THE COEFFICIENT BODY OF BELL REPRESENTATIONS OF FINITELY CONNECTED PLANAR DOMAINS

MOONJA JEONG AND MASAHIKO TANIGUCHI

Abstract. In this note, we determine the homotopy type of the coefficient body of Bell representations of non-degenerate n-connected planar domains with n ≥ 3. Also, by considering the isomorphism classes of rational functions, we get the precise number of those corresponding to Bell representations wth the same set of critical values. Further, the case of those with the same set of critical points is discussed.

1. Introduction

In this paper, a non-degenerate n-connected planar domain is a subdomain Ω of the Riemann sphere ℂ̂ such that ℂ̂ − Ω consists of exactly n connected components each of which contains more than one point. We also assume that n ≥ 2.

Then we know that every such Ω has a canonical representation as in the following theorem, which is called a Bell representation of it.

Theorem 1.1 ([6]). Every non-degenerate n-connected planar domain with n ≥ 2 is mapped biholomorphically onto a domain W_{a,b} defined by

\{z ∈ ℂ | |z + ∑_{k=1}^{n−1} \frac{a_{k}}{z − b_{k}} | < 1\}

with suitable complex vectors a = (a_{1}, a_{2}, ⋯, a_{n−1}) and b = (b_{1}, b_{2}, ⋯, b_{n−1}).

This theorem is considered as a natural generalization of the classical Riemann mapping theorem for simply connected planar domains. Importance of such representations consists in such a fact that every domain W_{a,b} has algebraic kernel functions. To be precise, the function

f_{a,b}(z) = z + ∑_{k=1}^{n−1} \frac{a_{k}}{z − b_{k}}

is a proper holomorphic mapping from W_{a,b} onto the unit disc U which is algebraic. Hence Bell’s result in [1] implies the following

Proposition 1.2. Every non-degenerate n-connected planar domain is biholomorphic to a domain with algebraic Bergman kernel and algebraic Szegő kernel.

Date: Jan.2, 2004.

1991 Mathematics Subject Classification. Primary 32G10, 32G15, 30C40; Secondary 30C20, 30F60.

Key words and phrases. conformal representation, rational function, Hurwitz space.

The first author was supported by Grant R04-2003-000-10045-0 from the Korea Science and Engineering Foundation.

Here it is important to know the locus of the complex vectors (a, b) which correspond to non-degenerate n-connected planar domains.

Definition 1.3. For every n ≥ 2, let B_{n} in ℂ^{2n−2} be the set of all complex vectors

(a, b) = (a_{1}, a_{2}, ⋯, a_{n−1}, b_{1}, b_{2}, ⋯, b_{n−1})

in ℂ^{2n−2} such that the corresponding domains

W_{a,b} = \{z ∈ ℂ | |z + ∑_{k=1}^{n−1} \frac{a_{k}}{z − b_{k}} | < 1\}

are non-degenerate n-connected planar domains.

We call B_{n} the coefficient body for non-degenerate n-connected canonical domains.

In this note, we will investigate the geometric structure of B_{n}.

2. A modified representation

To clarify the structure of the coefficient body, it is more convenient to consider the following modification. In the sequel, we assume that n > 2, since B_{2} and B_{2}^{∗} are explicitly known (cf. [7]).

Definition 2.1. We set

B_{n}^{∗} = {(a_{1}, ⋯, a_{n−1}, b) | (a_{1}^{2}, ⋯, a_{n−1}^{2}, b) ∈ B_{n}},

and call it the modified coefficient body.

Clearly, B_{n}^{∗} is contained in

(ℂ^{∗})^{n−1} × F_{0,n−1}ℂ,

where

F_{0,n−1}ℂ = {(z_{1}, ⋯, z_{n−1}) ∈ ℂ^{n−1} | z_{j} ≠ z_{k} if j ≠ k}.

Also it is invariant under the symmetry

S_{k} : (a_{1}, ⋯, a_{k}, ⋯, a_{n−1}, b) ↦ (a_{1}, ⋯, −a_{k}, ⋯, a_{n−1}, b)

of ℂ^{2n−2} for every k. And B_{n} can be identified with the quotient space of B_{n}^{∗} by the action of the group G =< S_{1}, ⋯, S_{n−1} > generated by these symmetries. Thus B_{n}^{∗} is 2^{n−1} sheeted holomorphic covering of B_{n} with the covering transformation group G.

Next, note that B_{n}^{∗} is circular in the following sense.

Proposition 2.2. For every (a, b) ∈ B_{n}^{∗} and every θ ∈ ℝ, e^{iθ}(a, b) ∈ B_{n}^{∗}.

Proof. If (a, b) ∈ B_{n}^{∗}, then letting

g_{a,b}(z) = z + ∑_{k=1}^{n−1} \frac{a_{k}^{2}}{z − b_{k}} ,

W_{a,b}^{∗} = {z ∈ ℂ | |g_{a,b}(z)| < 1} is a non-degenerate n-connected domain. Hence

W_{e^{iθ}a,e^{iθ}b}^{∗} = \{z ∈ ℂ | |z + ∑_{k=1}^{n−1} \frac{(e^{iθ}a_{k})^{2}}{z − e^{iθ}b_{k}} | < 1\}

= {z ∈ ℂ | |e^{iθ}g_{a,b}(e^{−iθ}z)| < 1} = e^{iθ}W_{a,b}^{∗},

which is biholomorphic to W_{a,b}^{∗}, is a non-degenerate n-connected planar domain.

Another important property is ”star-shapedness” of B_{n}^{∗}.

Proposition 2.3. For every (a, b) ∈ B_{n}^{∗} and every 0 < r ≤ 1, r(a, b) ∈ B_{n}^{∗}.

Proof. Let (a, b) ∈ B_{n}^{∗}, and g_{a,b}(z) be as in the previous proof. And for every 0 < r < 1, set

W_{a,b}^{r} = {|g_{a,b}(z)| < 1/r}.

It contains W_{a,b}^{∗}. And for every connected component F of the preimage g_{a,b}^{−1}(E_{r}) of E_{r} = {w ∈ ℂ||w| ≥ 1/r}, g_{a,b}(z) gives a homeomorphic map of F onto E_{r}. Since E_{r} contains more than a point, so does each F. Hence W_{a,b}^{r} is also a non-degenerate n-connected domain.

Since rW_{a,b}^{r} is biholomorphic to W_{a,b}^{r}, rW_{a,b}^{r} is also non-degenerate n-connected domain. Furthermore, since

rg_{a,b}(z/r) = z + ∑_{k=1}^{n−1} \frac{(ra_{k})^{2}}{z − rb_{k}} = g_{ra,rb}(z),

W_{ra,rb}^{∗} = rW_{a,b}^{r} is a non-degenerate n-connected domain and therefore r(a, b) ∈ B_{n}^{∗}.

We now have the following property of B_{n}^{∗} and B_{n}.

Theorem 2.4. B_{n}^{∗} and hence B_{n} are domains and have the same homotopy type as that of

(S^{1})^{n−1} × F_{0,n−1}ℂ.

Corollary 2.5. The modified coefficient body B_{n}^{∗} is a circular domain homeomorphic to B_{n}.

Remark 2.6. The fundamental group of F_{0,n−1}ℂ is called the pure braid group, and its structure is well-known. See for instance [3].

The above theorem follows from the following two lemmas.

Lemma 2.7. The coefficient body B_{n} is the set of all (a, b) such that

f′_{a,b}(z) = 0

has 2n − 2 solutions c_{1}, ⋯, c_{2n−2} counted with multiplicities such that

|f_{a,b}(c_{j})| < 1

for every j. The set B_{n}^{∗} is characterized in the same way.

In particular, B_{n} and B_{n}^{∗} are open subsets of ℂ^{2n−2}.

Proof. f_{a,b}(z) has exactly 2n − 2 finite critical points c_{1}, ⋯, c_{2n−2}, i.e. zeros of f′_{a,b}(z), counted with multiplicities, and (a, b) belongs to B_{n}, i.e.

{z ∈ ℂ | |f_{a,b}(z)| < 1}

is a non-degenerate n-connected domain if and only if, for every connected component F of the preimage

f_{a,b}^{−1}(E)

of E = {w ∈ ℂ | |w| ≥ 1}, f_{a,b}(z) gives a homeomorphic map of F onto E. Hence we have the first assertion. The case of B_{n}^{∗} is similar.

Next, since c_{j} varies continuously with respect to (a, b), B_{n} and B_{n}^{∗} are open subsets of ℂ^{2n−2}.

Next set

ρ(b) = min_{j≠k} |b_{j} − b_{k}|.

And for a sufficiently small ϵ > 0 with ϵ ≤ 1/(6n), we set

B_{n}^{ϵ} = \{(a, b) ∈ ℂ^{2n−2} | ρ(b) > 0, |b_{k}| ≤ 1/2, 0 < |a_{k}| ≤ ϵ\sqrt{ρ(b)}, 1 ≤ k ≤ n − 1\}.

Note that ρ(b) ≤ 1.

Lemma 2.8. B_{n}^{∗} has the same homotopy type as that of B_{n}^{ϵ}.

Proof. First we show that

B_{n}^{ϵ} ⊂ B_{n}^{∗}

Suppose that (a, b) ∈ B_{n}^{ϵ}. If we set

C_{k} = {z ∈ ℂ | |b_{k} − z| = ϵρ(b)}

then z ∈ C_{k} implies that |z| ≤ \frac{2}{3}, and

|b_{j} − z| ≥ (1 − ϵ)ρ(b) > ρ(b)/2

for every j ≠ k, and hence

|g_{a,b}(z)| ≤ |z| + ∑_{j=1}^{n−1} |\frac{a_{j}^{2}}{z − b_{j}} |

≤ \frac{2}{3} + \frac{ϵ^{2}ρ(b)}{ϵρ(b)} + (n − 2)\frac{ϵ^{2}ρ(b)}{ρ(b)/2}
= \frac{2}{3} + (1 + (2n − 4))ϵ < 1.

On the other hand, if we set

C̃_{k} = {|b_{k} − z| = |a_{k}^{2}|/2}

then |a_{k}^{2}|/2 < ϵ^{2}ρ(b), and z ∈ C̃_{k} implies that

|g_{a,b}(z)| ≥ \frac{|a_{k}^{2}|}{|z − b_{k}|} − |z| − ∑_{j≠k} |\frac{a_{j}^{2}}{z − b_{j}} |

≥ 2 − \frac{2}{3} − (n − 2)\frac{ϵ^{2}ρ(b)}{ρ(b)/2}
= 2 − \frac{2}{3} − (2n − 4)ϵ^{2} > 1.

Thus

{z ∈ ℂ | |g_{a,b}(z)| = 1}

has a component in

{z ∈ ℂ | |a_{k}^{2}|/2 < |z − b_{k}| < ϵρ(b)},

and W_{a,b}^{∗} is disjoint from {|z − b_{k}| ≤ |a_{k}^{2}|/2}, for every k, which implies that W_{a,b}^{∗} is non-degenerate and n-connected.

Next for every (a_{0}, b_{0}) ∈ B_{n}^{∗} with a_{0} = (a_{1,0}, ⋯, a_{n,0}) and b_{0} = (b_{1,0}, ⋯, b_{n,0}), let ℓ_{a_{0},b_{0}} be the ray

{(ra_{0}, rb_{0}) | 0 < r ≤ 1}.

Then by Proposition 2.3, ℓ_{a_{0},b_{0}} ⊂ B_{n}^{∗}. Also since ρ(rb_{0}) = rρ(b_{0}), we conclude that

|ra_{k,0}| = r|a_{k,0}| = ϵ′\sqrt{ρ(rb_{0})},

where

ϵ′ = \sqrt{r}|a_{k,0}|/\sqrt{ρ(b_{0})},

which in turn tends to 0 as r does.

Now, fix an ϵ > 0 with ϵ ≤ 1/(6n). Then, (ra_{0}, rb_{0}) ∈ B_{n}^{ϵ} for every sufficiently small r. Hence we can construct a deformation retraction

r_{ϵ} : B_{n}^{∗} → B_{n}^{ϵ},

by mapping the point (a_{0}, b_{0}) to the nearest point in B_{n}^{ϵ} along ℓ_{a_{0},b_{0}}. This retraction is clearly the identity on B_{n}^{ϵ}, and we conclude the assertion.

Here we give typical examples of points in B_{3}. Consider the case that

f(z) = f_{4a^{2},4a^{2},b,−b}(z) = z + \frac{4a^{2}}{z − b} + \frac{4a^{2}}{z + b}

with a, b ∈ ℂ − {0}. Then Lemma 2.7 implies the following theorem.

Theorem 2.9. The complex vector (4a^{2}, 4a^{2}, b, −b) belongs to B_{3} if and only if

|b^{2} + 4a^{2} + 4a(a^{2} + b^{2})^{1/2}| · |b^{2} − 2a^{2} + 2a(a^{2} + b^{2})^{1/2}|^{2} < |b|^{4}

where the same value of (a^{2} + b^{2})^{1/2} is taken in each term.

Proof. Since f′(z) has 4 roots

(b^{2} + 4a^{2} + 4a(a^{2} + b^{2})^{1/2})^{1/2},

Lemma 2.7 implies that (4a^{2}, 4a^{2}, b, −b) belongs to B_{3} if and only if

|b^{2} + 4a^{2} + 4a(a^{2} + b^{2})^{1/2}| |\frac{12a^{2} + 4a(a^{2} + b^{2})^{1/2}}{4a^{2} + 4a(a^{2} + b^{2})^{1/2}} |^{2}

= |b^{2} + 4a^{2} + 4a(a^{2} + b^{2})^{1/2}| |\frac{(3a + (a^{2} + b^{2})^{1/2})(−a + (a^{2} + b^{2})^{1/2})}{b^{2}} |^{2}

= |b^{2} + 4a^{2} + 4a(a^{2} + b^{2})^{1/2}| |\frac{b^{2} − 2a^{2} + 2a(a^{2} + b^{2})^{1/2})}{b^{2}} |^{2} < 1.

Example 2.10. Let a = 3/40 and b = 1/10. Since a and b satisfy the inequality in Theorem 2.9, (4a^{2}, 4a^{2}, b, −b) belongs to B_{3}. In fact,

\{±\frac{\sqrt{7}}{10}, ±\frac{\sqrt{2}}{20}i\}

is the set of critical points of f_{4a^{2},4a^{2},b,−b} and |f_{4a^{2},4a^{2},b,−b}| < 1 at each critical point.

3. Parametrization as the Hurwitz space

Sometimes, holomorphic functions are parametrized by the set of critical points, or that of the critical values, i.e. the images of critical points. Here we consider the parametrization using the critical values. Such a parametrization is usually considered for those functions in general position. In the sequel, we assume that n > 2.

Definition 3.1. Let Γ be the set of all points (a, b) of B_{n} such that the corresponding rational function f_{a,b} has a non-simple critical point or has a pair of critical points whose images are the same. We call Γ the collision locus.

Then for every point (a, b) in B_{n} −Γ, the rational function f_{a,b} has 2n−2 simple critical values. We denote the set of simple critical values of f_{a,b} by

S_{a,b} = {α_{1}, ⋯, α_{2n−2}},

where, letting {c_{j}}_{j=1}^{2n−2} be the set of the simple critical points of f_{a,b}, α_{j} = f_{a,b}(c_{j}) for every j. This set can be considered as a point in the unordered configuration space B_{0,2n−2}ℂ, i.e. the quotient space of F_{0,2n−2}ℂ by the symmetric group S_{2n−2}. Moreover by Lemma 2.7, we see that S_{a,b} is actually a point of the unordered configuration space B_{0,2n−2}U for the unit disc U (cf. [3]).

Thus we can define the projection

π_{S} : B_{n} − Γ → B_{0,2n−2}U

by setting

π_{S}(a, b) = S_{a,b}.

We have the following theorem about the projection π_{S}.

Theorem 3.2. The projection π_{S} is a

(2n − 2)! n^{n−3}

-sheeted proper holomorphic covering of B_{0,2n−2}U for every n > 2.

Remark 3.3. The number

\frac{(2n − 2)! n^{n−3}}{n!}

is called a Hurwitz number. See, for instance [5].

First recall that, for every point (a, b) ∈ B_{n} −Γ, the critical points c_{1}, ⋯, c_{2n−2} of f_{a,b} are the solutions of the algebraic equation

∏_{j=1}^{n−1}(z − b_{j})^{2} (1 − ∑_{k=1}^{n−1} \frac{a_{k}}{(z − b_{k})^{2}} ) = 0.

Hence c_{j} moves holomorphically with respect to (a, b). Since so does the image α_{j} of c_{j} for each j = 1, ⋯, 2n − 2, the map π_{S} is holomorphic.

Next we show by the following two lemmas that, for every point S in B_{0,2n−2}U, π_{S}^{−1}(S) consists of (2n − 2)! n^{n−3} points.

Definition 3.4. The marked Hurwitz space MH_{0,n}(1, ⋯, 1) of genus 0 and degree n with type (1, ⋯, 1) and with the ordered poles is the set of all isomorphism classes of rational functions in general position (i.e. with simple critical values) of degree n such that poles are simple and ordered. Here we say that two such rational functions f, g are isomorphic if there is a Möbius transformation A such that

f = g ∘ A

and A maps poles of f to those of g keeping the order. (Cf. [8].)

Lemma 3.5. B_{n} − Γ can be identified with the subset MH_{n}U of marked Hurwitz space MH_{0,n}(1, ⋯, 1), consisting of all isomorphism classes of rational functions whose critical values are in U, by the mapping ι which maps (a, b) to the isomorphism class of f_{a,b}.

Proof. By Lemma 2.7, every f = f_{a,b} with (a, b) ∈ B_{n} − Γ determines a point in MH_{n}U. Here we always assume that the order of poles is b_{1}, ⋯, b_{n−1}, ∞.

Next suppose that (a′, b′) is also in B_{n} − Γ. If g = f_{a′,b′} is in the isomorphism class of f, then there is a Möbius transformation A such that

f = g ∘ A

and since A maps poles of f to those of g keeping the order, A fixes ∞ and hence is affine, which we write as A(z) = pz + q. Then

z + ∑_{k=1}^{n−1} \frac{a_{k}}{z − b_{k}} = pz + q + ∑_{k=1}^{n−1} \frac{a′_{k}}{A(z) − b′_{k}} .

Hence A should be the identity map. This implies that

(a, b) = (a′, b′),

and hence ι is injective.

Finally, for every point P in MH_{n}U, take a representative (a rational function) f in this class. Then the poles of f are simple and ordered. By applying precomposition of a suitable Möbius transformation which sends ∞ to a pole if necessary, we may assume that f has the form

f(z) = az + b + ∑_{k=1}^{n−1} \frac{a_{k}}{z − b_{k}} .

Again by another precomposition of an affine transformation, we may assume that a = 1, b = 0, i.e. f = f_{a,b} with some (a, b) in B_{n} − Γ. Thus ι : B_{n} − Γ → MH_{n}U is surjective.

Now, fix a point S = {α_{j}}_{j=1}^{2n−2} in B_{0,2n−2}U. And fix a set of mutually disjoint cuts (simple smooth arcs) ℓ_{j} from α_{j} to a mutually distinct boundary point ω_{j} of U for every j. Here we assume that ω_{1}, ⋯, ω_{2n−2} are located with this order (with respect to the counter-clockwise direction) on the boundary ∂U of U.

Lemma 3.6. The number of points in π_{S}^{−1}(S) of S by π_{S} is always

(2n − 2)! n^{n−3}.

Proof. For every point (a, b) in π_{S}^{−1}(S), f_{a,b} gives a representative of the point ι((a, b)) in MH_{n}U over S. In other words, f_{a,b} gives an n-sheeted branched holomorphic covering of ℂ̂ by ℂ̂ with critical values S and ordered simple poles b_{1}, ⋯, b_{n−1}, ∞.

Recall that f = f_{a,b} also gives the branched covering of U by W_{a,b}. This covering can be reconstructed as follows: Set D = U − ∪_{j=1}^{2n−2} ℓ_{j}. Then the preimage f^{−1}(D) consists of n domains D_{k}, the order of which is naturally defined as follows: Let γ_{k} be the component of f_{a,b}^{−1}(∂U) surrounding the k-th pole. Then D_{k} is the component whose boundary contains the part of γ_{k} which is projected by f_{a,b} onto the subarc of ∂U from ω_{2n−2} to ω_{1} (which contains no ω_{j}).

Let ℓ_{j}^{k} be the ”slit” on D_{k} over ℓ_{j} (i.e. the part of the boundary corresponding to the preimage f^{−1}(ℓ_{j}) on D_{k}) for every k and j. Then each ℓ_{j}^{k} is divided by some critical point into two arcs, which can be considered as two sides of the ”slit” ℓ_{j}^{k}. And for every j, there is a pair, say {D_{k(j)}, D_{k′(j)}} such that sides of these ”slits” are glued ”crosswise” along ℓ_{j}^{k(j)} and ℓ_{j}^{k′(j)}. (Here two sides of every other ”slit” ℓ_{j}^{k} is glued trivially.) Hence we have a transposition σ_{j} = (k(j) k′(j)) of ordered n sheets at ℓ_{j} when we move counter-clockwise along ∂U for each j. Since W_{a,b} has exactly n boundary components,

σ_{2n−2} ∘ ⋯ ∘ σ_{1}

should be the identical permutation. And apply all such gluings as above, we can reconstruct the branched covering f : W_{a,b} → U.

Thus for every (a, b) in π_{S}^{−1}(S) and with fixed cuts {ℓ_{j}}, we have an ordered factorization of the identical permutation into 2n − 2 transpositions. And since W_{a,b} is connected, such transpositions generate the full symmetric group S_{n}.

Conversely, for every such an ordered factorization of the identical permutation, we can construct an n-sheeted branched covering of ℂ̂ by itself, and hence also of U by an n-connected domain W, having the set S as simple critical values. Then W has n boundary components, and hence by the argument in the proof of the main theorem in [6], we can find a point (a, b) in B_{n} −Γ such that W_{a,b} is biholomorphic to W and f_{a,b} belongs to the isomorphism class of the covering projection of the above covering. In other words, (a, b) ∈ π_{S}^{−1}(S). Also it is clear that different such factorizations give different branched covering structures, and hence different (a, b) in π_{S}^{−1}(S) by Lemma 3.5.

On the other hand, it is known (cf. [5]) that the number of such (transitive minimal) ordered factorizations of the identical permutation on {1, ⋯, n} into transpositions is

(2n − 2)! n^{n−3},

which shows the assertion.

Finally, we have

Lemma 3.7. π_{S} is locally biholomorphic, and evenly covered (i.e. for every point S ∈ B_{0,2n−2}U, there is a neighborhood V of S such that every component of π_{S}^{−1}(V) is biholomorphic to V).

Proof. Fix a point S in B_{0,2n−2}U and a point (a, b) in π_{S}^{−1}(S) arbitrarily. Then it is classically well-known (or can be shown by a standard arguments in the quasiconformal deformation theory) that we can find a neighborhood V of S and a holomorphic function ϕ of V into B_{n} such that

ϕ(S) = (a, b)

and for every (a′, b′) in ϕ(V), f_{a′,b′} gives the same factorization of the identical permutation as f_{a,b} does. Here if V is sufficiently small, we can consider the natural bijection between S and the set S′ of critical values of f_{a′,b′} for every (a′, b′) ∈ ϕ(V). And we take as the ”slits” ℓ′_{j} for S′ the image of ℓ_{j} by a self-diffeomorphism of U ∪ ∂U which is the identity outside mutually disjoint simply connected, relatively compact, neighborhoods of each α_{j} in U and induces the above bijection between S and S′.

Then from the construction, π_{S} ∘ ϕ is the identity. And since the number of points in the preimage π_{S}^{−1}(S) is a finite constant by above lemma, we conclude that π_{S} is locally biholomorphic, and also evenly covered.

Thus π_{S} gives an unbranched (2n − 2)! n^{n−3}-sheeted, holomorphic covering of B_{0,2n−2}U by B_{n} − Γ. In particular, it is proper, which completes the proof of Theorem 3.2.

Example 3.8. In the case n = 3, such ordered factorizations are

{(p q), (p q), (p r), (p r)},
{(p q), (p r), (p r), (p q)},
{(p q), (p r), (r q), (p r)},
{(p q), (p r), (q p), (q r)},

where we can take any bijection of {p, q, r} to {1, 2, 3}. Hence we have 4! different ordered (transitive minimal) factorizations of the identical permutations on {1, 2, 3}.

4. Parametrization by the critical points

We call the set of all points in (ℂ^{∗})^{n−1} × F_{0,n−1}ℂ such that the corresponding rational function f_{a,b} has a non-simple critical point the non-simple locus, and we denote it by Δ. Then Δ ⊂ Γ, and for every point (a, b) in (ℂ^{∗})^{n−1} × F_{0,n−1}ℂ − Δ, we denote the set of simple critical points of f_{a,b} by

C_{a,b} = {c_{1}, ⋯, c_{2n−2}}.

This set can be considered again as a point in the unordered configuration space B_{0,2n−2}ℂ. And we can define a holomorphic map

π_{C} : (ℂ^{∗})^{n−1} × F_{0,n−1}ℂ − Δ → B_{0,2n−2}ℂ

by setting

π_{C}(a, b) = C_{a,b}.

Similarly as in the previous section, we can show the following

Theorem 4.1. For every point C in B_{0,2n−2}ℂ, there are at most

\frac{(2n − 2)!}{n!}

preimages of C by π_{C}.

The number

\frac{(2n − 2)!}{n! (n − 1)!}

is called the n-th Catalan number. And the theorem can be shown by noting the following result.

Lemma 4.2 ([4], [9]). For every fixed C in B_{0,2n−2}ℂ, there are

\frac{(2n − 2)!}{n! (n − 1)!}

classes of rational functions of degree n which have C as the set of critical points.

Here two rational functions f and g are in the same class if there is a Möbius transformation A such that

f = A ∘ g.

But in this case, a class as above contains a rational function of the form

f(z) = z + ∑_{j=1}^{n−1} \frac{a_{j}}{z − b_{j}}

only if the image of ∞ is different from all critical values. And the image π_{C}(B_{n}−Δ) seems to be misterious.

Example 4.3. If we take

\{±(\sqrt[4]{3}/\sqrt{2})(1 ± i)\}

as the set of critical points (with n = 3), we have 2 = 4!/(3! 2!) different classes determined by

f_{1} = f_{−1,−1,1,−1} (= f_{−1,−1,−1,1})

and

f_{2} = f_{1,1,i,−i} (= f_{1,1,−i,i}).

But since |f_{1}((\sqrt[4]{3}/\sqrt{2})(1 + i))| > 1 and |f_{2}((\sqrt[4]{3}/\sqrt{2})(1 + i))| > 1 for instance, all of (−1, −1, 1, −1), (−1, −1, −1, 1), (1, 1, i, −i), and (1, 1, −i, i) do not belong to B_{3}.

On the other hand, if we take

\{±\frac{\sqrt{7}}{10}, ±\frac{\sqrt{2}}{20}i\}

as the set of critical points (with n = 3), we have 2 different classes determined by

f_{1} = f_{a,a,b,−b} (= f_{a,a,−b,b}) with a = \frac{9}{400}, b = \frac{1}{10}

and

f_{2} = f_{a′,a′,b′,−b′} (= f_{a′,a′,−b′,b′}) with a′ = \frac{1}{48}, b′ = \frac{1}{10}\sqrt{\frac{7}{6}}.

We already mentioned that |f_{1}| < 1 at each critical point in Example 2.10. Also |f_{2}| < 1 at each critical point and hence all of (a, a, b, −b), (a, a, −b, b), (a′, a′, b′, −b′), and (a′, a′, −b′, b′) belong to B_{3}.

Next, the rational function

f(z) = z + \frac{6z − 4}{(z − 1)^{2}} ,

has \{1, −1, 2 + \sqrt{3}, 2 − \sqrt{3}\} as critical points. Here since f(∞) = f(1), the class of f can contain no rational function f_{a,b} with (a, b) belonging even to (C^{∗})^{2} ×F_{0,2}ℂ.

Note that there is a natural smooth map of B_{n} to the (reduced) moduli space M_{0,0,n} of a non-degenerate n-connected planar domain, which is a real 3n − 6 dimensional variety if n > 2. (Recall that two points (a, b) and (a′, b′) are mapped to the same point of M_{0,0,n} if there is a biholomorphic map F of W_{a,b} onto W_{a′,b′}.)

Hence the preimages of a generic point of M_{0,0,n}, which we have called a leaf of the coefficient body B_{n} in [7], has a positive real dimension. And Theorem 3.2 and Theorem 4.1 imply the following

Corollary 4.4. There is a non-trivial real parameter family F of Bell representations in a single leaf such that F is disjoint from Γ, and that π_{C} and π_{S} are injective on F.

References

1. S. Bell, Finitely generated function fields and complexity in potential theory in the plane, Duke Math. J., 98 (1999), 187–207.

2. S. Bell, A Riemann surface attached to domains in the plane and complexity in potential theory, Houston J. Math., 26 (2000), 277–297.

3. S. Birman, Braids, Links, and Mapping Class Groups, Ann. Math. Studeis, 82, Princeton, 1975.

4. L. R. Goldberg, Catalan numbers and branched coverings by the Riemann sphere, Adv. in Math. 85. (1991), 129–144.

5. I. P. Goulden and D. M. Jackson, Transitive factorisation into transpositions and holomorphic mappings on the sphere, Proc. AMS 125. (1997), 51–60.

6. M. Jeong and M. Taniguchi, Bell representation of finitely connected planar domains, Proc. AMS., 131 (2003), 2325–2328.

7. M. Jeong and M. Taniguchi, Algebraic kernel functions and representation of planar domains, J. Korean Math. Soc., 40 (2003), 447–460.

8. S. Natanzon, Hurwitz spaces, London Math. Soc. Lecture Note Ser., 287, (2001), 165–177.

9. I. Scherbak, Rational functions with prescribed critical points, Geom. funct. anal. 12, (2002), 1365–1380.

Department of Mathematics, The University of Suwon, Suwon P.O.Box 77, Kyungkido, 440-600, Korea

E-mail address: mjeong@suwon.ac.kr

Department of Mathematics, Graduate school of Science, Kyoto University, Kyoto 606, Japan

E-mail address: tanig@kusm.kyoto-u.ac.jp