ERGODIC THEORY ON MODULI SPACES

WILLIAM M. GOLDMAN

Dedicated to the memory of Peter Greenberg and Russell Mills

Abstract. Let M be a compact surface with χ(M) < 0 and let G be a compact Lie group whose Levi factor is a product of groups locally isomorphic to SU(2) (for example SU(2)). Then the mapping class group Γ_{M} of M acts on the moduli space X(M) of flat G-bundles over M (possibly twisted by a fixed central element of G). When M is closed, then Γ_{M} preserves a symplectic structure on X(M) which has finite total volume on M. More generally, the subspace of X(M) corresponding to flat bundles with fixed behavior over ∂M carries a Γ_{M}-invariant symplectic structure. The main result is that Γ_{M} acts ergodically on X(M) with respect to the measure induced by the symplectic structure.

Contents

1. Introduction 2 1.1. Statement of results 2 1.2. The Narasimhan-Seshadri foliation of the universal moduli

space 3

1.3. Moduli spaces over surfaces with boundary 4 1.4. Parabolic structures 5 1.5. Further speculations 5 1.6. Outline of proof 7 1.7. Acknowledgements 10 2. Preliminaries 10 2.1. Moduli spaces 10 2.2. Twisted moduli spaces 12

Date: October 12, 1997. 1991 Mathematics Subject Classification. 57M05 (Low-dimensional topology),

22D40 (Ergodic theory on groups).

Key words and phrases. Fundamental group of surface, mapping class group, Dehn twist, symplectic structure, character variety, moduli spaces, stable holomorphic vector bundle, Teichmüller space, parabolic structure.

The author gratefully acknowledges support from National Science Foundation grants DMS-8613576, DMS-8902619, DMs-9 504764, the Alfred P. Sloan Foundation, and the University of Maryland Institute for Advanced Computer Studeies.

2.3. Ergodic decomposition 13 2.4. Surfaces with boundary 13 3. Decompositions of surfaces and character varieties 14 3.1. Nonseparating curves 15 3.2. Separating curves 16 4. Three-holed spheres 17 5. The one-holed torus 17 5.1. Trace coordinates 18 5.2. Action of a Dehn twist 18 5.3. The symplectic structure 20 6. The four-holed sphere 20 6.1. Trace coordinates 20 6.2. Action of a Dehn twist 22 6.3. Relationship with the one-holed torus 23 7. Pants decompositions 25 7.1. The character mapping of a pants decomposition 26 7.2. Automorphisms of pants decompositions 27 7.3. Relation to the work of Jeffrey-Weitsman 29 8. Conclusion of the proof 30 9. Appendix: Modifications for product groups 31 10. Appendix: Effective actions of automorphisms 32 10.1. Effectiveness on nonabelian representations 32 10.2. Hyperelliptic representations 32 References 33

1. Introduction

1.1. Statement of results. Let M be a closed oriented surface of genus g > 1 and let π denote its fundamental group. The mapping class group of M is by definition π_{0} Diff(M). According to J. Nielsen, the mapping class group is isomorphic to the outer automorphism group Γ_{M} = Out(π). Let G be a Lie group and let Hom(π, G) denote the space of homomorphisms π → G. Let Hom(π, G)/G denote the space of G-conjugacy classes of homomorphisms π → G. We may geometrically identify Hom(π, G)/G with the moduli space of isomorphism classes of flat principal G-bundles over M. Then Γ_{M} acts on Hom(π, G)/G. When G is a semisimple group then this action preserves a natural symplectic structure and hence a volume form (Goldman [12]). Although Hom(π, G)/G is compact, it has singularities and the symplectic structure is singular along the singular stratum; by Huebschmann [19], the measure of Hom(π, G)/G is nonetheless finite.

For many purposes it is useful to consider more general twisted moduli spaces, where the relation in the fundamental group is “twisted” by an element ζ_{0} of the center of G. For example for G = SU(2), the space Hom(π, G)/G identifies with the moduli space of semistable holomorphic rank two vector bundles with fixed determinant of even degree and the −I-twisted moduli space Hom_{−}(π, G)/G identifies with the moduli space of semistable holomorphic rank two vector bundles with fixed determinant of odd degree. Furthermore Hom_{−}(π, G)/G is smooth. (See §2.2 for the precise definition).

Theorem 1.1. Let G be a compact Lie group locally isomorphic to a product of copies of SU(2) and U(1). Then for every ζ_{0} ∈ Z(G), the mapping class group Γ_{M} is ergodic on Hom_{ζ_{0}}(π, G)/G.

Corollary 1.2. Let G = SU(2). Then the actions of Γ_{M} on Hom(π, G)/G and Hom_{−}(π, G)/G are ergodic.

This corollary was the original result of the paper. As we finished writing the paper, it became clear how to extend the techniques to cover the hypotheses above. The modifications necessary to prove Theorem 1.1 for products are given in §9.

Conjecture 1.3. Γ_{M} is ergodic on Hom_{ζ_{0}}(π, G)/G for any compact Lie group and ζ_{0} ∈ Z(G).

1.2. The Narasimhan-Seshadri foliation of the universal moduli space. Corollary 1.2 has a more geometric interpretation involving the Riemann moduli space M_{g} of curves of genus g. Let T_{g} denote the Teichmüller space of M; then M_{g} equals the quotient T_{g}/Γ_{M}. The fiber product

B_{g} = (T_{g} × Hom(π, G)/G)/Γ_{M}

is the total space of a bundle over M_{g}. The fiber over a point of M_{g} corresponding to a Riemann surface M consists of the Narasimhan-Seshadri moduli space of rank 2 semistable vector bundles over M with trivial determinant bundle. (See [36].) The foliation of T_{g} × Hom(π, G)/G by leaves T_{g} × {[ϕ]} is invariant under Γ_{M} and hence defines a foliation F of B_{g} transverse to the fibration over M_{g}. Corresponding to the Γ_{M}-invariant measure on Hom(π, G)/G is an invariant transverse measure to F. Similarly there is a foliated bundle (B′_{g}, F′) over M_{g} with fiber the moduli space of (−I)-twisted representations of π_{1}(M).

Theorem 1.4. The foliation F of B_{g} (respectively F′ of B′_{g}) is ergodic.

1.3. Moduli spaces over surfaces with boundary. Theorem 1.1 is proved by inductively gluing together “smaller surfaces,” which are necessarily compact surfaces-with-boundary. The inductive process involves the following generalization of Theorem 1.1. The character variety of M is defined to be

X(M) = Hom(π_{1}(M), G)/G

if M is path-connected and the Cartesian product

X(M) = ∏_{M_{i}∈π_{0}(M)} X(M_{i})

in general. If M is a surface-with-boundary, then the inclusions of the boundary components ∂_{i}M ↪ M induce a restriction map

∂^{#} : X(M) ⟶ X(∂M) = ∏ X(∂_{i}M)

In this way we regard X(M) as a family of relative character varieties over X(∂M). Since each X(∂_{i}M) identifies with the set [G] of conjugacy classes in G, the base of this family is the product of copies of [G], one for each boundary component. Explicitly, choose γ_{1}, …, γ_{m} ∈ π_{1}(M) corresponding to the boundary components. For each γ_{i} choose a conjugacy class C_{i} ∈ G. Then the relative character variety (defined with respect to

C = (C_{1}, …, C_{m}) ∈ X(∂M)),

equals

X_{C}(M) = {[ρ] ∈ Hom(π, G)/G | ρ(γ_{i}) ∈ C_{i}, ∀i = 1, …, m}

The mapping class group Γ_{M} of M equals the group of relative isotopy classes of diffeomorphisms M ⟶ M restricting to the identity on ∂M. The mapping class group identifies with the group of equivalence classes of automorphisms of π_{1}(M) preserving the conjugacy class of each cyclic subgroup ⟨γ_{i}⟩ corresponding to a boundary component.

Theorem 1.5. Let G be a compact Lie group locally isomorphic to a product of copies of SU(2) and U(1). Then the mapping class group Γ_{M} is ergodic on X_{C}(M).

This theorem includes Theorem 1.1 as the special case when M is a once-punctured surface and the boundary holonomy maps to the conjugacy class {ζ_{0}}.

1.4. Parabolic structures. Our methods apply equally well to compact surfaces M with finitely many punctures. The moduli spaces of representations of π_{1}(M) in SU(2) are moduli spaces of holomorphic vector bundles with parabolic structure. Fixing the conjugacy class of the holonomy of each component of the boundary amounts to a parabolic structure. For fixed boundary structure, the moduli space has the structure of a projective variety, as in Mehta-Seshadri [33]. For the symplectic geometry in this case, see Alekseev-Malkin [2], Biswas-Guruprasad [4], King-Sengupta [28], Huebschmann [19], and Jeffrey [21],[22]. The theorem we prove for compact surfaces applies to these moduli spaces as well; we simply state the result (which is just 1.5 translated into the language of holomorphic vector bundles with parabolic structure):

Theorem 1.6. Let M be a compact Riemann surface with m distinguished points p_{1}, …, p_{m}. Fix a rank-two parabolic structure P on M at p_{1}, …, p_{m}. Let Γ denote the group of isotopy classes of diffeomorphisms of M which fix each p_{i}. Then Γ is ergodic on the moduli space of semistable rank two vector bundles with parabolic structure P over M.

1.5. Further speculations. The space Hom(π, G)/G is a “nonabelian” extension of the space Hom(π, U(1)) of 1-dimensional unitary characters of π_{1}(M). (Alternatively this space is the first cohomology group of M with circle coefficients.) This compact 2g-dimensional torus identifies with the Jacobi variety of holomorphic line bundles of degree 0 when M is provided with a complex structure. The action of Γ_{M} on Hom(π, U(1)) factors through the action of the integral symplectic group Sp(ℤ^{2g}) on the torus ℝ^{2g}/ℤ^{2g}. This action is ergodic, although it is quite far from being effective. The action of Γ_{M} on the SU(2)-moduli space is effective (Theorem 10.1).

Several other actions of the mapping class group are known to be ergodic. According to Masur [31],[32], the action of the mapping class group on the Thurston space of measured foliations MF is ergodic. Indeed, the action on the product MF × MF is ergodic. These actions are analogous to the actions of a lattice subgroup of a semisimple Lie group on the boundary of the corresponding symmetric space (Compare Ivanov [20]). However, these ergodic actions do not have finite invariant measure. Theorem 1.1 seems to be the first geometrically natural example of an effective ergodic action of the mapping class group with finite invariant measure. (Effectiveness of the action is proved in §10.) The referee has pointed out that every group Γ has a “Bernoulli type” ergodic action on {0, 1}^{Γ} which is effective with finite invariant measure.

Related is the Out(π)-action on the Hom(π, G)/G when G = PSL(2, ℝ) (see [14]). One component of this space consists of flat structures on the PSL(2, ℝ)-bundle corresponding to the unit tangent bundle of M. This component consists of embeddings of π onto discrete subgroups of PSL(2, ℝ) and naturally identifies with the Teichmüller space T_{g} of M. The action of Out(π) on this component is properly discontinuous. (Its quotient is the Riemann moduli space M_{g} discussed above.) The proper discontinuity of this action markedly contrasts the ergodicity of Out(π) on the space of SU(2)-representations.

Conjecture 1.7. Let G be a compact Lie group. For each G-invariant symmetric bilinear form B : 𝔤 × 𝔤 ⟶ ℝ, let ω_{B} be the corresponding exterior 2-form on Hom(π, G)/G. Then the subalgebra of the de Rham algebra Ω^{∗}(Hom(π, G)/G) consisting of Out(π)-invariant differential forms on Hom(π, G)/G is the algebra generated by all ω_{B}.

(Although other Out(π)-invariant cohomology classes in

H^{∗}(Hom(π, G)/G)

exist, it seems difficult to represent them by invariant differential forms.)

Zimmer [43] has defined a notion of algebraic hull for smooth ergodic actions on smooth manifolds. This is the smallest algebraic subgroup of the general linear group for which an invariant measurable reduction of structure group exists. Since the action is symplectic the algebraic hull of this action lies in the symplectic group Sp(ℝ^{6g−6}). We conjecture that this algebraic hull actually equals the symplectic group.

Individual elements of Γ_{M} are ergodic on Hom(π, U(1)). In particular a mapping class ϕ ∈ Γ_{M} is ergodic on the torus Hom(π, U(1)) if and only if the induced homology map H_{1}(ϕ) ∈ Sp(ℤ^{2g}) is hyperbolic. Can a cyclic subgroup of Γ_{M} be ergodic on Hom(π, G)/G when G = SU(2)? Another interesting question is whether the action of Γ_{M} on the tangent bundle of Hom(π, G)/G is ergodic. Brown [5, 6] gives examples of hyperbolic mapping classes which do not act ergodically on moduli spaces of the once-punctured torus.

For G abelian, the action of Γ_{M} factors through the homomorphism

Γ_{M} ≅ Out(π) ⟶ Aut(H^{∗}(M; ℤ)) ≅ Sp(ℤ^{2g})

The Torelli subgroup is defined as the kernel of this homomorphism.

Conjecture 1.8. Let M be a closed surface. Then the action of the Torelli subgroup of Γ_{M} on X is ergodic.

The proof of Theorem 1.5 involves a careful analysis of the action of a Dehn twist on X. It is known that the Dehn twists generate Γ. The proof analyzes the functions invariant under certain subgroups generated by Dehn twists. In particular there are noncompact subgroups of Γ whose action on X is not ergodic. Scot Adams has observed that this implies that the action of Γ is not mixing, since mixing is preserved upon passage to noncompact subgroups and mixing implies ergodic.

On the other hand, these actions are weak-mixing: An ergodic action with finite invariant measure of Γ on X is weak-mixing if the diagonal action of Γ on X × X is ergodic. Now

Hom(π, G_{1} × G_{2}) = Hom(π, G_{1}) × Hom(π, G_{2})

and, more generally, if ζ_{1} ∈ Z(G_{1}) and ζ_{2} ∈ Z(G_{2}), then the twisted moduli spaces satisy

Hom_{(ζ_{1},ζ_{2})}(π, G_{1} × G_{2}) = Hom_{ζ_{1}}(π, G_{1}) × Hom_{ζ_{2}}(π, G_{2})

The character varieties decompose similarly. Since the class of groups G satisfying the hypotheses of Theorem 1.5 is closed under Cartesian products, Γ_{M} is ergodic on X × X and hence Γ_{M} is weak-mixing on X.

As the referee observed, the conclusion of weak-mixing is interesting given the structure of the proof. The proof reduces the action of Γ_{M} to irrational rotations on a torus, actions which are ergodic but not weak-mixing. (Most proofs of weak-mixing involve some hyperbolicity, which is noticeably absent here. In fact, one conclusion of [5, 6] is that “hyperbolic” mapping classes need not display hyperbolicity in their action on the moduli spaces.) What seems to be important here is the size and complexity of the group Γ_{M}.

1.6. Outline of proof. Let α ⊂ M be a simple closed curve corresponding to an element α ∈ π_{1}(M). Let G = SU(2). The map

Hom(π, G) ⟶ [−2, 2]
ρ ⟼ tr(ρ(α))

is invariant under inner automorphisms of G acting on Hom(π, G) and thus defines a function s_{α} on the quotient

X(M) = Hom(π, G)/G

which is a symplectic manifold (up to a singular set of measure zero). The corresponding Hamiltonian flow ξ_{α} has closed orbits on X(M), almost every one of which is a circle. By [13], this flow lifts to a flow on Hom(π, G) which can be described as follows: Split M along α to obtain a surface-with-boundary M|α with an identification map:

i_{α} : M|α ⟶ M

Reidentify the representation ρ over π_{1}(M|α) (which is mapped injectively into π_{1}(M)) by a one-parameter subgroup of G centralizing ρ(α).

Also associated to α is the Dehn twist τ_{α} ∈ Γ_{M} about α. The Dehn twist is represented by a diffeomorphism of M supported in a tubular neighborhood of α in M. The induced map on X(M) leaves invariant the function s_{α}. Almost every level set of s_{α} is a closed orbit of ξ_{α}, upon which τ_{α} acts by an irrational rotation. An irrational rotation of the circle is ergodic with respect to Lebesgue measure. Disintegrating the measure on X(M) over s_{α} gives measures on the level sets of s_{α} invariant under τ_{α}. In particular the action of τ_{α} on almost every level set of s_{α} is ergodic with respect to this measure.

These constructions extend easily to a collection of disjoint simple loops α = {α_{1}, …, α_{n}} on M. Assume no α_{i} is null-homotopic and no two α_{i} are homotopic. The map

s_{α} : Hom(π, G)/G ⟶ [−2, 2]^{n}

[ρ] ⟼ [tr(ρ(α_{1}) ; ⋮ ; tr(ρ(α_{n})]

is a moment map for a Hamiltonian ℝ^{n}-action ξ_{α} on X(M). For each i = 1, …, n let τ_{i} ∈ Γ_{M} correspond to the Dehn twist about α_{i}. The subgroup Γ_{α} ⊂ Γ_{M} generated by τ_{1}, …, τ_{n} is free abelian with basis τ_{1}, …, τ_{n}. Let X′(M|α) denote the subset of X(M|α) such that χ(α_{+}) = χ(α_{−}) where α_{+}, α_{−} ∈ π_{1}(M|α) correspond to i_{α}^{−1}(α) ⊂ ∂M|α. The identification map i_{α} induces a map

(i_{α})_{∗} : X(M) ⟶ X(M|α)

with image X′(M|α). Disintegrate the measure μ corresponding to symplectic volume on X(M) as a direct integral

μ = ∫_{[−2,2]^{n}} μ_{t} dμ_{α}(t)

of measures μ_{t} on the level sets s_{α}^{−1}(t) with respect to the pushforward measure μ_{α} = (s_{α})_{∗}μ on [−2, 2]^{n}..

The first ingredient of the proof involves a detailed analysis of the cases when M is a one-holed torus or a four-holed sphere. For a surface with nonempty boundary the restriction map ∂^{#} : X(M) ⟶ X(∂M) associates to an equivalence class [ρ] of a representation ρ : π_{1}(M) ⟶ G the product

{[ρ|_{π_{1}(∂_{i}M)}]}_{1≤i≤m} ∈ ∏_{i=1}^{m} X(∂_{i}M) = X(∂M)

taken over the m components ∂_{1}(M), … ∂_{m}(M) of ∂M. The boundary data in X(∂M) is recorded by a collection C = (C_{1}, …, C_{m}) of conjugacy classes in G. The fibers of ∂^{#} are the relative character varieties X_{C}(M) of M. When M is either a one-holed torus or a four-holed sphere, X_{C}(M) are two-dimensional; in trace coordinates they are cubic surfaces in affine space. If γ ⊂ M is a simple closed curve, and τ is the corresponding Dehn twist, then the ergodic decomposition of τ acting on X_{C}(M) is given by the trace mapping

s_{γ} : X(M) ⟶ [−2, 2]

This trace mapping exhibits the cubic surface X_{C}(M) as a pencil of ellipses s_{γ}^{−1}(x), for x ∈ [−2, 2]. Let γ_{1}, γ_{2} ⊂ M be a pair of homotopically nontrivial curves and let F_{1}, F_{2} be the corresponding pair of foliations of X_{C}(M) by ellipses. If γ_{1} and γ_{2} intersect, then F_{1} and F_{2} span all of X(M) in the following sense: every measurable subset of X(M) which is a both a union of leaves of F_{1} and a union of leaves of F_{2} must be null or conull.

The second ingredient in the proof is the following: Theorem 1.9. Let α be a maximal collection of homotopically distinct and nontrivial simple closed curves on M. For almost every t ∈ [−2, 2]^{n}, s_{α}^{−1}(t) is an orbit of the Hamiltonian ℝ^{n}-action diffeomorphic to an n-torus and Γ_{α} is ergodic on s_{α}^{−1}(t). For every Γ_{α}-invariant measurable function f : X(M) ⟶ ℝ, there exists a measurable function f′ : X′(M|α) ⟶ ℝ such that f = f′ ∘ (i_{α})_{∗} almost everywhere.

The functions s_{α} define coordinates on X(M). By applying Dehn twists about curves α, we show that any invariant measurable function is a function only of the coordinates s_{α}.

The two ingredients combine to prove the theorem. Consider a maximal collection α = {α_{1}, …, α_{N}} of null-homotopic simple closed curves, no two of which are homotopic. α corresponds to a “pants decomposition” of M, since each component of M − α is a “pants,” a three-holed sphere. The total number of curves in the decomposition equals

N = 3g − 3 + 2m

where g equals the genus of M and m equals the number of boundary components. By Theorem 1.9, a Γ_{M}-invariant function depends only on the trace functions s_{α_{i}} for i = 1, …, N.

Eliminate each variable s_{α_{i}} individually: For each α_{i}, consider an embedding of a surface A, homeomorphic to a one-holed torus or to a four-holed sphere, such that α_{i} ⊂ int(A) and ∂A ⊂ α. The embedding f : int(A) ↪ M induces an embedding of mapping class groups f_{∗} : Γ_{A} ⟶ Γ_{M}. The corresponding mapping of character varieties X(M) ⟶ X(A) is Γ_{A}-equivariant respecting f_{∗}. Since Γ_{A} is ergodic on the relative character varieties X_{C}(A), an invariant function depends on s_{α_{1}}, …, s_{α_{i−1}}, s_{α_{i+1}}, …, s_{α_{N}}: a Γ_{M}-invariant subset is (up to a null set) a union of level sets of s_{α_{i}}. This argument applied to each α_{i} as i = 1, …, N, implies that a Γ_{M}-invariant function is almost everywhere constant, or alternatively that a Γ_{M}-invariant subset is either null or conull.

1.7. Acknowledgements. Conversations with S. Adams, R. Benedetto, M. Boyle, S. Donaldson, D. Fried, H. Furstenberg, D. Heicklen, L. Jeffrey, D. Kazhdan, C. McMullen, J. Millson, W. Neumann, D. Rudolph, G. Stuck, J. Weitsman, S. Wolpert, and R. Zimmer were useful in understanding this subject. I especially wish to thank S. Adams and G. Stuck for comments on preliminary versions. Critical comments by S. Adams were particularly useful in clarifying ideas in the proof. I apologize for the long delay in this manuscript, the results of which were obtained in 1982 (and announced in [12]). Finally I would like to thank the referee for several helpful suggestions.

2. Preliminaries

2.1. Moduli spaces. In this section we summarize, for the reader’s convenience, basic facts about moduli spaces of representations. For details and proofs we refer to [7], [12],[16], [27],[29], [34],[35],[38].

Let K be a compact Lie group. K may be given the structure of a real algebraic Lie group, that is, a Zariski-closed subgroup of GL(n, ℝ). Let K_{ℂ} denote its complexification, which is an algebraic subgroup of GL(n, ℂ). For any finitely generated group π, the set Hom(π, K_{ℂ}) has the structure of a complex affine variety, that is, the set of complex points of an affine scheme defined over ℤ. The real variety Hom(π, K) is the subset consisting of ℝ-points.

Let G be a reductive algebraic group defined over a field k; for our purposes G will either be K or K_{ℂ} and k = ℝ or k = ℂ, respectively. An explicit description of Hom(π, G) involves a choice of generators. If γ_{1}, …, γ_{N} ∈ π is a generating set, then the evaluation map

Hom(π, G) ⟶ G^{N}
ρ ⟼ (ρ(γ_{1}), …, ρ(γ_{N}))

is injective and its image consists of all

(g_{1}, …, g_{N}) ∈ G^{N}

satisfying

R(g_{1}, …, g_{N}) = I (2.1)

for each relation R satisfied by the generating set γ_{1}, …, γ_{N}:

R(γ_{1}, …, γ_{N}) = I

The equations (2.1) are polynomial equations in the matrix entries of (g_{1}, …, g_{N}) and thus the image of the evaluation map is Zariski-closed in G^{N}. Furthermore two different generating sets define isomorphic affine variety structures on Hom(π, G).

The Cartesian product Aut(π)×Aut(G) acts on Hom(π, G) by composition, preserving the algebraic structure. In particular G acts on Hom(π, G) by inner automorphisms. In general the quotient space Hom(π, G)/G is not an algebraic variety. Rather there is a G-invariant Zariski-open subset Hom(π, G)^{−} (consisting of the semistable points) such that the orbit space

Hom(π, G)^{−}/G

is the set of k-points of an affine variety

Hom(π, G)//G

(the character variety) whose coordinate ring equals the subring of Ginvariant functions in the coordinate ring of Hom(π, G). In practice, Hom(π, G) will consist of semisimple (“completely reducible”) representations. In the case of a compact group K, every representation is semisimple and the map Hom(π, K)/K ⟶ Hom(π, K_{ℂ})/K_{ℂ} induced by inclusion K ⟶ K_{ℂ} is a bijection onto a component of the set of ℝ-points.

Since the subgroup Inn(π) of Aut(π) consisting of the inner automorphisms π → π preserves each G-orbit on Hom(π, G), it acts trivially on the quotient space

Hom(π, G)/G.

Hence the action of Aut(π) induces an action of the quotient group Out(π) = Aut(π)/ Inn(π) on Hom(π, G)/G.

When π is the fundamental group of a closed orientable surface of genus g > 1, there is the presentation

π = ⟨A_{1}, B_{1}, …, A_{g}, B_{g} | [A_{1}, B_{1}] … [A_{g}, B_{g}] = I⟩

where [X, Y] = XY X^{−1}Y^{−1}. Then Hom(π, G) identifies with the set of all

(a_{1}, b_{1}, …, a_{g}, b_{g}) ∈ G^{2g}

satisfying the equation (in G):

[a_{1}, b_{1}] … [a_{g}, b_{g}] = I (2.2)

2.2. Twisted moduli spaces. Consider an element ζ_{0} ∈ G in the center Z(G) of G. For simplicity suppose that Z(G) is discrete. As π admits the presentation

π = ⟨A_{1}, B_{1}, … A_{g}, B_{g} | [A_{1}, B_{1}] … [A_{g}, B_{g}] = I⟩

representations

ϕ ∈ Hom(π, G)

correspond to 2g-tuples (a_{1}, b_{1}, …, a_{g}, b_{g}) ∈ G^{2g} satisfying (2.2) by

a_{i} = ϕ(A_{i}), b_{i} = ϕ(B_{i})

A ζ_{0}-twisted representation is defined to be a 2g-tuple

(a_{1}, b_{1}, …, a_{g}, b_{g}) ∈ G^{2g}

satisfying

[a_{1}, b_{1}] … [a_{g}, b_{g}] = ζ_{0}.

Denote the subspace of G^{2g} consisting of ζ_{0}-twisted representations by Hom_{ζ_{0}}(π, G). Since ζ_{0} is central, the diagonal action of G by inner automorphisms preserves Hom_{ζ_{0}}(π, G) and we denote the quotient by Hom_{ζ_{0}}(π, G)/G.

Alternatively Hom_{ζ_{0}}(π, G) relates to the quotient group G/Z(G) as follows. The quotient homomorphism Π : G ⟶ G/Z(G) induces a map

Π_{∗} : Hom_{ζ_{0}}(π, G) ⟶ Hom(π, G/Z(G)) (2.3)

whose image is open and closed in Hom(π, G/Z(G)). The restriction of Π_{∗} to Hom_{ζ_{0}}(π, G) is a covering space. Every fiber of this map admits a simply transitive action of the abelian group Hom(π, Z(G)) under multiplication of homomorphisms: if ζ ∈ Hom(π, Z(G)) and ϕ ∈ Hom_{ζ_{0}}(π, G), then

ζ · ϕ : γ ↦ ζ(γ)ϕ(γ)

also lies in Hom_{ζ_{0}}(π, G). Therefore (2.3) is a principal Hom(π, Z(G))- bundle.

2.3. Ergodic decomposition. The basic tool for proving ergodicity is the following general lemma from measure theory:

Lemma 2.1. Let (X, 𝔅, μ) be a standard measure space and Y, Z be Borel spaces and F : X ⟶ Y a measurable map. Suppose that Γ is a group of automorphisms of (X, 𝔅, μ) such that F is Γ-invariant. Let μ_{y} be the measures on F^{−1}(y) obtained by disintegrating μ over F. Suppose that the action of Γ is ergodic on (F^{−1}(y), μ_{y}) for F_{∗}μalmost every y ∈ Y. Let h : X ⟶ Z be an Γ-invariant measurable function. Then there exists a measurable function H : Y ⟶ Z such that h = H ∘ F almost everywhere.

The proof follows readily from disintegration theory, as in Furstenberg [10] (Theorem 5.8). The decomposition of the measure μ on X into Γ-ergodic components is precisely the disintegration of μ over h. (See also [11], §3.5 of Rudolph [40], Varadarajan [41],§4 and also §13 of [8].) Recall that this means that there is a family μ_{y} of measures, parametrized by y ∈ Y, such that μ_{y} is a measure on F^{−1}(y) and a measure F_{∗}μ (the pushforward of μ by F), such that for every measurable function f on Y,

∫_{X} fdμ = ∫_{Y} (∫_{F^{−1}(y)} fdμ_{y}) d(F_{∗}μ)

In that case μ is said to be the direct integral of the measures μ_{y} and is written

μ = ∫_{Y} μ_{y}d(F_{∗}μ)

2.4. Surfaces with boundary. Let M be a compact surface of genus g ≥ 0 with boundary components C_{1}, …, C_{m}. Suppose that

χ(M) = 2 − 2g − m < 0.

For each j = 1, …, m choose a basepoint x_{j} ∈ C_{j}. The inclusion C_{j} ↪ M induces a monomorphism π_{1}(C_{j}, x_{j}) ⟶ π_{1}(M, x_{j}) and so we regard each fundamental group π_{1}(C_{j}) as an infinite cyclic subgroup of π_{1}(M). Choose a basepoint x_{0} in M and paths from x_{0} to x_{j}. Denote by σ_{j} the corresponding generator of π_{1}(C_{j}) ⊂ π_{1}(M). We systematically confuse simple closed curves on M with based loops and their homotopy classes in the fundamental group. In particular choose a collection

σ_{1}, …, σ_{m}, α_{1}, β_{1}, …, α_{g}, β_{g}

of simple loops such that the intersection of any two of them is x_{0}. Then π_{1}(M) admits a presentation:

π_{1}(M) ≅ ⟨σ_{1}, …, σ_{m}, α_{1}, β_{1}, …, α_{g}, β_{g} | σ_{1} … σ_{m}[α_{1}, β_{1}] … [α_{g}, β_{g}] = I⟩

For a compact surface M with nonempty boundary, its mapping class group Γ_{M} is the group of connected components of the group of diffeomorphisms of M whose restriction to ∂M is the identity. That is, Γ_{M} consists of equivalence classes of diffeomorphisms of M which pointwise fix the boundary. Two such diffeomorphisms are equivalent if they are joined by an isotopy fixing ∂M pointwise. Alternatively, Γ_{M} is the subgroup of Out(π) preserving the conjugacy class of each π_{1}(C_{j}) ⊂ π_{1}(M).

Let G = SU(2) and write X(M) = Hom(π, G)/G. Let μ be the measure on X(M) induced by the symplectic structure. Consider the map

∂_{j}^{#} : X(M) ⟶ ℝ
[ρ] ⟼ tr(ρ(σ_{j}))

and let

∂^{#} = (∂_{1}^{#}, …, ∂_{m}^{#}) : X(M) ⟶ ℝ^{m}.

For almost every t ∈ ℝ^{m}, the map q is submersive on ∂^{#}^{−1}(t) and the disintegration of μ with respect to q is a smooth measure μ_{t} on the submanifold ∂^{#}^{−1}(t). Here is the slightly more precise reformulation of Theorem 1.5 in the form which we shall prove:

Theorem 2.2. For each t, the action of Γ_{M} on the fiber ∂^{#}^{−1}(t) is ergodic with respect to μ_{t}. For every Γ_{M}-invariant measurable function f : X(M) ⟶ ℝ, there exists a (Lebesgue-) measurable function f̃ : ℝ^{m} ⟶ ℝ such that f = f̃∘ q almost everywhere.

3. Decompositions of surfaces and character varieties

Let C ⊂ M be a homotopically nontrivial simple closed curve. Let M|C be the surface M split along C, defined as the surface-with-boundary M|C for which there is an identification map i_{C} : M|C ⟶ M satisfying:

• The restriction of i_{C} to i_{C}^{−1}(M − C) (which contains the interior of M|C) is a diffeomorphism;

• i_{C}^{−1}(C) consists of two components C_{+}, C_{−} ⊂ ∂(M|C), to each of which the restriction of i_{C} is a diffeomorphism onto C.

The character variety X(M|C) is “simpler” than X(M) and X(M) is reconstructedfrom X(M|C). There are two cases, depending on whether C is nonseparating (M|C is connected) or C separates (M|C is disconnected), respectively.

3.1. Nonseparating curves. Suppose that M|C is connected. Then M is obtained by gluing two boundary components C_{+}, C_{−} of A = M|C. The identification map i_{C} : A ⟶ M is injective along A − (C_{−} ∪ C_{+}) and is two-to-one along

i_{C}^{−1}(C) = C_{−} ∪ C_{+}

Then π_{1}(M) is obtained from π_{1}(A) by an HNN-construction: π_{1}(M) is the quotient of the free product π_{1}(A) ⨿ ⟨τ⟩ of π_{1}(A) with a cyclic group ⟨τ⟩ ≅ ℤ by the normal subgroup generated by the set

{ι_{−}(γ)^{−1}τι_{+}(γ)τ^{−1} | γ ∈ π_{1}(C)} ⊂ π_{1}(A) ⨿ ⟨τ⟩}

where

ι_{±} : π_{1}(C) ⟶ π_{1}(C_{±}) ⟶^{(i_{C})_{∗}} π_{1}(M)

denote the embeddings induced by inclusion C ↪ C_{±} ↪ M. The new generator τ corresponds to a loop in M which crosses C. Its preimage in A = M|C is an arc α with one endpoint on C_{+} and one endpoint on C_{−}. Choose for a basepoint x_{0} the endpoint of α on C_{−}

Let γ_{−} be the element of π_{1}(A, x_{0}) corresponding to C_{−}. Let γ_{+} denote the element of π_{1}(A, x_{0}) corresponding to the loop α^{−1} ⋆ C_{+} ⋆ α where C_{+} is regarded as a loop based on the endpoint of α on C_{+}. A representation ρ : π_{1}(A) ⟶ G extends to a representation of π_{1}(M) if and only if there exists a t ∈ G such that

ρ(γ_{−}) = tρ(γ_{+})t^{−1} (3.1)

that is, if ρ(γ_{−}) is conjugate to ρ(γ_{+}). Since two elements of G are conjugate if and only if they have the same trace, the image of the restriction map X(M) ⟶ X(A) consists of all [ρ] such that if

tr(ρ(γ_{−})) = tr(ρ(γ_{+})).

The fiber of X(M) ⟶ X(A) identifies with the set of all t ∈ G (corresponding to ρ(τ)) satisfying (3.1). The set of all such t is a rightcoset of the centralizer Z(ρ(γ_{−})). In particular Z(ρ(γ_{−})) acts simply transitively on the fibers of X(M) ⟶ X(A) by the action defined by

ζ · ρ : \begin{cases}α & ⟼ ρ(α) for α ∈ π_{1}(A) \\ γ & ⟼ ρ(γ)ζ \end{cases}

3.2. Separating curves. Let M be a compact surface decomposed into closed subsurfaces A and B along a simple closed curve C = A∩B. Assume that χ(A) < 0 and χ(B) < 0. Then the inclusions

\begin{matrix}C & ⟶ & A \\ ↓ & & ↓ \\ B & ⟶ & M \end{matrix}

induce monomorphisms

π_{1}(C) ⟶ π_{1}(A)
↓  ↓
π_{1}(B) ⟶ π_{1}(M)

and π_{1}(M) is represented as the amalgamated free product

π_{1}(M) = π_{1}(A) ⨿_{π_{1}(C)} π_{1}(B)

corresponding to the above pushout diagram.

Define X(M; A, B, C) as the pullback in the diagram

X(C) ⟵  X(A)
↑  ↑
X(B) ⟵ X(M; A, B, C)

that is, X(M; A, B, C) consists of all pairs ([α], [β]) ∈ X(A) × X(B) such that

[α|_{π_{1}(C)}] = [α|_{π_{1}(C)}] ∈ X(C)

The natural map j : X(M; A, B, C) ⟶ X(M) is surjective: if [ρ] ∈ X(M), for a representation ρ, then the restrictions of ρ to π_{1}(A) (respectively π_{1}(B)) are equivalent to representations ρ_{A} (respectively ρ_{B}) such that [ρ_{A}] ∈ X(A) and [ρ_{B}] ∈ X(B) and ρ_{A}|π_{1}(C) = ρ_{B}|π_{1}(C). Let

([α], [β]) ∈ X(M; A, B, C)

and let ρ be a homomorphism such that

ρ|_{π_{1}(A)} ≡ α, ρ|_{π_{1}(B)} ≡ β.

Elements ζ of the centralizer Z = Z(ρ, C) of ρ(π_{1}(C)) act on the fiber j^{−1}([α], [β]) by the rule:

ζ · ρ : γ ⟼ \begin{cases}α(γ) & for γ ∈ π_{1}(A) \\ ζβ(γ)ζ^{−1} & for γ ∈ π_{1}(B) \end{cases}

where ρ ∈ j^{−1}([α], [β]).

4. Three-holed spheres

The basic building block of surfaces of negative Euler characteristic is the three-holed sphere (“pair-of-pants”). Let A, B, C denote its three boundary components. Its fundamental group admits the geometric presentation

π_{1}(M) = ⟨A, B, C | ABC = 1⟩ (4.1)

The following basic fact concerning representations of its fundamental group is discussed in [14] and the references given there:

Lemma 4.1. Let M be a surface of genus 0 with three boundary components A, B, C. Then the trace map

χ : X(M) ⟶ ℝ^{3}
[ρ] ⟼ [\begin{matrix}tr(ρ(A)) \\ tr(ρ(B)) \\ tr(ρ(C)) \end{matrix}]

identifies X(M) with the set

𝔅 = {(a, b, c) ∈ ℝ^{3} | −2 ≤ a, b, c ≤ 2, a^{2} + b^{2} + c^{2} − abc ≤ 4}

Characters of reducible representations form the boundary

∂𝔅 = {(a, b, c) ∈ ℝ^{3} | −2 ≤ a, b, c ≤ 2, a^{2} + b^{2} + c^{2} − abc = 4}

Since an element of G is determined up to conjugacy by its trace, the mapping χ is the restriction map X(M) ⟶ X(∂M). Lemma 4.1 asserts that this mapping is injective and hence the relative character varieties are trivial. The mapping class group Γ_{M} is trivial. This is the first – and the trivial — case of Theorem 2.2. (For groups with simple factors other than SU(2) and U 1, the relative character variety is nonatomic and hence an alternative method for proving Conjecture 1.3 is required. Compare §9.)

5. The one-holed torus

In this section we consider the case when M is a one-holed torus. Since its fundamental group is free of rank two, Lemma 4.1 gives explicit coordinates for X(M) as in the case of a three-holed sphere. We analyze the action of a Dehn twist on X(M), explicitly examining Theorem 2.2 in these coordinates. Applying Dehn twists about two loops generating π_{1}(M) demonstrates ergodicity of Γ_{M} on the relative character varieties.

5.1. Trace coordinates. Let M be a compact surface of genus 1 with one boundary component, a one-holed torus. Its fundamental group admits the (geometric) presentation

⟨X, Y, K | XY X^{−1}Y^{−1} = K⟩ (5.1)

where K corresponds to the generator of π_{1}(∂M). Its character variety, which parametrizes equivalence classes of representations ρ : π_{1}(M) ⟶ SL(2, ℂ), is isomorphic to affine space ℂ^{3}. Its global coordinates are given by:

x = tr(ρ(X)), y = tr(ρ(Y)), z = tr(ρ(XY)) (5.2)

An extremely important function on X(M) is

k = tr(ρ(K)) = x^{2} + y^{2} + z^{2} − xyz − 2

Characters of representations π_{1}(M) ⟶ SU(2) have coordinates satisfying

−2 ≤ x, y, z ≤ 2, x^{2} + y^{2} + z^{2} − xyz − 2 ≤ 2

and, for fixed k ∈ [−2, 2], comprise the relative character variety

X_{1}(k) = {(x, y, z) ∈ ℝ^{3} | −2 ≤ x, y, z ≤ 2, x^{2} + y^{2} + z^{2} − xyz − 2 = k} (5.3)

We denote the full character variety by

X_{1} = ⋃_{−2≤k≤2} X_{1}(k)

5.2. Action of a Dehn twist. The Dehn twist τ_{X} about X is given by the automorphism

X ↦ X, Y ↦ Y X

which induces the map

[ \begin{matrix}x \\ y \\ z \end{matrix} ] ↦ [ \begin{matrix}x \\ z \\ xz − y \end{matrix} ]

Clearly x and k are invariant functions under this transformation. For fixed values of x, k, the intersection

E_{x,k} := X_{1}(k) ∩ ({x} × ℝ^{2})

= \{x\} × \{(y, z) ∈ ℝ^{2} | \frac{2 + x}{4}(y + z)^{2} + \frac{2 − x}{4}(y − z)^{2} = 2 + k − x^{2}\}

is an ellipse preserved by τ_{X}.

The change of coordinates (which is linear in y, z)

x̃ = x
ỹ = \frac{\sqrt{2 − x} + \sqrt{2 + x}}{2}y + \frac{\sqrt{2 − x} − \sqrt{2 + x}}{2}z
z̃ = \frac{\sqrt{2 − x} − \sqrt{2 + x}}{2}y + \frac{\sqrt{2 − x} + \sqrt{2 + x}}{2}z

transforms E_{x,k} to the circle

E_{x,k} = {(ỹ, z̃) ∈ ℝ^{2} | ỹ^{2} + z̃^{2} = 2 + k − x^{2}}

τ_{X} transforms to the orthogonal rotation of E_{x,k}

τ_{X} : [x̃ ; ỹ ; z̃] ⟼ [\begin{matrix}1 & 0 & 0 \\ 0 & cos(θ) & sin(θ) \\ 0 & − sin(θ) & cos(θ) \end{matrix}] [x̃ ; ỹ ; z̃]

by angle

θ = cos^{−1}(a/2).

In particular whenever cos^{−1}(x/2) is an irrational multiple of π, the automorphism τ_{X} is ergodic on E_{x,k}. Since cos^{−1}(x/2)/π is irrational for almost every x ∈ [−2, 2], a τ_{X}-invariant measurable function f : X_{1} ⟶ ℝ of (x, y, k) must almost everywhere equal a function depending only on x, k.

Similarly, apply Dehn twists τ_{Y} about Y. Then f must be a function of k and y alone. The foliations defined by level sets of x, y span the level set X_{1}(k): every subset of X_{1}(k) which is almost everywhere a union of level sets of x and level sets of y is either null or conull. This establishes Theorem 2.2 in the case of a one-holed torus.

Think of this as “eliminating the variable y.” Let η : X_{1} ⟶ [−2, 2] denote the mapping defined by the coordinate y. Then the function f is almost everywhere constant along the level set η^{−1}(y) for almost every y ∈ [−2, 2]. The measure μ on X_{1} is the direct integral

∫_{[−2,2]} μ_{y}d(η_{∗}μ)

of the disintegrated measures μ_{y} on the fibers η^{−1}(y) with respect to the pushforward measure η_{∗}μ on [−2, 2]. Since τ_{X} is ergodic on η^{−1}(y) with respect to μ_{y}, Lemma 2.1 implies that every τ_{X}-invariant function factors through η.

5.3. The symplectic structure. The symplectic structure on each X_{1}(k) admits a simple expression in trace coordinates (Compare also Wolpert [42]). Clearly the mapping class group acts by polynomial automorphisms of affine space ℂ^{3} and since a complex polynomial with no zeroes is constant, the mapping class group acts by unimodular polynomial automorphisms. In particular the exterior 3-vector field

Ξ = \frac{1}{4π^{2}} \frac{∂}{∂x} ∧ \frac{∂}{∂y} ∧ \frac{∂}{∂z}

is invariant under Γ_{M}. (The coefficient 1/4π^{2} is included, as before, to give an integral cohomology class and to agree with Jeffrey-Weitsman.) Similarly the trace-commutator map k : ℂ^{3} ⟶ ℂ is also Γ_{M}-invariant, and so its exterior derivative

dk = (2x − yz)dx + (2y − zx)dy + (2z − xy)dz

is also Γ_{M}-invariant. The exterior 2-vector field

(5.4)  Ξ · dk = \frac{1}{4π^{2}} ( (2x − yz)\frac{∂}{∂y} ∧ \frac{∂}{∂z}

+ (2y − zx)\frac{∂}{∂z} ∧ \frac{∂}{∂y} (5.5)

+ (2z − xy)\frac{∂}{∂x} ∧ \frac{∂}{∂y} ) (5.6)

is also Γ_{M}-invariant and defines a cosymplectic structure on each relative character variety X_{1}(k).

6. The four-holed sphere

6.1. Trace coordinates. Next consider the four-holed sphere. Like the one-holed torus, its relative character varieties are two-dimensional. The one-holed torus doubly covers a disc with four branch points of order two; this branched covering leads to an embedding of the relative character variety of the one-holed torus in the relative character variety of the four-holed sphere with special boundary data (see §6.3 below). Their mapping class groups are commensurable. Analogous to the case of the one-holed torus, the mapping class group is ergodic on the relative character variety.

The fundamental group of the four-holed sphere admits the (geometric) presentation

⟨A, B, C, D | ABCD = I⟩

where A, B, C, D correspond to the four boundary components. By Magnus [30] (compare also [3],[16], [17], [18]), the seven functions a, b, c, d, x, y, z on X_{0}(a, b, c, d) defined by

tr(A) = a, tr(B) = b, tr(C) = c, tr(D) = d,
tr(AB) = x, tr(BC) = y, tr(CA) = z

satisfy the defining equation

(6.1)  (x^{2} + y^{2} + z^{2} + xyz) =
(ab + cd)x + (ad + bc)y + (ac + bd)z−(a^{2} + b^{2} + c^{2} + d^{2} + abcd − 4)

The character variety of the boundary consists of

(a, b, c, d) ∈ [−2, 2]^{4}

and for fixed boundary data (a, b, c, d), the relative character variety X_{0}(a, b, c, d) consists of all

(x, y, z) ∈ [−2, 2]^{3}

satisfying (6.1). Let Γ be the mapping class group of S. Theorem 2.2 asserts in this case, that for every (a, b, c, d) ∈ [−2, 2]^{4}, the action of Γ on X_{0}(a, b, c, d) is ergodic.

When x ≠ ±2, rewrite equation (6.1) as

(6.2) \frac{2 + x}{4}((y + z) − \frac{(a + b)(d + c)}{2 + x})^{2}+\frac{2 − x}{4}((y − z) − \frac{(a − b)(d − c)}{2 − x})^{2}

= \frac{(x^{2} − abx + a^{2} + b^{2} − 4)(x^{2} − cdx + c^{2} + d^{2} − 4)}{4 − x^{2}}

For fixed x with −2 < x < 2, the left-hand-side of (6.2) is a quadratic function of y, z with positive definite leading term. Since (a, b, x) and (c, d, x) are characters of representations into SU(2), both

x^{2} − abx + a^{2} + b^{2} − 4 < 0

and

x^{2} − cdx + c^{2} + d^{2} − 4 < 0

and the right-hand-side of (6.2) is nonnegative. It is positive whenever the representations of π_{1}(P) and π_{1}(P′) are irreducible. Rewrite (6.2) as

Q_{x}(y − y_{0}(x), z − z_{0}(x)) = \frac{(x^{2} − abx + a^{2} + b^{2} − 4)(x^{2} − cdx + c^{2} + d^{2} − 4)}{4 − x^{2}} (6.3)

where

y_{0}(x) = \frac{2}{4 − x^{2}} (2(ad + bc) − x(ac + bd))
z_{0}(x) = \frac{2}{4 − x^{2}} (2(ac + bd) − x(ad + bc))

and

Q_{x}(η, ζ) = \frac{2 + x}{4}(η + ζ)^{2} + \frac{2 − x}{4}(η − ζ)^{2}
= η^{2} + ζ^{2} − xηζ

is a positive definite quadratic form for all −2 < x < 2. This exhibits X_{0}(a, b, c, d) as a family of ellipses parametrized by (a, b, c, d, x).

6.2. Action of a Dehn twist. The Dehn twist τ_{X} about the separating curve homotopic to AB induces the automorphism of π_{1}(S) defined by:

A ↦ A
B ↦ B
C ↦ ABCB^{−1}A^{−1}
D ↦ ABDB^{−1}A^{−1}

The separating simple closed curves corresponding to AB, BC, and CA transform by:

AB ↦ AB
BC ↦ BABCB^{−1}A^{−1}
CA ↦ ABCB^{−1}

The transformation of X_{0}(a, b, c, d) induced by the Dehn twist τ_{X} preserves the boundary traces (a, b, c, d) and leaves invariant the function x : X_{0}(a, b, c, d) ⟶ [−2, 2] but maps

[\begin{matrix}y \\ z \end{matrix}] ↦ [\begin{matrix}(ad + bc) − x(ac + bd) + x(xy + z)) − y \\ (ac + bd) − (xy + z) \end{matrix}]

which in terms of y_{0}, z_{0} appears as:

[\begin{matrix}y \\ z \end{matrix}] ↦ [\begin{matrix}y_{0}(x) \\ z_{0}(x) \end{matrix}] + [\begin{matrix}x^{2} − 1 & x \\ −x & −1 \end{matrix}] · ([\begin{matrix}y \\ z \end{matrix}] − [\begin{matrix}y_{0}(x) \\ z_{0}(x) \end{matrix}])

This transformation is a rotation of angle 2 cos^{−1}(x/2) on the ellipse E_{x}(a, b, c, d) defined by (6.3) for given (a, b, c, d, x). In particular, for fixed boundary traces (a, b, c, d), and almost every x, the action of τ_{X} on the level set E_{x}(a, b, c, d) is ergodic.

The Dehn twist τ_{Y} about the separating curve homotopic to BC induces the automorphism of π_{1}(S) defined by:

A ↦ BCAC^{−1}B^{−1}
B ↦ B
C ↦ CABCB^{−1}A^{−1}
D ↦ BCDC^{−1}B^{−1}
AB ↦ BCAC^{−1}
BC ↦ BC
CA ↦ CBCAC^{−1}B^{−1}

The corresponding transformation of X_{0}(a, b, c, d) leaves invariant the functions a, b, c, d as well as y. It maps the point with coordinates z, x by:

[\begin{matrix}z \\ x \end{matrix}] ↦ [\begin{matrix}(ba + cd) − y(bd + ca) + y(yz + x)) − z \\ (bd + ca) − (yz + x) \end{matrix}]

Just as for τ_{X}, the transformation induced by τ_{Y} on almost level set of y is ergodic. Hence any function invariant under both τ_{X} and τ_{Y} must be a function only of the boundary traces (a, b, c, d). This establishes 2.2 in the case M is a four-holed sphere.

As in the case of T_{1}, the exterior 2-vector field

\frac{1}{4π^{2}} ((2x + yz − (ab + cd))\frac{∂}{∂y} ∧ \frac{∂}{∂z}
+(2y + zx − (ad + bc))\frac{∂}{∂z} ∧ \frac{∂}{∂y}
+(2z + xy − (ac + bd))\frac{∂}{∂x} ∧ \frac{∂}{∂y} )

defines invariant cosymplectic structures on the relative character variety X_{0}(a, b, c, d) for each (a, b, c, d).

6.3. Relationship with the one-holed torus. The character varieties of the one-holed torus T_{1} and the four-holed sphere S_{4} are intimately related by the elliptic involution on the torus. Consider the presentation (5.1) of π_{1}(T_{1}). Let ι denote the elliptic involution of T_{1}, which induces the automorphism of π_{1}(T_{1}):

ι_{∗}(X) = X^{−1},  ι_{∗}(Y) = Y^{−1},  ι_{∗}(K) = (Y X)^{−1}K(Y X)

Let O denote the quotient orbifold: the surface underlying O is a 2disc. O has three singularities, each a branch point of order two. Its fundamental group admits the presentation

π_{1}(O) = ⟨A′, B′, C′, D′ | (A′)^{2} = (B′)^{2} = (C′)^{2} = A′B′C′D′ = I⟩

and the covering projection ϕ : T_{1} ⟶ O induces the homomorphism ϕ_{∗} of orbifold fundamental groups

X ⟼ A′B′, Y ⟼ B′C′

and ϕ_{∗}(K) = (C′B′A′)^{2} = A′D′^{2}A′^{−1}.

Every semistable representation ϕ : π_{1}(T_{1}) ⟶ PSL(2, ℂ) arises from a representation π_{1}(O) ⟶ PSL(2, ℂ). (Recall ϕ is semistable if the image of any lift ϕ̃ : π_{1}(T_{1}) ⟶ SL(2, ℂ) is a direct sum of irreducible representations. Compare §2.1.) Therefore the representation variety of π_{1}(T_{1}) embeds in the representation variety of π_{1}(O). A lift ϕ̃ always exists, since π_{1}(T_{1}) is free; however a representation of π_{1}(O) to SL(2, ℂ) will almost never exist, due to 2-torsion. Specifically, if ϕ̃: π_{1}(O) ⟶ SL(2, ℂ) is a homomorphism, then

ϕ̃(A), ϕ̃(B), ϕ̃(C)

have order two in SL(2, ℂ) and must equal ±I; this gives very few representations.

To embed the full character variety of π_{1}(T_{1}) inside a character variety related to π_{1}(O), consider the following ℤ/2-central extension π̃ ⟶ π_{1}(O):

π̃ = ⟨Ã, B̃, C̃, D̃, ζ | [Ã, ζ] = [B̃, ζ] = [C̃, ζ] = ÃB̃C̃D̃ = ζ^{2} = I,
Ã^{2} = B̃^{2} = C̃^{2} = ζ⟩

The center of π̃ is generated by ζ. The quotient homomorphism π̃ ⟶ π_{1}(O) maps

Ã ↦ A
B̃ ↦ B
C̃ ↦ C
D̃ ↦ D

The homomorphism Ψ : π_{1}(T_{1}) ⟶ π̃ defined by:

Ψ(X) = ÃB̃, Ψ(Y) = B̃C̃

maps K to ζÃD̃^{2}Ã^{−1} Furthermore if ρ : π_{1}(T_{1}) ⟶ SL(2, ℂ) is a semistable representation, then there exists a representation ρ̃: π̃ ⟶ SL(2, ℂ) such that ρ = ρ̃ ∘ Ψ. (Compare Theorem 10.2.)

Consider an irreducible representation ρ̃: π̃ ⟶ SL(2, ℂ). Then ρ̃(ζ) is a central element of SL(2, ℂ) (Schur’s lemma), either ±I. If ρ̃(ζ) = I, then, as mentioned above, ρ̃ maps the generators Ã, B̃, C̃ to ±I, and the representation is reducible, a contradiction. Hence ρ̃(ζ) = −I and ρ̃(Ã), ρ̃(B̃), and ρ̃(C̃) are elements of SL(2, ℂ) having order four. Thus

tr(ρ̃(Ã)) = tr(ρ̃(B̃)) = tr(ρ̃(C̃)) = 0

Let Φ : π_{1}(S) ⟶ π̃ denote the homomorphism defined by

Φ(A) = Ã
Φ(B) = B̃
Φ(C) = C̃
Φ(D) = D̃

then the character (ã, b̃, c̃, d̃; x̃, ỹ, z̃) of ρ̃ ∘ Φ satisfies

ã = b̃ = c̃ = 0

Applying (6.1), the character variety has coordinates d̃, x̃, ỹ, z̃ given by

tr(ρ̃(D̃)) = d̃, tr ρ̃(ÃB̃) = x̃, tr ρ̃(B̃C̃) = ỹ, tr ρ̃(C̃Ã) = z̃

satisfying

x̃^{2} + ỹ^{2} + z̃^{2} + x̃ỹz̃ = 4 − d̃^{2}

There is an isomorphism between the character varieties of π_{1}(T_{1}) and π̃ as follows. The coordinates x, y, z, k on X(π_{1}(T_{1})) given by (5.2) relate to the coordinates x̃, ỹ, z̃, d̃ by

x = tr(ϕ(X)) = tr(ϕ̃ ∘ Ψ(X)) = tr(ϕ̃ ∘ Ψ(ÃB̃)) = x̃,
y = tr(ϕ(Y)) = tr(ϕ̃ ∘ Ψ(Y)) = tr(ϕ̃ ∘ Ψ(B̃C̃)) = ỹ,
z = tr(ϕ(XY)) = tr(ϕ̃ ∘ Ψ(Z)) = tr(ϕ̃ ∘ Ψ(ζÃC̃)) = −z̃,
k = tr(ϕ(K)) = tr(ϕ̃ ∘ Ψ(K)) = tr(ϕ̃ ∘ Ψ(ζÃ(D̃)^{2}Ã^{−1})) = −(d^{2} − 4)

Thus the relative character varieties of the one-holed torus T_{1} embeds in the relative character varieties of a four-holed sphere S.

7. Pants decompositions

The Fenchel-Nielsen coordinates on Teichmüller space T_{g} describe T_{g} explicitly as a cell. This description starts from a decomposition of M into pants, (three-holed spheres). The N curves in the decomposition define a map from T_{g} to ℝ^{N} the fibers of which are submanifolds of dimension N. Indeed, as shown by Wolpert [42], the map T_{g} ⟶ ℝ^{N} is a moment map for a Hamiltonian ℝ^{N}-action and its fibers are precisely the orbits. (See Abikoff [1], Wolpert [42] and Ratcliffe [39],§9.)

An analogous picture for X(M) has been extensively studied in a series of papers by Jeffrey-Weitsman [23], [24],[25]. A pants decomposition P of M determines a map X(M) ⟶ ℝ^{N} whose generic fiber is an N-torus. While global topological coordinates are impossible since X(M) is compact, as a measure space X(M) is a T^{N}-bundle over an N-cell.

M is a compact surface of genus g with m boundary components. Its Euler characteristic equals χ(M) = 2 − 2g − m. Decompose M into pants, surfaces of genus 0 with three boundary curves each. Denote by P the collection of pants in the decomposition; since the Euler characteristic of a pants equals −1, the cardinality of P is 2g − 2 + m. The union B of all ∂P_{i} (i = 1, …, 2g − 2 + m) consists of

N = 3g − 3 + 2m

disjoint simple closed curves. (If χ(M) ≤ −2, then we may assume that each union of an adjacent pair of subsurfaces is itself a genus 0 subsurface with four boundary components.)

7.1. The character mapping of a pants decomposition. For each b_{i} ∈ B, consider the mapping

s_{i} : X(M) ⟶ ℝ
[ρ] ⟼ tr ρ(b_{i})

and let s_{P} denote the mapping

X(M) ⟶ ℝ^{N}

defined by s_{P} = (s_{1}, …, s_{N}).

Lemma 7.1. The image ℑ_{P} of s_{P} consists of all t ∈ [−2, 2]^{N} such that

t_{i}^{2} + t_{j}^{2} + t_{k}^{2} − t_{i}t_{j}t_{k} ≤ 4 (7.1)

for each P_{l} ∈ P such that ∂P_{l} = {b_{i}, b_{j}, b_{k}}.

Proof. The character (t_{i}, t_{j}, t_{k}) of the restriction of ρ to π_{1}(P_{l}) lies in X(P_{l}) and so by Lemma 4.1, satisfies (7.1). Conversely suppose that t ∈ [−2, 2]^{N} satisfies (7.1) for each triple (i, j, k) corresponding to a pants P_{l} ∈ P. Choose a representation ρ_{l} : π_{1}(P_{l}) ⟶ G with character (t_{i}, t_{j}, t_{k}); then for each b_{i} = ∂P_{l} ∩ ∂P_{q}, there is g_{i} such that ρ_{l}(b_{i}) = g_{i}ρ_{q}(b_{i})g_{i}^{−1}. Continue this for all the pants in P.

Apply Lemma 4.1 to the mapping s_{P}. The equivalence class of a representation of each π_{1}(P_{l}) is determined by the three boundary traces t_{i}, t_{j}, t_{k}. The fiber s_{P}^{−1}(t) corresponds to the various amalgamations of the representations of the π_{1}(P_{l}) and π_{1}(P_{q}), where P_{l} and P_{q} are adjacent subsurfaces in the pants decomposition. Let

b_{j} = ∂P_{l} ∩ ∂P_{q} ∈ B;

then the inclusions b_{j} ↪ P_{l} and b_{j} ↪ P_{q} induce monomorphisms π_{1}(b_{j}) ⟶ π_{1}(P_{l}) and π_{1}(b_{j}) ⟶ π_{1}(P_{l}) respectively. Denote the elements of π_{1}(P_{l}) and π_{1}(P_{q}) corresponding to a generator of π_{1}(b_{j}) by γ_{j} as well. By van Kampen’s theorem, the fundamental group of P_{l} ∪ P_{q} admits the decomposition

π_{1}(P_{l} ∪ P_{q}) = π_{1}(P_{l}) ⨿_{γ_{j}} π_{1}(P_{q}).

Let ρ_{l} and ρ_{q} be representations of π_{1}(P_{l}) and π_{1}(P_{q}) respectively in G such that ρ_{l}(γ_{j}) = ρ_{q}(γ_{j}). Elements of the centralizer Z(ρ(γ_{j})) of ρ(γ_{j}) determine equivalence classes [ρ] of representations ρ : π_{1}(P_{l} ∪ P_{q}) ⟶ G such that [ρ] restricts to [ρ_{l}] on π_{1}(P_{l}) and to [ρ_{q}] on π_{1}(P_{q}) respectively. For every ζ ∈ Z(ρ(γ_{j})),

γ ⟼ \begin{cases}ρ(γ) & for γ ∈ π_{1}(P_{l}) \\ ζρ(γ)ζ^{−1} & for γ ∈ π_{1}(P_{q}) \end{cases}

defines a representation

ζ · ρ : π_{1}(P_{l} ∪ P_{q}) ⟶ G (7.2)

Moreover (7.2) defines an action of Z(ρ(γ_{j})) on X(P_{l} ∪ P_{q}). This action is trivial on the center Z(G) = {±I}. The resulting action of Z(ρ(γ_{j}))/Z(G) is effective if the restrictions of ρ to π_{1}(P_{l}) and π_{1}(P_{q}) are irreducible. If ρ(γ_{j}) ≠ ±I, then Z(ρ(γ_{j})) is the unique one-parameter subgroup (a maximal torus in SU(2)) containing ρ(γ_{j}). Thus the product over B

∏_{i=1}^{N} Z(ρ(γ_{i})) ≈ T^{N}

is isomorphic to an N-torus. The resulting T^{N}-action on the fiber s_{P}^{−1}(t) is simply transitive for almost every t ∈ ℑ_{P}.

7.2. Automorphisms of pants decompositions. The stabilizer Γ_{P} of P in Γ_{M} is generated by the Dehn twists τ_{i} about b_{i}. This free abelian group has basis τ_{1}, …, τ_{N}. Evidently s_{P} : X(M) ⟶ ℑ_{P} is Γ_{P}-invariant. We summarize the preceding results:

Lemma 7.2. Let t ∈ ℑ_{P}. Suppose that t_{j} ≠ ±2 for j = 1, …, N. Let θ_{j} = cos^{−1}(t_{j}/2). Then there is a Γ_{P}-equivariant homeomorphism

h : s_{P}^{−1}(t) ⟶ T^{N}

such that for ξ ∈ s_{P}^{−1}(t), and for all (n_{1}, …, n_{N}) ∈ ℤ^{n},

h : τ_{1}^{n_{1}} … τ_{N}^{n_{N}} ξ ⟼ [ e^{in_{1}θ_{1}}h_{1} ; ⋮ ; e^{in_{N} θ_{N}} h_{N} ]

where

h(ξ) = [ h_{1} ; ⋮ ; h_{N} ]

Furthermore h maps the measure on s_{P}^{−1}(t) obtained by disintegrating the symplectic volume on X over the map s_{P} : X ⟶ ℝ^{N} to a multiple of Haar measure on T^{N}.

The mapping s_{P} : X(M) ⟶ ℝ^{N} is a moment mapping for an ℝ^{N}-action (compare Goldman [13], Jeffrey-Weitsman [23]). It is a submersion except on the subset consisting of [ρ] ∈ X(M) such that s_{j}(ρ) = ±2 for some j = 1, …, N. Its orbits are tori of dimension N. The isotropy group of the orbit in the fiber s_{P}^{−1}(t) is the lattice

{(n_{1} sin^{−1}(t_{1}/2), …, n_{N} sin^{−1}(t_{N}/2)) | n_{1}, …, n_{N} ∈ ℤ}

After discarding from X(M) a null set, we obtain a set X′(M) such that the restriction s_{P} : X′(M) ⟶ [−2, 2]^{N} satisfies:

• The fibers s_{P}^{−1}(t) are all N-dimensional tori;

• The action of Γ_{P} on s_{P}^{−1}(t) is ergodic. Specifically let

X_{ℚ} = \{[ρ] ∈ X(M) | \frac{cos(s_{j}([ρ]))}{2} ∈ ℚ\}

For each [ρ] ∉ X_{ℚ}, the mapping s_{P} is submersive (by the above paragraph) and hence the fiber s_{P}^{−1}(t), where t = s_{P}([ρ]) is a smooth manifold of dimension N invariant under the locally free Hamiltonian ℝ^{N}- action. Since s_{P}^{−1}(t) is compact, this action is transitive and the fiber is a torus. Furthermore, since [ρ] ∉ X_{ℚ}, the action of Γ_{P} is a Cartesian product of irrational rotations. Therefore the action of Γ_{P} on s_{P}^{−1}(t) is ergodic. Since X_{ℚ} is a countable union of lower-dimensional subvarieties and the singular locus (which has zero measure), X_{ℚ} has measure zero. We therefore replace X(M) by the Γ_{P}-invariant subset X′(M) = X(M) − X_{ℚ} of full measure. Applying Lemma 2.1,

Lemma 7.3. Let f : X(M) ⟶ ℝ be a Γ_{P}-invariant measurable function. Then there exists a function F : ℑ_{P} ⟶ ℝ such that f = F ∘ s_{P} almost everywhere.

Theorem 1.9 follows readily since every system of curves α can be extended to a system of curves B arising from a pants decomposition.

7.3. Relation to the work of Jeffrey-Weitsman. Jeffrey and Weitsman consider the angular variable cos^{−1}(s_{j}/2) rather than the trace s_{j} itself. (See [23] and [24],§3.10.) The advantage of their approach is that the periods of the corresponding Hamiltonian flows are all equal. Hence these flows generate commuting circle actions. They combine to define an action of a compact Lie group (a torus). The disadvantage is that these actions are only defined on a dense open subset. Since the set upon which these periodic Hamiltonians are not defined has zero measure, the approach of Jeffrey and Weitsman would be perfectly appropriate here. The angular coordinates of [23],[24] are the compact analogue of length displacement functions on Teichmüller space. (Compare §1.12, §4.11 of [13]).

We can explicitly determine the pushforward (s_{P})_{∗}μ of the Liouville measure μ associated to the symplectic structure on X(M). To be consistent with Jeffrey-Weitsman [23], [24], [25], we consider the symplectic structure associated with the bilinear form

B : 𝔰𝔲(2) × 𝔰𝔲(2) ⟶ ℝ
(X, Y) ⟼ \frac{1}{4π^{2}} tr(XY)

which has integral cohomology class in X(M). Let γ ∈ π correspond to a homotopically nontrivial simple loop. Then the Hamiltonian flow corresponding to the function

h_{γ} : X(M) ⟶ ℝ
[ρ] ⟼ \frac{1}{π} cos^{−1} (\frac{tr(ρ(γ))}{2})

has period 1 (respectively 1/2) if γ represents a nonseparating (respectively separating) simple closed curve.

Define h_{P} : X(M) ⟶ ℝ^{N} by

h_{P}([ρ]) = (h_{b_{1}}, …, h_{b_{N}})

(Jeffrey and Weitsman consider this moment map and their work motivates our discussion.) The pushforward (h_{P})_{∗}(μ) then equals 2^{−r} times Euclidean Lebesgue measure on ℝ^{n}, where r equals the number of separating curves in B. By computing the derivative of h_{c}, the pushforward of μ under s_{P} is:

(s_{P})_{∗}(μ) = 2^{r}(2π)^{N} ∏_{i=1}^{N}(4 − s_{i}^{2})^{−1/2}μ(s)

where μ(s) is Euclidean Lebesgue measure on ℝ^{N}.

8. Conclusion of the proof

The proof of Theorem 2.2 concludes by considering Dehn twists about curves which cross curves in the pants decomposition. The reduction associated with a pants decomposition P combines with the analysis of the one-holed torus and four-holed spheres to deduce ergodicity of all of Γ_{M} as follows. Suppose that f : X(M) ⟶ ℝ is a measurable function invariant under Γ_{M}. Since f is invariant under the subgroup Γ_{P} ⊂ Γ_{M}, it follows from Lemma 7.3 that f depends only on the traces s_{1}, …, s_{N} of the curves in B. Precisely, there is a function F defined on the image of s_{P} such that

f([ρ]) = F (s_{1}([ρ]), …, s_{N}([ρ]))

The choice of the pants decomposition P is really quite arbitrary. We next apply Dehn twists not preserving the pants decomposition, to show that the function f is actually independent of the functions s_{j}. Choose a curve b_{j} ∈ B. Let P, P′ ∈ P be the subsurfaces whose boundaries contain b_{j} and let S = P ∪ P′ ⊂ M. We distinguish two cases:

• b_{j} is nonseparating, in which case P = P′ and A is a one-holed torus;

• b_{j} separates, in which case A is a four-holed sphere. In either case Γ_{A} embeds as a subgroup of Γ_{M}. The restriction map X(M) ⟶ X(A) is Γ_{A}-equivariant.

Suppose inductively on

j = 1, …, N

that F depends only on s_{j}, …, s_{N}. We show that F is independent of s_{j}. That is, disintegrate the measure on X(M) over the map (s_{1}, …, s_{j−1}) and suppose that F is almost everywhere constant on almost every level set of s_{k} for 1 ≤ k < j. The special cases of Theorem 2.2 for a one-holed torus or four-holed sphere (proved respectively in §5 and §6) imply that Γ_{A} is ergodic on almost every level set of the mapping (s_{1}, …, s_{j−1}, s_{j+1}, …, s_{N}). By the induction hypothesis, F is almost everywhere constant on almost every level set of (s_{j+1}, …, s_{N}).

Thus we have “eliminated the variable s_{j}” for j = 1, …, N and F is almost everywhere constant. It follows that Γ_{M} is ergodic on X(M).

9. Appendix: Modifications for product groups

We describe the steps necessary to modify the proof of Theorem 2.2 from G = SU(2) to the general case when G is a compact Lie group locally isomorphic to a product of copies of SU(2) and an abelian group. Such a group G admits a finite covering space

G̃ = SU(2)^{n} × U(1)^{m} ⟶ G

and it suffices to prove Theorem 2.2 for G̃.

The outline of the proof for the product group G̃ is exactly the same as for SU(2). For any group π,

Hom(π, G̃) = Hom(π, SU(2))^{n} × Hom(π, U(1))^{m}

Begin with a pair-of-pants M, with presentation of its fundamental group given by (4.1). A representation of π_{1}(M) into U(1) is given by (α, β, γ) ∈ U(1) satisfying αβγ = 1. The set of such triples is a 2-torus which we denote by 𝔱. A representation ρ : π_{1}(M) ⟶ G̃ is completely determined up to conjugacy by the characters of its factors

(χ(ρ_{1}), …, χ(ρ_{n}), ρ_{n+1}, …, ρ_{n+m}) ∈ 𝔅^{n} × 𝔱^{m}

in the notation of §4. As in §4, the restriction map

∂^{#} : X(M) ⟶ X(∂M)

is an embedding and the relative character varieties are points. (This property fails for other compact Lie groups, such as SU(n) for n ≥ 3.)

The proof of ergodicity when M is a one-holed torus or a four-holed sphere goes through as before. For simplicity we only consider the case of the one-holed torus T_{1}. The restriction map in this case is

∂^{#} : X(M) ⟶ 𝔅^{n} × 𝔱^{m}

The fibers of the character map associated to the interior curve X ⊂ M

x : X(M) ⟶ 𝔅^{n} × 𝔱^{m}

are tori of dimension n + m upon which the Dehn twist τ_{X} acts by rotation. For almost every t, the action of τ_{X} on x^{−1}(t) is an ergodic rotation. (Although for an ergodic rotation r of the circle T^{1}, the diagonal action T^{2} ⟶ T^{2} defined by (t_{1}, t_{2}) ↦ (rt_{1}, rt_{2}) is not ergodic, for almost every pair (r_{1}, r_{2}) of rotations, the rotation (t_{1}, t_{2}) ↦ (r_{1}t_{1}, r_{2}t_{2}) is ergodic.)

The rest of the proof is identical to the SU(2) case.

10. Appendix: Effective actions of automorphisms

In this section we prove that the action of the mapping class group is effective. This is the only place in the paper in which we use the complex character variety. We also include several related facts about character varieties of small surfaces.

10.1. Effectiveness on nonabelian representations.

Theorem 10.1. Let G be a compact nonabelian Lie group and let M be a closed surface of genus g > 2. Then Γ acts effectively on Hom(π, G).

Proof. Since G contains SU(2) or PU(2) and PU(2) is a quotient of SU(2), there is a Γ-equivariant open mapping

Hom(π, SU(2)) ⟶ Hom(π, G)

Thus it suffices to consider the case G = SU(2).

Suppose γ ∈ Γ acts as the identity on Hom(π, G), then it must act as the identity on the complexification Hom(π, G)_{ℂ} ≅ Hom(π, SL(2, ℂ)), which is an irreducible variety (Goldman [14], compare also Rapinchuk, Benyash-Krivetz, and Chernousov [38]). There is a Γ-equivariant embedding T_{g} ⟶ Hom(π, SL(2, ℂ)) of the Teichmüller space T_{g}. By assumption γ fixes every point of T_{M}. Since g > 2, then γ = 1.

If g = 2, the only mapping class which preserves every conformal structure on M must be the hyperelliptic involution, and indeed since every genus 2 Riemann surface is hyperelliptic, the action of Γ on Hom(π, G) is not effective.

10.2. Hyperelliptic representations. A closely related phenomenon involves representations of π_{1}(M) which are invariant under a hyperelliptic involution of M. Let M be a compact surface of genus g with m boundary components. An involution f : M ⟶ M is said to be hyperelliptic if the quotient surface is planar, that is, if the quotient orbifold is an m-holed sphere with 2 + 2g − m branch points of order two. Then a Riemann surface is hyperelliptic if it admits a hyperelliptic involution preserving the conformal structure. It is well known that every closed Riemann surface of genus at most 2 is hyperelliptic, and that every Riemann surface of genus 1 with at most two boundary components is hyperelliptic. Furthermore in these cases the hyperelliptic involution determines a central element F of the mapping class group.

Consider how the hyperelliptic mapping class acts on the character variety of M. Say that a representation

ρ ∈ Hom(π_{1}(M), G)

is hyperelliptic if there exists a hyperelliptic involution f of M such that

[ρ ∘ f_{∗}] = [ρ] ∈ ρ ∈ Hom(π_{1}(M), G)/G

The uniformization representation of a hyperelliptic Riemann surface is hyperelliptic. The Fuchsian representations of π_{1}(M) in SL(2, ℂ) form the Teichmüller space T_{g}, which is fixed by f_{∗}. Since the action of f_{∗} on the variety of SL(2, ℂ)-characters X_{ℂ}(M) is algebraic, the subset fixed by f_{∗} is closed in the complex Zariski topology. Since T_{g} is dense in X_{ℂ}(M) with the complex Zariski topology, it follows that f_{∗} acts trivially on X_{ℂ}(M), and hence on X(M):

Theorem 10.2. Let M be a closed surface of genus 2, or an m-holed torus with m = 1, 2.

• Then every representation π_{1}(M) ⟶ G is invariant under the hyperelliptic involution.

• Let O be the quotient orbifold and let π_{1}(M) ↪ π_{1}(O) be the corresponding embedding of π_{1}(M) as an index-two subgroup of π_{1}(O). Then every representation π_{1}(M) ⟶ G extends to a representation π_{1}(O) ⟶ G.

This theorem is false if G is replaced by other groups (such as SU(3) or SL(3)).

From wmg Fri Apr 25 11:49:01 1997 To: annals@math.princeton.edu Status: RO

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

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