THE COMPLEX-SYMPLECTIC GEOMETRY OF SL(2, ℂ)-CHARACTERS OVER SURFACES

WILLIAM M. GOLDMAN

Dedicated to M.S. Raghunathan on his sixtieth birthday

Abstract. The SL(2, ℂ)-character variety X of a closed surface M enjoys a natural complex-symplectic structure invariant under the mapping class group Γ of M. Using the ergodicity of Γ on the SU(2)-character variety, we deduce that every Γ-invariant meromorphic function on X is constant. The trace functions of closed curves on M determine regular functions which generate complex Hamiltonian flows. For simple closed curves, these complex Hamiltonian flows arise from holomorphic flows on the representation variety generalizing the Fenchel-Nielsen twist flows on Teichmüller space and the complex quakebend flows on quasi-Fuchsian space. Closed curves in the complex trajectories of these flows lift to paths in the deformation space ℂℙ^{1}(M) of complexprojective structures between different ℂℙ^{1}-structures with the same holonomy (grafting). If P is a pants decomposition, then the trace map τ_{P} : X ⟶ ℂ^{P} defines a holomorphic completely integrable system. Furthermore, if Γ_{P} is the subgroup of Γ preserving P, then every Γ_{P}-invariant holomorphic function X ⟶ ℂ factors through τ_{P}. This holomorphic integrable system is related to the complex Fenchel-Nielsen coordinates on quasi-Fuchsian space QF(M) developed by Tan and Kourouniotis, and relate to recent formulas of Platis and Series on complex-length functions and complex twist flows on QF(M).

Contents

Introduction 2 1. Representation varieties and character varieties 4 2. The Hamiltonian vector field of a character function 8 3. Abelian Hamiltonian Actions 20 4. Deformation spaces of ℂℙ^{1}-structures 23 References 32

Date: September 2, 2003. The author gratefully acknowledges partial support from National Science Foun-

dation grant DMS-0103889.

Introduction

The SL(2, ℂ)-character variety of a closed surface M with fundamental group π is the space of equivalence classes of representations ϕ of the fundamental group π of M into SL(2, ℂ). The Teichmüller space 𝔗(M), the moduli space X_{U} of irreducible flat unitary SU(2)- bundles, the deformation space ℂℙ^{1}(M) of ℂℙ^{1}-structures on M, and the quasi-Fuchsian space QF(M) all lie in the smooth stratum X of the SL(2, ℂ)-character variety. These spaces all share several common features: a symplectic geometry derived from the topology of M, and a compatible action of the mapping class group Γ of M. This paper investigates these structures in terms of the Γ-invariant complex-symplectic structure on X.

A complex-symplectic structure on a complex manifold is a nondegenerate closed holomorphic exterior 2-form. The SL(2, ℂ)-character variety is an affine variety defined over ℂ whose set of smooth ℂ-points is a complex-symplectic manifold of complex dimension −3χ(M) whose complex-symplectic structure is invariant under Γ.

In [46], Fenchel and Nielsen develop a set of coordinates for 𝔗(M) based on hyperbolic geometry. This set of coordinates is based on a pants decomposition P = {α_{1}, …, α_{N}} on M, that is a set of disjoint simple closed curves α_{i} cutting M into three-holed spheres (“pants”). In a hyperbolic structure on M, the curves α_{i} are represented by disjoint simple closed geodesics and their lengths define a function

l_{P} : 𝔗(M) ⟶ (ℝ_{+})^{N}

which consitute half of the Fenchel-Nielsen coordinates on 𝔗(M). According to Wolpert [48], these functions Poisson-commute and define a completely integrable Hamiltonian system. In the language of classical mechanics, these are the action variables, for which the Fenchel-Nielsen twist vector fields define angle variables. In particular l_{P} is the moment map for a Hamiltonian ℝ^{N}-action. Choosing a section σ to l_{P} determines a symplectomorphim

(ℝ_{+})^{N} × ℝ^{N} ⟶ 𝔗(M)
(λ; t) ⟼ ξ_{t}σ(λ)

where ξ_{t} is the Hamiltonian ℝ^{N}-action defined by l_{P}. See Wolpert [46, 47, 48] for details.

SL(2, ℂ)-CHARACTER VARIETIES 3

This picture motivated the study of a symplectic geometry of moduli spaces Hom(π, G)/G developed in [9, 11]. The results there extend directly to the complex-symplectic geometry of deformation spaces Hom(π, G)/G where G is a complex Lie group with an Ad-invariant complex-orthogonal structure 𝔹 on its Lie algebra. In this paper we consider only the special case G = SL(2, ℂ) where 𝔹 is the trace form. Corresponding to an element α ∈ π is the function f_{α} : X ⟶ ℂ associating to the equivalence class of ϕ the trace of ϕ(α). When α is represented by a simple closed curve A, then the complex-Hamiltonian vector field Ham(f_{α}) generates a flow which is covered by a complex twist flow on Hom(π, SL(2, ℂ)). We study this flow, computing its periods, and relating it to the action on X of the Dehn twist about A. Following [9, 11], we relate the complex-symplectic geometry and the Hamiltonian actions to the Fenchel-Nielsen twist flows on 𝔗(M) and the complex earthquakes and bending deformations on ℂℙ^{1}(M) and QF(M).

In the presence of a conformal structure on M, the moduli spaces Hom(π, G)/G admit stronger structures (a Kähler structure when G is compact, and hyper-Kähler when G is complex), but these stronger structures fail to be Γ-invariant. However the symplectic (and complexsymplectic) structures are Γ-invariant. The symplectic structure defines a Γ-invariant measure. By [15] and Pickrell-Xia [38], the resulting measure is ergodic under Γ when G is a compact Lie group. This has the following consequence for the holomorphic geometry when G is complex:

Theorem. There are no nonconstant Γ-invariant meromorphic functions on X.

The proof uses the inclusion of the set X_{U} of irreducible unitary characters in X, and the ergodicity of the action of Γ on X_{U} ([15]). A key idea in the proof is the action of the subgroup Γ_{P} preserving a pants decomposition P of M. The group Γ_{P} is a free abelian group freely generated by the Dehn twists about the curves in P. This ℤ^{N}- action lies in a Hamiltonian ℝ^{N}-action. The map which associates to [ϕ] the collection of traces

X_{U} ⟶ ℝ^{N}

is a moment map for the ℝ^{N}-action and is also the ergodic decomposition for the ℤ^{N}-action. The holomorphic analog is:

Theorem. Every Γ_{P}-invariant meromorphic function on X factors through the map

X ⟶ ℂ^{N}

which associates to a character its values on the curves in P.

The complex twist flows have been extensively studied in the quasi-Fuchsian space QF(M), which is the open subset of the SL(2, ℂ)- character variety comprising equivalence classes of quasi-Fuchsian representations. In this case, the complex twist flows correspond geometrically to quake-bending pleated surfaces in quasi-Fuchsian hyperbolic 3manifolds, that is, composing Fenchel-Nielsen twist flows (earthquakes) with bending deformations (Epstein-Marden [7]). These quakebends are defined more generally for geodesic laminations, although we only consider deformations supported on simple closed curves here. The complex Fenchel-Nielsen coordinates of Kourouniotis [32, 33, 34]) and Tan [43] are holomorphic Darboux coordinates for the complex-sym-plectic structure. We recover results of Platis [39] expressing the symplectic duality between the complex twist flows and the complex length functions, and the formula of Series [42] for the derivative of a complex length function under a twist flow in terms of the complex-symplectic geometry of X.

More generally the complex twist flows are defined for ℂℙ^{1}-structures, that is, geometric structures with coordinates modelled on ℂℙ^{1} with coordinate changes in PSL(2, ℂ). Let ℂℙ^{1}(M) denote the deformation space of ℂℙ^{1}-structures on M. Using the local biholomorphicity of the holonomy mapping

hol : ℂℙ^{1}(M) ⟶ Hom(π, SL(2, ℂ))//SL(2, ℂ).

one obtains complex-symplectic structures, complex length functions, and Hamiltonian complex twist flows on ℂℙ^{1}(M). The complex twist flows on ℂℙ^{1}-structures can be described geometrically by inserting annuli into a ℂℙ^{1}-manifold split along a simple closed curve which is locally circular. This is a special case of the grafting construction considered in Tanigawa [44] and McMullen [36]. In particular closed curves in the complex trajectory of a complex twist flow lift to paths between different ℂℙ^{1}-structures with the same holonomy (Maskit and Hejhal, see Goldman [12]). The holomorphic properties of the grafting construction are discussed in McMullen [36], Tanigawa [44] and Scannell-Wolf [41].

1. Representation varieties and character varieties

Let M be a closed oriented surface with fundamental group π. Let Hom(π, SL(2, ℂ)) denote the complex affine variety consisting of homomorphisms π ⟶ SL(2, ℂ). (It is irreducible, by Goldman [13]; see Benyash-Krivets -Chernousov -Rapinchuk [2] for stronger more general results, and also Li [35].)

The group SL(2, ℂ) acts by conjugation on the affine algebraic set Hom(π, SL(2, ℂ)). Let Hom(π, SL(2, ℂ))/SL(2, ℂ) denote the set of SL(2, ℂ)-orbits of Hom(π, SL(2, ℂ)). Denote the orbit of a representation ϕ by

[ϕ] ∈ Hom(π, SL(2, ℂ)).

1.1. Stable and semistable points. A homomorphism

ϕ ∈ Hom(π, SL(2, ℂ))

is irreducible if it leaves invariant no proper linear subspace of ℂ^{2}. Equivalently, ϕ is irreducible if the corresponding projective action fixes no point in ℂℙ^{1}. Irreducible homomorphisms are the stable points of Hom(π, SL(2, ℂ)) (with respect to the SL(2, ℂ)-action). The subset of Hom(π, SL(2, ℂ))^{s} ⊂ Hom(π, SL(2, ℂ)) consisting of irreducible homomorphisms is Zariski-open and nonsingular. Hom(π, SL(2, ℂ))^{s} is a complex manifold of complex dimension 6g − 3 (see [9]).

More generally, a homomorphism is reductive if every invariant subspace possesses an invariant complement. Equivalently, ϕ is reductive if it is either reducible or fixes a pair of distinct points on ℂℙ^{1} or is central. A central homomorphism maps into the center {±𝕀} of SL(2, ℂ) and acts trivially on ℂℙ^{1}. Letting Fix(ϕ) denote the subset of ℂℙ^{1} fixed by ϕ, a homomorphism ϕ is reductive if and only if Fix(ϕ) equals ∅, a pair of distinct points, or all of ℂℙ^{1}. Reductive homomorphisms are the semistable points of Hom(π, SL(2, ℂ)) (with respect to the SL(2, ℂ)- action),comprising the subset

Hom(π, SL(2, ℂ))^{ss} ⊂ Hom(π, SL(2, ℂ)).

A semistable (respectively stable) orbit is the orbit of a semistable (respectively stable) point.

Let Hom(π, SL(2, ℂ))//SL(2, ℂ) denote the set of ℂ-points of the categorical quotient of the SL(2, ℂ)-action on Hom(π, SL(2, ℂ)), that is, the variety whose coordinate ring is the ring of invariants of SL(2, ℂ) acting on the coordinate ring ℂ[Hom(π, SL(2, ℂ))] of Hom(π, SL(2, ℂ)). The natural map

Hom(π, SL(2, ℂ))/SL(2, ℂ) ⟶ Hom(π, SL(2, ℂ))//SL(2, ℂ)

is surjective, but not injective. However Hom(π, SL(2, ℂ))//SL(2, ℂ) may be identified with the set of semistable orbits. Thus the inclusion

Hom(π, SL(2, ℂ))^{ss} ↪ Hom(π, SL(2, ℂ))

induces a bijection

Hom(π, SL(2, ℂ))^{ss}/SL(2, ℂ) ⟷ Hom(π, SL(2, ℂ))//SL(2, ℂ).

The group SL(2, ℂ) acts freely and properly on Hom(π, SL(2, ℂ))^{s}. The quotient X := Hom(π, SL(2, ℂ))^{s}/SL(2, ℂ) is thus a (6g−6)-dimensional complex manifold which embeds as a Zariski open subset of the categorical quotient Hom(π, SL(2, ℂ))//SL(2, ℂ). Thus the map

X = Hom(π, SL(2, ℂ))^{s}/SL(2, ℂ) ⟶ Hom(π, SL(2, ℂ))//SL(2, ℂ)

induced by Hom(π, SL(2, ℂ))^{s} ↪ Hom(π, SL(2, ℂ)) is an embedding onto an open subset. Thus X is a smooth irreducible complex quasiaffine variety which is dense in the quotient Hom(π, SL(2, ℂ))//SL(2, ℂ).

1.2. Symplectic geometry of deformation spaces. By the general construction of [9], X has a natural complex-symplectic structure Ω, which is Γ-invariant. Furthermore Γ is algebraic in the sense that there exists an algebraic tensor field on Hom(π, SL(2, ℂ)) inducing Ω. (See [9] for an explicit formula.)

The Zariski tangent space T_{ϕ}Hom(π, SL(2, ℂ)) is the space

Z^{1}(π, 𝔰𝔩(2, ℂ)_{Adϕ})

of 1-cocycles π ⟶ 𝔰𝔩(2, ℂ)_{Adϕ} and the tangent space T_{ϕ}(G · ϕ) of the G-orbit equals the subspace

B^{1}(π, 𝔰𝔩(2, ℂ)_{Adϕ}) ⊂ Z^{1}(π, 𝔰𝔩(2, ℂ)_{Adϕ})

of 1-coboundaries. (Compare Raghunathan [40].) The quotient vector space is the cohomology

H^{1}(π, 𝔰𝔩(2, ℂ)_{Adϕ})

which, under the de Rham isomorphism is isomorphic to the cohomology H^{1}(M; V_{ϕ}) where V_{ϕ} denotes the flat vector bundle over M corresponding to the π-module 𝔰𝔩(2, ℂ)_{Adϕ}.

Let 𝔹 : 𝔰𝔩(2, ℂ) × 𝔰𝔩(2, ℂ) ⟶ ℂ be the trace form of the standard representation on ℂ^{2}: if α, β ∈ 𝔰𝔩(2, ℂ), then the inner product is defined as

𝔹(α, β) := tr(αβ).

Since 𝔹 is Ad-invariant, it defines a bilinear pairing of π-modules

𝔰𝔩(2, ℂ)_{Adϕ} × 𝔰𝔩(2, ℂ)_{Adϕ} ⟶ ℂ

or equivalently flat vector bundles

V_{ϕ} × V_{ϕ} ⟶ ℂ.

Cup-product defines a bilinear pairing

Ω_{ϕ} : H^{1}(M; V_{ϕ}) × H^{1}(M; V_{ϕ}) ⟶ ℂ

with coefficients paired by 𝔹. Symmetry of 𝔹 implies that Ω_{ϕ} is skew-symmetric. Since 𝔹 is nondegenerate, Ω_{ϕ} is nondegenerate. It follows from [9] that Ω_{ϕ} can be expressed as an algebraic tensor on Hom(π, SL(2, ℂ)), and thus is a holomorphic exterior 2-form. By arguments of [9] or Karshon [29], Weinstein [45], Guruprasad-Huebschmann-Jeffrey-Weinstein [19], Ω is closed. Thus Ω is a nondegenerate closed holomorphic (2, 0)-form, that is a complex-symplectic structure.

1.3. The mapping class group. The automorphism group Aut(π) of π acts algebraically on Hom(π, SL(2, ℂ)) commuting with the action of SL(2, ℂ). The normal subgroup Inn(π) of inner automorphisms acts trivially on the quotient Hom(π, SL(2, ℂ))//SL(2, ℂ). The quotient Out(π) = Aut(π)/Inn(π), acts on Hom(π, SL(2, ℂ))//SL(2, ℂ), leaving invariant the subset X. Furthermore the mapping class group Γ := π_{0}(Diff^{+}(M)) of M is isomorphic to Out(π) by Nielsen [37]. The algebraic complex-symplectic structure on X is Γ-invariant.

1.4. Ergodicity and its holomorphic analog. The subset X_{U} of unitary characters is invariant under Γ. Furthermore the complexsymplectic structure restricts to a (real) symplectic structure on X_{U} which is the Kähler form for a Kähler structure on X_{U}. In particular the Lebesgue measure class on X_{U} is invariant under Γ. The main result of [15] is that this action is ergodic, that is every Γ-invariant measurable function on X_{U} is constant almost everywhere.

Ergodicity no longer holds for the SL(2, ℂ)-character variety X (see §4.3). However ergodicity on X_{U} does imply the following property of holomorphic functions on X:

Theorem 1.4.1. A Γ-invariant meromorphic function X ⟶^{h} ℂℙ^{1} is constant.

Proof. The restriction of h to X_{U} is a Γ-invariant measurable function, and by the main result of Goldman [15], h must be constant almost everywhere. Since h is continuous, it is constant.

Now we argue that in local holomorphic coordinates, X_{U} is equivalent to ℝ^{n} ⊂ ℂ^{n} and a holomorphic function constant on X_{U} must be globally constant on X. (Compare, for example, Lemma 1 of §2.3 of Platis [39]). For the reader’s convenience, we supply a brief proof. Let U be a nonempty open coordinate neighborhood with local holomorphic coordinates z = (z^{1}, …, z^{n}). In local holomorphic coordinates, X_{U} ∩ U is described by

z ∈ ℝ^{n} ⊂ ℂ^{n}

and h is given by a power series

h(z) = ∑_{k=0}^{∞} a_{k}z^{k}

which converges in the nonempty open set z(U) ∈ ℂ^{n}. Since the restriction of h to a z(U) ∩ ℝ^{n} is constant, a_{k} = 0 for k > 0 and thus h is constant on U. Since X is connected, analytic continuation implies that h must be constant. □

We do not know whether X − X_{U} admits Γ-invariant nonconstant meromorphic functions.

2. The Hamiltonian vector field of a character function

Corresponding to free homotopy classes α of closed curves on M are complex regular functions f_{α} : X ⟶ ℂ. The complex-symplectic structure associates to these functions complex Hamiltonian vector fields Ham(f_{α}), which generate holomorphic local flows on X. When α corresponds to a simple closed curve, we define complex twist flows on Hom(π, SL(2, ℂ)) which cover these holomorphic local flows on X. We begin with an preliminary section on the traces in SL(2, ℂ).

2.1. The variation of the trace function on SL(2, ℂ). Let

f : SL(2, ℂ) ⟶ ℂ

be the trace function f(P) = tr(P) and 𝔹 the trace form

𝔰𝔩(2, ℂ) × 𝔰𝔩(2, ℂ) ⟶ ℂ
(X, Y) ⟼ tr(XY).

As in Goldman [11] the differential of f and the orthogonal structure 𝔹 determines a variation function

F : SL(2, ℂ) ⟶ 𝔰𝔩(2, ℂ)

characterized by the identity

\frac{d}{dt}|_{t=0}f(P exp(tX)) = 𝔹(F (P), X).

This function is defined by:

F (P) = P − \frac{tr(P)}{2}𝕀

and corresponds to the composition of the inclusion

SL(2, ℂ) ↪ 𝔤𝔩(2, ℂ)

with orthogonal projection

𝔤𝔩(2, ℂ) ↪ 𝔰𝔩(2, ℂ)

(orthogonal with respect to the trace form 𝔹 on 𝔰𝔩(2, ℂ)). Invariance of the trace

f(QP Q^{−1}) = f(P)

and Ad-invariance of the orthogonal structure 𝔹

𝔹(Ad(Q)X, Ad(Q)Y) = 𝔹(X, Y)

implies Ad-equivariance of its variation:

(2.1)  F (QP Q^{−1}) = Ad(Q)F (P).

Taking Q = P shows that F (P) lies in the centralizer of P in 𝔰𝔩(2, ℂ). In particular the complex one-parameter subgroup of SL(2, ℂ)

(2.2)  ζ_{t} := exp (tF (P))

centralizes A in SL(2, ℂ).

For example, if P is the diagonal matrix

[\begin{matrix}λ & 0 \\ 0 & λ^{−1} \end{matrix}]

then f(P) = λ + λ^{−1} and

F (P) = \frac{λ − λ^{−1}}{2} [\begin{matrix}1 & 0 \\ 0 & −1 \end{matrix}]

with corresponding complex one-parameter subgroup:

(2.3)  ζ_{t} = [\begin{matrix}e^{t(λ−λ^{−1})/2} & 0 \\ 0 & e^{−t(λ−λ^{−1})/2} \end{matrix}] .

If P is ± the unipotent matrix

[\begin{matrix}1 & 1 \\ 0 & 1 \end{matrix}]

then f(P) = ±2 and

F (P) = [\begin{matrix}0 & 1 \\ 0 & 0 \end{matrix}]

with corresponding complex one-parameter subgroup:

(2.4)  ζ_{t} = [\begin{matrix}1 & t \\ 0 & 1 \end{matrix}]

The following lemma, whose proof is immediate from the above calculations, will be needed in the sequel. Recall that for any subset S ⊂ SL(2, ℂ), Fix(S) denotes the subset of ℂℙ^{1} fixed by S.

Lemma 2.1.1. If P ∈ SL(2, ℂ), then Fix(ζ_{t}) = Fix(P) or ζ_{t} = ±𝕀 2.2. Character functions. Let α ∈ π. The character function

Hom(π, SL(2, ℂ)) ⟶^{f̃_{α}} ℂ
ϕ  ⟼ f(ϕ(α))

is a regular function. Since f is a class function, f̃_{α} is invariant under SL(2, ℂ) and defines a regular function

f_{α} : X ⟶ ℂ.

Let [ϕ] ∈ X. Then the tangent space to X equals H^{1}(M; V_{ϕ}), so the Hamiltonian vector field Ham(f_{α}) associates to [ϕ] ∈ X a tangent vector in H^{1}(M; V_{ϕ}).

Cap product with the fundamental homology class [M] ∈ H_{2}(M; ℂ) defines the Poincaré duality isomorphism:

∩[M] : H^{1}(M; V_{ϕ}) ⟶^{≅} H_{1}(M; V_{ϕ}).

Choose a basepoint x_{0} ∈ M, an isomorphism of the fiber of V_{ϕ} over x_{0} with ℂ^{2}, and a representative holonomy homomorphism

ϕ_{0} : π_{1}(M; x_{0}) ⟶ SL(2, ℂ).

Let s_{0} ∈ S^{1} be a basepoint. Let α_{0} : (S^{1}, s_{0}) ⟶ (M, x_{0}) be a based loop in M corresponding to α. Let σ be the parallel section of the flat vector bundle α_{0}^{∗}V_{ϕ} over S^{1} which equals F (ϕ_{0}(α_{0})) at s_{0}. Then σ defines a V_{ϕ}-valued 1-cycle in M, with homology class

[σ] ∈ H_{1}(M; V_{ϕ}).

Lemma 2.2.1. The value of the Hamiltonian vector field Ham(f_{α}) at a point [ϕ] ∈ X is the vector in

T_{[ϕ]}X ≅ H^{1}(M; V_{ϕ})

corresponding to the Poincaré dual of the homology class of the V_{ϕ}- valued cycle σ:

∩[M] : H^{1}(M; V_{ϕ}) ⟶ H_{1}(M; V_{ϕ})
Ham(f_{α}) ⟼ [σ].

Using the above formula and the duality between cup-product and intersections of cycles, we obtain a formula for the Poisson bracket of trace functions ([11]):

Proposition 2.2.2. Let α, β be oriented closed curves meeting transversely in double points p_{1}, …, p_{k}. For each p_{i}, choose representatives

ϕ_{i} : π_{1}(M; p_{i}) ⟶ SL(2, ℂ).

Let α_{i} and β_{i} be the elements of π_{1}(M; p_{i}) representing α, β respectively. Then the Poisson bracket of functions f_{α}, f_{β} is:

{f_{α}, f_{β}} = Ω(Ham(f_{α}), Ham(f_{β}))

= ∑_{i=1}^{k} ϵ(p_{i}; α, β) 𝔹(F (ϕ_{i}(α_{i})), F (ϕ_{i}(β_{i})))

where ϵ(p_{i}; α, β) denotes oriented intersection number.

Corollary 2.2.3. If α, β are disjoint, then f_{α} and f_{β} Poisson-commute.

By arguments in [11] (based on a suggestion of S. Wolpert), the converse holds if one of α or β is simple. A purely topological proof of this fact has recently been given by Chas [5].

2.3. Complex twist flows. Suppose that α is represented by a simple closed curve A. Then a holomorphic ℂ-action on the representation variety Hom(π, SL(2, ℂ)) covers the complex Hamiltonian flow on X. The complex twist flow τ_{α} is the holomorphic action of ℂ of Hom(π, SL(2, ℂ)) defined as follows.

There are two cases, depending on whether A separates M or not. Denote by M|A the compact surface with boundary whose interior is homeomorphic to the complement M −A. Denote the two components of ∂(M|A) by A_{+} and A_{−}. The quotient map q : M|A ⟶ A results from identifying these components by a homeomorphism η : A_{+} ⟶ A_{−}.

2.3.1. The nonseparating case. If A is nonseparating, then M|A is connected, and choosing a basepoint in A_{+} we express π as an HNNextension

π_{1}(M|A) ⋆_{ι}

as follows. Choose a basepoint x_{0} ∈ A, and lift it to a basepoint x̃_{0} ∈ A_{+} ⊂ M|A. Let α_{+} denote the element of π_{1}(M|A; x̃_{0}) corresponding to A_{+}. Let β̃ denote a simple arc joining x̃_{0} to η(x̃_{0}) in M|A, and α_{−} the based loop β̃^{−1} · α_{+} · β̃ at the basepoint x̃_{0}. Then η corresponds to the isomorphism

ι : ⟨α_{+}⟩ ⟶ ⟨α_{−}⟩

between the two subgroups of π_{1}(M|A). Let N be the normal closure of the element

βα_{+}β^{−1}(α_{−})^{−1}

of π_{1}(M|A) ⋆ ⟨β⟩. Then the HNN-extension π_{1}(M|A)⋆_{ι} is defined as the quotient

π ≅ (π_{1}(M|A) ⋆ ⟨β⟩)/N.

As in [11], for any ϕ ∈ Hom(π, SL(2, ℂ)), and t ∈ ℂ, define

(2.5)  ξ_{t}(ϕ) : γ ⟼ \begin{cases}ϕ(γ) & if γ ∈ π_{1}(M|A) \\ ϕ(β) ζ_{t} & if γ = β. \end{cases}

where

ζ_{t} = exp (tF (ϕ(α_{+})))

is the one-parameter subgroup corresponding to ϕ(α_{+}).

2.3.2. The separating case. If A separates M, then let M_{+} and M_{−} denote the two components of M|A. Then π is the free product of subgroups corresponding to π_{+}(M_{+}) and π_{1}(M_{−}) respectively, amalgamated over the images ⟨α⟩ ⟶ π_{1}(M_{±}). Then as in [11], for any ϕ ∈ Hom(π, SL(2, ℂ)), t ∈ ℂ, define

ξ_{t}^{α}(ϕ) : γ ⟼ \begin{cases}ϕ(γ) & if γ ∈ π_{1}(M_{+}) \\ ζ_{t} ϕ(γ) ζ_{−t} & if γ ∈ π_{1}(M_{−}) \end{cases}

where

ζ_{t} = exp (tF (ϕ(α))

is the one-parameter subgroup corresponding to ϕ(α).

2.3.3. Irreducibility. To show that the Hamiltonian flows act on X, we need to show that the complex twist flows preserve the set of irreducible representations.

Lemma 2.3.1. Suppose that ϕ ∈ Hom(π, SL(2, ℂ))^{s}. Then

ξ_{t}^{α}(ϕ) ∈ Hom(π, SL(2, ℂ))^{s}.

Proof. Let M′ be a component of M|A. As above, choose basepoints and identifications to express π as an amalgamated free product or HNN construction with π_{1}(M′) ↪ π_{1}(M). Then

ξ_{t}^{α}ϕ(π) ⊃ ϕ(π_{1}(M′))

implies that if ϕ_{π_{1}(M′)} is irreducible, then so is ξ_{t}^{α}ϕ. Thus we may assume that ϕ_{π_{1}(M′)} is reducible but ϕ is irreducible.

Suppose first that A is nonseparating. We assume ξ_{t}^{α}ϕ is reducible and derive a contradiction. Let Φ = Fix(ϕ(π_{1}(M|A))). Since

ϕ(α_{+}) ∈ ϕ(π_{1}(M|A)),

the element ϕ(α_{+}) fixes Φ, and Lemma 2.1.1 implies that ζ_{t} fixes Φ. Since ξ_{t}^{α}ϕ(π) is reducible and is generated by ϕ(π_{1}(M|A)) and ϕ(β), it follows that

ξ_{t}^{α}ϕ(β) = ϕ(β)ζ_{t}

fixes no element of Φ. Since ζ_{t} fixes Φ, this implies that ϕ(β) fixes Φ, a contradiction.

Suppose finally that A separates M into two components M_{±}. By the remark above, we may assume that ϕ_{π_{1}(M_{±})} is reducible. Let

Φ_{±} = Fix(ϕ(π_{1}(M_{±}))).

We may assume that each Φ_{±} is nonempty but Φ_{+} ∩Φ_{−} = ∅. Now ϕ(α) fixes each Π_{±}, and Lemma 2.1.1 implies that ζ_{t} fixes each Π_{±}. Since ϕ(π_{1}(M_{+})) and ζ_{t}ϕ(π_{1}(M_{−}))ζ_{t}^{−1} generate ξ_{t}^{α}ϕ(π),

Fix(ξ_{t}^{α}ϕ(π)) = Fix(ϕ(π_{1}(M_{+}))) ∩ Fix(ζ_{t}ϕ(π_{1}(M_{−}))ζ_{t}^{−1})
= Fix(ϕ(π_{1}(M_{+}))) ∩ ζ_{t}Fix(ϕ(π_{1}(M_{−}))))
= Φ_{+} ∩ ζ_{t}Φ_{−}
= Φ_{+} ∩ Φ_{−} = ∅

as desired. □

2.4. Gauge-Theoretic Interpretation. These twist flows admit an interpretation in terms of flat connections. The character variety X identifies with the quotient of the space F^{irr}(E) of irreducible flat SL(2, ℂ)-connections on a principal SL(2, ℂ)-bundle E over M by the group G(E) of gauge transformations of E. (Since a principal SL(2, ℂ)- bundle over M which admits a flat connection is necessarily trivial, we may assume E is the product bundle.)

Let A ⊂ M be a simple closed curve, and choose a basepoint a_{0} ∈ A. Choose an orientation on A and let α be a based loop on M corresponding to the orientation on A.

Pull E back to a principal SL(2, ℂ)-bundle q^{∗}E over M|A by the quotient map q : M ⟶ M|A. Choose a point e_{0} in the fiber of E over a_{0}. Let a_{±} be the two elements of q^{−1}(a_{0}) in A_{±} respectively and e_{±} the elements in the fibers of q^{∗}E over a_{±} corresponding to e_{0}.

Define the twist flow ξ̃_{t} on F^{irr}(E) as follows. Let ∇ ∈ F^{irr}(E) be a flat connection on E. Parallel transport of e_{0} along α with respect to α defines a holonomy transformation ϕ_{0}(α_{0}) as above. Let ζ_{t} denote the corresponding one-parameter subgroup of SL(2, ℂ). There is a oneparameter family of gauge-transformations g_{t} of q^{∗}E supported in a collar neighborhood N of A_{+} ⊂ M|A assuming the “value” ζ_{t} on A_{+}.

Explicitly, choose a smooth embedding

ψ : [0, 1] × S^{1} ⟶ N ↪ M|A

mapping {1} × S^{1} diffeomorphically to A_{+}. Let r : [0, 1] ⟶ S^{1} denote the quotient mapping identifying 0, 1 ∈ [0, 1]. Parallel transport of e_{+} defines a trivialization of the pullback of E to [0, 1] × [0, 1] so that ψ^{∗}E identifies with the quotient of

[0, 1] × [0, 1] × SL(2, ℂ)

by the identification

(s, 0, h) ⟷ (s, 1, ϕ_{0}(α_{0})h).

Define g_{t} as the gauge transformation which is the identity map on the complement of N in M|A, and equals

(s, θ, h) ⟶ (s, θ, ζ_{s,t}h)

in this trivialization.

Since ζ_{t} centralizes the holonomy of q^{∗}∇ along A_{+}, the identification map η : A_{−} ⟶ A_{+} identifies the restrictions of (g_{t})^{∗}(q^{∗}∇) to A_{±}. Thus a unique flat connection ξ̃_{t}(∇) exists, satisfying

q^{∗}(ξ̃_{t}(∇)) = (g_{t})^{∗}(q^{∗}∇).

This is the orbit of the twist flow on flat connections. Clearly the gauge transformation g_{t} does not arise from a gauge transformation of E, unless the holonomy along A is ±𝕀. The orbit covers the orbit of the holonomy of ∇ under the twist flow in X ≅ F^{irr}(E)/G(E).

The flow ξ̃ on F^{irr}(E) depends only on the choice of the collar neighborhood N ⊂ M|A and ψ : [0, 1] × S^{1} ⟶ N.

2.5. Periods. The subspace Hom(π, SU(2))^{s} maps to a (real) symplectic submanifold X_{U} ⊂ X, and the corresponding real flows define Hamiltonian systems which have been studied in [11], and are closely related to periodic flows studied by Jeffrey-Weitsman [24, 25, 26, 27]. The orbits of these flows are all closed, although the period of the orbit varies with the value of f.

Proposition 2.5.1. Let [ϕ] ∈ X_{U} and let α ∈ π be represented by a simple loop A. If f_{α}([ϕ] ≠ ±2, the period of the trajectory of the flow of Ham(f_{α}) at [ϕ] equals

\frac{4π}{\sqrt{4 − f_{α}([ϕ])^{2}}}

if A is nonseparating and

\frac{2π}{\sqrt{4 − f_{α}([ϕ])^{2}}}

if A separates. If f_{α}([ϕ]) = ±2, then [ϕ] is fixed under the flow of Ham(f_{α}).

Proof. Let P ∈ SU(2). Apply an inner automorphism of SU(2) to assume that P is diagonal:

(2.6)  P = [\begin{matrix}e^{iθ} & 0 \\ 0 & e^{−iθ} \end{matrix}] .

Then

f(P) = 2 cos(θ), F (P) = [\begin{matrix}i sin(θ) & 0 \\ 0 & −i sin(θ) \end{matrix}]

with corresponding one-parameter subgroup

ζ_{t} = exp(tF (P)) = [\begin{matrix}e^{i sin(θ)t} & 0 \\ 0 & e^{−i sin(θ)t} \end{matrix}]

Since the PSL(2, ℂ)-action on Hom(π, SL(2, ℂ))^{s} is proper and free, the quotient map is a principal PSL(2, ℂ)-fibration:

PSL(2, ℂ) ⟶ Hom(π, SL(2, ℂ))^{s} ⟶ X.

Thus the trajectory of the flow ξ_{t}^{α} projects diffeomorphically to the trajectory of the flow of Ham(f_{α}) on X. The period equals the infimum of all t > 0 such that

ξ_{t}^{α}(ϕ) = ϕ.

If α is nonseparating, then (2.5) implies that T equals the infimum of all t > 0 such that

ζ_{t}ϕ(β) = ϕ(β).

By (2.2),

T = \frac{2π}{sin(θ)} = \frac{4π}{\sqrt{4 − f_{α}([ϕ])^{2}}}

If α separates, then (2.6) implies that T equals the infimum of all t > 0 such that

ζ_{t}ϕ(β)ζ_{−t} = ϕ(β).

for all β ∈ π_{1}(M_{−}). If

ϕ(β) ∈ {±𝕀} = center(SU(2))

for all β ∈ π_{1}(M_{−}), then ϕ(α) = 1. Since ϕ ∈ Hom(π, SL(2, ℂ))^{s}, there exists β ∈ π such that ϕ(β) ≠ 1. The period T is the infimum of all t > 0 such that ζ_{t} = ±𝕀. By (2.2),

T = \frac{π}{sin(θ)} = \frac{2π}{\sqrt{4 − f_{α}([ϕ])^{2}}}

□

2.6. Dehn twists. Closely related to the twist flows are the maps of X_{U} induced by Dehn twists. Recall that the Dehn twist about a simple loop A ⊂ M is the diffeomorphism supported in a tubular neighborhood of A. If

ψ : [0, 1] × S^{1} ↪ M
\{\frac{1}{2}\} × S^{1} ↪ A

is such a tubular neighborhood, then the Dehn twist τ_{A} about A is ψrelated to the diffeomorphism of [0, 1] × S^{1} (restricting to the identity on the boundary) defined by:

(s, e^{iθ}) ⟼ (s, e^{2πis}e^{iθ}).

The action of τ_{A} on the fundamental group is given by:

(τ_{A})_{∗} (ϕ) : γ ⟼ \begin{cases}ϕ(γ) & if γ ∈ π_{1}(M|A) \\ ϕ(β) ϕ(α) & if γ = β. \end{cases}

if A is nonseparating, and by:

(τ_{A})_{∗} (ϕ) : γ ⟼ \begin{cases}ϕ(γ) & if γ ∈ π_{1}(M_{+}) \\ ϕ(α)ϕ(γ)ϕ(α)^{−1} & if γ ∈ π_{1}(M_{−}) \end{cases}

if A separates.

Let P be as in (2.6). Comparing (2.6) with (2.2), ζ_{t} = P for

θ = sin(θ)t,

that is,

(2.7)  t = \frac{θ}{sin(θ)} = \frac{cos^{−1} (f(P)/2)}{2π\sqrt{4 − f(P)^{2}}}.

Combining (2.7), (2.7) with (2.7) implies:

Proposition 2.6.1. Let α ∈ π correspond to a simple closed curve A ⊂ M. The map (τ_{A})_{∗} on Hom(π, SL(2, ℂ)) induced by Dehn twist about A equals the time-t map of the flow generated by Ham(f_{α}), where

t = \frac{2 cos^{−1}(f_{α}([ϕ])/2)}{\sqrt{4 − f_{α}([ϕ])^{2}}}.

In terms of a parametrization ℝ/ℤ ⟶ X_{U} of this trajectory, (τ_{A})_{∗} acts by translation of

\frac{θ}{2π} = \frac{cos^{−1} (f(P)/2)}{2π}

which has infinite order for almost every value of f(A).

By reparametrizing the flow, we obtain Hamiltonian flows whose time-one map is the identity map or the Dehn twist. Jeffrey-Weitsman [25] (§5.1), consider flows of Hamiltonians given by invariant functions

θ(P) = cos^{−1} (\frac{f(P)}{2})

which define S^{1}-actions, but are undefined at P = ±𝕀. Thus their flows are only defined on the on the dense open subset where ϕ(P) is not central. On the other hand their flows are periodic with period 2π if A is nonseparating, and period π if A separates. (Actually they work with the symplectic form which is 1/(4π^{2}) of ours, to obtain a 2-form with integral cohomology class. Thus their periods are 1/(2π) if A is nonseparating and 1/(4π) if A separates.)

2.6.1. Orbits of complex twist flows. On the SL(2, ℂ)-character variety X, the Hamiltonian vector field Ham(f_{α}) generates a holomorphic ℂaction. (2.4) implies that if f(A) = ±2 and A ≠ ±𝕀 (that is, A is ±-unipotent), then the trajectory defines an injective holomorphic map

(2.8)  ℂ ⟶ X
t ⟼ ξ_{t}^{α}([ϕ]).

If f(A) ≠ ±2, then (2.3) implies that ζ_{t} = 1 whenever t ∈ t_{0}ℤ where

(2.9)  t_{0} := \frac{4πi}{λ − λ^{−1}} = \frac{4πi}{(f(A)^{2} − 4)^{1/2}}

(which is well-defined only up to sign). In this case the trajectory of [ϕ] is the image of the holomorphic embedding

(2.10)  ℂ^{∗} ⟶ X
e^{z} ⟼ ξ_{(2z / (λ−λ^{−1}))}^{α} ([ϕ]).

2.6.2. Dehn twists. At a point [ϕ] ∈ X where f_{α}(ϕ) = ±2, then either

• ϕ(α) = ±𝕀 is a central element and [ϕ] is fixed under the entire ℂ-action, or

• ϕ(α) is ± a parabolic element and (2.4) implies ϕ(α) = ± exp F (α). In the latter case the embedding (2.8) is equivariant with respect to the action of ℤ by translation on ℂ and the ℤ-action generated by (τ_{A})_{∗} on X.

Suppose that f_{α}([ϕ]) ≠ ±2. In that case ζ_{t} = P precisely when t ≡ t_{1}( mod t_{0}), where

t_{1} = \frac{log(λ)}{λ − λ^{−1}} = \frac{log ((f(A) ± (f(A)^{2} − 4)^{1/2})/2)}{(f(A)^{2} − 2)^{1/2}}.

(The two choices for (f(A)^{2} − 4)^{1/2} differ by sign but determine equal values for t_{1}.) In that case the embedding (2.10) is equivariant with respect to the actions generated by multiplication by

λ = \frac{(f(A) ± (f(A)^{2} − 4)^{1/2})}{2}

on ℂ^{∗} and by (τ_{A})_{∗} on X.

2.6.3. Fenchel-Nielsen twist flows for G = SL(2, ℝ). Since the complex trace form 𝔹 on 𝔰𝔩(2, ℂ) restricts to the trace form on 𝔰𝔩(2, ℝ), the complex-symplectic structure Ω on X restricts to the symplectic structure on 𝔗(M) defined by the trace form of SL(2, ℝ). By [9] this symplectic structure equals (-2) the Weil-Petersson Kähler form.

Inside the complex twist flows are the Fenchel-Nielsen twist flows when the holonomy is hyperbolic. Suppose that ϕ ∈ Hom(π, SL(2, ℝ)) is a discrete embedding (that is, a Fuchsian representation). The subset of X consisting of equivalence classes of Fuchsian representations corresponds bijectively to the space of marked hyperbolic structures on M, that is, the Teichmüller space 𝔗(M). In that case, for every 𝕀 ≠ α ∈ π, the element ϕ(α) ∈ SL(2, ℝ) is hyperbolic. Geometrically it is a transvection along a geodesic γ, and is conjugate to a diagonal matrix

P = ± [\begin{matrix}e^{l/2} & 0 \\ 0 & e^{−l/2} \end{matrix}] .

l is the distance ϕ(α) moves points along γ. In the hyperbolic structure on M, the element α corresponds to a homotopy class of closed loops; γ corresponds to the unique closed geodesic A freely homotopic to a free loop in α and l is the length of this geodesic.

The character function is related to the length function by:

f = 2 cosh(l/2).

Thus the invariant function l (which is only defined on the subset of SL(2, ℝ) consisting of hyperbolic elements) defines functions

l_{α} : 𝔗(M) ⟶ ℝ_{+}.

As in [11], the corresponding variation is

L(P) = ± [\begin{matrix}1 & 0 \\ 0 & −1 \end{matrix}] .

with corresponding one-parameter subgroup

ζ_{t} = [\begin{matrix}e^{t} & 0 \\ 0 & e^{−t} \end{matrix}] ,

the one-parameter subgroup of transvections moving at speed 2 along the geodesic.

The inner product of these infinitesimal transvections can be computed as follows. The matrices

L_{1} = [\begin{matrix}0 & 1 \\ 1 & 0 \end{matrix}] , L_{2} = [\begin{matrix}− sin(ψ) & cos(ψ) \\ cos(ψ) & sin(ψ) \end{matrix}]

represent infinitesimal transvections about axes which intersect with angle ψ and

𝔹(L_{1}, L_{2}) = tr(L_{1}L_{2}) = 2 cos(ψ).

Thus (as in §3.9 of [11]), Wolpert’s cosine formula 2.6.2, follows from Proposition 2.2.2.

If α corresponds to a simple closed curve A, then the corresponding twist flow η_{t}^{α} corresponds to the Fenchel-Nielsen twist flow on 𝔗(M) defined geometrically as follows. (Compare Abikoff [1], Buser [4] and Harvey [20].) Represent A by the (necessarily simple) closed geodesic. Split M along A, and identify the boundary components A_{+}, A_{−} of M|A by a translation of length t in the positively oriented direction. This is a well-defined path of marked hyperbolic structures in 𝔗(M); see Wolpert [46] for details. In [46], Wolpert proves that the Fenchel-Nielsen twist flow is Hamiltonian with respect to the Weil-Petersson Kähler form on 𝔗(M), with Hamiltonian potential the length function l_{α} [47].

The Fenchel-Nielsen twist flow η^{α} and the twist flow ξ^{α} are reparametrizations of one another. Specifically,

ξ_{t}^{α} = η_{sinh(l_{α})}^{α}.

As in [9] (see also [11]), the Weil-Petersson Kähler form equals −1/2 the symplectic structure defined by the trace form 𝔹 restricted to SL(2, ℝ), implying the Fenchel-Nielsen flow is Hamiltonian. Since the Fenchel-Nielsen twist flow is Hamiltonian for the length function l_{α}, the derivative of the length function along the twist vector field equals the Poisson bracket {l_{α}, l_{β}}, which can be computed by Proposition 2.2.2. We thus obtain Wolpert’s Derivative Formula:

Theorem 2.6.2 ((Wolpert [47, 46]). Let α, β ∈ π where α is represented by a simple closed curve. Then the derivative of the length function l_{β} with respect to the Fenchel-Nielsen twist flow equals the sum

∑_{i=1}^{k} cos(θ_{i})

where α and β are represented by closed geodesics, p_{1}, …, p_{k} are their intersection points, and θ_{i} is the angle from α to β at p_{i}.

In particular the action of the Dehn twist τ_{A} equals the time-t map of the Fenchel-Nielsen twist flow at time t = l/2.

3. Abelian Hamiltonian Actions

The classical Fenchel-Nielsen coordinates on 𝔗(M) can be interpreted as a moment map arising from a pants decomposition, that is, a maximal collection of disjoint simple closed curves P = {α_{1}, …, α_{N}} on M which are each homotopically nontrivial and mutually nonhomotopic. If M has genus g, then N = 3g − 3 and the length functions define a map

l_{P} = (l_{α_{1}}, …, l_{α_{N}}) : 𝔗(M) ⟶ (ℝ_{+})^{N}

which is the moment map for a Hamiltonian ℝ^{N}-action. This ℝ^{N}- action is proper and free, and choosing a cross-section (a left-inverse to l_{P})

σ : (ℝ_{+})^{N} ⟶ 𝔗(M)

defines a diffeomorphism

(3.1)  (ℝ_{+})^{N} × ℝ^{N} ⟶ 𝔗(M)
(λ, t) ⟼ η_{t_{1}}^{α_{1}} … η_{t_{N}}^{α_{N}} (σ(λ))

The map l_{P} defines the action variables λ while the coordinates t ∈ ℝ^{N} are the angle variables. Indeed, Wolpert shows [48] that this completely integrable system defines a symplectomorphism with (ℝ_{+})^{N} × ℝ^{N}: the Fenchel-Nielsen coordinates are Darboux coordinates with respect to the symplectic structure ω_{W P} defined by the Weil-Petersson Kähler form:

ω_{W P} = ∑_{i=1}^{N} dλ_{i} ∧ dt_{i}

This section extends this theory from SL(2, ℝ) to SL(2, ℂ).

3.1. Pants decompositions. By Corollary 2.2.3, the complex Hamiltonian vector fields of f_{α} and f_{β} Poisson commute if α and β are represented by disjoint curves. Thus for a family of disjoint simple closed curves

P = {A_{1}, …, A_{N}}

the corresponding complex twist flows of Hamf_{α_{i}} generate a ℂ^{n}-action. Suppose these curves are each homotopically nontrivial and mutually nonhomotopic. The resulting map

τ_{P} : X ⟶ ℂ^{N}

is a moment map for a complex-Hamiltonian action of ℂ^{N} on the complex-symplectic manifold X.

Suppose that P is maximal, that is N = 3g − 3, in which case each component

P ⊂ M|(∪_{i=1}^{N}A_{i})

is homeomorphic to a 3-holed sphere. In that case Fricke’s theorem [13, 16] implies that the generic inverse image (τ_{P})^{−1}(z) is a single ℂ^{N}-orbit. Specifically, let (C^{N})^{s} denote the subset consisting of z ∈ ℂ^{N} such that, for every (i, j, k) for which α_{i}, α_{j}, α_{k} bound a 3-holed sphere P_{ijk},

z_{α_{i}}^{2} + z_{α_{j}}^{2} + z_{α_{k}}^{2} − z_{α_{i}}z_{α_{j}}z_{α_{k}} ≠ 4.

This expresses the condition that the restriction ϕ|_{π_{1}(P_{ijk})} is irreducible; Fricke’s theorem asserts the triple

(f(ϕ(α_{i}), f(ϕ(α_{j}), f(ϕ(α_{k})) ∈ ℂ^{3}

determines the equivalence class of such an irreducible representation. If z ∈ (ℂ^{N})^{s}, the fiber (τ_{P})^{−1}(z) is a single ℂ^{N}-orbit.

The subgroup Γ_{P} of Γ preserving the pants decomposition is the free abelian group generated by the Dehn twists τ_{α_{i}} for i = 1, …, N.

The resulting ℤ^{n}-action lies in the ℂ^{N}-action by the formulas in §2.6. Namely, at a point [ϕ] ∈ X the orbit is an embedded product

∏_{i=1}^{N} G_{i}

where

G_{i} = \begin{cases}\{1\} & if α_{i}([ϕ]) = ±𝕀 \\ ℂ & if f_{α_{i}}([ϕ]) = ±2 and α_{i}([ϕ]) ≠ ±𝕀 \\ ℂ^{∗} & if f_{α_{i}}([ϕ]) ≠ ±2 \end{cases} .

For each i with with f_{α_{i}}(ϕ) ≠ 2, let

λ_{i} = \frac{f_{α_{i}}(ϕ) ± (f_{α_{i}}(ϕ)^{2} − 4)^{1/2})}{2}.

Then (k_{1}, …, k_{N}) ∈ ℤ^{N} acts by translation by k_{i} on the i-th factor G_{i} ≈ ℂ for each i with f_{α_{i}}(ϕ) = ±2 and by multiplication by λ_{i} on each factor G_{i} ≈ ℂ^{∗} with f_{α_{i}}(ϕ) ≠ ±2.

3.2. The moment map as an ergodic decomposition and its holomorphic extension. The action of Γ_{P} on X_{U} is discussed in [15]. In this case the restriction

f_{P} : X_{U} ⟶ [−2, 2]^{N}

is a moment map for a Hamiltonian ℝ^{n}-action whose orbits are all closed. These orbits are tori, invariant under the Γ_{P} ≅ ℤ^{N}-action. Almost every fiber has dimension N, and the action is equivalent to an ℤ^{N}-action on T^{N} by translation. For almost all level sets, this action is ergodic. It follows that the moment map f_{P} for the Hamiltonian ℝ^{N}-action is also the ergodic decomposition for the ℤ^{N}-action ([15], Theorem 2.2):

Proposition 3.2.1. Let h : X_{U} ⟶ ℝ be a Γ_{P}-invariant measurable function. Then there exists a measurable function ψ : [−2, 2]^{N} ⟶ ℝ such that h = ψ ∘ f_{P} almost everywhere.

Theorem 3.2.2. Let h : X ⟶ ℂℙ^{1} be a meromorphic function which is invariant under Γ_{P}. Then there exists a meromorphic function H : ℂ^{P} ⟶ ℂℙ^{1} such that h = H ∘ τ_{P}.

Proof. Let G denote the complex-Hamiltonian ℂ^{P}-action on X and let G_{U} denote the Hamiltonian ℝ^{P}-action on X_{U}. By Proposition 3.2.1, the measurable function h must be almost everywhere constant on the G_{U}-orbits on X_{U}. Thus for each t ∈ τ_{P}(X_{U}), the function h is constant on the preimage τ_{P}^{−1}(t) ∩ X_{U} and:

• The preimages of the restriction of τ_{P} to X_{U} are the G_{U}-orbits in X_{U};

• Each G_{U}-orbit in τ_{P}^{−1}(t) is ℂ-Zariski dense in its G-orbit;

• G acts transitively on τ_{P}^{−1}(t).

Hence h is constant on each τ_{P}^{−1}(t). for t ∈ τ_{P}(X_{U}).

Now we find a holomorphic section of τ_{P}. There exists a connected open neighborhood W of X_{U}, a Zariski-closed subset Z ⊂ X of positive codimension, and a holomorphic map

W − Z ⟶^{σ} τ_{p}p^{−1}(W − Z)

so that the composition

W − Z ⟶^{σ} τ_{p}p^{−1}(W − Z) ⟶^{τ_{P}} W − Z

equals the identity. Then the meromorphic function

h − h ∘ σ ∘ τ_{P}

on τ_{p}p^{−1}(W − Z) vanishes on each fiber τ_{p}p^{−1}(t) for t ∈ W − Z ∩ X_{U}. Since W − Z ∩ X_{U} is totally real, it follows that h − h ∘ σ ∘ τ_{P} vanishes on all of W − Z. Since W − Z is dense in W, this function vanishes on W. Since W is nonempty and open, this function vanishes on all of X. Thus h is constant on the fibers of τ_{P} as desired. □

4. Deformation spaces of ℂℙ^{1}-structures

A ℂℙ^{1}-structure on M is a geometric structure with local coordinate charts mapping to ℂℙ^{1} with coordinate changes in the group PSL(2, ℂ). The holonomy mapping hol maps the deformation space ℂℙ^{1}(M) locally biholomorphically to the character variety X − X_{U}. Therefore Ω induces a complex-symplectic structure hol^{∗}Ω on ℂℙ^{1}(M). On the other hand, ℂℙ^{1}(M) is classically the total space of an affine bundle over 𝔗(M) whose associated vector bundle is the cotangent bundle T^{∗}𝔗(M). The canonical exact complex-symplectic structure on T^{∗}𝔗(M) defines a complex-symplectic structure on ℂℙ^{1}(M). Kawai [30] has proved this complex-symplectic structure equals hol^{∗}Ω:

Theorem. (Kawai) The complex-symplectic structure on ℂℙ^{1}(M) induced from the holomorphic cotangent bundle structure on T^{∗}𝔗(M) equals the complex-symplectic structure induced from the complex-symplectic structure Ω on X by

ℂℙ^{1}(M) ⟶^{hol} Hom(π, SL(2, ℂ))//SL(2, ℂ).

The complex twist flows on X lift to complex twist flows on ℂℙ^{1}(M). We relate the complex twist geometry on the open subset QF(M) ⊂ ℂℙ^{1}(M) comprising quasi-Fuchsian structures to the quakebends studied by Epstein-Marden [7], Kourouniotis [32, 33, 34], McMullen [36], Platis [39], Series [42], and Tanigawa [44].

4.1. ℂℙ^{1}-structures. Let M be a smooth surface. A ℂℙ^{1}-atlas on M consists of

• An open covering U of M by coordinate patches;

• For each coordinate patch U ∈ U, a coordinate chart

ψ_{U} : U ⟶ ℂℙ^{1}

which is a diffeomorphism of U onto its image ψ_{U}(U);

• For each connected open subset C ⊂ U ∩V of the intersection of coordinate patches U, V ∈ U, a transformation g_{C} ∈ PSL(2, ℂ) such that

g_{C} ∘ ψ_{U}|_{U∩V} = ψ_{V} |_{U∩V} .

The coordinate change g_{C} is unique. Such atlases are partially ordered by inclusion. A ℂℙ^{1}-structure on M is a maximal ℂℙ^{1}-atlas. A ℂℙ^{1}-manifold is a manifold with a ℂℙ^{1}- structure.

Suppose M and M′ are ℂℙ^{1}-manifolds. A mapping ϕ : M ⟶ M′ is a ℂℙ^{1}-mapping if and only if for each coordinate chart (U, ψ_{U}) in M and each coordinate chart (U′, ψ′_{U′}) in M′ the composition

(ψ′_{U′}) ∘ ϕ ∘ (ψ_{U})^{−1} : ψ_{U}(U ∩ ϕ^{−1}(U′)) ⟶ ψ′_{U′}(ϕ(U) ∩ U′)

extends to an element of PSL(2, ℂ) on each component of ψ_{U}(U ∩ ϕ^{−1}(U′). A ℂℙ^{1}-mapping is necessarily a local diffeomorphism.

If f : M ⟶ M′ is a local diffeomorphism, then every ℂℙ^{1}-structure on M′ determines a unique ℂℙ^{1}-structure on M such that f is a ℂℙ^{1}- mapping. In particular every covering space of a ℂℙ^{1}-manifold is a ℂℙ^{1}-manifold, and its group of covering transformations is realized by a group of ℂℙ^{1}-automorphisms.

If M is a simply-connected ℂℙ^{1}-manifold, then any chart on M extends to a globally defined developing map

dev : M ⟶ ℂℙ^{1}

which is a ℂℙ^{1}-mapping. This mapping is unique up to left-composition with transformations in PSL(2, ℂ).

We denote by Σ the Riemann surface whose underlying manifold is M for which dev is holomorphic.

In general the universal covering space M̃ of a ℂℙ^{1}-manifold admits a developing map into ℂℙ^{1}. The group π_{1}(M) of covering transformations of M̃ ⟶ M admits a representation ρ : π_{1}(M) ⟶ PSL(2, ℂ) such that the diagram

M̃ ⟶^{dev} ℂℙ^{1}
γ↓  ↓^{ρ(γ)}
M̃ ⟶^{dev} ℂℙ^{1}

commutes for every γ ∈ π_{1}(M). The representation ρ is called the holonomy representation. The developing pair (dev, ρ) is unique up to the action of PSL(2, ℂ) (by left-composition with dev and conjugation on ρ). The developing map globalizes the coordinate charts and the holonomy representation globalizes the coordinate changes.

4.1.1. Deformation spaces. Let S be a fixed topogical surface with fundamental group π = π(S). A marked ℂℙ^{1}-structure on S consists of a ℂℙ^{1}-manifold M and a homeomorphism f_{M} : S ⟶ M. Two marked ℂℙ^{1}-structures (M, f_{M}) and (M′, f_{M′}) are equivalent if and only if there exists a ℂℙ^{1}-isomorphism ϕ : M ⟶ M′ such that

ϕ ∘ f_{M} ≃ f_{M′}.

Let ℂℙ^{1}(S) denote the set of equivalence classes of marked ℂℙ^{1}-structures on S. Holonomy defines a mapping

hol : ℂℙ^{1}(S) ⟶ Hom(π, SL(2, ℂ))//SL(2, ℂ).

As in [14] (see also Earle [6], Hubbard [22] and Kapovich [28]), there is a natural topology on ℂℙ^{1}(S) such that hol is a local homeomorphism. Furthermore Gallo-Kapovich-Marden [8] proved that the image of hol is X − X_{U} − X_{D}, where X_{D} ⊂ X − X_{U} is the closed subset consisting of equivalence classes [ϕ] of representations leaving invariant a pair of points of ℂℙ^{1} (these correspond to irreducible representations for which an index-two subgroup acts reducibly).

4.2. Holonomy and the conformal structure. Since PSL(2, ℂ) acts holomorphically on ℂℙ^{1}, a ℂℙ^{1}-atlas on M is a holomorphic atlas, that is, an atlas for a complex structure on M. Thus underlying every ℂℙ^{1}-manifold is a Riemann surface. Recording the complex structure underlying a ℂℙ^{1}-structure is a map

ℂℙ^{1}(S) ⟶^{Π} 𝔗(M).

A projective structure on a Riemann surface Σ is a ℂℙ^{1}-structure whose underlying complex structure is Σ.

Lemma 4.2.1. Π is holomorphic.

Proof. We recall the classical description of ℂℙ^{1}(S) as an affine bundle Q(M) over 𝔗(M) whose associated vector bundle equals the cotangent bundle T^{∗}(𝔗(M)). (See for example §2.3 of Earle [6], Hubbard [22] or Gunning [18]). Fix a Riemann surface Σ homeomorphic to M and a projective structure s_{0} on Σ. Given any other projective structure on Σ, its developing map is a holomorphic map from the universal covering Σ̃ to ℂℙ^{1}. Its Schwarzian derivative (in the local projective coordinates defined by s_{0}) is a holomorphic quadratic differential which is invariant under π, and therefore defines a holomorphic quadratic differential on Σ. Furthermore this quadratic differential completely determines the developing map up to an element of PSL(2, ℂ) of ℂℙ^{1}. Thus the fiber Π^{−1}([Σ]) admits a simply transitive action of the complex vector space H^{0}(Σ; O(κ^{2})) comprising holomorphic quadratic differentials on Σ. The composition law for the Schwarzian derivative implies that changing the origin s_{0} effects a translation of Q(Σ), so Π^{−1}(Σ) is an affine space modelled on H^{0}(Σ; O(κ^{2})), which is the cotangent space T_{[Σ]}^{∗}𝔗(M) of 𝔗(M) at [Σ].

In particular ℂℙ^{1}(S) inherits a complex structure, making it a holomorphic affine bundle over 𝔗(M). In particular the projection Π is holomorphic in this complex structure.

By Lemma 2 of Earle [6] or Hubbard [22], the holonomy mapping hol is a local biholomorphism, and therefore this complex structure on ℂℙ^{1} induced by hol is the above complex structure. Thus Π is holomorphic with respect to the complex structure induced by hol. □

Kawai’s theorem implies that Π is a holomorphic Lagrangian fibration. As described in Hubbard [22], the differential dΠ identifies with the map

T_{s_{0}}ℂℙ^{1}(S) ≅ H^{1}(Σ; 𝔰𝔩(2, ℂ)) ⟶ H^{1}(Σ; O(κ^{−1})) ≅ T_{[Σ]}𝔗(M)

induced by the short exact sequence of sheaves

(4.1)  0 ⟶ 𝔰𝔩(2, ℂ) ⟶^{i} O(κ^{−1}) ⟶^{D_{3}} O(κ^{2}) ⟶ 0.

Here D_{3} is the map of holomorphic sheaves given in local holomorphic coordinates by

h(z) \frac{∂}{∂z} ⟼ h′′′(z) dz^{2}.

Its kernel 𝔰𝔩(2, ℂ) is the locally constant sheaf of locally projective vector fields (whose coefficients in affine coordinates are quadratic). The monomorphism i regards a locally projective vector field as a holomorphic vector field. Then (4.1) induces the exact sequence

0 ⟶ H^{0}(Σ, O(κ^{2})) ⟶^{δ} H^{1}(Σ; 𝔰𝔩(2, ℂ)) ⟶^{i_{∗}} H^{1}(Σ; O(κ^{−1})) ⟶ 0.

If α ∈ H^{0}(Σ, O(κ^{2})) is a holomorphic quadratic differential and β ∈ H^{1}(Σ; 𝔰𝔩(2, ℂ)), then

(4.2)  ⟨δ(α), β⟩ = α · i_{∗}(β)

where the first pairing ⟨, ⟩ is the pairing on H^{1}(Σ; 𝔰𝔩(2, ℂ)) induced by the complex Killing form 𝔹 on 𝔰𝔩(2, ℂ) and the second pairing is the Serre duality pairing

H^{0}(Σ, O(κ^{2})) × H^{1}(Σ, O(κ^{−1})) ⟶ H^{1}(Σ, O(κ)) ≅ ℂ.

This discussion implies that Π is a Lagrangian fibration. Suppose

β, γ ∈ ker(dΠ) = δ(H^{0}(Σ, O(κ^{2})) = ker(i_{∗})

Then γ = δ(α) for some α ∈ H^{0}(Σ, O(κ^{2})), and (4.2) implies ⟨γ, β⟩ = 0 since i_{∗}β = 0. Therefore ker(dΠ) is isotropic. Since its dimension is half of that of H^{1}(Σ; 𝔰𝔩(2, ℂ)), the fibers of Π are Lagrangian. For further details, see Kawai [30].

4.2.1. Meromorphic functions on ℂℙ^{1}(S).

Proposition 4.2.2. There exist nonconstant Γ-invariant meromorphic functions on ℂℙ^{1}(S).

Proof. The Riemann moduli space of curves is the quotient 𝔗(M)/Γ, which by Knudsen [31], is a quasiprojective variety. Thus 𝔗(M)/Γ admits nonconstant meromorphic functions and 𝔗(M) admits nonconstant Γ-invariant meromorphic functions.

Let ψ be a nonconstant Γ-invariant meromorphic function on 𝔗(M). Composing with the projection

Π : ℂℙ^{1}(S) ⟶ 𝔗(M)

provides a nonconstant Γ-invariant meromorphic function ψ ∘ Π on ℂℙ^{1}(S). □

4.3. Quasi-Fuchsian space. A ℂℙ^{1}-structure is Fuchsian if and only if a developing map embeds M̃ in a hyperbolic plane H_{ℂ}^{1} ⊂ ℂℙ^{1}. Necessarily the holonomy representation is conjugate to a Fuchsian representation, that is, a discrete embedding π ↪ PU(1, 1) (where PU(1, 1) is the stabilizer in PSL(2, ℂ) of the Poincaré disc H_{ℂ}^{1}). However by a construction of Maskit and Hejhal (see also Goldman [12]) there exist ℂℙ^{1}-structures with Fuchsian holonomy with surjective developing maps. A quasi-Fuchsian ℂℙ^{1}-structure is a ℂℙ^{1}-structure topologically conjugate to a Fuchsian structure, that is, there exists a homeomorphism h of ℂℙ^{1} such that

(h ∘ dev, h ∘ ρ ∘ h^{−1})

is the developing pair for a Fuchsian structure. The limit set Λ of ρ(π) is the Jordan curve h(∂H_{ℂ}^{1}). The developing image dev(M̃) is one component of the complement ℂℙ^{1} − Λ.

Quasi-Fuchsian space QF(M) is the subset of ℂℙ^{1}(S) corresponding to quasi-Fuchsian structures. The holonomy representation ρ is a quasi-Fuchsian representation. The holonomy mapping

hol : QF(M) ⟶ X

embeds QF(M) as the open subset of X consisting of conjugacy classes of quasi-Fuchsian representations. Quasi-Fuchsian space QF(M) is both the open subset of ℂℙ^{1}(S) consisting of quasi-Fuchsian structures and the open subset of X − X_{U} consisting of characters of quasi-Fuchsian representations.

The conformal structures of the quotients of ℂℙ^{1} − Λ by ϕ(π) determine an ordered pair 𝔗(M) × 𝔗̄(M); the first parameter is the marked conformal structure underlying the ℂℙ^{1}-structure. The simultaneous uniformization theorem of Bers [3] asserts that the corresponding map

QF(M) ⟶^{≈} 𝔗(M) × 𝔗̄(M)

is a biholomorphism. It is evidently Γ-equivariant.

Since QF(M) ↪ ℂℙ^{1}(S), Corollary 4.2.2 implies that nonconstant Γ-invariant meromorphic functions exist on QF(M).

4.4. Twist flows of ℂℙ^{1}-structures and grafting. Let α ∈ π. Then the composition

ℂℙ^{1}(S) ⟶^{hol} X ⟶^{f_{α}} ℂ

defines a holomorphic function on ℂℙ^{1}(S) and its complex-Hamiltonian Ham(f_{α} ∘hol) is a holomorphic vector field on ℂℙ^{1}(S). If α corresponds to a simple closed curve A, then there is a complex twist flow ξ̃_{t}^{α} on ℂℙ^{1} covering the complex twist flow generated by the vector field Ham(f_{α}) on X.

This twist flow is defined geometrically on Fuchsian structures as follows (see [12], Kourouniotis [32, 33, 34]). Let s_{0} be a Fuchsian ℂℙ^{1}- structure, so a developing map embeds M̃ as a geometric disc Δ ⊂ ℂℙ^{1}. Then the simple closed curve A ⊂ M can be represented by a simple closed geodesic with respect the Poincaré metric induced from that of Δ. Let ℓ(A) denote the Poincaré length of this geodesic. A lift Ã of A to M̃ develops to a circular arc orthogonal to ∂dev(M̃). The split surface M|A inherits a ℂℙ^{1}-structure whose boundary components develop to circular arcs. To define s_{t} := ξ̃^{α}_{t}(s_{0}), insert an annulus A_{θ} with ℂℙ^{1}- structure into M|A. The angular parameter θ is the imaginary part Im(t). The annulus is a θ-annulus in the sense of §2.12 of [12]. Choose a holomorphic universal covering map

E : ℂ ⟶ ℂℙ^{1} − Fix(ϕ(α))

which is periodic with period 2πi. (When Fix(ϕ(α)) = {0, ∞}, then this is just the exponential map exp : ℂ ⟶ ℂ^{∗}. For general A, we compose exp with a projective transformation taking {0, ∞} to Fix(ϕ(α)).)

Define the θ-strip

S_{θ} := ℝ + i[0, θ] ⊂ ℂ

with ℂℙ^{1}-structure induced by E. The developing map E for S_{θ} is equivariant with respect to the ℤ-action on S_{θ} generated by translation by ℓ(A) and the ℤ-action on ℂℙ^{1} − Fix(ϕ(α)). The θ-annulus A_{θ} is defined as the quotient ℂℙ^{1}-manifold of S_{θ}.

The grafted manifold M(t) is obtained by inserting a θ-annulus into M|A. Choose one component of A_{−} ⊂ ∂(M|A) and attach A_{θ} to A_{−} to obtain a ℂℙ^{1}-manifold homeomorphic to M|A with one boundary component A′_{+} corresponding to the component A_{+} of ∂(M|A) and another boundary component A′ corresponding to the other component of ∂(A_{θ}). Identify A′_{+} to A′ by a transvection of displacement Re(t).

When θ = 0, this is just the Fenchel-Nielsen twist deformation, obtained from M|A by identifying the two components of ∂(M|A) by a transvection of displacement Re(t).

Let P ∈ SL(2, ℂ). These holomorphic flows on Hom(π, SL(2, ℂ)) cover holomorphic flows on X which have been extensively studied on quasi-Fuchsian space QF(M) ⊂ X. On QF(M) these flows geometrically correpond to the quakebends or complex earthquakes discussed by Epstein-Marden [7], Kourouniotis [32, 33, 34], McMullen [36], Platis [39], Series [42], and Tanigawa [44]. In particular we obtain the following result of Platis [39]:

Theorem 4.4.1 ((Platis [39], Theorem 7, §2.2). The complex twist vector field ξ̃ on QF(M) is Hamiltonian with respect to the complex length function l^{ℂ}.

Just as the geodesic length function l on the hyperbolic subset of SL(2, ℝ) and the angle function θ on SU(2) − {±𝕀} are more geometrically natural than the trace functions, we consider the complex length “function” on SL(2, ℂ) defined by:

2 cosh (\frac{l^{ℂ}(P)}{2}) = f(P)

or equivalently

l^{ℂ}(P) = 2 log (\frac{(f(P) ± (f(P)^{2} − 4)^{1/2})}{2}).

(This differs from Tan [43] whose complex length is half of ours. Our definition is consistent with Kourouniotis [32, 33, 34], Platis [39] and Series [42].) This function takes values in (ℂ/4πiℤ)/{±1}, since the logarithm is well-defined only modulo 4πi and the choice of square root introduces an ambiguity of sign.

Choose the branch l̃_{ℂ} of the complex length which is positive on the subset of hyperbolic elements in SL(2, ℝ); since QF(M) is simply connected

[ϕ] ⟼ (l̃^{ℂ}(ϕ(α_{1})), …, l̃^{ℂ}(ϕ(α_{N})))

defines a single-valued function

l^{ℂ}|_{P} : QF(M) ⟶ ℂ^{N}.

The restriction of l_{P}^{ℂ} to 𝔗(M) is the length function l_{P} : 𝔗(M) ⟶ (ℝ_{+})^{N} defined by (3.1). The fibers of l^{ℂ}|_{P} are orbits of the complex-Hamiltonian ℂ^{N}-action having l^{ℂ}|_{P} as moment map. Tan [43] and Kourouniotis [34] show that there exists a section σ : ℂ^{N} ⟶ QF(M) and a neighborhood of (ℝ_{+})^{N} × {0} in ℂ^{N} × ℂ^{N} which maps biholomorphically to QF(M).

Korouniotis [33] and Series [42] derive formulas for the derivative of the complex length functions along quakebend flows. We briefly sketch how their formulas can be derived from Proposition 2.2.2 as a Poisson bracket, referring to [33, 42] for details.

Let P ∈ SL(2, ℂ) be loxodromic with invariant axis a_{P}. Then the variation function L^{ℂ} associated to the invariant function l^{ℂ} and the complex-orthogonal structure 𝔹(X, Y) = tr(XY) maps a diagonal matrix to

D_{0} := ± [\begin{matrix}1 & 0 \\ 0 & −1 \end{matrix}]

(which is not uniquely defined without restrictions to make the function l^{ℂ} single-valued). By (2.1), the element of PSL(2, ℂ) corresponding to L^{ℂ}(P) is the involution fixing a_{P}. If P_{1}, P_{2} ∈ PSL(2, ℂ) are loxodromic, with complex distance z between their invariant axes, then a simple calculation implies that 𝔹(L^{ℂ}(P_{1}), L^{ℂ}(P_{2})) equals cosh(z). Applying Proposition 2.2.2 to the complex length functions l_{β}^{ℂ} and l_{α}^{ℂ}, where α corresponds to a simple closed curve, we obtain the following extension of Theorem 2.6.2:

Theorem 4.4.2 ((Korouniotis [33], Series [42]). Let α, β ∈ π where α is represented by a simple closed curve A. Then the derivative of the complex length function l_{β}^{C} on QF(M) with respect to the quakebend flow with respect to A equals the sum

∑_{i=1}^{k} cosh(d_{i})

where α and β are represented by closed geodesics, p_{1}, …, p_{k} are their intersection points, and d_{i} the complex distance between the axes of ϕ_{i}(α_{i}) to ϕ_{i}(β_{i}) at p_{i}, where ϕ_{i} is a representative homomorphism from π_{1}(M; p_{i}) and α_{i}, β_{i} are elements of π_{1}(M; p_{i}) representing α and β.

See Korouniotis [33] and Series [42] for details.

Tan [43] and Kourouniotis [32, 33, 34] extend Fenchel-Nielsen coordinates to complex Fenchel-Nielsen coordinates on QF(M). (See also Series [42] for another exposition.)

References

[1] Abikoff, W., “The Real Analytic Theory of Teichmüller Space,” Lecture Notes in Mathematics 820 (1980) Springer-Verlag, Berlin and New York.

[2] Benyash-Krivets, V., Chernousov, V., and Rapinchuk, A.S., Representation varieties of the fundamental groups of compact orientable surfaces, Israel J. Math. 93 (1996), 29–71.

[3] Bers, L., Simultaneous uniformization, Bull. A.M.S. 66 (1960), 94–97.

[4] Buser, P., “Geometry and Spectra of Compact Riemann Surfaces,” Progress in Mathematics 106, Birkhäuser Boston (1992).

[5] Chas, M., Combinatorial Lie bialgebras of curves on surfaces, Topology (to appear).

[6] Earle, C., On variation of projective structures, in “Riemann surfaces and related topics: Proceedings of the 1978 Stony Brook Conference”, Ann. Math. Study 97 (1981), Princeton Univ. Press, 87–99.

[7] Epstein, D. and Marden, A., Convex hulls in hyperbolic space, in “Analytical and Geometric Aspects of Hyperbolic Space,” London Math. Soc. Lecture Notes series 111, Cambridge Univ. Press (1987).

[8] Gallo, D., Kapovich, M. and Marden, A., The monodromy groups of Schwarzian equations on closed Riemann suraces, Ann. Math. 151 (2000), 625–704.

[9] Goldman, W., The symplectic nature of fundamental groups of surfaces, Adv. Math. 54 (1984), 200–225.

[10] , Representations of fundamental groups of surfaces, in “Geometry and Topology, Proceedings, University of Maryland 1983–1984”, J. Alexander and J. Harer (eds.), Lecture Notes in Mathematics 1167 (1985), 95–117, Springer-Verlag.

[11] , Invariant functions on Lie groups and Hamiltonian flows of surface group representations, Inv. Math. 85 (1986), 1–40.

[12] , Projective structures with Fuchsian holonomy, J. Diff. Geom. 25 (1987), 297–326.

[13] , Topological components of spaces of representations, Inv. Math. Vol. 93 No. 3, (1988), 557–607.

[14] , Geometric structures and varieties of representations, in “The Geometry of Group Representations,” Contemp. Math. 74, Amer. Math. Soc. (1988), 169–198.

[15] , Ergodic theory on moduli spaces, Ann. Math. 146 (1997), 1–33.

[16] , An exposition of results of Fricke, (in preparation).

[17] and Millson, J., The deformation theory of representations of fundamental groups of compact Kähler manifolds, Publ. Math. I.H.E.S. 67 (1988), 43–96.

[18] Gunning, R., Affine and projective structures on Riemann surfaces, in “Riemann surfaces and related topics: Proceedings of the 1978 Stony Brook Conference”, Ann. Math. Study 97 (1981), Princeton Univ. Press, 225–244.

[19] Guruprasad, K. Huebschmann, J., Jeffrey, L., and Weinstein, A., Group systems, groupoids and moduli spaces of parabolic bundles, Duke Math. J. 89 (1997), 377412.

[20] Harvey, W. J., Spaces of Discrete Groups, in “Discrete Groups and Automorphic Functions,” Academic Press (1977), 295–347.

[21] Hitchin, N. J., The self-duality equations on Riemann surfaces, Proc. London Math. Soc. (3)55 (1987), 59–126.

[22] Hubbard, J. H., The monodromy of projective structures, in “Riemann surfaces and related topics: Proceedings of the 1978 Stony Brook Conference”, Ann. Math. Study 97 (1981), Princeton Univ. Press, 257–277.

[23] Huebschmann, J., On the variation of the Poisson structures of certain moduli spaces, Math. Ann. 319 (2001), no. 2, 267-310.

[24] Jeffrey, L. and Weitsman,J., Bohr-Sommerfeld orbits in the moduli space of flat connetions and the Verlinde dimension formula, Commun. Math. Phys. 150 (1992), 595–620.

[25] , Toric structures on the moduli space of flat connections on a Riemann surface: volumes and the moment map, Adv. Math. 106 (2) (1994), 151–168.

[26] , Torus actions, moment maps, and the symplectic geometry of flat connections on a two-manifold, Cont. Math. 175 (1994), 149 –159.

[27] , Torus actions and the symplectic geometry of flat connections on 2manifolds, in “Topology, geometry and field theory,” 49–58, World Sci. Publ., River Edge, NJ (1994).

[28] Kapovich, M., “Hyperbolic manifolds and Kleinian groups,” Progress in Math. (2001) Birkhäuser.

[29] Karshon, Y., An algebraic proof for the symplectic structure of moduli space, Proc. Amer. Math. Soc. 116 (1992), no. 3, 591–605.

[30] Kawai, S., The symplectic nature of the space of projective connections on Riemann surfaces, Math. Ann. 305 (1996), 161–182.

[31] Knudsen, F., The projectivity of the moduli space of stable curves, I. Math. Scand. 39 (1976), 19–66, II. Math. Scand. 52 (1983), 161–199, III. Math. Scand. 52 (1983), 200–212.

[32] Kourouniotis, C., Bending in the space of quasi-Fuchsian structures, Glasgow Math. J. 33 (1991), 41–49.

[33] , The geometry of bending quasi-Fuchsian groups, in “Discrete groups and geometry (Birmingham 1991), 148–164, London Math. Soc. Lecture Notes 173, Cambridge Univ. Press, Cambridge (1992).

[34] , Complex length coordinates for quasi-Fuchsian groups, Mathematika 41 (1994), 173-188.

[35] Li, J., The space of surface group representations, Manuscripta Math. 78 (3) (1993), 223–243.

[36] McMullen, C., Complex earthquakes and Teichmüller theory, J. Amer. Math. Soc. 11 (1998), 283–320.

[37] Nielsen, J., Untersuchungen zur Topologie der geschlossenen zweiseitigen Flächen I, Acta Math. 50 (1927), 189–358.

[38] Pickrell, D. and Xia, E., Ergodicity of Mapping Class Group Actions on Representation Varieties, I. Closed Surfaces, Comment. Math. Helv. 77 (2001), 339–362.

[39] Platis, I., Complex symplectic geometry of quasi-Fuchsian space, Geometriae Dedicata 87 (2001), 17 –34.

[40] Raghunathan, M., “Discrete Subgroups of Lie Groups,” Springer-Verlag (1972).

[41] Scannell, K., and Wolf, M., The grafting map of Teichmüller space, J. Amer. Math. Soc. (2002).

[42] Series, C., An extension of Wolpert’s derivative formula, Pac. J. Math. 197 (1) (2001), 223–239.

[43] Tan, S., Complex Fenchel-Nielsen coordinates for quasifuchsian structures, Int. J. Math. 5 (2) (1994), 239–251.

[44] Tanigawa, H., Grafting, harmonic maps and projective structures, J. Diff. Geom. 47 (1997), 399–419.

[45] Weinstein, A., The Symplectic Structure on Moduli Space, in “The A. Floer Memorial Volume,” ed. H. Hofer, C. Taubes, A. Weinstein, E. Zehnder. Birkhauser, Basel. 1995, Progress in Math., Vol. 13, pp. 627635.

[46] Wolpert, S., The Fenchel-Nielsen deformation, Ann. Math. 116 (1982), 501– 528.

[47] , On the symplectic geometry of deformations of a hyperbolic surface, Ann. Math. 117 (1983), 207–234.

[48] , On the Weil-Petersson geometry of the moduli space of curves, Amer. J. Math. 107 (1985), 969–997.

Mathematics Department, University of Maryland, College Park, MD 20742 USA

  E-mail address: wmg@math.umd.edu