The Hurwitz space of a Bell representation

Moonja Jeong and Masahiko Taniguchi

1 The coefficient body

We say that a bounded domain D in ℂ is an n-ply connected non-degenerate planar domain if the boundary of D in ℂ consists of n simple closed curves. Bell proposed in [1] a family of simple such domains, and we have shown in [4] that every non-degenerate n-ply connected planar domain with n > 1 is mapped biholomorphically onto a domain W_{a,b} proposed by Bell, which is defined by

W_{a,b} = \{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}). We call such a domain W_{a,b} a Bell representation of W. (Also see [5].)

Definition 1.1 Let B_{n} be the locus in ℂ^{2n−2} consisting of all points (a, b) such that the corresponding domains W_{a,b} are non-degenerate n-ply connected planar domains. We call this locus B_{n} the coefficient body of degree n.

Also 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 (of degree n).

It is obvious that B_{n} and B_{n}^{∗} are contained in the product space (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}

is the configuration space of n − 1 points in ℂ. And B_{n}^{∗} is a 2^{n−1}-sheeted smooth holomorphic covering space of B_{n}.

In the sequel, we assume that n > 2, since B_{2} and B_{2}^{∗} are explicitly known. Note that B_{n}^{∗} is circular, i.e. for every point (a, b) ∈ B_{n}^{∗} and every θ ∈ ℝ, e^{iθ}(a, b) ∈ B_{n}^{∗}. Also for every point (a, b) ∈ B_{n}^{∗} and every 0 < r ≤ 1, r(a, b) ∈ B_{n}^{∗}.

Theorem 1.1 ([6]) The modified coefficient body B_{n}^{∗} is a circular domain homeomorphic to B_{n}, and B_{n} has the same homotopy type as that of (ℂ^{∗})^{n−1}× F_{0,n−1}ℂ.

Remark The fundamental group of F_{0,n−1}ℂ is called the pure braid group, and its structure is well-known. See for instance [2].

2 The Hurwitz spaces

Now, rational functions can be parametrized also by the set of critical values.

Definition 2.1 Let Γ be the set of all points (a, b) of B_{n} such that the corresponding rational function

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

has either a non-simple critical point or has a pair of critical points whose images by f_{a,b} are the same. We call Γ the collision locus of B_{n}.

Then for every point (a, b) in B_{n} −Γ, the rational function f_{a,b} has 2n−2 critical values, which we denote by

S_{a,b} = {α_{1}, ⋯, α_{2n−2}}.

This set can be considered as a point in the unordered configuration space B_{0,2n−2}ℂ, the quotient space of F_{0,2n−2}ℂ by the symmetric group. Actually, we know that S_{a,b} is a point of the unordered configuration space B_{0,2n−2}U of 2n − 2 points on the unit disc U. Hence 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 2.1 ([6]) The projection π_{S} gives a (2n − 2)! n^{n−3}-sheeted proper holomorphic covering of B_{0,2n−2}U by B_{n} − Γ for every n > 2.

We can see this theorem by regarding B_{n} − Γ as a Hurwitz space. First we intruduce a conceptually natural kind of such spaces.

Let f_{1} and f_{2} be rational functions. We say that f_{1} and f_{2} determine the same covering structure if there are Möbius transformations γ and δ of ℂ̂ such that f_{2} = h ∘ f_{1} ∘ g^{−1}. We denote by C_{f} the covering structure determined by f, and call the set of all covering structures determined by rational functions g which are quasiconformally equivalent to f the prime Hurwitz space of f, which is denoted by H^{#}(f). Here we say that f and g are quasiconformally equivalent if there are quasiconformal self-maps φ and ψ of ℂ̂ such that g = ψ ∘ f ∘ φ^{−1}.

We introduce the prime Hurwitz distance d_{H^{#}} on H^{#}(f) by setting

d_{H^{#}}(C_{f_{1}}, C_{f_{2}}) = inf_{φ} log K(φ),

where the infimum is taken over all quasiconformal maps φ of ℂ̂ which satisfy ψ ∘ f_{2} ∘ φ^{−1} = f_{1} with suitable quasiconformal maps ψ of ℂ̂.

Proposition 2.1 d_{H^{#}} is actually a distance and complete on H^{#}(f).

Proof. If d_{H^{#}}(C_{f_{1}}, C_{f_{2}}) = 0, then there are sequences {ψ_{n}} and {φ_{n}} of quasiconformal maps of ℂ̂ fixing 0, 1, ∞ such that ψ_{n} ∘ f_{2} ∘ φ_{n}^{−1} ∈ C_{f_{1}} for every n, and K(φ_{n}) = K(ψ_{n}) tend to 1 as n tend to ∞. Then ψ_{n} and φ_{n} converge to Möbius transformations γ and δ, respectively, which shows that d_{H^{#}} is actually a distance.

Similarly, let {C_{f_{n}}} be a Cauchy sequence with respect to d_{H^{#}}. Then by a standard argument and taking subsequences if necessary, we can find sequences {ψ_{n}} and {φ_{n}} of quasiconformal maps of ℂ̂ fixing 0, 1, ∞ such that ψ_{n} ∘ f_{1} ∘ φ_{n}^{−1} ∈ C_{f_{n}} for every n, and quasiconformal maps ψ and φ of ℂ̂, such that f_{∞} = ψ ∘ f_{1} ∘ φ^{−1} is a rational function and K(φ_{n} ∘ φ^{−1}) and K(ψ_{n} ∘ ψ^{−1}) tend to 1 as n tend to ∞. This implies that C_{f_{n}} converges to C_{f_{∞}} with respect to d_{H^{#}}, which shows completeness.

But a finer equivalence relation is often considered.

Definition 2.2 We say that rational functions f_{1} and f_{2} determine the same isomorphism class if there is a Möbius transformation γ such that f_{2} = f_{1}∘γ. We call the set of all isomorphism classes I_{g} of rational functions g which are quasiconformally equivalent to f the Hurwitz space of f, and denote it by H(f).

Here, we distinguish the value ∞ and assume that ψ(∞) = ∞ for every quasiconformal map ψ appeared in quasiconformal equivalence relation g = ψ ∘ f ∘ φ^{−1}

Example 1 The set H_{0,n}[n] of genus 0 and degree n with type (n), consisting of all isomorphism classes of polynomials of degree n in general position, is the Hurwitz space H(f) for any such an f.

The set H_{0,n}[1^{n}] of genus 0 and degree n with type 1^{n} = (1, ⋯, 1), consisting of all isomorphism classes of rational functions of degree n in general position with n simple poles, is the Hurwitz space H(f) for any such an f.

Remark The coefficient representation for H(f) is not faithful in general. Also see [10].

We can define the normalized Hurwitz distance d_{H} on H(f) by setting

d_{H}(I_{f_{1}}, I_{f_{2}}) = inf_{φ} log K(φ),

where the infimum is taken over all quasiconformal maps φ of ℂ̂ satisfying f_{1} = ψ ∘ f_{2} ∘ φ^{−1} with quasiconformal maps ψ of ℂ̂ normalized as follows: Let n be the degree of f. Prescribe 2n + 1 distinct points in ℂ̂ including ∞ and assume that each ψ fixes three of them including ∞.

Then similarly as in the proof of the previous proposition, we have

Proposition 2.2 d_{H} is actually a distance and complete on H(f).

Remark Historically, the Hurwitz space of a rational function f is defined algebraically. Natanzon showed that such Hurwitz spaces are the same as the set Top(f) of all isomorphism classes of rational functions g which are topologically equivalent to f, i.e. there are self-homeomorphisms φ and ψ of ℂ̂ such that ψ(∞) = ∞ and g = ψ ∘ f ∘ φ^{−1} (cf. [7]).

Finally we consider the marked Hurwitz space MH_{0,n}[1^{n}] of all isomorphism classes of rational functions of degree n in general position with n ordered simple poles. Then another Hurwitz space MH_{n}U in MH_{0,n}[1^{n}], consisting of all isomorphism classes of those with critical values, all of which are contained in U, is a complete metric space with another normalized Hurwitz metric

d_{MH,U}(I_{f_{1}}, I_{f_{2}}) = inf_{φ} log K(φ),

where the infimum is taken over all quasiconformal maps φ of ℂ̂ preserving the order of poles and satisfying f_{1} = ψ ∘ f_{2} ∘ φ^{−1} with quasiconformal selfmaps ψ of ℂ̂ which satisfy ψ(U) = U and fix points ±1 and ∞.

Theorem 2.2 ([6]) B_{n} − Γ can be identified with MH_{n}U.

3 The synthetic Teichmüller spaces

To recover B_{n} from B_{n} − Γ, we need, for instance, to compactify the marked Hurwitz spaces MH_{0,n}[1^{n}].

Remark The Hurwitz spaces of a rational function can be compactified naturally. See [3] and [8]. We call these compactifications as the DENT compactification. But they have the singularities in general. And the boundary part of B_{n} − Γ corresponding to Γ in the DENT compactification of the marked Hurwitz space MH_{0,n}[1^{n}] may be different from Γ.

Definition 3.1 Let f = f_{a,b} correspond to a, b ∈ B_{n}. Then the full deformation set F D(f) of f is the set of all meromorphic functions g on ℂ such that there are quasiconformal self-maps φ of ℂ which fix 0 and 1, and satisfy the qc-L^{∞} condition:

D_{f}(g; φ) = ∥f − g ∘ φ∥_{∞} (= sup_{ℂ}|f − g ∘ φ|) < ∞.

For every pair of functions f_{1} and f_{2} in F D(f), we set

d(f_{1}, f_{2}) = inf (log K(φ_{1} ∘ φ_{2}^{−1}) + ∥f_{1} ∘ φ_{1} − f_{2} ∘ φ_{2}∥_{∞}) ,

where the infimum is taken over all quasiconformal maps φ_{1} and φ_{2} of ℂ which fix 0 and 1, and satisfy the qc-L^{∞} conditions D_{f}(f_{j}; φ_{j}) < ∞. This d is actually a distance, and F D(f) equipped with this distance is a complete metric space. We call the distance d defined above the synthetic Teichmüller distance on F D(f). The space F D(f) equipped with this synthetic Teichmüller distance is called the full synthetic deformation space of f and is denoted as F SD(f).

Remark The synthetic deformation space is firstly defined for an entire functions. See [9]. The results for structurally finite entire functions such as in [9] can be formulated and proved also for structurally finite meromorphic functions.

Theorem 3.1 For every f = f_{a,b} with (a, b) ∈ B_{n}, F SD(f) is the set Rat_{n}[1^{n}] consisting of all rational functions of degree n with simple poles which include ∞.

Proof. First, it is easy to see that every g ∈ Rat_{n}[1^{n}] belongs to F SD(f).

On the other hand, let g be a meromorphic function belonging to F SD(f). Take a quasiconformal map φ of ℂ such that D_{f}(g; φ) < +∞. Then

lim_{z→∞} g(z) = ∞,

i.e., g has a pole at ∞, and hence is a rational function.

Next for a sufficiently small ϵ > 0, take discs

D_{k} = {|z − b_{k}| < ϵ} (k = 1, ⋯, n − 1)

D_{n} = {|z| > 1/ϵ} ∪ {∞}.

Here we may assume that f(D_{k}) is disjoint from

{|z| ≤ 2D_{f}(g; φ)},

and that the winding number of the image f(C_{k}) around 0 is −1 for every k = 1, ⋯, n, where C_{k} is the boundary of D_{k}. Then the assumption implies that g(φ(D_{k})) is disjoint from {|z| ≤ D_{f}(g; φ)} and that the winding number of g(φ(C_{k})) around 0 is −1 for every k = 1, ⋯, n. Thus φ(D_{k}) contains no zeros and a single simple pole for every k, which implies that the degree of g is n.

Now for our purpose, we consider the marked full synthetic deformation space MF SD(f) of a given f = f_{a,b} with (a, b) ∈ B_{n}, by keeping the order of simple poles as before, which is again a complete metric space.

Corollary 1 MF SD(f) is the set MRat_{n}[1^{n}] of all rational functions of degree n with ordered simple poles which include ∞, and hence is independent of the choice of f.

The coefficient body B_{n} is a subset of MF SD(f), and the relative topology on B_{n} is the same as the relative topology induced from that of (ℂ^{∗})^{n−1} × F_{0,n−1}ℂ.

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

[3] S. Diaz, On the Natanzon-Turaev compactification of the Hurwitz space, Proc. Amer. Math. Soc. 130, (2001), 613–618.

[4] M. Jeong and M. Taniguchi, Bell representation of finitely connected planar domains, Proc. Amer. Math. Soc. 131, (2003), 2325–2328.

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

[6] M. Jeong and M. Taniguchi, The coefficient body of Bell representations of finitely connected planar domains, J. Math. Anal. Appl., 295 (2004) 620–632

[7] S. M. Natanzon, Hurwitz spaces, Topics on Riemann surfaces and Fuchsian Groups, London Math. Soc. Lecture Notes 287, (2001), 165– 177.

[8] S. M. Natanzon and V. Turaev, A compactification of the Hurwitz spaces, Topology 38, (1999), 889–914.

[9] M. Taniguchi, Synthetic deformation spaces of an entire function, Contemporary Math. 303, (2002), 107–136.

[10] M. Taniguchi, Covering structure and dynamical structure of a structurally finite entire function, Kodai Math. J., to appear.

Moonja JEONG  Masahiko TANIGUCHI
Department of Mathematics  Department of Mathematics
University of Suwon  Kyoto University
Hwasung-si, Kyunggi-do, KOREA Kyoto 606, JAPAN