Annales Academiæ Scientiarum Fennicæ Series A. I. Mathematica Volumen 19, 1994, 247–258

THE MIXED ELLIPTICALLY FIXED POINT PROPERTY FOR KLEINIAN GROUPS

Rubén A. Hidalgo

Universidad Técnica “Federico Santa María”, Departamento de Matemática Avenida España 1680, Valparaiso, Chile; rhidalgo@trantor.mat.utfsm.cl

Abstract. We define the mixed elliptically fixed point (MEFP) property for Kleinian groups. Such a property is a set of possibilities for the locations on the Riemann sphere of the fixed points of elliptic elements in Kleinian groups. We show that the MEFP property is invariant under the operations given by the Maskit–Klein combination theorems. As a consequence, we obtain that finitely generated function groups satisfy such a property. We also show that geometrically finite Kleinian groups satisfy the MEFP property. Examples of Kleinian groups where such a property does not hold are provided.

1. Introduction

We consider a set of possible locations for the fixed points of elliptic elements in Kleinian groups. We call such a set of possibilities the mixed elliptically fixed point (MEFP) property for Kleinian groups. We show that the MEFP property is invariant under the operations of the Klein–Maskit combination theorems (Theorems C1 and C2). This result and Maskit’s classification theorem of finitely generated function groups given in [3] imply that these groups satisfy the MEFP property (Theorem A).

The fact that geometrically finite Kleinian groups have finite-sided convex fundamental polyhedron in the hyperbolic three space implies the desired result for this class of groups (Theorem B).

In general, a Kleinian group G does not always satisfy the MEFP property. We provide a couple of groups for which this property fails. In fact, finite normal extensions of Kleinian groups satisfying the MEFP property do not necessarily satisfy such a property (see the remark at the end of Section 5). Corollaries 1 and 2 assert that this is not the case if we consider finite extensions of finitely generated function groups.

The MEFP property can be used to study the problem of lifting conformal automorphisms of Riemann surfaces to the region of discontinuity of Kleinian groups uniformizing it. In fact, this work started from an attempt to give necessary and sufficient conditions for a group H of conformal automorphisms of a closed Riemann surface S to be lifted to the region of discontinuity of some Schottky group G uniformizing S. Corollary 1 of this paper has been applied in that direction [2].

1991 Mathematics Subject Classification: Primary 30F40.

This work is organized as follows. In Section 2 we recall some definitions which can also be found in [4]. In Section 3 we define the MEFP property. We give examples where such a property holds and provide two examples where it does not. In Section 4 we recall the Klein–Maskit combination theorems and Maskit’s classification theorem of finitely generated function groups [3]. In Section 5 we state the main theorems and corollaries of this work. They indicate that the MEFP property is invariant under the combination theorems and that a Kleinian group which is either a finitely generated function group or a geometrically finite Kleinian group satisfies the MEFP property. We apply these results to finite extensions of finitely generated function groups (Corollaries 1 and 2). In Section 6 we prove Theorems C1, C2 and A. In Section 7 we prove Theorem B.

2. Preliminaries and definitions

A Kleinian group G is a subgroup of the group M of Möbius transformations (also called fractional linear transformations), which acts discontinuously on some part of the Riemann sphere. The group M is canonically isomorphic to the projective group P SL(2, C). The (open) set of points where G acts discontinuously is denoted by Ω(G) and called the region of discontinuity of G. The complement of Ω(G) is called the limit set of G and denoted by Λ(G).

If T is a connected component of the region of discontinuity of the Kleinian group G, we say that T is a component of G.

Kleinian groups act in a natural way as orientation-preserving isometries of the hyperbolic three-space H^{3} = { (z, t); z ∈ C, t > 0 }. The Riemann sphere Ĉ can be thought of as the boundary of this space.

A Kleinian group G is said to be a function group if there exists a G-invariant component of its region of discontinuity. If G is also finitely generated, we say that G is a finitely generated function group.

A function group F for which the limit set is contained in some Jordan curve γ is called a quasi-Fuchsian group. A group G (necessarily a Kleinian group) which contains a quasi-Fuchsian group F of index two is called a Z/2Z-extension of the quasi-Fuchsian group F. In case we can choose the curve γ as an analytic circle, we call F a Fuchsian group.

Geometrically finite Kleinian groups are Kleinian groups with a finite-sided convex fundamental polyhedron in H^{3} . We assume standard results about these groups; see for instance [4].

Degenerated groups are function groups with the property that the region of discontinuity is connected, simply-connected and hyperbolic, that is, its universal covering is the hyperbolic plane. We remark at this point that no explicit example is known of such a finitely generated group.

Kleinian groups which belong to the next families are called basic groups.

(1) finite groups;

(2) Euclidean groups;

(3) Kleinian groups with limit sets consisting of exactly two points;

(4) finitely generated quasi-Fuchsian groups;

(5) finitely generated degenerated groups.

3. The MEFP property on Kleinian groups

Definition (The MEFP property). A Kleinian group G is said to satisfy the mixed elliptically fixed point (MEFP) property if for any elliptic transformation h ∈ G either:

(i) both fixed points of h belong to the region of discontinuity of G, Ω(G); (ii) there exists a loxodromic transformation g in G commuting with h; (iii) there exists a parabolic transformation p in G sharing one fixed point with

h, and the other fixed point of h is in Ω(G); (iv) there exist parabolic transformations p and q in G, each one sharing a fixed

point with h; or

(v) G contains a degenerated subgroup H containing h. Remarks. (1) In case (v), one of the fixed points of h, say x, necessarily

belongs to the region of discontinuity of H and the other, say y, belongs to the limit set of H. In such a case, there exists neither a parabolic transformation in G fixing y nor a loxodromic transformation commuting with h [1]. (2) In a Kleinian group G satisfying the MEFP property, all the above possibilities (i)– (v) can occur for different elliptic elements of G. Examples of groups of these types can be easily constructed from the Klein–Maskit combination theorems (once Theorems C1 and C2 have been proved).

Examples of Kleinian groups having the MEFP property are:

(a) finite groups,

(b) Euclidean groups,

(c) Kleinian groups with limit sets consisting exactly of two points,

(d) quasi-Fuchsian groups,

(e) degenerated groups, and

(f) torsion-free Kleinian groups. Groups of type (a), (b) and (c) are known as the elementary groups. They are

easily seen necessarily to satisfy the MEFP property. The fixed points of elliptic elements in quasi-Fuchsian groups are always points in the region of discontinuity. In particular, they satisfy the above property. Cases (e) and (f) are trivial. As a consequence the basic groups satisfy the MEFP property. This observation will be used to prove Theorem A. The following are examples of Kleinian groups which do not satisfy MEFP property.

(g) This example is an easy modification of Maskit’s example in [6] of a Web group which cannot be constructed from the combination theorems. Let H and H^{∗} be finitely generated Fuchsian groups of the first kind, acting in the unit disc △. Assume H to be a purely hyperbolic group and △/H^{∗} to be a Riemann surface of signature (g, 1; v), where g ≥ 1 and 2 ≤ v < ∞. Assume that the origin, then ∞, is not an elliptic fixed point in H^{∗}. Consider the transformation a_{t}(z) = tz, where t is a non-zero complex number, and let H_{t} = a_{t}Ha_{t}^{−1} . Let G_{t} be the group generated by H_{t} and H^{∗}. For sufficiently small ∥t∥ the group G_{t} can be formed from H_{t} and H^{∗} by Combination Theorem 1 (see Section 4); for each such t, G_{t} is the free product of H_{t} and H^{∗} and does not contain parabolic elements. It can be shown that there exists a ray arg(t) = θ_{0} on which every element of G_{t} is either loxodromic or elliptic. Assume θ_{0} = 0, that is, t is a real number. Let T be the set of all t such that G_{t′} can be formed by the free combination in Maskit’s sense, for all t′ ≤ t. It follows from Combination Theorem 1 that T is an open set. Let t_{0} = lub T and let G = G_{t_{0}} . The group G is the free product of H_{t_{0}} and H^{∗}, and every element of G is either loxodromic or elliptic. A fundamental set for G is the union of a fundamental set of H^{∗} in the region {∥z∥ > 1} and a fundamental set of H_{t_{0}} in the disc {∥z∥ < t_{0}}. Any component of G is conjugate to either {∥z∥ < t_{0}} or {∥z∥ > 1}. G is actually a Web group (see [4] for a definition). Any elliptic element of H^{∗} has one fixed point in the region {∥z∥ > 1}, that is, it is a regular point of the group G. The other fixed point is a limit point of G. Since G has neither parabolic elements nor degenerated subgroups, G cannot satisfy the MEFP property.

(h) The following example is an infinitely generated Kleinian group. Let G be the group generated by the transformations A, B_{1} , B_{2} , …, where

A = ( \begin{matrix}i & 0 \\ 0 & −i \end{matrix} ) , B_{n} = ( \begin{matrix}4n & (16n^{2} − 1)/4 \\ 4 & 4n \end{matrix} ) .

The group G is a Kleinian group without parabolic elements as a consequence of the Maskit–Klein combination theorems [4] (also Section 4). The elliptic element A has one fixed point, the origin, in the region of discontinuity of G and the other fixed point, ∞, belongs to the limit set of G. The group G does not contain degenerated subgroups. In particular, G does not satisfy the MEFP property.

4. The Klein–Maskit combination theorems and function groups

Let H be a subgroup of the Kleinian group G. A set T is called precisely invariant under H if:

(i) h(T) = T for every h in H; and (ii) g(T) ∩ T = ϕ for all g in G − H.

For a cyclic subgroup H of G, a precisely invariant topological disc B is the interior of a closed topological disc, with closure B̄, where B̄ − Λ(H) is precisely invariant under H, and (B̄ − Λ(H)) ⊂ Ω(G). Here Λ(H) represents the limit set of the subgroup H.

Let γ_{i} be a sequence of loops and let z be a point on the Riemann sphere. We say that the sequence of loops γ_{i} nests about z if the loop γ_{i} separates z from the loop γ_{i−1} and any sequence of points q_{i} ∈ γ_{i} converges to z. We permit the point z to belong to some γ_{i_{0}} , in which case z must belong to γ_{i}, for all i ≥ i_{0} .

Combination Theorem 1. Let G_{1} , G_{2} be Kleinian groups with a common subgroup H. Assume the group H is either trivial, or cyclically generated by an elliptic or a parabolic transformation. For i = 1, 2, let B_{i} be a precisely invariant topological disc under H in G_{i}. Assume that B_{1} and B_{2} have a common boundary γ and B_{1} ∩ B_{2} = ϕ. Let G be the group generated by G_{1} and G_{2} ; then

(1) G is Kleinian.

(2) G = G_{1} ∗_{H} G_{2} , that is, G is the free product of G_{1} and G_{2} with an amalgamated subgroup H.

(3) For z ∈ Λ(G) where z is not a limit point of a conjugate of either G_{1} or G_{2} , there exists a sequence {j_{n}} of elements of G so that {j_{n} (γ)} nests about z.

(4) If γ is precisely invariant under H in either G_{1} or G_{2} , then every elliptic or parabolic element in G lies in a conjugate of either G_{1} or G_{2} .

Combination Theorem 2. Let G_{1} be a Kleinian group and let H_{1} and H_{2} be subgroups of G_{1} . Assume that the groups H_{1} and H_{2} are either trivial, or cyclically generated by an elliptic or parabolic transformation. For i = 1, 2, let B_{i} be a precisely invariant topological disc under H_{i} in G_{1} , and let γ_{i} be the boundary of B_{i}. We assume that g(B̄_{1}) ∩ B̄_{2} = ϕ, for all g in G_{1} . Let G_{2} = ⟨f⟩ be a cyclic group generated by f, where f(γ_{1}) = γ_{2} , f(B_{1}) ∩ B_{2} = ϕ, and f ∘ H_{1} ∘ f^{−1} = H_{2} . Let G be the group generated by G_{1} and G_{2} ; then (1) G is Kleinian.

(2) G = G_{1}∗_{G_{2}} , that is, every relation in G is a consequence of the relations in G_{1} and the relations f ∘ H_{1} ∘ f^{−1} = H_{2} .

(3) For every point z ∈ Λ(G) where z is not a limit point of a conjugate of either G_{1} or G_{2} , there exists a sequence {j_{n}} of elements of G so that {j_{n} (γ_{1})} nests about z.

(4) If γ_{1} is precisely invariant under H_{1} in G_{1} , then every elliptic or parabolic element in G lies in a conjugate of G_{1} .

From now on, we assume the following additional conditions on the parabolic cyclic groups in the combination theorems. (I) In Combination Theorem 1, assume that if H is a cyclic group generated by

a parabolic transformation, H must also be its own normalizer in either G_{1} or G_{2} . This means that conclusion (4) of Combination Theorem 1 can be applied.

(II) In Combination Theorem 2, assume that if H_{1} and H_{2} are cyclic groups generated by parabolic transformations, then each of them is its own normalizer in G_{1} . This means that conclusion (4) of Combination Theorem 2 can be applied.

A proof of the combination theorems can be found in Chapter VII of [4]. Another proof of these theorems (as established above) is given in [3].

Classification Theorem [3]. Let G be a finitely generated function group. Then G can be constructed from the basic groups using Combination Theorems 1 and 2 under the assumptions (I) and (II).

5. Main results

In this section we establish the main theorems of this work and two consequential corollaries.

Theorem A. Finitely generated function groups satisfy the MEFP property.

Theorem B. Geometrically finite Kleinian groups satisfy the MEFP property.

Remark. In Theorem A the condition “finitely generated” is necessary (cf. Example (h)). Since geometrically finite Kleinian groups do not contain degenerated groups [3], the case (v) in the definition of the MEFP property cannot occur in these cases.

Examples of geometrically finite function groups without parabolic elements are the finite extensions of Schottky groups (see [5] for a definition of Schottky groups).

Corollary 1. Let G be a finite extension of a Schottky group and let h be any elliptic transformation in G. Then the only possibilities for the fixed points of h are given by (i) and (ii) in the definition of the MEFP property.

Corollary 2. Let G̃ be a finite extension of a finitely generated function group G. Then G̃ satisfies the MEFP property.

Proof. Let h be an elliptic element in G̃, and let △ be a G-invariant component of Ω(G̃) = Ω(G). We have two possibilities for the image of △ under the transformation h.

Case 1. h(△) = △. In this case the group G_{1} , generated by G and h, is also a function group. Since G_{1} is a finite extension of a finitely generated group, it is also finitely generated. Theorem A now implies that G_{1} satisfies the MEFP property. Since G_{1} is a subgroup of G̃ of finite index, it is easy to see that h must satisfy some of the conditions in the definition of the MEFP property.

Case 2. h(△) ≠ △. Let us denote by △_{1} the image of △ under the transformation h. Then △_{1} is another component of the group G̃. Let F be the subgroup of G̃ keeping △ invariant. Necessarily, F also keeps △_{1} invariant. Now F is a finitely generated Kleinian group with two invariant components. It is well known [5] that such a group is a quasi-Fuchsian group. In particular, the components are simply connected with a quasi-circle as a common boundary. The group G̃ is then a Z/2Z -extension of the finitely generated quasi-Fuchsian group F. As a consequence, G̃ is a geometrically finite Kleinian group. It follows from Theorem B that G̃ satisfies the MEFP property.

Remark. Finite normal extensions of finitely generated Kleinian groups with the MEFP property do not generally satisfy such a property. As an example, consider a normal torsion-free subgroup F of finite index (Selberg’s lemma) of the group G constructed in (h). The group F trivially satisfies the MEFP property, since it is torsion-free.

The next two theorems show the invariance of the MEFP property under the Klein–Maskit combination theorems. They will be used to prove Theorem A.

Theorem C1. Under the same hypotheses as in Combination Theorem 1, assume that G_{1} and G_{2} satisfy the MEFP property. Then the group G, generated by G_{1} and G_{2} , satisfies the MEFP property.

Theorem C2. Under the same hypotheses as in Combination Theorem 2, assume that G_{1} satisfies the MEFP property. Then the group G, generated by G_{1} and G_{2} = ⟨f⟩, satisfies the MEFP property.

6. Proof of Theorems C1, C2 and A

Proof of Theorem C1. Let t be an elliptic transformation of G. Conclusion (4) in Combination Theorem 1 implies that t is conjugate to an elliptic transformation h in G_{1} or G_{2} . Without loss of generality, we may assume h to be in G_{1} . Let us observe that t satisfies any of the conditions of the definition of the MEFP property if and only if h does. In particular, we only need to check that h satisfies any of these conditions. Denote the fixed points of h by x and y. We have that either (1) h is not a G-conjugate of any element of H, or

(2) h is a G-conjugate of some element in H. Case (1) Let us assume h is not a G-conjugate of any element of H. Since B̄_{1}

is a precisely invariant topological disc under H in G_{1} and h is not a G-conjugate of any element of H, the fixed points of h necessarily belong to B_{2} − ∪{g(B̄_{1}); g ∈ G_{1}}. In particular, x and y are regular points of G_{2} . Assume x and y to be limit points of G_{1} . Then, by the MEFP property of G_{1} , either there exists a loxodromic transformation g in G_{1} with x and y as fixed points, or there exist parabolic transformations P and Q in G_{1} with fixed points x and y, respectively, or there exists a degenerated subgroup H_{1} of G_{1} containing h. Since G_{1} is a subset of G, this case is proved. Let us assume x and y to be regular points of G_{1} . Conclusion (3) of Combination Theorem 1 implies that x and y are regular points of G. Now assume that one of the fixed points of h, say x, is a regular point of G_{1} and the other is a limit point of G_{1} . Again, conclusion (3) of Combination Theorem 1 implies that x is a regular point of G. The MEFP property of G_{1} implies that either there exists a parabolic transformation Q in G_{1} with y as a fixed point, or there exists a degenerated subgroup H_{1} of G_{1} containing h (y is then a limit point of H_{1} ). Since G_{1} is contained in G, we are done.

Case (2) Assume h is G-conjugated to some element of H. In this case H must be a cyclic group generated by an elliptic transformation. Let j be a generator of the group H. Without loss of generality, we may assume h to be some power of j. It is easy to check that a non-trivial power of j satisfies some of the conditions given in the definition of the MEFP property if and only if j does, so we assume h = j, with x and y as its fixed points. Clearly, one of the fixed points of j, say x, belongs to B_{1} and the other belongs to B_{2} . In this case x is a regular point of G_{1} and y is a regular point of G_{2} . Since G_{1} and G_{2} satisfy the MEFP property, we only have the following possibilities: (a1) y is a regular point of G_{1} , or (a2) there exists a parabolic transformation g_{1} ∈ G_{1} with y as a fixed point, or (a3) G_{1} contains a degenerated subgroup H_{1} containing j; and either (b1) x is a regular point of G_{2} , or (b2) there exists a parabolic transformation g_{2} ∈ G_{2} with x as fixed point, or (b3) G_{2} contains a degenerated subgroup H_{2} containing j.

We have nine possibilities, and since G_{1} and G_{2} are subsets of G, the following lemma concludes the proof.

Lemma 1. (1) (a1) implies that y is either a regular point of G, or there exists a loxodromic transformation in G commuting with j. (2) (b1) implies that x is either a regular point of G, or there exists a loxodromic transformation in G commuting with j.

Proof. The proofs of (1) and (2) are the same. We only prove part (1). Let us assume y is a limit point of G. Conclusion (3) of Combination Theorem 1 implies that there exists a sequence of G-translates of the loop γ nesting about y. Since y belongs to B_{2} , it follows that there exists g_{1} in G_{1} − H such that y belongs to g_{1}(B_{1}). In this case, g_{1}^{−1}(y) belongs to B_{1} and it is a fixed point of g_{1}^{−1} ∘ j ∘ g_{1} ∈ G_{1} . Since B_{1} is precisely invariant under H in G_{1} and g^{−1}(y) is different from y, we have g^{−1}(y) = x and g^{−1}(x) = y, that is, g_{1}(x) = y and g_{1}(y) = x. In particular, x is also a limit point of G and g_{1} has order two. Applying the same argument to x, we can find an element g_{2} in G_{2} − H such that g_{2}(x) = y, g_{2}(y) = x and g_{2} has order two. Consider the transformation g = g_{2} ∘ g_{1} . It can be seen that g is necessarily a loxodromic transformation in G with x and y as fixed points. In particular, g commutes with j.

Proof of Theorem C2. Let t be an elliptic transformation of G. Conclusion (4) of Combination Theorem 2 implies that t is conjugated to an elliptic transformation h in G_{1} . As observed in the proof of Theorem C1, we may assume t = h. Denote by x and y the fixed points of the transformation h. We have the following possibilities for h; either

(1) h is not G-conjugated to any element in H_{1} , or (2) h is G-conjugated to some element in H_{1} . (In this case H_{1} is necessarily a

cyclic group generated by an elliptic transformation.)

Case (1) Assume h is not a G-conjugate to any element of H_{1} . Since B̄_{1} is precisely invariant under H_{1} , B̄_{2} is precisely invariant under H_{2} and h is not a G-conjugate to any element in H_{1} , the points x and y must belong to Ĉ − { ∪(g(B̄_{1}) ∪ g(B̄_{2})); g ∈ G_{1} }. Assume now x and y to be limit points of G_{1} . The MEFP property of G_{1} asserts that either there exists a loxodromic transformation g in G_{1} commuting with h, or there are parabolic transformations P and Q in G_{1} with x and y as fixed points, respectively, or there exists a degenerated subgroup H_{0} in G_{1} containing h. Since G_{1} is a subset of G, this case is proved. Assume now x and y to be regular points of G_{1} . Since x and y are not in B̄_{1} ∪ B̄_{2} , they cannot be fixed points of f. In particular, they are regular points of G_{2} . Conclusion (3) (of Combination Theorem 2) implies that x and y are regular points of G, and this case is clear. Assume now that one of the fixed points of h, say y, is a limit point of G_{1} and x is a regular point of G_{1} . The MEFP property of G_{1} implies that either there is a parabolic transformation g in G_{1} with y as a fixed point, or there exists a degenerated subgroup H_{0} in G_{1} containing h. Since x does not belong to B̄_{1} ∪ B̄_{2} , the point x cannot be a fixed point of f. In particular, x is a regular point of G_{2} . Again, conclusion (3) of Combination Theorem 2 implies that x is a regular point of G. In this case either condition (iii) or (v), in the definition of the MEFP property, holds for the group G.

Case (2) Assume h is G-conjugated to some element of H_{1} . In this case H_{1} is the cyclic group generated by the element j. As in case (2) in the proof of Theorem C1, we may assume h = j. As before, x and y denote the fixed points of j. One of these points must belong to B_{1} , say x. Since f ∘ H_{1} ∘ f^{−1} = H_{2} , we have f(y) belonging to B_{2} . Note that the fixed points of f ∘ j ∘ f^{−1} are f(x) and f(y). In particular, x and f(y) are regular points of G_{1} . The MEFP property of G_{1} implies the following (only) possibilities: (a1) y is a regular point of G_{1} , or (a2) there exists a parabolic transformation g_{1} ∈ G_{1} with y as a fixed point, or (a3) there exists a degenerated subgroup H_{0} in G_{1} containing j.

Case (a3) is clear, since G_{1} is contained in G. From now on, assume that there is no degenerated subgroup of G containing j. In particular, there is no degenerated subgroup of G containing f ∘ j ∘ f^{−1} . The MEFP property of G_{1} implies that: (b1) f(x) is a regular point of G_{1} , or (b2) there is a parabolic transformation g_{2} ∈ G_{1} with f(x) as a fixed point. In

this case x is also a parabolic fixed point.

The next lemma concludes the proof, with the observation that the four possibilities (a1) and (b1); (a1) and (b2); (a2) and (b1); (a2) and (b2) each gives one of the possibilities in the definition of the MEFP property.

Lemma 2. (1) (a1) implies that either y is a regular point of G or there exists a loxodromic transformation g in G commuting with j. (2) (b1) implies that either x is a regular point of G or there exists a loxodromic transformation g in G commuting with j.

Proof. Note that (2) is equivalent to the following: (2′) (b1) implies that either f(x) is a regular point of G or there exists a loxodromic transformation g in G commuting with f ∘ j ∘ f^{−1} .

The proof of (2) is the same as the proof of (1), so we only prove (2). Assume x is a limit point of G. Since f belongs to G, the point f(x) is also a limit point of G. Conclusion (3) of Combination Theorem 2 and the fact that f(x) is a regular point of G_{1} imply that there exists g_{1} ∈ G_{1} − H_{1} ∪ H_{2} such that f(x) belongs to g_{1}(B_{1}). The fact that B_{1} is precisely invariant under H_{1} in G_{1} and the fixed point g_{1}^{−1}(f(x)) of the transformation k = g_{1}^{−1} ∘ f ∘ j ∘ f^{−1} ∘ g_{1} belongs to B_{1} , implies that k is in H_{1} . Since the transformation k is not the identity and H_{1} is a cyclic group generated by j, the fixed points of k must be x and y. The only possibility for this to happen is g_{1}(x) = f(x) and g_{1}(y) = f(y). We consider the transformation g = f^{−1} ∘ g_{1} which is loxodromic with x and y as fixed points, so commuting with j.

Proof of Theorem A. The classification theorem asserts that any finitely generated function group is constructed from the basic groups using Klein–Maskit combination theorems. Now, Theorems C1 and C2 and the fact that the basic groups satisfy the MEFP property imply the desired result.

7. The MEFP property and geometrically finite Kleinian groups

In this section we prove Theorem B. Let us start with some properties of geometrically finite Kleinian groups which will be used in the proof.

The equivalence of the following statements is shown in Chapter VI of [4].

(1) G is geometrically finite.

(2) Any convex fundamental polyhedron for G in H^{3} is a finite-sided one.

(3) Any limit point of G is either a point of approximation or a double-cusped parabolic fixed point or a rank-two parabolic fixed point. Proof of Theorem B. If G is torsion free, there is nothing to check. Assume

now G has torsion. Let h ∈ G be any elliptic element with fixed points x and y. Case (1) x and y are regular points of G, in which case we are done. Case (2) x or y is a limit point. Without loss of generality we can assume

that y is a limit point. Let j ∈ G be be a primitive elliptic element with y as a fixed point.

We claim the following :

(i) Either j(x) = x or there is a parabolic element in G with y as a fixed point. (ii) If g(y) = y, for g in G, then either g is conjugate to a power of j in G, or g

is a loxodromic element with x and y as fixed points, or there is a parabolic element with y as its fixed point. In fact, (i) if we assume j(x) ≠ x, the commutator [j, h] = j ∘ h ∘ j^{−1} ∘ h^{−1} is

a parabolic element in G with y as a fixed point. (ii) Let g be an element in G such that g(y) = y. Assume y is not a fixed point of any parabolic element of G. The only possibility for g is to be either an elliptic or a loxodromic element. By our assumption on y, necessarily g(x) = x, otherwise [g, j] will be a parabolic element of G with y as a fixed point. At this point, g is either a power of j or a loxodromic element with x and y as fixed points. This ends the proof of our claim.

From now on we assume there is no parabolic element of G with y as a fixed point. Let L be the hyperbolic geodesic in H^{3} with x and y as its end points. Then j fixes L pointwise. Let P be a convex fundamental polyhedron for G. Since y is a limit point which is not a parabolic fixed point, y must be a point of approximation for G (see p. 128 in [4]). This implies that y cannot be in the closure of P (see p. 122 of [4]). As a consequence, we can find a sequence of points y_{n} ∈ L, converging to y, all of them non-equivalent points by G, and a sequence g_{n} ∈ G, g_{n} ≠ g_{m} ∘ j^{l}, for all l and n ≠ m, such that g_{n}(y_{n}) = z_{n} ∈ P̄, where P̄ denotes the Euclidean closure of P. Let us consider a subsequence such that z_{n} converges, say to z, g_{n}(y) converges, say to u, and g_{n}(x) converges, say to t. In this way, the points u and t are limit points of the group G. Since z_{n} ∈ P̄, we must also have z ∈ P̄. We have two possibilities for z, that is, z is either a regular point of G, or z is a parabolic fixed point (see p. 128 in [4]). It is clear that z_{n} is an elliptic fixed point; in fact z_{n} = g_{n} ∘ j ∘ g_{n}^{−1}(z_{n}). This implies that z_{n} must be on some edge of P. Since P has a finite number of edges, we can assume all the points z_{n} on the same edge of P. Let M be the geodesic in H^{3} containing this edge. In particular, z belongs to the closure of M.

We claim that z is a regular point of G. In fact, since z belongs to the boundary of the polyhedron P, either z is a regular point of G or z is a parabolic fixed point. If we assume z to be a parabolic fixed point, then the stabilizer of z in G (denoted by stab_{G}(z)) is a Euclidean group [4]. All points in H^{3} are regular points of the group G, so z must be one of the end points of M. The discreteness of G implies the existence of a horoball H contained in H^{3} which is precisely invariant under stab_{G}(z) in G. In particular, z belongs to the boundary of H. The geodesic M must intersect H, and such an intersection is an arc of geodesic with one end point being z. Since the sequence of points z_{n} are in M and they converge to z, we can assume without loss of generality that all z_{n} belong to such an intersection. The linear transformations g_{n} ∘ j ∘ g_{n}^{−1} have z_{n} as fixed points, so g_{n} ∘j ∘g_{n}^{−1}(H)∩H ≠ ϕ. By the definition of H, we must have g_{n} ∘j ∘g_{n}^{−1}(H) = H and g_{n} ∘j ∘g_{n}^{−1}(z) = z. The additional condition that g_{n} ∘j ∘g_{n}^{−1}(z_{n}) = z_{n} implies g_{n} ∘ j ∘ g_{n}^{−1}(w) = w, for all w in the geodesic M. In particular, g_{n}(L) = M. Let us consider the linear transformations h_{n} = g_{n} ∘g_{n+1}^{−1} . These transformations have as fixed points the end points of M, in particular z, so they are not parabolic transformations. Since the transformation h_{n} cannot act as the identity on M by the choice of the sequence g_{n}, it must be loxodromic. This contradicts the fact that z is a parabolic fixed point and that G is a discrete group. In particular, z must be a regular point of G.

Let us consider the geodesics L_{n} = g_{n}(L) through z_{n} having end points g_{n}(x) and g_{n}(y). Since g_{n}(x) and g_{n}(y) converge to t and u, respectively, L_{n} must converge either to a point or to the geodesic with end points u and t. If L_{n} converges to a point, we necessarily have u = t = z. This is a contradiction to the fact that z is a regular point and u = t is a limit point for G. The other possibility is that L_{n} converges to a geodesic γ, with end points u and t. In this case, since the end points of γ are limit points of G and z is a regular point of G, we must have z in γ ∩ H^{3} . Any neighborhood of z contains z_{n}, for n sufficiently large. We know z is a regular point, so there exists a neighborhood U of it which is precisely invariant under stab_{G}(z), which is known to be finite. We can then assume without loss of generality that z_{n} belongs to U, g_{n} ∘ j ∘ g_{n}^{−1}(z) = z, and g_{n}∘j∘g_{n}^{−1} = h, for all n. In this case, we must have (g_{m}^{−1}∘g_{n})∘j∘(g_{m}^{−1}∘g_{n})^{−1} = j. Since g_{n} ∘ j ∘ g_{n}^{−1}(z_{n}) = z_{n}, g_{n} ∘ j ∘ g_{n}^{−1}(z) = z and z_{n} ≠ z, for all n, we must have g_{n} ∘ j ∘ g_{n}^{−1}(w) = w for all w in γ. In particular, g_{n} ∘ j ∘ g_{n}(t) = t and g_{n} ∘ j ∘ g_{n}^{−1}(u) = u. It follows that {g_{n}(x), g_{n}(y)} = {t, u}. The fact that t ≠ u and g_{n}(x) converges to t implies that g_{n}(x) = t and g_{n}(y) = u for n sufficiently large. We may assume that the above holds for all n. The last statement implies that g_{m}^{−1} ∘ g_{n}(x) = x and g_{m}^{−1} ∘ g_{n}(y) = y for all n, m. The transformations g_{m}^{−1} ∘ g_{n} also keep L invariant, and for n ≠ m this transformation cannot be the identity on L. It follows that g_{m}^{−1} ∘ g_{n} is a loxodromic element in G with x and y as fixed points.

References

[1] Anderson, J.: Mixing elements into Kleinian groups. - Ph.D. Thesis, S.U.N.Y. at Stony Brook, 1991.

[2] Hidalgo, R. A.: On Schottky groups with automorphism. - Ph.D. Thesis, S.U.N.Y. at Stony Brook, 1991.

[3] Maskit, B.: Decomposition of certain Kleinian groups. - Acta Math. 130, 1973, 243–263.

[4] Maskit, B.: Kleinian groups. - Grundlehren Math. Wiss. 287, Springer-Verlag, Berlin– Heildelberg–New York, 1988.

[5] Maskit, B.: On extended quasifuchsian groups. - Ann. Acad. Sci. Fenn. Ser. A I Math. 15, 1990, 53–64.

[6] Maskit, B.: Intersections of component subgroups of Kleinian groups. - In: Discontinuous groups and Riemann surfaces. Proceedings of the conference held in University Park, Md., 1973; L. Greenberg, editor. Ann. of Math. Stud. 79, Princeton Univ. Press, Princeton, N.J.

Received 4 March 1992