Notes on projective structures and Kleinian groups

Katsuhiko Matsuzaki and John A. Velling ^{∗}

1 Introduction

Throughout this paper, C will denote the complex plane, Ĉ = C ∪ {∞} the number sphere, and D = {z : |z| < 1} ⊂ C the unit disk. We use PSL(2, C) = SL(2, C)/ ± id for the group of Möbius transformations of Ĉ. With Γ an arbitrary Fuchsian group, possibly having elliptic elements, let R be the hyperbolic orbifold D/Γ.

A projective structure P = (M, f) on R is a representation (the monodromy representation) M : Γ → PSL(2, C) and a locally univalent holomorphic M-equivariant map (the developing map) f : D → Ĉ, so that f ∘ γ = M(γ) ∘ f for all γ ∈ Γ (see [2, ch.9]). Let M(Γ) denote the image of Γ by M. The kernel of a projective structure means ker(M), and a projective structure is called faithful if its kernel is trivial. Two projective structures (M_{1}, f_{1}) and (M_{2}, f_{2}) are said to be equivalent if and only if there is a g ∈ PSL(2, C) so that f_{1} = g ∘ f_{2} (and hence M_{1} = gM_{2}g^{−1}). It is well known that the space of equivalence classes of projective structures is in one–to–one correspondence with the affine space of holomorphic quadratic differentials on R, Q(R).

We choose an origin in the space of quadratic differentials by fixing an equivalence class of projective structures. In this case the Fuchsian equivalence class will be denoted by 0 ∈ Q(R), making Q(R) a vector space. Now the mapping from equivalence classes of projective structures to quadratic differentials is readily expressed: P = (M, f) corresponds to the quadratic differential S_{f} ∈ Q(R), where S_{f} is the Schwarzian derivative of f : D → Ĉ. This indeed determines a map from the equivalence classes of projective structures to Q(R); for any g ∈ PSL(2, C), S_{g∘f} = S_{f}, implying that equivalent projective structures correspond to the same quadratic differential. This map is actually a bijection (see either [2, ch.9] or [4, §II.3]), and we will identify equivalent projective structures and use the identification with Q(R) implicitly. Thus P_{Q} means a representative of the equivalence class of projective structures corresponding to Q. Note that ker(M_{Q}) depends only on Q, not on a choice of representative.

∗Partially supported by PSC-CUNY grant #661365.

Letting ρ_{R}|dz| denote the metric of curvature −1 on R induced by projection of that on D, where ρ_{D}(z) = \frac{2}{(1 − |z|^{2})}, we norm Q(R) by ∥Q∥ =

sup_{D} |Q(z) · ρ_{D}^{−2}(z)| = sup_{R} |Q(z) · ρ_{R}^{−2}(z)|. We denote by Q^{∞}(R) ⊂ Q(R) the space of norm-bounded quadratic differentials on R, and the corresponding projective structures will be called bounded projective structures. Unless otherwise stated, all quadratic differentials considered will be presumed bounded. Nehari [18] showed that if ∥Q∥ ≤ \frac{1}{2} then f_{Q} is univalent, while it follows from a standard theorem of Kraus [12] that {Q : f_{Q} is univalent} is a closed subset of \{Q : ∥Q∥ ≤ \frac{3}{2}\}.

The behavior of f_{Q} when it is not univalent is not well understood. Gunning [3] showed that for compact R either f_{Q} maps D onto Ĉ or else f_{Q} is a covering map of a domain Ω. Kra [8], [9] added that M_{Q}(Γ) acts discontinuously on Ω = f_{Q}(D) if and only if f_{Q} is a covering map of Ω and extended these results to finite area R in the case when Q ∈ Q^{∞}(R). Whether or not this is the case clearly depends only on Q.

Given these considerations, for an arbitrary hyperbolic Riemann surface, we define three classes of projective structures on R, listed in order of decreasing size:

1. \underline{bounded discrete projective structures} (or simply discrete projective structures, if boundedness is either assumed or dropped) — D(R) = {P_{Q} : Q ∈ Q^{∞}(R), M_{Q}(Γ) is discrete },

2. \underline{bounded Kleinian projective structures} (or Kleinian projective structures) — K(R) = {P_{Q} ∈ D(R) : M_{Q}(Γ) has a nonempty region of discontinuity in Ĉ}, 3. \underline{bounded covering projective structures} (or covering projective struc-

tures) — S(R) = {P_{Q} ∈ K(R) : f_{Q} is a covering map of its image and M_{Q}(Γ) acts discontinuously there }.

It follows that the space of faithful covering projective structures corresponds precisely to those Q ∈ Q^{∞}(R) for which f_{Q} is univalent. Note that both Maskit [13] and Hejhal [5] have given examples of projective structures where M_{Q}(Γ) is discrete but f_{Q} is not a covering map. Hence, in general, S(R) is a proper subset of K(R). More recently Kra [10], following Hejhal [6], completely classified the geometrically finite isolated points in S(R).

Assuming for the moment that R is compact, Kra and Maskit showed in [11] that S(R) ⊂ Q^{∞}(R) is compact. Shiga showed in [20] that Int(K(R)) ∩ S(R) coincides with the Bers embedding, centered at R, of the Teichmüller space of R. In theorem 3 and 4, we extend these results in non-compact cases.

The second author would like to thank the International Center for Theoretical Physics in Trieste, Italy, for their generous hospitality during part of the work contained herein.

2 Bounded covering projective structures with distinct kernels are separated

Throughout this section we will only be concerned with bounded projective structures.

In personal conversation, Fred Gardiner asked whether or not quadratic differentials whose corresponding classes of projective structures are covering and have distinct kernels are in distinct components of S(R). We answer this in the positive with theorem 1 of this section, under rather loose conditions on the hyperbolic geometry of R. These conditions include R having finite hyperbolic area. But first we set the stage for the theorem.

Definition. We say a Fuchsian group Γ is maximal if it is not properly contained in any other Fuchsian group. The corresponding R will also be called maximal.

It is immediate that all maximal Fuchsian groups are of the first kind. The following proposition simplifies considerations of S(R) for R maximal.

Proposition 1: If R is maximal then for any covering projective structure P_{Q} on R, f_{Q}(D)/M_{Q}(Γ) is conformally equivalent to R.

Proof: We have the following commutative diagram:

D ⟶  f_{Q}(D)
↓  ↓
R ⟶ f_{Q}(D)/M_{Q}(Γ)

If R is maximal then f_{Q}(D)/M_{Q}(Γ) ≅ R. □

Lemma 1. If Q ∈ S(R), then M_{Q}(Γ) is a non-elementary Kleinian group whenever

1. R has finite hyperbolic area, or

2. R is maximal and has cusps.

Proof: In case (1) the proof may be found in [2] or [8, th. 1]. For case (2), by proposition 1 we have f_{Q}(D)/M_{Q}(Γ) ≅ R. Here, if M_{Q}(Γ) is elementary, the planar surface f_{Q}(D) must have a cusp. The following proposition shows that |Q(z) · ρ^{−2}(z)| is not bounded. Thus P_{Q} ∉ S(R). □

Proposition 2. Let Ω be a planar domain with cusps and f : D → Ω a locally univalent covering map. Then |S_{f} · ρ_{D}^{−2}| is unbounded on D.

Proof: By conjugation we may assume that 0 ∈ Ĉ \ Ω is a cusp, that the group

Γ′ = ⟨z ↦ \frac{(1 + \frac{i}{π})z − 1}{z − (1 − \frac{i}{π})}⟩

is a subgroup of the deck group under which a horodisk B tangent to the unit circle at 1 is precisely invariant, and that B/Γ′ is a neighborhood of the cusp under via f. Then there is a conformal map h on some neighborhood of 0 such that h(0) = 0 and h ∘ f(z) = exp \frac{z + 1}{z − 1} on B.

By the Cayley identity, we have

|S_{h∘f}(z) · ρ_{D}^{−2}(z)| = |S_{h}(f(z)) · ρ_{Ω}^{−2}(f(z)) + S_{f}(z) · ρ_{D}^{−2}(z)|.

As z → 1 radially in B, the left-hand side of this equality, which is equal to

|\frac{2}{(z − 1)^{4}} \frac{(1 − |z|^{2})^{2}}{4} |,

goes to ∞. At the same time, on the right-hand side, S_{h}(f(z)) is bounded near 0, and

lim_{f(z)→0} ρ_{Ω}^{−2}(f(z)) = 0.

Hence we have |S_{f}(z) · ρ_{D}^{−2}(z)| → ∞ as z → 1 radially in B, and |S_{f} · ρ_{D}^{−2}| is unbounded. □

We now have

Theorem 1. Let R satisfy one of the conditions of lemma 1. If P_{Q_{1}}, P_{Q_{2}} are covering projective structures corresponding to Q_{1}, Q_{2}, and ker(M_{Q_{1}}) ≠ ker(M_{Q_{2}}), then P_{Q_{1}} and P_{Q_{2}} are in different components of S(R).

The proof requires several facts …

Lemma 2. If Q ∈ S(R), and α ∈ Γ \ ker(M_{Q}) then there is some β ∈ Γ such that M_{Q}(α) and M_{Q}(β) generate a non-elementary discrete group.

Proof of Lemma 2: For any β ∈ Γ, the discreteness of the group generated by M_{Q}(α) and M_{Q}(β) is given, as Q ∈ S(R). Since M_{Q}(Γ) is non-elementary (lemma 1), for any α ∈ Γ with M_{Q}(α) ≠ id there is some β ∈ Γ \ ker(M_{Q}) such that the fixed points of M_{Q}(α) and of M_{Q}(β) are distinct. For such β, M_{Q}(α) and M_{Q}(β) generate a non-elementary group (see, for example, [15, p.23]). □

Now, for any α, β ∈ Γ, let S(R, α, β) = {Q ∈ S(R) : ⟨M_{Q}(α), M_{Q}(β)⟩ is not elementary }.

Lemma 3. For any α, β ∈ Γ, S(R, α, β) is both open and closed in S(R) .

Proof of Lemma 3: That S(R, α, β) is open is seen as follows. Let Q_{0} ∈ S(R, α, β). Since ⟨M_{Q_{0}}(α), M_{Q_{0}}(β)⟩ is non-elementary, it contains a Schottky group of rank 2, i.e. there exist α′, β′ ∈ ⟨α, β⟩ ⊂ Γ so that ⟨M_{Q_{0}}(α′), M_{Q_{0}}(β′)⟩ = M_{Q_{0}}(α′) ∗ M_{Q_{0}}(β′) is Schottky. Since small deformations of Schottky groups are also Schottky [15], there is an ϵ > 0 such that if ∥Q − Q_{0}∥ < ϵ then ⟨M_{Q}(α′), M_{Q}(β′)⟩ is also a Schottky group of rank 2. Hence when ∥Q − Q_{0}∥ < ϵ and Q ∈ S(R) we have ⟨M_{Q}(α), M_{Q}(β)⟩ is non-elementary. This implies that Q ∈ S(R, α, β).

To show that S(R, α, β) is also closed, proving the lemma, we use the ensuing lemma.

Lemma.[7] The algebraic limit of a sequence of non-elementary discrete groups with a bounded number of generators is also a non-elementary discrete group.

Thus we assume {Q_{n}} is a sequence in S(R, α, β) with Q_{n} → Q as n → ∞. Since the ⟨M_{Q_{n}}(α), M_{Q_{n}}(β)⟩ are non-elementary and ⟨M_{Q_{n}}(α), M_{Q_{n}}(β)⟩ → ⟨M_{Q}(α), M_{Q}(β)⟩ algebraically, we conclude that ⟨M_{Q}(α), M_{Q}(β)⟩ is a nonelementary discrete group, i.e. Q ∈ S(R, α, β). □

We are now ready to prove our theorem. Proof of Theorem 1: Suppose P_{Q_{1}} and P_{Q_{2}} are discrete projective structures representing Q_{1} and Q_{2}, respectively, such that ker(M_{Q_{1}}) ≠ ker(M_{Q_{2}}). Without loss of generality let α, β ∈ Γ satisfy the following conditions: α ∈ ker(M_{Q_{2}}) \ ker(M_{Q_{1}}) and M_{Q_{1}}(α), M_{Q_{1}}(β) generate a non-elementary discrete group. This is always possible by lemma 2 and the hypotheses of our theorem.

By lemma 3, whether Q ∈ S(R, α, β) or not is determined on components of S(R). Since by our choice of α, β we have Q_{1} ∈ S(R, α, β) while Q_{2} ∉ S(R, α, β), it follows that Q_{1} and Q_{2} are in distinct components of S(R). □

It is interesting to ask to what extent ker(M_{Q}), as a subgroup of Γ, determines the component of S(R) in which Q lies. We cannot answer this at present, so we instead ask to what extent does ker(M_{Q}) determine the conformal equivalence class of f_{Q}(D). One sees easily that if Q ∈ S(R) has trivial kernel then f_{Q}(D) is simply connected and conformally equivalent to D. More generally we show

Theorem 2. If ker(M_{Q}) is such that D/ker(M_{Q}) is conformally a plane domain (i.e. simple closed curves separate), then f_{Q}(D) is conformally equivalent to D/ker(M_{Q}).

Proof: We let Γ′ be a Fuchsian extension of Γ such that D/Γ′ = R′ is conformally the surface f_{Q}(D)/M_{Q}(Γ), and ker′ the kernel of the monodromy map from Γ′. Let R̂ = D/ker(M_{Q}), and R̂′ = f_{Q}(D) is conformally D/ker′. We have that the diagram

D

↙ ^{/ker}  ↘ ^{/ker′}
R̂ ⟶^{f̂_{Q}} R̂′
↓^{/(Γ/ker)}  ↓^{/(Γ′/ker′)}
R  ⟶^{f̃_{Q}}  R′

commutes (see [11]).

We will establish the theorem by assuming that f̂_{Q} is not a conformal equivalence and showing that R̂ must have a nonseparating simple closed curve. To do this we use

Lemma 4. There exists γ′ ∈ ker′ \ Γ corresponding to a simple closed curve on R̂′.

Proof of Lemma 4: The γ′ ∈ ker′ corresponding to simple closed curves on R̂′ generate ker′, if such a γ′ does not exist then ker′ < Γ, whence ker′ = ker and f̂_{Q} is a conformal equivalence. □

Let Ω_{i} (i = 1, 2) be two sheets of R̂ over R̂′, with cuts A_{i} and B_{i} to be determined. The first criterion for this choice is that A_{1}^{+} (B_{1}^{+}) are identified with A_{2}^{−} (B_{2}^{−}, respectively) by the covering, a simple closed curve corresponding to γ′ of lemma 4 crosses from A_{1}^{−} to A_{2}^{+}, and B_{i} is the image of A_{i} by some non-trivial element of Γ/ker. See figure 1.

We choose the B_{i}, moreover, so that simple arcs α_{i}^{±} may be taken from A_{i}^{±} to B_{i}^{±} in Ω_{i}, with the same endpoints on the respective cuts. By monodromy this is always possible. The curve α_{2}^{+}(α_{1}^{−})^{−1} is a closed curve in R̂. See figure 2. It does not separate R̂, as we may connect any two points in Ω_{i} \ α_{i}^{+} or Ω_{i}\α_{i}^{−} by a simple arc in Ω_{i} since neither α^{+} nor α^{−} disconnects Ω_{i}, and using the action of the infinite group Γ/ker we may connect any Ω_{i} to any other Ω_{j} through a finite number of other Ω_{k} while staying away from α_{2}^{+}(α_{1}^{−})^{−1}. □ Figure 1:

Figure 2:

3 Boundedness of S(R)

By considering the boundedness of S(R) for more general R, we extend the Kra–Maskit result mentioned earlier.

Theorem 3. Let R be maximal. Then S(R) is bounded in Q^{∞}(R) precisely when the lengths of simple closed geodesics on R are bounded away from 0.

Proof: For a covering map f_{Q} : D → Ω ⊂ Ĉ, with Q ∈ S(R), let us consider the injectivity radius. Assume that the lengths of simple closed geodesics on R are bounded away from 0. Then the injectivity radius of the universal covering map π : D → R is bounded away from 0 for any z ∈ D except in the cusped regions, and so is f_{Q}. Since Q ∈ Q^{∞}(R), again by [9, lem. 1], f_{Q} is univalent on the cusped regions.

Hence we know there exists a positive constant δ such that the injectivity radius of f_{Q} for any Q ∈ S(R) is larger than δ at any z ∈ D. By the Kra– Maskit lemma [11, lem. 5.1a] we have ∥Q∥ ≤ 6 tanh^{−2} δ, so that S(R) is indeed bounded.

Conversely, let {γ_{n}} be a sequence of simple closed geodesics on R whose lengths ℓ(γ_{n}) → 0 as n → ∞. For each n, we construct a special Kleinian group G_{n}. Cutting R along γ_{n} and using the combination theorems [15], we construct G_{n}, with an invariant component Ω_{n} of the region of discontinuity such that Ω_{n} is conformally the normal cover of R corresponding to γ_{n} and R = Ω_{n}/G_{n}.

Let f_{n} be the locally univalent covering map D → Ω_{n} and set Q_{n} = S_{f_{n}}. They belong to Q(R). Furthermore, since

inf_{z∈D}\{ injectivity radius of f_{n} at z\} = \frac{ℓ(γ_{n})}{2},

Q_{n} is actually in Q^{∞}(R).

Now using [11, lem. 5.1b], ∥Q_{n}∥ → ∞ as n → ∞, since ℓ(γ_{n}) → 0. □

Remark: The above condition on R has arisen recently in related settings ([16], [17], and [19], for example).

Corollary. Compactness of S(R) is equivalent to R having finite hyperbolic area. (Recall the topology of S(R).)

Proof: Since S(R) is closed [8], and as the ability of R to cover only finitely many other Riemann surfaces (see [10]) implies that S(R) is bounded (of course, finite area is used here), the result follows as Q^{∞}(R) is finite dimensional. □

The following proposition shows that the presence of cusps does allow, however, for the existence of unbounded covering projective structures.

Proposition 4. If R has cusps there are necessarily unbounded covering projective structures on R.

Proof: Let R be conformally R′ \ {p}, where R′ = U/Γ′, U ⊂ Ĉ is a holomorphic universal cover of R′, and Γ′ is a subgroup of PSL(2, C). Now suppose U ∋ 0 ↦ p ∈ R′ via the universal covering of R′. In this case the universal covering of U \ Γ′(0) by D is the developing map f of a discrete projective structure on R. The kernel of the monodromy group here is the normalizer of a parabolic element in Γ corresponding to the cusp on R. That |S_{f} · ρ^{−2}| is unbounded on D was shown in proposition 2. □

4 The structure of Int(D(R))

Shiga [20] studied the structure of Int(K(R)) for a compact hyperbolic surface. We extend this by considering the case where R = D/Γ is of finite type, possibly with cone points, and studying Int(D(R)). Though our argument is the same as Shiga’s, we present it in full for the reader’s convenience.

Lemma 5. For R = D/Γ arbitrary, if Q ∈ Int(D(R)), then M_{Q} is a type preserving isomorphism.

Proof: For Q ∈ Q^{∞}(R), the homomorphism M_{Q} preserves parabolic elements and the type of elliptic elements. Thus if there exists some Q_{0} ∈ Int(D(R)) such that M_{Q_{0}} is not a type preserving isomorphism, then there exists a hyperbolic element γ ∈ Γ such that tr^{2}(M_{Q}(γ)) is a non-constant holomorphic function of Q ∈ Q^{∞}(R) and M_{Q_{0}}(γ) is either elliptic or parabolic. In either case, since tr^{2}(M_{Q}(γ)) is an open mapping, there is a Q_{1} ∈ Int(D(R)) near to Q_{0} such that M_{Q_{1}}(γ) is elliptic with infinite order. But a discrete subgroup of PSL(2, C) cannot have such elements, yielding a contradiction. □

Lemma 6. Let R = D/Γ have finite hyperbolic area. For Q ∈ Int(D(R)), M_{Q}(Γ) has a non-empty region of discontinuity in Ĉ. Thus Int(D(R)) = Int(K(R)).

Proof: Assume Q_{0} ∈ Int(D(R)) is such that M_{Q_{0}}(Γ) is not Kleinian. Since M_{Q_{0}}(Γ) is finitely generated, it is Mostow–Sullivan rigid [21, th. 5]. Take a small ball B with center at Q_{0} in Int(D(R)). By Lemma 5, we can define a family {M_{Q} ∘ M_{Q_{0}}^{−1}} of type-preserving isomorphisms depending holomorphically on Q ∈ B. Then by Bers [1] (see also Shiga [20, th. 1]), we see M_{Q_{0}}(Γ) and M_{Q}(Γ) are quasiconformally equivalent for each Q ∈ B \ {Q_{0}}. But the rigidity of M_{Q_{0}}(Γ) implies that the representations M_{Q} and M_{Q_{0}} are actually conformally equivalent. This means that Q = Q_{0}, which is impossible. □

Theorem 4. For R of finite area, Int(D(R)) ∩ S(R) = T(R), where T(R) is the Bers embedding, centered at R, of the Teichmüller space of R.

Proof: The inclusion Int(D(R)) ∩ S(R) ⊃ T(R) is clear. For the other direction, let Q be a point in Int(D(R)) ∩ S(R). By the above two lemmas, we know Q ∈ Int(K(R)) and M_{Q} is a type preserving isomorphism. So by a theorem of Maskit, [14, th. 6], M_{Q}(Γ) is quasi-Fuchsian or totally degenerate without accidental parabolics, and the developing map f_{Q} is univalent.

By the same reasoning as given by Shiga in [20, th. 2], we see that M_{Q}(Γ) cannot be totally degenerate. Hence M_{Q}(Γ) is quasi-Fuchsian, and Q ∈ T(R) as desired. □

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Katsuhiko Matsuzaki: Department of Mathematics, Tokyo Institute of Technology, Meguro, Tokyo 152 JAPAN email: matsuzak@math.titech.ac.jp

John A. Velling: Department of Mathematics, Brooklyn College, Brooklyn, New York 11210 USA email: jvelling@bklyn.bitnet