Rigidity of real-analytic actions of $SL(n,\Z)$ on $\T^n$: A case of realization of Zimmer program

We prove that any real-analytic action of $SL(n,\Z), n\ge 3$ with standard homotopy data that preserves an ergodic measure $\mu$ whose support is not contained in a ball, is analytically conjugate on an open invariant set to the standard linear action on the complement to a finite union of periodic orbits.


Introduction. Formulation of results
Let G be a semisimple Lie group whose simple factors all have real rank greater than one, or an irreducible lattice in such a group. "Zimmer program" first formulated in 1986 [9] and modified in the 1990s to take into account examples described in [5], aims at proving that volume preserving actions of G by diffeomorphisms of compact manifold M are "essentially algebraic". This means that M splits into disjoint union of open G-invariant subsets U 1 , . . . , U n and a nowhere dense closed subset F such that the restriction of the action to each of the open sets is smoothly conjugate to the restriction of a certain standard algebraic action (homogeneous or affine) to an open dense G-invariant subset.
Without attempting an overview of results in the direction of Zimmer program let us point out that so far they have been either negative (such as non-existence of actions in low dimension), or perturbative (local differentiable rigidity of algebraic actions), or subject to dynamical restrictions such as existence of Anosov elements.
In this note we present what is, to the best of our knowledge, the first positive result free of such restrictions. Its principal limitation is that we consider real-analytic, rather than differentiable actions with the pay-off that the conjugacy is also real-analytic. On the other hand, instead of preservation of volume we make a weaker assumption of existence of an invariant measure with a "homotopically large" support.
Let us consider the torus T n . The group SL(n, Z) acts on T n by automorphisms which are projections of the linear maps in R n . We will call this action of SL(n, Z) on T n the standard action and denote it ρ 0 . We will call Date: February 15, 2010. *) Based on research supported by NSF grant DMS-0803880. **) Partially supported by the Center for Dynamics and Geometry at Penn State. the corresponding action on Z n the standard homotopy data. Since SL(n, Z) is a lattice in the simple connected Lie group SL(n, R) of real rank n − 1 we assume n ≥ 3. Let ρ be an action of a group Γ ⊂ SL(n, Z) by diffeomorphisms of T n . Induced action ρ * on Z n = π 1 (T n ) is called the homotopy data of ρ. We will say that ρ has standard homotopy data if ρ * is the restriction of the standard homotopy data to Γ.
This note is dedicated to the proof of the following result: ρ be a C ω (real-analytic) action of Γ on T n with standard homotopy data, preserving an ergodic measure µ whose support is not contained in a ball. Then: (1) There is a finite index subgroup Γ ′ ⊂ Γ, a finite ρ 0 -invariant set F and a bijective real-analytic map (2) The map H −1 can be extended to a continuous (not necessarily invertible) map P : T n → T n such that ρ • P = ρ 0 • P . Moreover, for any x ∈ F , pre-image P −1 (x) is a connected set.
Since Lebesgue measure λ is the only non-atomic invariant measure for ρ 0 we deduce that µ = H * λ and we have the following corollary Corollary 1.2. The measure µ is given by a real-analytic density on an open dense subset of its support. Furthermore, it is the only ergodic ρ-invariant measure whose support is not contained in a ball.
The map H is the inverse of the conjugacy between the action ρ on an open set and the standard action in the complement to finitely many periodic orbits.
Thus ρ is obtained by "blowing up" finitely many periodic orbits of ρ 0 and leaving the rest the same up to a real-analytic time change. Possibility of such non-trivial blow-ups on the torus is an open question. Constructions from [5] produce real-analytic actions with blowups on some manifolds other than the torus, e.g by gluing in a projective space RP (n − 1) by a σ-process or glueing two such projective spaces and thus attaching a kind of handle to the torus. The same construction produces a real-analytic action on the torus with an open round hole that obviously cannot be extended to a realanalytic action inside the hole. C 0 extension is possible but whether it can be extended in a smooth way is an open question.

Proof of Theorem 1.1
We will assume throughout this section that Γ is a finite index subgroup of SL(n, Z), n ≥ 3.
A particular case of [8,Theorem,6.10] asserts that a Γ action on T n with standard homotopy data preserving a measure µ with full support is essentially semi-conjugate to the standard linear action. The following proposition is an improvement of that statement in two respects: (i) we put a weaker condition on µ and (ii) we assert that µ is absolutely continuous. To achieve that we rely on results from [6] about actions of Cartan (maximal rank semisimple abelian) subgroups of SL(n, Z). Let Γ be a finite index subgroup of SL(n, Z).
Proposition 2.1. Let ρ be a C 1+α , α > 0 Γ action on T n with standard homotopy data. Assume that ρ preserves an ergodic measure µ whose support is not contained in a ball. Then µ is absolutely continuous and there is a finite index subgroup Γ ′ of Γ and a continuous map h : T n → T n such that for every γ ∈ Γ ′ .
Proof. The scheme of the proof is as follows: First we construct a measurable map h defined µ almost everywhere and satisfying (2.1). Then we show that h extends to a continuous map defined on supp µ. Next we show that the image of this map is a complement to a finite set and (2.2) holds. Finally we show that h extends uniquely from supp µ to the whole torus.
Once we get this we argue as follows. Let ν = h * µ; the measure ν is invariant and ergodic with respect to the linear Γ ′ action, hence it is either Lebesgue or atomic. It will not be atomic (see bellow) hence ν is Lebesgue measure λ, this means that µ is a large measure as defined in [6], i.e h * µ = λ. Since for any Cartan subgroup C ⊂ SL(n, R) large invariant measure is unique and absolutely continuous by [6], if we take a Cartan subgroup C ⊂ Γ ′ (which can be done since Γ ′ is of finite index in SL(n, Z)), measure µ must coincide with this large measure and absolute continuity and uniqueness follow.
Step 1. We shall use the following consequence of Zimmer's cocycle superrigidity (see the proof of the Main Theorem in [2]): Let A be a set of full µ measure where the equality (2.1) holds for h 0 . Without loss of generality we may assume that A is invariant with respect to ρ(γ) for every γ ∈ Γ ′ .
Step 2. Now we prove that φ extends to a continuous map supp µ → R n .
Let γ ∈ Γ ′ be a hyperbolic matrix so that ρ 0 (γ) is an Anosov linear map. Then there is a continuous map φ γ : Hence it is enough to see that v ′ is constant µ-a.e. Let L C be the set where |v ′ (x)| < C, taking C large L C has measure as close to 1 as wanted. Call f = ρ(γ) and take x ∈ L C such that x returns infinitely many times to L C in the future and the past. This set has full µ measure in L C by Poincaré recurrence. Then Hence φ = φ γ − c ′ µ a.e. and φ extends continuously to the support of µ; we will still denote this extended map by φ. Let h : supp(µ) → T n , h(x) = x + φ(x), then we get that h • ρ(γ) = ρ 0 (γ) • h for every γ ∈ Γ ′ since h = h 0 on a set of full µ measure. So, let us forget about h 0 and work with h. Notice that for every hyperbolic γ ∈ Γ ′ there is h γ homotopic to the identity such that h γ • ρ(γ) = ρ 0 (γ) • h γ and arguing as above we get that h = h γ + c ′ for some c ′ ∈ (I − Γ) −1 (Z n ) on the support of µ. Observe also that changing h γ with h γ + c ′ we get that h γ + c ′ also conjugates ρ(γ) with γ. Hence we may assume already that h and h γ coincide on supp µ for every γ ∈ Γ ′ hyperbolic. Since hyperbolic elements generate a finite index subgroup of Γ ′ we obtained desired map h : supp µ → T n .
Step 3. Let ν = h * µ be the push-forward of the measure µ. Obviously ν is invariant and ergodic with respect to ρ 0 restricted to Γ ′ . The only ergodic invariant measures for the Γ ′ linear action ρ 0 are Lebesgue measure and measures supported on periodic orbits. Let us show that ν cannot have finite support. If this were the case, the support of µ would belong to the union of pre-images of finitely many points. Take an element γ ∈ Γ ′ inside a Cartan subgroup. Then we have that h = h γ and hence the pre-images of h are inside the pre-images by h γ . Now, the results in [6] gives us that the pre-image of every point by h γ is inside a ball, indeed the pre-image of every point is the intersection of nested cubes. Hence the support of the measure is inside a finite disjoint union of cubes. Now, it is not hard to see that this finite disjoint union of cubes fit inside a ball, which implies that the support of µ is inside a ball, a contradiction.
Thus we get that ν is Lebesgue measure and hence for every Anosov element of ρ 0 , µ is a large measure as defined in [6].
Step 4. Let us see that h can be extended to the rest of the torus conjugating the whole action. Let us make the following observations. Let R be the set of regular points which we may assume that are regular for all hyperbolic elements of the action. It has full µ measure. Then for every x in R and for every γ ∈ Γ belonging to a Cartan subgroup we have that W s γ (x) ⊂ supp(µ). Let U be a connected component of the complement of supp(µ). Let us see that for every γ ∈ Γ an element in a Cartan subgroup C γ , h γ (U ) is a point and moreover this point does not depend on γ. Fix first γ and take a point x ∈ U . We know from [6] that x lies inside the intersection of a family of nested cubes C n such that C n = (h γ ) −1 h γ (x) and h γ (C n ) is a family of cubes such that h γ (C n ) = h γ (x). The boundary of the cubes C n are formed by pieces of stable manifolds of regular elements with respect to different elements of C γ . Hence the boundary of C n is in supp(µ) and since U is connected and U ∩ supp(µ) = ∅ we get that U ⊂ C n . In particular h γ (U ) = h γ (x). So the whole U is collapsed by h γ into a single point. Let us take now another Cartan element γ ′ . Then h γ ′ (U ) = h γ ′ (x) also. Now, we want to proof that h γ (x) = h γ ′ (x). But the boundary of U is in the support of µ and on the support of µ we know that h γ , h γ ′ and h coincide. Hence Since Cartan subgroups generate SL(n, Z) this finishes the proof. The next step in the proof of Theorem 1.1 is finding a periodic orbit for the action ρ. Recall that by the proof of Proposition 2.1 Step 2, we may assume that the semiconjugacy h coincides with h η for every η ∈ Γ ′ . Hence, if p is a proper periodic point for a Cartan action as in Theorem 2.4 then p = h −1 (h(p)). Call q = h(p) and observe that since q is periodic for a linear Anosov map it has to be a rational point, hence it is periodic for the whole linear SL(n, Z) action. Finally, since h • ρ(γ) = ρ 0 (γ) • h for every γ ∈ Γ ′ we have that p is periodic for the ρ action restricted to Γ ′ . So, we may take a finite index subgroup Γ ′′ ⊂ Γ ′ and assume that p is fixed. We will assume that Γ ′ already equals Γ ′′ . Since the derivative cocycle (homomorphism) at p is nontrivial by Theorem 2.4, Margulis super-rigidity Theorem implies that the derivative at p coincides with the linear one up to some conjugacy that we may assume to be trivial by taking an appropriate coordinate chart at p. The next step is a linearization of our action in a neighborhood of the periodic point p. This is the only place where we use real analyticity of the action. We shall use the local linearization theorem of G. Cairns and E. Ghys for real analytic actions, [1].
Theorem 2.5. Let Γ be any irreducible lattice in a connected semi-simple Lie group with finite center, no non-trivial compact factor group and of rank greater than 1. Every C ω -action of Γ on (R m , 0) is linearizable.
In particular, Theorem 2.5 applies to Γ ′ , a finite index subgroup of SL(n, Z). In this case the linear action is conjugate to the standard action. So this theorem gives us a real analytic map H : U → T n where U ⊂ T n is a neighborhood of q such that H • ρ 0 (γ)(x) = ρ(γ) • H(x) for every γ ∈ Γ ′ and for every x where the above equality make sense. H(q) = p and we may assume without loss of generality that D q H = Id.
To finish the proof of Theorem 1.1 it is sufficient to show that the semiconjugacy h coincides with H −1 and hence it is analytic and invertible in a neighborhood V of p. Let us see that H −1 coincides with h in a neighborhood of p. Let γ ∈ Γ ′ be an element in a Cartan subgroup C γ and assume its stable manifold for the linear element is one dimensional. Call W s γ (p) the stable manifold of p for ρ(γ) and E s γ the stable space for ρ 0 (γ). C γ acts locally transitively on W s γ (p) and on q + E s γ . By uniqueness of the invariant manifolds, we have that H(q + E s γ ∩ B ε (q)) ⊂ W s γ (p). We know from the results in [6] where γ ∈ Γ ′ ranges over the elements belonging to a Cartan subgroup and with one-dimensional stable manifold is dense in B ε (q). This is because the projective action on S n−1 is minimal and A corresponds with the orbit of a point in S n−1 , namely the orbit of the direction associated to the stable manifold of one of this γ's. Hence H(A) is dense in B δ (p). On the other hand h and H −1 coincide on H(A) hence by continuity they coincide on B δ (p) which finishes the proof of statements (1) and (2) of Theorem 1.1.
To prove statement (3) notice that SL(n, Z) is generated by its maximal Cartan subgroups. For any such subgroup C the semi-conjugacy P conjugates restriction of ρ to C with an affine action ρ C with standard homotopy data (see [4] for a detailed proof). Elements of such an action ρ C are compositions of automorphisms and rational translations. In particular, ρ C preserves the finite set F ix C of fixed points of ρ 0 restricted to C. Since the intersection of those fixed point sets for different Cartan subgroups is the identity, this implies that the only affine action of SL(n, Z) with standard homotopy data is ρ 0 .
Thus for any Cartan subgroup C, the action ρ preserves the set P −1 (F ix C ) and hence their intersection P −1 (0). This implies that ρ C coincides with ρ 0 for any Cartan subgroup C and statement (3) follows.

Remarks and open problems
With a more careful analysis of the structure of finite index subgroups of SL(n, Z) one can very likely prove that the semiconjugacy P also serves as a semiconjugacy between the whole action ρ and an affine actionρ 0 with the standard homotopy data. Basically the question reduces to consideration of finite order elements of Γ that are not products of elements of Cartan subgroups.
A much more interesting question concerns restrictions of ρ 0 to infinite index subgroups whose linear representations are rigid, such as integer lattices in other higher rank simple Lie groups or irreducible representations of SL(n, Z) into SL(N, Z) for large N . While Cairns-Ghys Theorem 2.5 is available in those settings, the other key ingredient, a weaker form of rigidity for maximal abelian subgroups like in [3,6], is missing. Progress in this direction beyond the Cartan case has been achieved recently in [7] but the strong simplicity condition of that paper is not satisfied in the interesting cases mentioned above. Furthermore, in some cases, such as the representation of SL(n, Z) into SL(n 2 − 1, Z) given by the adjoint action on traceless matrices, there are no Anosov elements altogether and one should hope to tie elements of rigidity for partially hyperbolic abelian subgroups together to produce rigidity for the whole action.
A more immediate and probably more accessible issue is extension of our results to the differentiable case. Here the situation is reversed: rigidity for Cartan actions is proven but local linearization is not, and it may not even be true. Rigidity for Cartan actions provides extensive information about the semi-conjugacy P . Already for a Cartan subgroup, it is smooth in Whitney sense on sets whose measure is arbitrary close to full measure, see [7]. Those sets include grids of codimension one smooth submanifolds that divide the space into "boxes", most of them small. Superimposing those pictures for different Cartan subgroups provides such grids in a dense set of directions. However, certain elements of uniformity needed to conclude smoothness on an open set, are lacking.