Action of the Johnson-Torelli group on Representation Varieties

Let \Sigma be a compact orientable surface with genus g and n boundary components B = (B_1,..., B_n). Let c = (c_1,...,c_n) in [-2,2]^n. Then the mapping class group MCG of \Sigma acts on the relative SU(2)-character variety X_c := Hom_C(\pi, SU(2))/SU(2), comprising conjugacy classes of representations \rho with tr(\rho(B_i)) = c_i. This action preserves a symplectic structure on the smooth part of X_c, and the corresponding measure is finite. Suppose g = 1 and n = 2. Let J be the subgroup of MCG generated by Dehn twists along null homologous simple loops in \Sigma. Then the action of J on X_c is ergodic for almost all c.


Introduction
Let Σ = Σ g,n be a compact oriented surface of genus g with n boundary components B = {B 1 , ..., B n }. Let π = π 1 (Σ) denote its fundamental group. The mapping class group MCG consists of isotopy classes of orientation-preserving homeomorphisms of Σ which pointwise fix each B i . Alternatively, MCG is the image under the quotient homomorphism Aut(π) −→ Out(π) := Aut(π)/Inn(π) of the subgroup Aut(π, B) of all automorphisms of π that preserve the set B of conjugacy classes of the cyclic subgroups π 1 (B i ) ⊂ π and correspond to orientation-preserving homeomorphisms.
Let G be a Lie group. Then G acts on Hom(π, G) by conjugation. Let X(G) = Hom(π, G)/G. Let C = {C 1 , · · · , C n }, where C i ⊆ G is a conjugacy class for 1 ≤ i ≤ n. Then the relative representation variety is Hom C (π, G) = {ρ ∈ Hom(π, G) : ρ(B j ) ∈ C j , for 1 ≤ j ≤ n}. The group G acts on Hom C (π, G) by conjugation and the moduli space is the quotient X C (G) = Hom C (π, G)/G. The group Aut(π, B) acts on π, preserving B. Hence it acts on Hom C (π, G). Furthermore the action descends to a MCG-action on X C (G). The moduli space X C (G) has an invariant dense open subset X U C (G) which is a smooth manifold. This subset has an MCG-invariant symplectic structure ω, hence, a natural smooth MCG-invariant measure µ [3,5].
Denote by S the set of homotopy classes of simple closed curves on Σ and by J ⊆ S the null homologous (in H 1 (Σ, Z)) subset. The group MCG is generated by Dehn twists τ a along simple loops in S. Denote by J ⊆ MCG the subgroup generated by Dehn twists along simple loops in J and by T ⊆ MCG the subgroup generated by Dehn twists τ a for a ∈ J and products τ a τ −1 b , where a and b are disjoint but homologous simple loops in S.
When n ≤ 1, T is the Torelli group, i.e. the subgroup of MCG acting trivially on H 1 (Σ, Z) [8]. Johnson constructed epimorphisms for n = 1 and define the kernels to be J [7,6]. For n > 1, our definition of T relates to the functorial Torelli group (see [11,12]). The ergodicity of the MCG-action on X C (SU(2)) was proved in [2,4]. See [10,9] for similar results when G is a general compact group. Here we prove the following ergodicity result: Theorem 1.1. Suppose g = 1 and n = 2. Then the J -action on X C (SU(2)) is ergodic for generic C 1 and C 2 .
The moduli space X C (SU(2)) possesses a symplectic structure. The group J is generated by simple loops described above. The same simple loops also correspond to fundamental group elements. These Dehn twist actions embed into the Hamiltonian vector field flows of the trace functions on these corresponding fundamental group elements. It is then a routine matter to produce a set of such Hamiltonian vector fields, whose flows are locally transitive on an open dense (Zariski) subset U ⊆ X C (SU(2)). However since X C (SU(2)) is a real variety (SU(2) < SL(2, C) is a real form), V = X C (SU(2)) \ U is of R-codimension 1.
In other words, V may contain "walls" between components of U. To prove ergodicity of the J -action, we analyze the vector fields along V explicitly. This then requires an explicit computation of the symplectic form with the aid of a computer. The inability to carry out these explicit computations for curves of higher genuses and/or with more punctures is the main obstacle in generalizing Theorem 1.1 to these curves.

Trace functions and Hamiltonian flows
This section summarizes some needed results from [4]. Let X be a symplectic manifold and f : X → R a smooth function. Denote by H(f ) the associated Hamiltonian vector field. Proposition 2.1. Let X be a connected symplectic manifold and let F be a set of real smooth R-valued functions on X such that at every point x ∈ X, the differentials df (x), for f ∈ F , span the cotangent space T * x (X). Then the group generated by the Hamiltonian flows of the vector fields H(f ), for f ∈ F , acts transitively on X.
Proof. The proof is a straightforward application of the implicit function therem; see Lemma 3.2 in [4].
Let G = SL(2, C) and C = {C 1 , · · · , C n } be a family of conjugacy classes in G such that C i is non-parabolic for each 1 ≤ i ≤ n. Let c = (c 1 , · · · , c n ) ∈ C n such that c i = tr(A) ∈ C for all A ∈ C i . Then the representation variety is equivalently defined as In this setting, if α ∈ π is a homotopy class of based loops, then t α , the trace function of α on X C , is defined as Since the function SL(2, C) Proposition 2.2. Let α be a simple separating curve on Σ with Dehn twist τ α . Let ψ : X C → R be a measurable function invariant under the cyclic group (τ α ) * . Then ψ is almost everywhere invariant under the flow of H(t α ).
For the rest of the paper, we shorten X C (SU(2)) (resp. X U C (SU(2))) to X c (resp. X U c ).

Ergodicity
For g = 0 and n = 4 or for g = 1 and n = 2, the fundamental group π is isomorphic to the free group of three generators where F i corresponds to a simple closed curve on Σ. By convention, we also use elements in π to denote curves they represent on Σ.
3.1. The 4-holed sphere. Suppose g = 0 and n = 4. The boundary components of Σ are The fundamental group π is isomorphic to F 3 with the isomorphism Then for c = (c 1 , c 2 , c 3 , c 4 ) ∈ I 4 , X c is a compact component of F −1 (c).
The Johnson kernel J for the 4-holed torus is trivial as any nontrivial Dehn twist must be along simple curves that separate the four boundary components into pairs. However one may study a different group action as follows: Fix the boundary components into two pairs {B 1 , B 2 } and {B 3 , B 4 }. Let J ′ ⊆ S be the subset containing all the curves separating Σ into two pairs of pants containing {B 1 , B 2 } and {B 3 , B 4 }, respectively. Let J ′ ⊆ MCG be the subgroup generated by Dehn twists along elements in J ′ . In this section, we study the J ′ -action on X U c . This problem is interesting in its own right and instructive in the study of the J -action on X U c when Σ is the 2-holed torus.
The symplectic bi-vector field relating to ω is By convention, we use F i to also denote a simple closed curve it represents. The Dehn twist along the simple closed curve F 2 F 3 takes the simple closed curve F 1 F 2 to a simple closed curve F 0 . Denote by τ 12 , τ 0 the Dehn twists along F 1 F 2 and F 0 , respectively. Let Γ = τ 12 , τ 0 . Let M be the space of measurable functions X → R. The trace functions of F 1 F 2 , F 0 are, respectively, Let H i = W (dp i ) be the Hamiltonian vector field (notice that, to conserve notation, the subscript index i may mean either a number or a pair of numbers). Let G i = G(p i ) where G(p i ) is the group generated by the Hamiltonion flow of H(p i ). Let G be the group generated by A direct calculation shows that   is zero in R[K]/(k, s); that is, H 12 (s) ∈ (k, s).
We now compute a Gröbner basis for (k, s) ⊆ R[K]. A direct computation shows that the residue of H 12 (s) is not zero. This implies that H 12 (s) ∈ (k, s). This implies that for a generic c ∈ I 4 , H 12 is not tangent to V . Proof. Suppose f ∈ M Γ . Then f ∈ M G . For almost all c ∈ I 4 , the set Q = X U c ∩ V has measure zero and divides X U c into a finite number of components. Let A ⊆ X U c \ Q be a connected component. By Lemma 3.2, the fibres of p 12 , p 0 are tangent to each other at v only if v ∈ Q. Hence, by Proposition 2.1, f must be constant almost everywhere on A.
Lemma 3.4 implies that there is a Zariski dense subset of Q upon which H 12 is not tangent to Q. Hence there exists a smooth vector field (namely H 12 ) that flows across Q between adjacent components. This implies that if v 0 , v 1 ∈ X U c \ Q, then there exists g ∈ G such that g(v 0 ) = v 1 . Since X U c is smooth and connected, f is constant almost everywhere on X U c \ Q. Since X U c is open and dense in X c and Q has measure zero, the theorem follows.
Remark 3.6. The moduli space X c is the subspace of X defined by k s = c 1 + c 2 and k p = c 1 c 2 .
With the aid of a computer, one may compute a Gröbner basis for (k, s) and show that the residue of H 12 (s) is not zero. Hence H 12 (s) ∈ (k, s). This implies that for a generic c ∈ I 2 , H 12 is not tangent to s.
It so happens that s = s 1 s 2 is reducible with two factors. Hence one may compute the Gröbner basis (k, s 1 ) and (k, s 2 ) and then compute the residues in each cases.
Proposition 3.11. The Γ-action on X U c is ergodic for almost every c ∈ I 2 .
Proof. Suppose f ∈ M Γ . Then f ∈ M G . For almost all c ∈ I 2 , the set Q = X U c ∩ V has measure zero and divides X U c into a finite number of components. Let A ⊆ X U c \ Q be a connected component. By Lemma 3.8 and Proposition 2.1, f is constant almost everywhere on A.
Lemma 3.10 implies that there is a Zariski dense subset of Q upon which H 12 is not tangent to Q. Hence there exists a smooth vector field (namely H 12 ) in G 12 that flows across Q between adjacent components. This implies that if v 0 , v 1 ∈ X U c \ Q, then there exists g ∈ G such that g(v 0 ) = v 1 . Since X U c is smooth and connected, f is constant almost everywhere on X U c \ Q. Since Q has measure zero, the theorem follows.
Theorem 1.1 follows as Γ ⊆ J and X U c is open and dense in X c .