Cocycle Rigidity and Splitting for some Discrete Parabolic Actions

We prove trivialization of the first cohomology with coefficients in smooth vector fields, for a class of $\mathbb Z^2$ parabolic actions on $(SL(2, \mathbb R)\times SL(2, \mathbb R))/\Gamma$, where the lattice $\Gamma$ is irreducible and co-compact. We also obtain a splitting construction involving first and second coboundary operators in the cohomology with coefficients in smooth vector fields.


INTRODUCTION
Classification of cohomological obtructions for the horocycle flow provided a tool for proving cohomological rigidity for the R 2 parabolic action on SL(2, R) × SL(2, R)/Γ, [8]. This is the action on SL(2, R) × SL(2, R)/Γ induced by left translations by the elements of the two-dimensional subgroup 1 t 0 1 × 1 s 0 1 where s, t ∈ R. The main ingredient in the proof of cohomological rigidity in [8] is the complete and detailed description of cohomological obstructions for the horocycle flow from [5]. Moreover, in [2] the first author constructed a splitting in cohomology with coefficients in smooth vector fields. The splitting consists essentially in obtaining inverse operators for the first and second coboundary operators over the given action. The splitting construction in [2] uses in an essential way another part of the result proved in [5], namely the fact that the space of distributional obstructions to solving the cohomological equation over the horocycle flow in each irreducible representation is finite dimensional. An important property of the splitting constructed in [2] is that it is tame: inverses constructed for the first and second coboundary operators are tame maps with respect to the collection of Sobolev norms. A map is tame if the loss of the regularity for the image, for data of any given regularity, is fixed. Tame splitting construction lead to a perturbation type result in [2] for the parabolic R 2 action defined above. 1 Namely, for Γ an irreducible lattice in SL(2, R) × SL(2, R), it is proved in [2] that the R 2 parabolic action on SL(2, R) × SL(2, R)/Γ is transversally locally rigid: the action sits within a smooth finite dimensional family of homogeneous actions such that every sufficiently small transversal perturbation of the family intersects the smooth conjugacy class of the parabolic action. The finite dimensional family of homogeneous actions is precisely defined by the finite dimensional cohomology with coefficients in vector fields, over the parabolic R 2 action. In this paper instead of the R 2 parabolic action on SL(2, R)×SL(2, R)/Γ, we consider its Z 2 subactions. As before, Γ is assumed throughout to be an irreducible co-compact lattice in SL(2, R) × SL(2, R). A representative example from the class of these Z 2 parabolic actions is the time-one Z 2 parabolic action, namely the Z 2 action generated by left translations by the following elements: For the horocycle map (the time-one map of the horocycle flow) on SL(2, R)/Γ, the cohomological obstructions have been completely described by the second author in [12]. It is proved in [12] that the invariant distributions are a complete set of cohomological obstructions. However, unlike the case of the unipotent flow, the space of invariant distributions for the horocycle map in each irreducible representation has countable infinite dimension. For functions for which all the obstructions vanish, the cohomological problem is solved completely. However, the estimates for the solution of the cohomological equation obtained in [12] are not tame.
In what follows we show vanishing of distributional obstructions for Z 2 parabolic actions, which implies trivial first cohomology with coefficients in smooth functions. Further, even though the space of invariant distributions in each irreducible representation is infinite dimensional, we prove existence of a splitting in cohomology by exploiting the decay in the space invariant distributions obtained in [12]. The splitting we obtain is not tame, but this is only due to the lack of tame estimates for the solution of the first coboundary problem. We comment on possible improvement of nontame to tame estimates following Conjecture 5.1. Since the action involves unipotent maps, we extend these cohomological results to cohomology with coefficients in vector fields. At the end we state a conjecture concerning perturbations of Z 2 parabolic actions. Let U 1 = U × 0 and U 2 = 0 × U be the two commuting stable unipotent vector fields on G. The left action α by a lattice in the upper unipotent subgroup of G can be viewed as where (n, m) ∈ SZ × T Z, with S, T ∈ R + fixed, and x ∈ G/Γ. The first coboundary operators L 1 and L 2 are

Results.
In Section 3, we prove that the first cohomology over the discrete parabolic action α on G/Γ is indeed trivial (i.e. contains only constant cocycles), and we obtain Sobolev estimates of the transfer function with respect to the given data.
Theorem 1.1. Let Γ ⊂ G be as above. For any r ≥ 0, there is a constant C r,Γ > 0 such that for any f, g ∈ C ∞ (G/Γ) with zero average and that satisfy L 1 g = L 2 f , there is a solution P ∈ C ∞ (G/Γ) such that L 1 P = f and L 2 P = g, and P r ≤ C r,Γ f 3r+6 .
In Section 4, we prove the splitting for the first cohomology with coefficients in smooth functions: Theorem 1.2. Let Γ ⊂ G be as above. For any r ≥ 0, there is a constant C r,Γ > 0 such that for any f, g, φ ∈ C ∞ (G/Γ) with zero average that satisfy In Section 5, the above results are extended to cohomology with coefficients in smooth vector fields.

PRELIMINARIES
2.1. Representation Spaces. Let L 2 (G/Γ) be the space of complex-valued square integrable functions with respect to the SL(2, R)×SL(2, R)-invariant volume form for G/Γ. The left regular representation L 2 (G/Γ) of SL(2, R)× SL(2, R) decomposes into irreducible, unitary representations as follows: The Lie algebra of sl(2, R) has basis elements consisting of the generators for the stable and unstable horocycle flows and the geodesic flow. The Lie algebra of G is sl(2, R) × sl(2, R), which has the basis The Laplacian determined by this basis is the elliptic element of the enveloping algebra of sl(2, R) × sl(2, R) defined as . The elements of sl(2, R) are skew-adjoint, and thus △ is an essentially selfadjoint operator. Let △ 1 = −(U 2 + X 2 + V 2 ). It will sometimes be useful to use the equivalence of operators (I + △) r ≈ (1 + △ 1 , I) r + (I, I + △ 1 ) r , for any r ≥ 0.
The Sobolev space of order r > 0 of a unitary representation space H of G is the Hilbert space W r (H) ⊂ H defined to be the maximal domain determined by the inner product We denote the Casimir operators in the first and second components of SL(2, R) × SL(2, R) by 1 and 2 , respectively. Because Γ is irreducible, each Casimir operator has a spectral gap in the sense that spec( j ) ∩ R + has a lower bound µ 0 > 0, for j = 1, 2 (see Theorem 15 of [8]).

Invariant Distributions.
We now define the invariant distributions for the map e U in a representation space H of SL(2, R). The space of smooth functions in H is defined analogously, as in [5]. The Sobolev space of order r > 0 is the Hilbert space W r (H) ⊂ H defined to be the maximal domain determined by the inner product The space of invariant distributions for the map e U in the representation space H of SL(2, R) is defined by The invariant distributions for the time-one map are all finite regularity. See Theorem 1.1 of [12] for a more precise description than what we need here. In what follows, we will restrict our attention to the cannonical Z 2 action when T = S = 1, as the results for the SZ × T Z action follow completely analogously. Set As before, consider operators L 1 and L 2 on L 2 (G/Γ) Iterating the annihilation and creation operators from sl(2, R) gives a standard orthogonal basis {u µ,j } j for each irreducible representation H µ of SL(2, R). Taking tensor products, we use the orthogonal basis {u µ,j ⊗ u θ,k } j,k for the irreducible representation space

2.4.
Cohomology over Z k actions. To ease notation for this subsection, we write M := G/Γ. Let be a smooth Z k action on a smooth manifold M, preserving a smooth volume form.
Then we have the cohomology sequence: where the operators δ 1 v and δ 2 v are defined as follows. .
If the manifold M is homogeneous and the action ρ is affine, then for g ∈ Z k , the map ρ(g) * is linear. Since M is parallelizable, if there is a basis of vector fields in which all ρ(g) * are upper triangular with ones on the diagonal, then computing the cohomology H 1 ρ (Z k , V ect ∞ (M)) can be inductively reduced to computing the first cohomology with coefficients in smooth functions where the operators δ 1 and δ 2 are defined by: We say the first cohomology over ρ with coefficients in C ∞ (M) trivializes if it is isomorphic to R, i.e. if every f ∈ C ∞ (M) with zero average is in the image of δ 1 .
We say the first cohomology over ρ with coefficients in V ect ∞ (M) trivializes if it is the same as the cohomology over ρ with coefficients in constant vector fields.
3. PROOF OF THEOREM 1.1 The argument that follows does not change with the scaling of S, T for the lattice action SZ×T Z by upper unipotent elements described in formula (1). We therefore take S, T = 1 for simplicity.
We cite the following well-known Howe-Moore Ergodicity theorem, [3]. Then there is some 0 < r < ∞ such that I(H) ⊂ W −r (H).
Our approach follows [8] and [11]. As in [11], for each D ∈ I(H µ ) and similarly for each D ∈ I(H θ ), defineD 2 by Now extendD j linearly to finite sums of simple tensors, for j = 1, 2. In formula (12) below,D j is extended to be defined on infinite sums of simple tensors.
For any DefineD 2 analogously. The operatorsD j map C ∞ functions to C ∞ functions.
Proof. We will carry the argument forD 1 , and the case forD 2 will follow in the same way. Using the finite regularity condition from Lemma 3.1 and the fact that {u θ,k } k is an orthogonal basis, we have that for any r ≥ 0, . Now Formula (25) of [5] shows that be nonzero functions of zero average, and suppose f, g satisfy L 2 f = L 1 g. Then for all D ∈ I(H µ ) (resp. D ∈ I(H θ )), we haveD 1 (f ) = 0 (resp.D 2 (g) = 0).
Proof. Let f be as in (11). From the definitions of (4) and (10), we have the relations Then the assumption L 2 f = L 1 g implies L 2 (D 1 (f )) = 0. We will use the following result from [12] in the proof of Theorem 1.  [12] shows that C r,T,H is uniformly bounded in T in a neighborhood of T = 1. This fact will be used in Proposition 5.2.
Proof of Theorem 1.1: We use the representation parameters ν µ := √ 1 − 4ℜµ and ν θ := √ 1 − 4ℜθ, and we now show that j∈Aµ c j,k u µ,j ∈ C ∞ (G/Γ). Using formula (36) of [5], we have that for any s ≥ ℜν 2 , Then D j∈Aµ c j,k u µ,j = j∈Aµ D (c j,k u µ,j ) for all k. Then applying Lemma 3.3, we have that for all D ∈ I(H µ ), Now because {u θ,k } k∈A θ is a basis for H θ , we conclude that for all k, Recall that 1 has a spectral gap at zero, so Theorem 1.2 of [12] together with (16) and (17) show that there is some P µ,θ k ∈ H µ such that Observe that Then for every r ≥ 0, Using the relation Because u 1 is ergodic, it follows that L 2 P − g = C on G/Γ, for some constant C ∈ C. Therefore, all irreducible components of L 2 P − g are zero except possibly the trivial one. Note g has zero average, which implies C = 0, and we conclude the proof of Theorem 1.1.

PROOF OF THEOREM 1.2
In this section we use the ideas from [2] to construct the splitting. However, the important difference is that the space of invariant distributions is infinite dimensional in each irreducible component for the horocycle map. There is no a priori reason why the same strategy as in [2] would work in the case of an infinite dimensional space of invariant distributions. However, in this particular situation we may use the fast decay of distributions at infinity, which is proven in [12].  [12] shows there is some C r,H > 0 such that We prove Theorem 1.2 in one irreducible component at a time, and then we are able to glue the estimates together using the existence of a spectral gap for irreducible lattices of SL(2, R) × SL(2, R).
Proof. For each n ∈ Z, let γ n ∈ C ∞ (H µ ) be dual to D n . By Lemma B.1 in [12], we can label the distributions D n and choose the γ n so that γ n r ≤ C r n r . ( By (20), it follows that so that R is bounded on simple tensors. Extend R linearly to finite sums of simple tensors in H. Then let f ∈ C ∞ (H µ ⊗ H θ ), and write its decomposition with respect to the basis {u µ,j ⊗ u θ,k } j,k , as in (11). Define Then by triangle inequality and (21), we have Multiplying and dividing the summand by (1 + j 2 + k 2 ), Cauchy-Schwarz gives (ii) Again, let f be as in (11). By the proof of Part (i), the series defining Rf converges unconditionally in W r (H µ ⊗ H θ ), and by Lemma 3.2,D 1 m is a bounded operator on this space. Then we can interchangeD 1 m with the sums and concludẽ Now because D m is dual to γ m , we get (iii) Let f ∈ Ann( {D 1 n } n∈Z ) be as in (11). Again, we may interchange sums by Part (i) and by (16) to get Proof of Theorem 1.2 : With this, the same argument used to prove Theorem 3.2 of [2] proves Theorem 1.2. We provide it here for the convenience of the reader.
Define R ⊥ (f ) = f − Rf. Then property (ii) impliesD 1 n (R ⊥ f ) = 0 for every n. Then property (i) and the argument below (17) gives a solution P ∈ C ∞ (H) satisfying L 1 P = R ⊥ f, and Moreover, so by property (iii) and (i v), Because L 1 (L 2 P − g) ∈ Ann({D 1 n } n∈Z ), we again use the argument below (17) to conclude The estimate of L 1 P − f r follows analogously. Now we glue estimates from each irreducible component, as in equation (19). This concludes the proof of Theorem 1.2.

COHOMOLOGY WITH COEFFICIENTS IN VECTOR FIELDS, SPLITTING AND CONJECTURE ON RIGIDITY OF PERTURBATIONS
The motivation for studying the first cohomology over Z 2 parabolic actions on SL(2, R) × SL(2, R)/Γ is to understand the local structure about these actions. For the corresponding continuous time action, which is in fact a maximal unipotent abelian action on SL(2, R) × SL(2, R)/Γ, the local picture has been understood to large extent in [2]. Namely, the continuous time action is transversally locally rigid. A similar picture for the discrete time sub-action of the continuous time unipotent action would be a stronger result.
A general approach towards understanding perturbations of an action is to study the first cohomology over the the given Z 2 action ρ with coefficients in vector fields, rather than only the cohomology with coefficients in smooth functions (see Subsection 2.4). This is because the formal tangent space at ρ to the space of all smooth Z 2 actions on the given manifold M is precisely the space of smooth cocycles over ρ with coefficients in vector fields. Essentially, the first cohomology with coefficients in vector fields can be thought of as the first approximation to the set of actions in a small neighborhood of the given action, modulo smooth conjugacy classes. Now, if it turns out that the cohomology H 1 ρ (Z 2 , V ect ∞ (M)) is finite dimensional and well understood, and if there is a splitting in cohomology which is tame, then it may be possible to carry out an iterative procedure (similar to classical KAM iterative scheme, or proofs of generalized implicit function theorems) to obtain a result on perturbations of the action. For smooth Lie group actions a general result of this type is obtained in [1]. So far a similar approach has not been applied to discrete abelian group actions with non-trivial but finite dimensional H 1 ρ (Z 2 , V ect ∞ (M)) and the examples in this paper may be the first ones amenable to this technique. We remark further that for parabolic actions one does not have rich geometric structure as for partially hyperbolic actions, so to study general perturbations of such actions the only successful approach so far has been the one mentioned above: via the first cohomology and an iterative procedure.
In this section we show that this approach is very likely to be successful in the context of parabolic Z 2 actions on SL(2, R) × SL(2, R)/Γ: we prove that the first cohomology with coefficients in vector fields is indeed finite dimensional and fairly simple to describe. After that, using the results from previous sections we show that there is a splitting in cohomology via an explicit construction. Finally, based on these results we state a conjecture on the local structure about these Z 2 parabolic actions.

Action induced operators on vector fields.
If v is a vector in R 3 , let v τ be its transpose. Let , f (x)) τ be given in (25). Then Lemma 5.1 will follow immediately from Claim 5.1. Similarly, This concludes the proof of Lemma 5.1.
The action induced first coboundary operators on vector fields in sl(2, R)× sl(2, R) are described in Section 2.4, equation (7). Let Therefore the cohomology over the given Z 2 parabolic action with coefficients in constant vector fields is 4-dimensional and the cohomology classes are parametrized by cocycles (F, Y ) where F ∈ RU 2 + RV 1 and Y ∈ RU 1 + RV 2 .
The proof of Proposition 5.1 will be apparent from the proof of Proposition 5.2, so we defer it until then. Proposition 5.2 shows that the first cohomology with coefficients in smooth vector fields reduces to the first cohomology with constant vector fields. Notice that by integrating each coefficient over G/Γ, we obtain a constant coefficient vector field. For a smooth vector field F we denote by AveF the constant vector field obtained from F by taking averages of all coordinate functions of F in the basis of sl(2, R) × sl(2, R) described above. We say (AveF, AveY ) is in the trivial cohomology class if (28) is satisfied for (AveF, AveY ).
Assume also that (AveF, AveY ) is in the trivial cohomology class. Then there exist H,F ,Ỹ ∈ Vect ∞ (G/Γ) such that F = L 1 H +F and Y = L 2 H +Ỹ , and for every r > 0 and s ≥ 27r + 130 there is constant C s,r,S,T,Γ > 0 such that the following estimates hold: F r ≤ C s,r,S,T,Γ Φ s , Ỹ r ≤ C s,r,S,T,Γ Φ s , and H r ≤ C s,r,S,T,Γ F s .
Moreover, for (S, T ) in an ǫ neighborhood of (1, 1), the constants C s,r,S,T,Γ are uniformly bounded in S, T .
Proof. We first prove the claim for T = S = 1. Recall that u 1 := e U 1 and u 2 := e U 2 . Using the coordinates (25), let F = (h 1 , We consider the first three coordinates. Then L 2 f 1 − L 1 f 2 = φ 3 . Then Theorem 1.2 implies that if Avef 1 = Avef 2 = 0 then there is some P ∈ C ∞ (G/Γ) and a constant C r,Γ > 0 such that Next, By (30), there is someφ 3 ∈ C ∞ (G/Γ) such that where φ 3 r ≤ C r,Γ φ 3 3r+10 for all r ≥ 0. Therefore, Applying Theorem 1.2 again shows that if Aveg 2 = 0 then there is some Q ∈ C ∞ (G/Γ) such that Notice that we did not need to assume that Aveg 1 = 0 because we have a freedom to choose P up to a constant, so we chose it so that Ave(g 1 − P • u 1 ) = 0. Now By (31), where φ 2 r ≤ C r,Γ φ 2 −φ 3 3r+10 for all r ≥ 0. Then By Theorem 1.2 again, if Aveh 2 = 0 (and after adjusting Q by a constant so that Then repeating the process for the bottom three coordinates, we find a vector field H and a remainderF ∈ Vect ∞ (G/Γ) such that The following estimates are satisfied: where s ≥ 3(3(3r + 10) + 10) + 10 = 27r + 130. Lastly and in a completely analogous manor, the same holds for the maps e T U 1 and e SU 2 . The fact that the constant C s,r,S,T,Γ > 0 is uniformly bounded in (S, T ) in an ǫ > 0 neighborhood of (1, 1) follows from the estimates in the proof of Theorem 1.2 of [12].
Proof of Proposition 5.1: We first consider the case when (F, Y ) is a (constant coefficient) cocycle. Now Φ = 0, so (29) implies (27) after all the coefficients are substituted by constants, namely that in this casef 2 = f 3 = g 2 = g 3 = 0. When (F, Y ) reduces to a coboundary then the proof above shows that the necessary conditions for that are: f 1 = h 3 = g 3 = f 3 = h 2 = g 2 = f 2 = f 4 = 0. This implies (28) and concludes the proof of Proposition 5.1.
We remark that if the estimates in the splitting construction in Proposition 5.2 were tame i.e. if the loss of regularity forF ,Ỹ and H was fixed and independent of r, for every given regularity r, then the conjecture would follow by the similar approach as in [2], adapted to the discrete case. Now the fact that the estimates in Proposition 5.2 are not tame comes directly from not having tame estimates for the solution of the coboundary equation for functions over the discrete unipotent map on SL(2, R)/Γ, obtained in [12].
The authors believe that obtaining tame estimates for the solution of the coboundary equation for functions over the discrete unipotent map on SL(2, R)/Γ is a challenging problem. The second author obtained the non tame estimates through an analysis in irreducible models [12]. It is not clear to the authors how to substantially improve these estimates, however a more precise analysis may be possible. Flaminio-Forni proved tame estimates for the cohomological equation of the horocycle flow in [5], but their method does not seem to work for the horocycle map. The important point for the argument of Flaminio-Forni is that the restriction of the generator for the horocycle flow to each irreducible component H µ of L 2 (SL(2, R)/Γ) takes a relatively simple form in terms of the bases {u µ,k } k . This allowed them to determine the invariant distributions by their values on basis elements in H µ , and then construct a Green's operator. The case of the time-one map is different. The operator e U is a complicated infinite dimensional matrix in each basis {u µ,k } k , and it becomes unclear how to proceed. Another approach is to obtain tame estimates for cohomological equations that are more similar to that of the horocycle flow. For example, it is possible to derive the cohomological equation for the time-1 horocycle map from the cohomological equations (U + in)f = g, for n ∈ Z.