infinitely many g ∈ G. In contrast, the action is dissipative on A if there is a measurable fundamental set for the action of G in A, that is there is a measurable set F ⊂ A such that F contains exactly one point from each orbit Gz, z ∈ A. Conservative and dissipative action are not very compatible: if G acts conservatively and no point of A is fixed by some g ≠ id in G, then there cannot be a measurable fundamental set for the action on A if μ(A) > 0. Under fairly general conditions, it is possible to divide X into measurable disjoint pieces A and B, X = A ∪ B, such that the action is conservative on A and dissipative on B. This result seems to be generally known but the only reference I know is [1] whose proof works if the set of points x ∈ X fixed by some g ≠ id in G is a nullset.

Let G be a discrete group of Möbius transformations of the n-sphere S^{n} and let m be the n-dimensional Hausdorff measure on S^{n}. In this case Sullivan [4] identified the conservative part as the horospheric limit set ℋ (G). Recall that a Möbius group of S^{n} also acts on the open unit (n + 1)-ball B^{n+1} and that a horoball at x ∈ S^{n} is an open (n + 1)-ball B properly contained in B^{n+1} such that ∂B is tangent to S^{n} at x. Now we can define that a point x ∈ S^{n} is in the horospheric limit set ℋ (G) if any horoball at x contains an infinite number of points from any orbit Gz, z ∈ B^{n+1} .

1991 Mathematics Subject Classification: Primary 30F40; Secondary 28D99.

We will use the following variation of the horospheric limit set: We define the big horospheric limit set ℋ_{b}(G) as the set of points x ∈ S^{n} such that, given z ∈ B^{n+1} , then there is some horoball B at x such that B ∩ Gz is infinite.

We will extend Sullivan’s result for general conformal measures. A conformal (G-)measure of dimension δ on S^{n} is a finite Borel measure satisfying the transformation rule

μ(gX) = ∫_{X} |g′|^{δ} dμ

for g ∈ G and Borel subsets X of S^{n}; here δ is the dimension of μ. Conformal measures are sometimes called Patterson–Sullivan measures and their general theory was developed in [5] (cf. also Nicholls [3]). Typically, they are supported by the limit set L(G) of G but of course we can extend by zero to all of S^{n}. For instance, the Hausdorff n-measure of S^{n} is a conformal measure and, more generally, if A ⊂ S^{n} is G-invariant and the δ -dimensional Hausdorff-measure m_{δ} of A is finite, then m_{δ} is a conformal measure.

Sullivan extended a method of Patterson in [5] and showed that there is a non-trivial conformal measure supported by L(G) for any infinite discrete Möbius group; the dimension of this measure is the exponent of convergence δ_{G} of G. The measure is unique in many cases, and for the so-called convex cocompact groups the δ_{G} -dimensional Hausdorff measure of L(G) is a conformal measure. Using conformal measures, it is possible to extend the measure theoretical aspects of the theory of Möbius groups of first kind (i.e. L(G) = S^{n}) to the groups of the second kind (i.e. L(G) ≠ S^{n}).

We claim that the conservative part is the big horospheric limit set ℋ_{b}(G). The ideas of the proof are much the same as in [4] but there are some complications, for instance we had to move to the big horospheric limit set.

The set 𝒢 = ℋ_{b}(G) \ ℋ (G) is very similar to the “Garnett points” of [4] since, if x ∈ 𝒢 and z ∈ B^{n+1} , there is a unique horoball B_{x} at x such that Gz ∩ B is finite if B is any smaller horoball at x and infinite if B is any larger horoball at x; in contrast, we recall that x is a Garnett point if x ∈ 𝒢 and any smaller horoball B ⊂ B_{x} does not contain any points of Gz. We denote the set of Garnett points by 𝒢_{0} ; unlike ℋ (G) and ℋ_{b}(G) it might depend on the orbit Gz but we assume a fixed reference point z and define 𝒢_{0} using it.

Part of Sullivan’s proof (due to John Garnett) of [4] was to show that the ndimensional Hausdorff measure m of 𝒢_{0} vanishes; the same proof would also show that m(𝒢) = 0. So Sullivan’s original theorem could have been formulated as well using the big horospheric limit set. Unfortunately, the proof of the null-measure of Garnett points does not seem to generalize for general conformal measures (although it is possible to prove that μ(𝒢 \ 𝒢_{0}) = 0, see the Appendix).

We recall that a Möbius transformation g of S^{n} can be classified as elliptic (including the identity map), loxodromic or parabolic: g is elliptic if g can be conjugated by a Möbius transformation of S^{n} onto R̄^{n} to an orthogonal linear map of R^{n}, g is loxodromic if g can be conjugated to a map of the form x ↦ λβ where λ > 1 and β is an orthogonal linear map, and g is parabolic if g can be conjugated to a map of the form x ↦ β(x) + a where a ∈ R^{n} and β is an orthogonal linear map such that β(a) = a. We consider here only non-elementary Möbius groups defined as discrete Möbius groups such that the limit set contains more than two points.

Our aim is to prove

Theorem 1. Let G be a non-elementary Möbius group on S^{n} and let μ be a conformal G-measure on S^{n}. Then the action of G is conservative on the big horospheric limit set ℋ_{b}(G) and dissipative on S^{n} \ ℋ_{b}(G). If 𝒞 and 𝒟 are G-invariant subsets of S^{n} and G acts conservatively on 𝒞 and dissipatively on 𝒟, then 𝒟 ⊂ S^{n} \ ℋ_{b}(G) and 𝒞 ⊂ ℋ_{b}(G) up to μ-nullsets and atoms at parabolic fixed points.

Remark. So we have found the dissipative and conservative parts and they are unique up to nullsets and atoms at parabolic fixed points.

Note that a loxodromic fixed point cannot be an atom of μ since the dimension δ of μ is positive for non-elementary G ([5]) and since |g′(v)| ≠ 1 at a fixed point v of a loxodromic g. Thus the stabilizer G_{v} = {g ∈ G : g(v) = v} of an atom of μ can contain only parabolic and elliptic elements and if G_{v} is purely elliptic, it is finite by [6, Corollary C] and in this case v must be put into the dissipative part. Consequently, if G_{v} is infinite and v is an atom, then G_{v} must contain parabolic elements and obviously the orbit Gv can be put at will either to the conservative or the dissipative part.

We start the proof of Theorem 1 with the following characterization ℋ_{b}(G). Here |g′(z)| is the operator norm of the derivative. The same proof shows that z ∈ ℋ (G) if and only if there are g_{i} ∈ G such that |g′_{i}(z)| → ∞, cf. (3).

Lemma 2. A point z ∈ S^{n} is in the big horospheric limit set ℋ_{b}(G) if and only if there are m > 0 and a sequence of distinct g_{i} ∈ G such that |g′_{i}(z)| ≥ m.

Proof. We will prove that if g_{i} ∈ G are distinct, then g_{i}^{−1}(0) are in some horoball at z if and only if there is m > 0 such that |g′_{i}(z)| ≥ m.

Suppose that h_{i} ∈ G are distinct. Then |h_{i}(0)| → 1 by discreteness and a simple calculation shows that setting d_{i} = 1 − |h_{i}(0)|, then h_{i}(0) are in some horoball at z if and only if

(1)  |z − h_{i}(0)| ≤ a\sqrt{d_{i}}

for some suitable positive constant a.

Let g_{i} = h_{i}^{−1} . We claim that (1) is true for some a > 0 if and only if

(2)  |g′_{i}(z)| ≥ m

for some m > 0.

The plan of the proof is simple. We first show that |g′_{i}| = |σ′_{i}| when σ_{i} is the reflection on the isometric sphere S_{i} of g_{i}. Let τ_{i} be the center and r_{i} the radius of the isometric sphere S_{i}. We show that r_{i} is approximately \sqrt{2d_{i}}. Remembering that |σ′_{i}(w)| = r_{i}^{2}|w − τ_{i}|^{−2} , to conclude that (1) and (2) are equivalent we will show that |z − h_{i}(0)| is approximately |z − τ_{i}|.

Let L_{i} be the hyperbolic line segment joining 0 and h_{i}(0). The isometric sphere of g_{i} is the n-sphere S_{i} orthogonal to S^{n} such that S_{i} intersects L_{i} orthogonally at the hyperbolic midpoint c_{i} of L_{i}. We can see this as follows. Let σ_{i} be the reflection on S_{i}. Then σ_{i}h_{i}(0) = 0 and both σ_{i} and h_{i} preserve S^{n}. However, the only Möbius transformations preserving 0 and S^{n} are orthogonal linear maps of R^{n}. Thus |(σ_{i}h_{i})′| = 1. Since also |(g_{i}h_{i})′| = 1, it follows that |g′_{i}| = |σ′_{i}|, and so we can use σ_{i} in (2) instead of g_{i}. In particular, S_{i} is the isometric sphere of g_{i}, i.e. the n-sphere S such that |g′_{i}| = 1 on S.

Let the hyperbolic metric of B^{n+1} be given by the element of length 2|dx|/(1− |x|^{2}). Then the hyperbolic distance is d(0, x) = log(1 + |x|)/(1 − |x|)). Thus

log \frac{1 + |c_{i}|}{1 − |c_{i}|} = \frac{1}{2} log \frac{1 + |h_{i}(0)|}{1 − |h_{i}(0)|}

implying

1 − |c_{i}| = a_{i}\sqrt{1 − |h_{i}(0)|} = a_{i}\sqrt{d_{i}}

where a_{i} → \sqrt{2} as i → ∞. We also obtain

|c_{i} − h_{i}(0)| = |h_{i}(0)| − |c_{i}| = (1 − |c_{i}|) − (1 − |h_{i}(0)| = a′_{i}\sqrt{d_{i}}

where a′_{i} → \sqrt{2} as i → ∞.

The center τ_{i} of the isometric circle is mapped by σ_{i} onto ∞ and so |σ′_{i}(τ_{i})| = |g′_{i}(τ_{i})| = ∞ and so also g_{i}(τ_{i}) = ∞. Let ϱ be the reflection on S^{n}. Then ϱ interchanges 0 and ∞ and commutes with Möbius transformations preserving S^{n}. Thus 0 = ϱg_{i}(τ_{i}) = g_{i}ϱ(τ_{i}) and so ϱ(τ_{i}) = h_{i}(0). Since |h_{i}(0)| → 1,

|h_{i}(0) − τ_{i}| = b_{i}(1 − |h_{i}(0)|) = b_{i}d_{i}

where b_{i} → 2 as i → ∞. It follows that the radius of the isometric circle

r_{i} = |c_{i} − τ_{i}| = |c_{i} − h_{i}(0) + h_{i}(0) − τ_{i}| = a′′_{i}\sqrt{d_{i}}

where a′′_{i} → \sqrt{2} as i → ∞. Similarly,

|z − τ_{i}| = |z − h_{i}(0) + h_{i}(0) − τ_{i}| = |z − h_{i}(0)| + δ_{i}

where |δ_{i}| ≤ b_{i}d_{i}.

We can now estimate the derivative

(3)  |g′_{i}(z)| = |σ′_{i}(z)| = \frac{r_{i}^{2}}{|z − τ_{i}|^{2}} = \frac{(a′′_{i})^{2}d_{i}}{(|z − h_{i}(0)| + δ_{i})^{2}} .

The conclusion is now immediate. If (1) is true, then (2) is true for big i with m = a^{−2} . If (2) is true, then (1) is true for big i with a = 2/\sqrt{m}.

We return to the proof of the theorem. If x is an atom of μ, then by the above Remark, x cannot be fixed by a loxodromic g ∈ G and if x is fixed by a parabolic g ∈ G, then we can put the orbit of x at will either to the conservative or to the dissipative part. Consequently, to prove the theorem we can assume that no atom of μ is fixed by an element of G of infinite order. Under this assumption, we will prove:

A. S^{n} \ ℋ_{b}(G) has a measurable fundamental set. B. The action of G is conservative on ℋ_{b}(G). C. If 𝒞 ⊂ S^{n} is G-invariant and G acts conservatively on 𝒞 , then 𝒞 ⊂

ℋ_{b}(G) up to a nullset.

D. If 𝒟 ⊂ S^{n} is G-invariant and G is dissipative on 𝒟, then 𝒟 ⊂ S^{n}\ℋ_{b}(G) up to a nullset.

Proof of A. Denote 𝒟 = S^{n} \ ℋ_{b}(G). If z ∈ 𝒟, then Γ_{z} = {g ∈ G : |g′(z)| ≥ |h′(z)| for all h ∈ G} is finite and non-empty by Lemma 2. Thus the set

(4)  F_{0} = {x ∈ 𝒟 : |f′(x)| ≤ 1 for all f ∈ G}

which is also the union ⋃_{z∈𝒟} Γ_{z}z, is a measurable set containing a finite and nonzero number of points from each orbit Gz, z ∈ 𝒟. It is not difficult to extract a measurable fundamental set from F_{0} , for instance as follows. Let {e_{i}}_{i>0} , be a dense set of S^{n} and define measurable F_{1} ⊂ F_{0} by

F_{1} = {x ∈ F_{0} : |x − e_{1}| ≤ |z − e_{1}| for all z ∈ Gx ∩ F_{0}}.

Even this may fail to give a fundamental set but we continue and set F_{2} = {x ∈ F_{1} : |x − e_{2}| ≤ |z − e_{2}| for all z ∈ Gx ∩ F_{1}}, etc. The intersection ⋂_{i} F_{i} will be a measurable fundamental set for 𝒟.

Proof of B. Unless the action of G on ℋ_{b}(G) is conservative, there is a set A ⊂ ℋ_{b}(G) of positive measure such that μ(A ∩ gA) > 0 for only finitely many g ∈ G. Applying Lemma 4 below, we can find measurable B ⊂ A of positive measure and a finite subgroup H ⊂ G such that the stabilizer G_{x} = {g ∈ G : g(x) = x} = H for every x ∈ B and that gB ∩ hB = ∅ if g, h ∈ G and gH ≠ hH. In view of Lemma 2, and possibly by making B smaller but still of positive measure, we can assume that there is m > 0 such that, for each z ∈ B, |g′(z)| ≥ m for an infinite number of g ∈ G. It follows that we can find distinct g_{11}, …, g_{1n_{1}} ∈ G such that if

E_{1} = {z ∈ B : |g′_{1i}(z)| ≥ m for some i, 1 ≤ i ≤ n_{1}},

then μ(E_{1}) > \frac{1}{2}μ(B). Since |g′(z)| ≥ m for an infinite number of g ’s, we can continue and find for each k elements g_{k1}, …, g_{kn_{k}} ∈ G for i ≤ n_{k} such that g_{ki} = g_{pr} only if k = p and i = r and such that if

E_{k} = {z ∈ B : |g′_{ki}(z)| ≥ m for some i, 1 ≤ i ≤ n_{k}}

then μ(E_{k}) > \frac{1}{2}μ(B). Let E_{ki} = \{z ∈ B : |g′_{ki}(z) ≥ m\} so that E_{k} = ⋃_{i} E_{ki}. We can now estimate when #H is the number of elements of H

\begin{aligned}∞ & > μ( ⋃_{g∈G} gB) = \frac{1}{#H} ∑_{g∈G} μ(gB) ≥ \frac{1}{#H} ∑_{k,i} μ(g_{ki}B) \\ & ≥ \frac{1}{#H} ∑_{k,i} μ(g_{ki}E_{ki}) ≥ \frac{1}{#H} ∑_{k,i} mμ(E_{ki}) ≥ \frac{m}{#H} ∑_{k} μ(E_{k}) = ∞ \end{aligned}

since μ(E_{k}) ≥ \frac{1}{2}μ(B) > 0. This contradiction implies B.

Proof of C. Suppose that G is conservative on 𝒞 and that μ(𝒞 \ℋ_{b}(G)) > 0. We derive a contradiction from this.

By A, there is a fundamental set F for S^{n} \ ℋ_{b}(G). Replacing F by some gF, g ∈ G, we can assume that μ(𝒞 ∩ F) > 0. Let A = 𝒞 ∩ F. By Lemma 3 below, there are B ⊂ A and a finite subgroup H ⊂ G such that μ(B) > 0 and that g(x) ≠ x if x ∈ B and g ∈ G \ H. However, if x ∈ B ∩ gB for some g ∈ G, then x must be fixed by g since B is a subset of the fundamental set. Hence g is in the finite set H, contradicting the conservativity in 𝒞 .

Proof of D. Suppose that G is dissipative on 𝒟 and Z = ℋ_{b}(G) ∩ 𝒟 has positive μ-measure. Since the action is dissipative on 𝒟 and hence on Z, there is a fundamental set F_{1} for Z. Then μ(F_{1}) > 0 and hence by B, μ(F_{1} ∩ gF_{1}) > 0 for infinitely many g ∈ G. Pick one such g_{1} and set F_{2} = F_{1} ∩ g_{1}F_{1} . Find then g_{2} ≠ g_{1} such that F_{3} = F_{2} ∩ g_{2}F_{2} has positive μ-measure. Continuing in this manner we find a sequence of distinct g_{i} ∈ G and a decreasing sequence F_{1} ⊃ F_{2} ⊃ ⋯ of sets of positive μ-measure such that μ(F_{i} ∩ g_{i}F_{i}) > 0. Since F_{i} are subsets of the fundamental set F_{1} , F_{i} ∩ g_{i}F_{i} consists of fixed points of g_{i}. If g_{i} is of infinite order, then g_{i} fixes at most two points and neither of them is an atom by our assumptions. It follows that each g_{i} is elliptic and similarly, the group G_{i} generated by g_{1}, …, g_{i}, all of whose elements fix points of F_{i}, must be purely elliptic. Thus ⋃_{i} G_{i} would be a purely elliptic infinite group and this is impossible by [6].

We still need to prove two lemmas. As above μ is a conformal measure such that no parabolic or loxodromic fixed point is an atom of μ.

Lemma 3. Let A ⊂ S^{n} be measurable with positive μ-measure. Then there are a finite subgroup H ⊂ G and measurable B ⊂ A of positive measure such that g(x) = x for every x ∈ B and g ∈ H but that g(x) ≠ x if g ∈ G \ H.

Proof. Remove first all fixed points of loxodromic or parabolic elements of G. Since these points are not atoms of μ and their number is countable, still μ(A) > 0. Thus if x ∈ A and g(x) = x for some g ∈ G, then g is elliptic.

In the following X_{F} = {x ∈ S^{n} : g(x) = x for all g ∈ F } when F ⊂ G is a subgroup. We claim that there is a maximal finite subgroup H ⊂ G such that μ(X_{H} ∩ A) > 0. If there is no such maximal group, we can find a sequence H_{1} ⊂ H_{2} ⊂ ⋯ with proper inclusions of finite subgroups of G such that μ(X_{H_{i}} ∩A) > 0. Now H = ⋃_{i} H_{i} is a purely elliptic discrete subgroup and hence, by [6, Corollary C], H is finite. This is a contradiction and hence there is such a maximal H.

Let B = X_{H} ∩ A. If g ∈ G \ H and x ∈ B and g(x) = x, then the group F generated by g and H is purely elliptic (since x is not a parabolic or loxodromic fixed point) and hence F is finite as we saw. So μ(X_{F} ∩ B) = 0 by maximality of H. Removing a countable number of nullsets of this form, we obtain the set B with the required properties.

Lemma 4. Suppose that A ⊂ S^{n} is of positive μ-measure and that μ(A ∩ gA) > 0 for only finitely many g ∈ G. Then there are a finite subgroup H ⊂ G and a measurable subset B ⊂ A of positive μ-measure such that g(x) = x for all x ∈ B and g ∈ H but that gB ∩ B = ∅ for all g ∈ G \ H.

Proof. Let B ⊂ A and H ⊂ G be the set and subgroup given by Lemma 3. Since G is countable and each g ∈ G is absolutely continuous with respect to μ, we can obtain by removing a countable number of nullsets that μ(B ∩ gB) > 0 if and only if B ∩ gB ≠ ∅ whenever g ∈ G.

Let G_{0} = {g ∈ G : μ(B ∩ gB) > 0} = {g ∈ G : B ∩ gB ≠ ∅}. It is a finite union of cosets gH. Choose representatives g_{1}, …, g_{k} from each coset. Pick x ∈ B such that μ(U) > 0 for every neighborhood U of x in B. Now g_{i}(x), i ≤ k, are all distinct and hence it is possible to choose a neighborhood U of x such that g_{i}U are disjoint. If U ∩ gU ≠ ∅, then g ∈ G_{0} and hence gU = g_{i}U for some i. This is possible only if g ∈ H and so U ∩ gU = ∅ for all g ∈ G \ H. Thus the lemma is true if U is substituted for B.

Remark. It may be useful to see to what extent the proof depends on special properties of Möbius transformations. Proof of A works generally. Let X be a separable metric space, μ a finite Borel regular measure on X and G a countable group of absolutely continuous measurable bijections of X; we need the Radon– Nikodym derivative for g ∈ G, and hence we still need to assume, for instance, that that the family of balls of X satisfy the assumptions of the Vitali covering theorem [2, 2.8.16]. Interpret |g′(x)| as the Radon–Nikodym derivative of g and the formula (4) gives a measurable set F_{0} containing a finite and nonzero number from each orbit Gz when z ∈ 𝒟 and 𝒟 is the set of x ∈ X such that the Radon– Nikodym derivative |g′(x)| exists for all g ∈ G and E_{m} = {g ∈ G : |g′(x)| ≥ m} is finite for every m > 0; this latter condition could even be relaxed to the form that E_{m} is finite and non-empty for some m > 0. The proof of A is valid and extracts a measurable fundamental set from F_{0} .

In contrast, to show that the action is conservative on X \ 𝒟 and that the division to conservative and dissipative parts is unique up to nullsets, requires in addition two more specific assumptions: The set of points fixed by some g ∈ G of infinite order is a nullset, and every subgroup of G consisting of elements of finite order is finite. With these assumptions our proof is valid.

Appendix. We now give the promised proof that μ(𝒢 \ 𝒢_{0}) = 0.

Let Z = 𝒢 \ 𝒢_{0} . We will show that the action is dissipative on Z. It follows by Theorem 1 that Z is up to a nullset a union of atoms at parabolic fixed points. However, a parabolic fixed point is not a point of Z as one easily sees.

We use here the method in Sullivan’s paper [4]. Notice that if x ∈ Z, then there is a unique horoball B_{x} at x such that Gz ∩ ∂B_{x} ≠ ∅ but that Gz ∩ B_{x} is non-empty and finite; here z is the reference point used to define 𝒢_{0} . Let Γ_{x} = ∂B_{x} ∩ Gz which is finite and non-empty. If F ⊂ G is finite and non-empty, set

D_{F} = {x ∈ Z : Γ_{x} = F }.

The sets D_{F} , as F varies over finite and non-empty subsets of G, form a countable family of measurable, mutually disjoint sets covering Z. Obviously, g(D_{F}) = D_{gF} and hence it is possible to find a countable family ℱ of finite and non-empty subsets of G such that the countable union ∪{D_{F} : F ∈ ℱ } will be a measurable fundamental set for Z.

References

[1] Agard, S.: Mostow rigidity on line: A survey. - In: Holomorphic Functions and Moduli II, edited by D. Drasin et al., MSRI Publications 11, Springer-Verlag 1988, 1–12.

[2] Federer, H: Geometric Measure Theory. - Springer-Verlag, 1969.

[3] Nicholls, P.J.: The Ergodic Theory of Discrete Groups. - London Math. Soc. Lecture Note Ser. 143, Cambridge University Press, 1989.

[4] Sullivan, D.: On the ergodic theory at infinity of an arbitrary discrete group of hyperbolic motions. - In: Riemann Surfaces and Related Topics: Proceedings of the 1978 Stony Brook Conference, edited by I. Kra and B. Maskit, Ann. of Math Studies 97, Princeton University Press, 1981, 465–496.

[5] Sullivan, D.: The density at infinity of a discrete group of hyperbolic motions. - Inst. Hautes Études Sci. Publ. Math. 50, 1979, 171–202.

[6] Waterman, P.L.: Purely elliptic Möbius groups. - In: Holomorphic Functions and Moduli II, edited by D. Drasin et al., MSRI Publications 11, Springer-Verlag, 1988, 173–178.

Received 18 January 1996