Ergodicity of group actions and spectral gap, applications to random walks and Markov shifts

Let $(X, \cal B, \nu)$ be a probability space and let $\Gamma$ be a countable group of $\nu$-preserving invertible maps of $X$ into itself. To a probability measure $\mu$ on $\Gamma$ corresponds a random walk on $X$ with Markov operator $P$ given by $P\psi(x) = \sum_{a} \psi(ax) \, \mu(a)$. A powerful tool is the spectral gap property for the operator $P$ when it holds. We consider various examples of ergodic $\Gamma$-actions and random walks and their extensions by a vector space: groups of automorphisms or affine transformations on compact nilmanifolds, random walk in random scenery on non amenable groups, translations on homogeneous spaces of simple Lie groups, random walks on motion groups. The spectral gap property is applied to obtain limit theorems, recurrence/transience property and ergodicity for random walks on non compact extensions of the corresponding dynamical systems.


Introduction
Let (X, B, ν) be a metric space endowed with its Borel σ-algebra B and a probability measure ν, and let Γ be a countable group of Borel invertible maps of X into itself which preserve ν.
Let µ be a probability measure on Γ such that the group generated by A := supp(µ) is Γ. We consider the random walk on X defined by µ, with Markov operator P given by P ψ(x) = a∈A ψ(ax) µ(a), x ∈ X. (1) These data, i.e., the probability space (X, ν), the group Γ acting on (X, ν) and the probability measure µ on Γ, will be denoted by (X, ν, Γ, µ).
The operator P is a contraction of L p (X, ν), ∀p ≥ 1, and it preserves the subspace L 2 0 (X, ν) of functions ϕ in L 2 (X, ν) such that ν(ϕ) = 0. P is said to be ergodic if the constant functions are the only P -invariant functions in L 2 (X, ν).
Our aim is to consider some examples of ergodic actions and extensions of these actions by a vector space. We will use a strong reinforcement of the ergodicity, the spectral gap property for the operator P when it holds and we will develop some of its consequences. Let us recall its definition and related notions.
We recall that a unitary representation ρ of a group Γ in a Hilbert space H is said to contain weakly the identity representation if there exists a sequence (x n ) in H with x n = 1 such that, for every γ ∈ Γ, lim n ρ(γ)x n − x n = 0. See [BeHaVa08] for this notion.
Recall also that Γ is said to have property (T) if, when the identity representation is weakly contained in a unitary representation ρ of Γ, then it is contained in ρ.
The natural action of Γ on L 2 0 (X) defines a unitary representation ρ 0 of Γ in L 2 0 (X). Property (SG) implies that the identity representation of Γ is not weakly contained in ρ 0 . The converse is true if, for every k > 0, (supp(µ)) k generates Γ (see below Corollary 3.12). Property (SG) depends only on the support of µ.
For a countable group Γ acting measurably on a probability measure space (X, ν) where ν is Γ-invariant, according to [FuSh99] the Γ-action on (X, ν) is said to be strongly ergodic if ν is the unique Γ-invariant continuous positive normalized functional on L ∞ (X, ν). Property (SG) implies strong ergodicity, hence ergodicity of the action of Γ on X.
Our framework will be essentially algebraic. As examples, we study the action of groups of automorphisms or affine transformations on tori and compact nilmanifolds, and translations on homogeneous spaces of simple Lie groups. In Section 1 we show for nilmanifolds that the ergodicity of P follows from the ergodicity of its restriction to the maximal torus quotient. In Section 2, we recall property (SG) for subgroups of SL(d, Z) acting on T d , as well as recent results on property (SG) for the nilmanifolds. In Section 3 we consider random walks on non compact extensions of dynamical systems and apply property (SG) to recurrence and ergodicity. The last section is devoted to examples.
The authors thank Bachir Bekka for useful discussions.

Ergodicity of a group of affine transformations on nilmanifolds
In this section, we consider groups of affine transformations Γ on compact nilmanifolds X. In order to obtain ergodicity of Markov operators on X, as described in the introduction, we study the question of ergodicity of the action of Γ.
Let N be a connected, simply connected, nilpotent Lie group and D a lattice in N, i.e. a discrete subgroup D such that the quotient X = N/D is compact. If L 1 , L 2 are two subgroups of N, we denote by [L 1 , L 2 ] the closed subgroup generated by the elements {n 1 n 2 n −1 1 n −1 2 , n 1 ∈ L 1 , n 2 ∈ L 2 }, L ′ := [L, L] the derived group of L, e the neutral element of N. The descending series of N is defined by where N ℓ+1 := [N ℓ , N], for ℓ ≥ 0, with N 0 = N.
The elements g ∈ N act on N/D by left translation: nD ∈ N/D → gnD. We say that τ is an automorphism of the nilmanifold N/D if τ is an automorphism of the group N such that τ D = D. The group of automorphisms of N/D is denoted by Aut(N/D). The action of τ ∈ Aut(N/D) on N/D is nD → τ (n)D. An affine transformation γ of N/D is a map of the form: (2) nD → γ(n)D = α γ τ γ (n)D, with α γ ∈ N and τ γ ∈ Aut(N/D).
Let Γ be a group of affine transformations of the nilmanifold. The measure m on N/D deduced from a Haar measure on N is Γ invariant. The group Γ acts on the quotients N ℓ /N ℓ ∩ D, 0 ≤ ℓ ≤ k + 1, and in particular on the factor torus T = N/N 1 .D. When Γ is a group of automorphisms, ergodicity of the action on the torus is equivalent to the fact that every non trivial character has an infinite Γ-orbit.
When Γ is generated by a single automorphism (or more generally by an affine transformation), W. Parry has proved ( [Pa69], [Pa70]) that the ergodicity of the action on the quotient T implies the ergodicity of the action on the nilmanifold. We will show (Theorem 1.4) that the analogous statement holds for a group of affine transformations.
Notations: For a given group Γ of affine transformations of N/D,Γ denotes the subgroup of Aut(N/D) generated by {τ γ , γ ∈ Γ}, where τ γ is the automorphism associated with γ as in (2). We denote by N ℓ e the Lie algebra of N ℓ and by dτ e the linear map tangent at e to an automorphism τ of N.
We will use the following lemmas.
Lemma 1.1. (cf. CoGu74) If Γ is a subgroup of GL(R d ) such that the eigenvalues of each element of Γ has modulus 1, then there is a Γ-invariant subspace W = {0} of R d such that the action of Γ on W is relatively compact. If Γ is a subgroup of GL(Z d ), the action of Γ on W is that of a finite group of rotations and reduces to the identity for γ in a subgroup Γ 0 of finite index in G.
Proof. We extend the action of Γ to C d . LetW be a subspace of C d which is different from {0} and invariant by Γ on which the action of Γ is irreducible. Let (e i ) be a basis ofW , and let E ij be the maps defined by E ij (e k ) = δ kj e i , ∀k. We denote byτ the endomorphism corresponding to the action of τ ∈ Γ onW .
The action of Γ onW being irreducible, by Burnside's theorem there are constants b k and elements τ k of Γ such that E ji = k b kτk . The coefficients of the transformations τ satisfy then: Therefore sup τ ∈Γ |a ij (τ | < ∞, which implies the relative compactness of the action of Γ onW , as well on W , the Γ-invariant subspace of R d generated by {ℜe v, v ∈W }. Now assume that Γ is a subgroup of GL(Z d ). The symmetric functions of the eigenvalues of γ in Γ take values in Z and remain bounded when γ runs in Γ. This implies that the set of the characteristic polynomials of the elements γ is finite. If λ is an eigenvalue of γ the set (λ n ) n∈Z is finite and therefore λ is a root of the unity. The order of these roots remains bounded on Γ. This implies the last assertion.
Lemma 1.2. If a group Γ of affine transformations on T d has an invariant square integrable function f non a.e. constant, then it has an invariant function which is a non identically constant trigonometric polynomial. If the action of Γ is ergodic, every eigenfunction is a trigonometric polynomial.
The subspaces W k of L 2 generated by e 2πi<p,.> , for p ∈ R k , have a finite dimension, are pairwise orthogonal and invariant by each γ ∈ Γ. The orthogonal projections of f on these subspaces give Γ-eigenfunctions with the same eigenvalue as for f . This shows the existence of a non constant eigenfunction (invariant if f is invariant) which is a trigonometric polynomial. If the group Γ acts ergodically, only one of these projections is non null. Hence f is a trigonometric polynomial.
Lemma 1.3. If a group of affine transformations Γ of a torus T d is ergodic, then every subgroup Γ 0 of Γ with finite index is also ergodic on T d .
Proof. Let Γ 0 be a subgroup of Γ with finite index. As the action of Γ is ergodic, the σ-algebra of the Γ 0 -invariant subsets is an atomic finite σ-algebra whose elements are permuted by γ ∈ Γ. From Lemma 1.2 there exists a non constant trigonometric polynomial which is invariant by Γ 0 . This polynomial should be measurable with respect to the σ-algebra of the Γ 0 -invariant subsets which is atomic. The connectedness of the torus implies that it is constant.
Ergodicity of a group of affine transformations Theorem 1.4. Let Γ be a group of affine transformations on N/D. If its action on the torus quotient N/N 1 .D is ergodic, then every eigenfunction for the action of Γ on N/D factorizes into an eigenfunction on N/N 1 .D. In particular, the action of Γ is ergodic on N/D if and only if its action on the quotient N/N 1 .D is ergodic.
Proof. We follow essentially the method of W. Parry ( [Pa69]). We make an induction on the length k of the descending central series of N. The property stated in the theorem is clearly satisfied if k = 0.
The induction assumption is that, for every group of affine transformations of N/D, ergodicity of the action on N/N 1 .D implies ergodicity of the action on N/N k .D and every eigenfunction for the action on N/N k .D factorizes through N/N 1 .D. (The quotient (N/N k )/(N/N k ) ′ can be identified with the quotient N/N 1 .) Remark that, for every subgroup Γ 0 with a finite index in a group Γ of affine transformations on N/D, the action of Γ 0 on N/N 1 .D is ergodic if the action of Γ on N/N 1 .D is also ergodic from Lemma 1.3. The induction assumption implies then that Γ 0 acts ergodically on N/N k .D and that every eigenfunction for the action of Γ 0 on N/N k .D factorizes through N/N 1 .D.
Let f ∈ L 2 (N/D) be a Γ-eigenfunction, i.e. such that for complex numbers β(γ) of modulus 1, We are going to show that f factorizes into an eigenfunction on the quotient N/N k .D and therefore, by the induction hypothesis, into an eigenfunction on N/N 1 .D.
The proof is given in several steps. a) We denote by Z the center of N. We have N k ⊂ Z ∩ N k−1 . The torus H := Z ∩ N k−1 /Z ∩ N k−1 ∩ D acts by left translation on N/D and its action commutes with the translation by elements of N. Let Θ be the group of characters of H. The space L 2 (N/D) decomposes into pairwise orthogonal subspaces V θ , where θ belongs to Θ and V θ stands for the subspace of functions which are transformed according to the character θ under the action of H.
We decompose f into components in the subspaces V θ . By (4), we have, for every γ ∈ Γ, the orthogonal decomposition of f : We will show that the components f θ , hence also f , are invariant by translation by the elements of N k .
From now on, we consider Γ 0 and the component f θ which we denote by f for simplicity.
From what precedes, we have an ergodic action of a group of affine transformations Γ 0 , a character θ ∈ Θ such that τ γ θ = θ, ∀γ ∈ Γ 0 , and a function f satisfying (denoting by x an element of N/D): We can assume that θ is non trivial on N k . By replacing N/D by N/H 0 .D, where H 0 is the connected component of the neutral element of Ker θ, we can also assume that N k /N k ∩ D has dimension 1.
The previous equations imply that |f | is γ-invariant for every γ ∈ Γ 0 and N kinvariant. Therefore the function |f | is a.e. equal to a constant that we can assume to be 1.
The function θ is continuous and θ(e) = 1; therefore θ(g) = 0 on a neighborhood of e. The invariance of the measure implies: Denote by G the subgroup of N k−1 defined by (the equality in (7) follows from the equality in Cauchy-Schwarz inequality). Denote by G 0 the connected component of the neutral element in G.
Therefore g t is in G, for every t, hence g ∈ G 0 . The analysis is the same for W 2 . Consider the quotient N k−1 /Z ∩ N k−1 . As G 0 contains exp W 1 and exp W 2 , the differentials of the automorphisms τ ∈ Γ 0 have only eigenvalues of modulus 1 for their action on N k−1 e /Z e ∩ N k−1 e . By Lemma 1.1, there exists then in N k−1 e /Z e ∩ N k−1 e a subspace W 3 on which the action of the transformations which are linear tangent to the automorphisms ∈ Γ 0 is compact. As the automorphisms τ preserve a lattice, their action on W 3 has a finite order. The subgroupΓ 1 which leaves fixed the elements of W 3 has a finite index iñ Γ 0 .
Let g be an element of N k−1 in exp W 3 . For every τ γ ∈Γ 1 , there exists g 0 ∈ Z ∩N k−1 such that τ γ g = g 0 g. Equation (10) reads . The function f (g.) f (.) is an eigenfunction for every γ ∈ Γ 1 and is invariant by h ∈ N k . It factorizes into an eigenfunction for Γ 1 on N/N k .D. Either its integral is 0 (if for γ ∈ Γ 1 the corresponding eigenvalue is = 1) or it is invariant by Γ 1 . In the latter case, as by the induction hypothesis ergodicity holds for the action of Γ 1 on N/N k .D, the function f (g.) f (.) is equal to a constant which has modulus 1 (since |f | = 1).
We have therefore |θ(g)| = 0 or 1. By a continuity argument |θ(g)| = 1, which implies that g ∈ G. Using as above a one parameter subgroup, we obtain that g ∈ G 0 . This gives a contradiction, since G 0 ⊂ Z ∩ N k−1 . d) Let (h t ) be a one parameter subgroup of N and let g be in G 0 . We have from (9): By continuity, θ(h t ) is different from zero in a neighborhood of t = 0. The previous relation implies θ(h t gh −1 t g −1 ) = 1 in a neighborhood of t = 0 and, G 0 being connected, is equal to 1 everywhere. As [N, G 0 ] = N k , this shows that the character θ is identically equal to 1 on N k and therefore the announced factorization property is satisfied.
Remark There are compact nilmanifolds N/Γ for which the group Aut(N/Γ) is non ergodic (cf. [DiLi57]). This contrasts with the case of Heisenberg nilmanifolds, for which there is a large group of automorphisms.

An example
Now we give an example of nilmanifold with an ergodic group Γ of automorphisms such that each automorphism in Γ is non ergodic.

Construction on the torus
Examples of groups of matrices such that each of them has an eigenvalue equal to 1 can be constructed by action on the space of quadratic forms. We explicit the example in dimension 2.
Let A := a b c d ∈ GL(2, R), with eigenvalues λ 1 , λ 2 . The matrix corresponding to the action of A on the vector space of symmetric 2 × 2 matrices M → AMA t is whose eigenvalues are detA, λ 2 1 , λ 2 2 . The vector (2b, d − a, −2c) t is an eigenvector for q(A) with eigenvalue detA, and is invariant by q(A) if detA = 1.
The restriction to SL(2, Z) of the map A → q(A) defines an isomorphism onto a discrete subgroup Λ 0 of automorphisms of SL(3, Z) whose each element is non ergodic (each element q(A) leaves fixed a non trivial character of the torus T 3 ), but which acts ergodically on T 3 , since the orbits of the transposed action on Z 3 \ {0} are infinite.

Extension to a nilmanifold
Now we extend the action of Λ 0 to a nilmanifold. This group is ergodic by Theorem 1.4. Let us consider the real Heisenberg group H 2d+1 of dimension 2d + 1, d ≥ 1, identified with the group of matrices (d + 2) × (d + 2) of the form:  where x and y are respectively line and column vectors of dimension d, z a scalar and I d the identity matrix of dimension d. The law of group in H 2d+1 can be defined by:

The spectral gap property
We will now describe some classes of examples where property (SG) is satisfied.

Tori
As a basic example where property (SG) is valid, let us consider as in [FuSh99] (See also [Gu06]) the d-dimensional torus X = T d endowed with the Lebesgue measure, and the action of SL(d, Z) on T d by automorphisms. The Lebesgue measure is preserved. Every γ ∈ SL(d, Z) acts by duality on Z d by γ t . We denote by µ t the push-forward of a probability measure µ on SL(d, Z) by the map γ → γ t .
Proposition 2.1. Let µ be a probability measure on SL(d, Z) such that supp(µ t ) has no invariant measure on the projective space P d−1 . Let P be the Markov operator on T d defined by Then the corresponding contraction Π 0 on L 2 0 (X) satisfies r(Π 0 ) < 1. Proof. The Plancherel formula gives an isometry I between L 2 0 (T d ) and Suppose r(L) = r(Π 0 ) = 1 and let e iθ be a spectral value. Then two cases can occur. Either there exists a sequence Since e iθ L * is a contraction on ℓ 2 (Z d \ {0}), its fixed points are also fixed points of its adjoint e −iθ L, hence Lf = e iθ f . It follows that it suffices to consider the first case. The condition lim n Lf n − e iθ f n 2 = 0 implies lim n Lf n 2 = lim n e iθ f n 2 = 1 and also lim The inequality In other words, if ν n denotes the probability measure on ℓ 2 (Z d \ {0}) with density |f n | 2 , we have in variational norm: Let ν n be the projection of ν n on P d−1 and ν a weak limit of ν n . By (13), we have , which contradicts the hypothesis on supp(µ t ).
Remarks 1) The hypothesis on the support of µ t is satisfied if the group generated by supp(µ) has no irreducible subgroup of finite index. This is a consequence of the following fact observed by H. Furstenberg: if a linear group has an invariant measure on P d−1 , then either it is bounded or it has a reducible finite index subgroup (See [Zi84], p. 39, for a proof).
2) The above corresponds to a special case in the characterization of property (SG) given in [BeGu11], Theorem 5, for affine maps of T d . In particular ifμ is a probability measure on the group Aut(T d ) ⋉ T d and µ is its projection in Aut(T d ), property (SG) forμ acting on T d is valid if the group generated by supp(µ) is non virtually abelian and its action on R d is Q-irreducible.

Nilmanifolds
As in Section 1, let N be a simply connected nilpotent group, D a lattice in N, X = N/D the corresponding nilmanifold, and T the maximal torus factor. Let µ be a probability measure on Aut(X) ⋉ N, µ its projection on Aut(T ) ⋉ T . Then it is shown in [BeGu11], Theorem 1, that if the convolution operator on L 2 0 (T ) associated with µ satisfies property (SG), then the same is true for the operator on X associated with µ.
In view of the torus situation described above, this gives various examples of measures µ where property (SG) is valid. If N is a Heisenberg group, more precise results are available, which will be recalled in Section 4.

Simple Lie groups
Let us consider a non compact simple Lie group G and let ∆ be a lattice in G, i.e. a discrete subgroup such that X = G/∆ has finite volume for the Haar measure v. Let µ be a probability measure on G. It follows from Theorem 6.10 in [FuSh99] that, if µ is not supported on a coset of a closed amenable subgroup of G, then property (SG) is valid for the action of µ on X.

Compact Lie groups
We take X = SU(d), ν the Haar measure on X. Then it is known (See [GaJaSa99], [BoGa10]) that for d ≥ 3, if Γ ⊂ SU(d) is a countable dense subgroup such that the coefficients of every element of Γ are algebraic over Q and µ generates Γ, then the natural representation of Γ in L 2 0 (X) does not contain weakly Id Γ (cf. Definitions 0.1).
In particular there are dense free subgroups of SU(d) as above. Also, if X = SO(d) and d ≥ 5, there are countable dense subgroups of SO(d) which have property (T ).
Let A be a finite set of generators for Γ and µ a probability with supp(µ) = A. Then property (SG) is valid for the convolution action of µ on X, since Γ is ergodic on (X, ν), a consequence of the density of Γ.

Applications of the spectral gap property
3.1. Extensions of group actions and random walks. As in the introduction, let X be a metric space, Γ a countable group of invertible Borel maps of X into itself which preserve a probability measure ν on X, and µ a probability measure on Γ with finite support such that A := supp(µ) generates Γ as a group. We assume that the action of Γ on (X, ν) is ergodic. We will use both notations: a f (ax) µ(a) or f (ax) dµ(a).
We consider also the bilateral shift onΩ := A Z still denoted by σ. It preserves the product measureP = µ ⊗Z .
Proof. The dual operator of the composition by σ 1 on L 2 (P 1 ) is To prove the ergodicity of σ 1 , it suffices to test the convergence of the means and f on Ω depends only on the first p coordinates, for some p ≥ 0. Setting Ergodicity of P , hence ofP , implies the convergence of the means 1 The centering condition of the displacement is assumed, i.e. −1 x)). We denote byΓ the group of Borel maps of X × V generated byÃ = {ã, a ∈ A}. The action ofΓ preserves the fibering of X × V over X, and the projection of X × V on X is equivariant with respect to the action of Γ on X. We have a homomorphism γ → γ fromΓ to Γ which mapsã to a, for every a ∈ A.
In other words, using the displacement (c a (x), a ∈ A), we can extend the action of the group Γ on X to the action of the groupΓ generated by the mapsã, a ∈ A, on X ×V . Clearly the mapsγ ∈Γ commute with the translations on the second coordinate on X × V by elements of V and therefore are of the formγ( The displacement satisfies the cocycle property (for Γ) if the value of the sum in (15) depends only on the value of the product γ = a r ...a 1 in Γ.
It should be noticed that this cocycle property in general is not satisfied, since the value of the sum in (15) depends in the general case on the "path" (a 1 , ..., a r ). This The cocycle property then trivially holds in Γ. This is also the case if c a (x) is a limit of coboundaries.

Extension of the random walk
We consider the random walk on E := X × V defined by the probability measure µ and the mapsã. Its Markov operatorP extends the Markov operator P of the random walk on X given by (1) and is defined by  (x, a). Under spectral gap conditions on certain functional spaces, it is possible to develop a detailed study of the iterationP n of P (See for example [Gu06] when the functional spaces are Hölder spaces). In the framework of the present paper, no regularity is assumed. We supposed only that the displacement consists in bounded Borel maps.
The set Y (resp. Y × V ) can be identified with the space of trajectories of the Markov chain defined by P (resp.P ). With the notation of (15) we have Hence, S n (y) appears as a Birkhoff sum over (Y, σ 1 ). The iterates ofσ on Y × V read:σ n (y, v) = (σ n 1 y, v + S n (y)), ∀n ≥ 1.
Also if we denote byμ the push-forward of µ by the map a →ã, we can express the iterateP n ofP asP whereμ n is the n-fold convolution product ofμ by itself.
Here we are interested in the asymptotic properties ofσ n and (S n (y)) with respect to the measures P 1 ⊗ℓ and P 1 under the condition that P has "nice" spectral properties on X (see below). The L 2 -spectral gap condition can be compared to the so-called Doeblin condition for the Markov operator P .
, and where σ is now acting on the bilateral spaceΩ.
We will need to analyzeP n using methods of Fourier analysis. Hence we are led to introduce a family of operators P λ on L 2 (X), λ ∈ V , defined by We observe that, since sup a∈A c a ∞ = c < ∞, the above formula still makes sense if λ ∈ R d is replaced by z ∈ C d , and we obtain an operator valued holomorphic function z → P z satisfying, for any ϕ, ψ ∈ L 2 (X), This will allow us to use perturbation theory (See [GuHa88] for an analogous situation).
The V -valued function h(x) := a∈A c a (x)µ(a) belongs to L 2 0 (X), in view of the centering condition (14).
Since r(Π 0 ) < 1, the restriction of P − I to L 2 0 (X) is invertible, hence we can solve the equation (P − I)u = h, with u ∈ L 2 0 (X). The modified displacement c ′ (a, x) defined, for a ∈ Γ, by A basic tool for the study ofP n will be the analysis of the Fourier operators P λ , and in fact their spectral gap properties. Their family satisfies (as in [GuHa88] (Lemmas 1 and 2)): Lemma 3.2. For any ϕ ∈ L 2 (X), we have For λ small, P λ has a dominant eigenvalue k(λ) given by In order to analyze more closely the operators P λ , we introduce some definitions related to the aperiodicity of the displacement. We observe that the formulaã(x, z) = (ax, z e i( λ, c(a,x) −θ) ) defines an action ofΓ on X × T and that ergodicity of these actions for every λ = 0 implies condition (AP). Clearly (AP) implies (NR).
There are special cases (corresponding to functional of Markov chains) where the previous conditions can be easily verified.
Lemma 3.4. Assume that c a (x) = c(x) does not depend on a ∈ A, and that the R d -valued function c is bounded, not ν-a.e. 0 and satisfies the centering condition c(x) dν(x) = 0.

1) If
A ⊂ Γ has a symmetric subset B such that B 2 acts ergodically on (X, ν), then (NR) is satisfied.
2) If the measure c(ν) is not supported on a coset of a proper closed subgroup of R d , then (AP) is satisfied.
Since B 2 acts ergodically on (X, ν) we get, for λ = 0 and some c λ of modulus 1, e i( λ,c(x) = c λ . This means that c(x) belongs to the coset of the proper closed subgroup of R d defined by e i( λ,v = c λ , which contradicts the hypothesis.
3.2. Ergodicity, recurrence/transience. In this section, we study ergodicity, recurrence and transience of the extended dynamical systems considered above. First we recall briefly the notion of recurrence in the framework of dynamical systems.
Let (Y, λ, τ ) be a dynamical system with Y a metric space, λ a probability measure on Y and τ an invertible Borel map of Y into itself which preserves λ. We suppose the system ergodic. If ϕ is a Borel map from Y to R d , the ergodic sums S n ϕ(y) = n−1 k=0 ϕ(τ k y) define a "random walk in R d over the dynamical system" (Y, λ, τ ). The corresponding skew product is the dynamical system defined on (Y × R d , λ × ℓ) by the transformation τ ϕ : (y, v) → (τ y, v + ϕ(y)).
We say that y ∈ Y is recurrent if, for every neighborhood U of 0 in R d , n≥0 1 U (S n ϕ(y)) = +∞.
We say that y is transient if, for every neighborhood U of 0 in R d this sum is finite.
Since the set of recurrent points is clearly invariant and the system (Y, λ, τ ) is ergodic, every cocycle (S n ϕ) is either transient or recurrent.
For the sake of completeness, let us give a simple proof of the following known equivalence: Proposition 3.5. The recurrence of (S n ϕ) is equivalent to the conservativity of the system (Y × R d , λ ⊗ ℓ, τ ϕ ).
Proof. By definition the dynamical system (Y × R d , λ ⊗ ℓ, τ ϕ ) is conservative if, for every measurable set A in Y × R d with positive measure, for a.e. (y, v) ∈ A there exists n ≥ 1 such that τ n ϕ (y, v) = (τ n y, v + S n ϕ(y)) ∈ A.
We show conversely that this property holds if (S n ϕ) n≥1 is recurrent in the sense of the above definition.
We can suppose that A is included in Y × L, where L is a compact set in R d . One checks easily that the set B := {(y, v) ∈ A : ∀n ≥ 1, τ n ϕ (y, v) ∈ A} has pairwise disjoint images.
Using the recurrence of (S n ϕ), one can find for every ε > 0 a compact set K ε such that, for a set of measure ≥ 1 − ε of points y, the sums S n ϕ(y) belongs to K ε for infinitely many n (we use the fact that for a.e. y, the set {S n ϕ(y), n ≥ 1} has a finite accumulation point and that there is a neighborhood of this accumulation point in which S n ϕ(y) returns infinitely often).
In the transient case, the system (Y × R d , λ ⊗ ℓ, τ ϕ ) is dissipative, i.e. there exists a Borel subset B ⊂ Y × R d , such that Now we will study these properties of recurrence and transience in the case of the random walk (S n (ω, x)) over the dynamical system (Ω × X,P 1 , σ 1 ) and its extension (Ω × E,σ,P 1 ⊗ ℓ) defined at the beginning of this section.
1a) If the displacement (c a , a ∈ A) satisfies (NR), then ( 1 √ n S n (ω, x)) n≥1 converges in law with respect to P ⊗ ν to the centered normal law on V with non degenerate covariance Σ.

Proof. 1) We have
On the other hand, for any v ∈ V , by definition of Σ(v) and from (18) we have the martingale property and in particular Clearly Y k = Y 1 • σ k 1 and σ 1 preserves the measure P ⊗ ν. Ergodicity of σ 1 and the ergodic theorem imply P ⊗ ν-a.e. Brown's central limit theorem ( [Br71]) applies to (Y k ), and gives the CLT for (S n ) since the coboundary term u(a n ...a 1 x) − u(x) is bounded. So we get the convergence in law of 1 √ n S n (ω, x) with respect to P ⊗ ν to the centered normal law on V with covariance Σ. The non degeneracy of Σ follows from the formula Σ(v) = v, c ′ a (x) 2 dµ(a) dν(x) and Condition (NR). The statement 1b) (the convergence (19)) is proved in Lemma 3.8 below.
2a) Using the CLT as a recurrence criterion for the R 2 -valued Z-cocycle, (S n y) over the measure preserving transformation σ 1 (cf. [Sc98] or [Co99]), the recurrence property follows: 2b) We observe that the trajectories of the random walk on E = X × V defined by µ are given by X n (ω, x, v) = (σ n 1 (ω, x), v + S n (ω, x)).

By (20), for any relatively compact open set
In other words, the Markov kernelP on E satisfies Property R defined in [GuRa09]. Hence, using Proposition 2.6 in [GuRa09], the ergodicity of (Ω × E,σ,P 1 ⊗ ℓ) will follow if we show that the equationP h = h, for h ∈ L ∞ (ν ⊗ ℓ), has only constant solutions.
SinceP h = h, we have for any n ∈ N, ϕ ∈ L 2 (X), Lemma 3.7 below gives ϕ ⊗ f, h = 0, hence h is invariant by translation by v and defines an element h ∈ L ∞ (X, ν) with P h = h. Then we have a∈A h(ax)µ(a) = h(x), hence the invariance of h by a ∈ A. Since ν is Γ ergodic, h is constant ν-a.e. Therefore h is constant ν ⊗ ℓ-a.e. This proves 2b).
In particular by (19) the sequence (n d/2 P n ψ, ψ ) is bounded. If d > 2 the series ∞ n=0 | P n ψ, ψ | converges. Hence the result. 3) The assertion is shown in the proof of 2b). Now, under the assumption (AP) as in the theorem, we prove the lemmas used in the previous proof.
Proof. In the proof of Proposition 3.6 of [GuSt04], a Markov operator Q on X × R d which commutes with the R d -translations is considered and it is proved that lim n→∞ Q n (u ⊗ f ) 1 = 0 for Hölder continuous functions u in H ε (X) and f as above. The essential points of the proof are polynomial growth of R d as a group and a spectral gap property for the Q-action on function of the form u ⊗ λ where u ∈ H ε (X) and λ is a character of R d .
Here we observe that the adjoint operatorP * ofP on L 2 0 (E, ν ⊗ ℓ) is associated witȟ µ the symmetric of µ which has the same properties as µ as it was observed above.
The action ofP * is also well defined on the functions of the form u ⊗ λ where λ is a fixed character of V and u is in L 2 (X). It reduces there to the action of P * λ on L 2 (X) hence using b) ⇒ a) of Proposition 3.9 below we get that (AP) implies that P * λ has a spectral gap. Hence the lemma follows from the proof of Proposition 3.6 in [GuSt04] with Q =P * .
Lemma 3.8. Let α be a probability measure on X which has a L 2 -density with respect to ν. Letα be the probability measure α ⊗ δ 0 on X × V and letμ n := (2πn) d/2 (det Σ) Proof. Let ϕ be a function in L 2 (X) and f ∈ L 1 (V ) be such that its Fourier transform f (λ) = f (v) e i λ,v dℓ(v) has a compact support on V . Then, by the inversion formula we have: As in [Br68], p. 225, we test the convergence ofμ ϕ n (f ) =μ n (ϕ ⊗ f ) using functions f as above. We apply the method of [GuHa88] for proving the local limit theorem, giving only the main steps of the proof. According to b) ⇒ a) of Proposition 3.9 below, we observe that the operator P λ considered above satisfies r(P λ ) < 1 for λ = 0, in view of Condition (AP). Furthermore, by perturbation theory, for λ small enough, P λ has a dominant eigenvalue k(λ) and a corresponding one dimensional projection p λ such that: Also p λ , r λ depend analytically on λ. These facts will allow us to adapt the analogous proof of [GuHa88]. We writeP nα as follows: Since r(P λ ) < 1 for λ = 0, the integration can be reduced, in the limit, to a small neighborhood U of 0 in R d and it suffices to consider Using the spectral decomposition of P λ we see that lim n→∞ I n = lim n→∞ J n with Since lim n→∞ k(λ/ √ n) n = e − 1 2 Σ(λ) and lim λ→0 p λ (ϕ) = ν(ϕ), we get Proposition 3.9. Let Π 0 be the restriction of P to L 2 0 (X) and let P λ , λ ∈ V , λ = 0, be defined on L 2 (X) by (17) with c a (x) ∈ L ∞ (X) for every a ∈ supp(µ). Assume that r(Π 0 ) < 1. Then the following properties are equivalent: a) the spectral radius r(P λ ) of P λ acting on L 2 (X) is 1; b) the condition (AP) is not satisfied at λ: there exists a real θ and a measurable function α such that e i( λ, ca(x) −θ) = e i(α(ax)−α(x)) , ν − a.e. c) there exists θ λ ∈ R such that e i( λ, ca(x) −θ λ ) extends as a cocycle σ λ (γ, x) on Γ ×X, with values in the group of complex numbers of modulus 1, and the representation ρ λ of Γ on L 2 (X) contains Id Γ , where We begin as in the proof of Proposition 2.1. Assume r(P λ ) = 1 and let e iθ , (θ ∈ [0, 2π[) be a spectral value of P λ . Then, either the subspace Im(e iθ −P λ ) is not dense in L 2 (X) or there exists ϕ n ∈ L 2 (X), with ϕ n = 1, such that lim n P λ ϕ n − e iθ ϕ n 2 = 0.
On the other hand, the condition lim n P λ ϕ n , e iθ ϕ n = 1 can be written as lim n e i λ,ca ϕ n • a, e iθ ϕ n dµ(a) = 1, where for each a ∈ supp(µ) | e i λ,ca ϕ n • a, e iθ ϕ n | ≤ 1.
This means that e iθ is an eigenvalue of P λ , so that r(P λ ) = 1.
Remarks: 1) In general, it is non trivial to calculate the set of λ ∈ R d such that r(P λ ) = 1. However, Corollary 3.10 is useful for this question.
2) Also we observe that condition c) in the proposition implies that the action of Γ on X × T given by γ(x, t) = (γx, tσ λ (γ, x)) is not ergodic.
The definition of P λ , P λ ′ gives the following inequality: hence, since |c a (x)| is bounded by a constant c, Therefore, we have P λ − P λ ′ ≤ c|λ − λ ′ |, which implies that, if λ is fixed with r(P λ ) < 1, we have also r(P λ ′ ) < 1 for λ ′ sufficiently close to λ. Hence R µ is closed.
Assume now Condition (NR) is satisfied. Observe that P λ − P 0 ≤ c|λ|. Since r(Π 0 ) < 1, the spectrum of P = P 0 consists of {1} and a compact subset of the open unit disk. Hence P has a dominant isolated eigenvalue, which is a simple eigenvalue. By perturbation theory, this property remains valid in a neighborhood of 0.
Using Lemma 3.2 and the fact (noted in Theorem 3.6 part 1) that the covariance matrix Σ is non degenerate, the dominant spectral value k(λ) satisfies: r(P λ ) = |k(λ)| < 1, for λ small and = 0. Hence R µ ∩ W = {0} for some neighborhood W of 0, i.e. R µ is a discrete subgroup of R d .
Remark. If c a (x) takes values in Z d , the previous results have an analogue if we replace the space X × R d by E = X × Z d . The character λ ∈ V should be replaced by a character of Z d , i.e. λ ∈ T d , and the Lebesgue measure ℓ by the counting measure on Z d . This will be used in 4.3 below.
The following corollary makes explicit the result in Theorem 3.6 for a functional c(x) of a Markov chain.
Corollary 3.11. Assume that (X, ν, Γ, µ) satisfies property (SG), that c a (x) = c(x) does not depend on a ∈ A, and that the R d -valued function c is bounded and satisfies c(x) dν(x) = 0. Moreover, assume that A ⊂ Γ has a symmetric subset B such that B 2 acts ergodically on (X, ν). Then we have: 1) if d ≤ 2, we have P × ν-a.e. lim inf n→∞ S n (ω, x) = 0; 2) if the measure c(ν) is not supported on a coset of a proper closed subgroup of R d , -for d ≤ 2,σ is ergodic with respect to µ × ν × ℓ, -for d ≥ 3, the local limit theorem is valid for (S n (ω, x)) and lim n→∞ S n (ω, x) = +∞, P × ν-a.e.
Proof. The result follows from Lemma 3.4 and Theorem 3.6.
The arguments in the proof of the proposition give also the following corollary, which is a direct consequence of the main result of [JoSc87] (see also Theorem 6.3 in [FuSh99]).
Corollary 3.12. Assume supp(µ) is finite, generates Γ and the representation ρ 0 of Γ in L 2 0 (X) does not contain weakly Id Γ . Let Γ * be the group of characters of Γ and Γ * X be the subset of elements of Γ * contained in the natural representation of Γ in L 2 (X, ν). Then Γ * X is a finite subgroup of Γ * . The measure µ satisfies property (SG) if and only if supp(µ) is not contained in a coset of the subgroup ker χ for some χ ∈ Γ * X , χ = 1. In particular, if (supp(µ)) k generates Γ for any k > 0, then (SG) is satisfied.
Also it is clear that Γ * X is a subgroup of Γ * . To obtain that Γ * X is closed in Γ * , we note that if χ ∈ Γ * satisfies for some sequence (ϕ n ) with |ϕ n (x)| = 1, χ(γ) = lim n ϕ n (γx)/ϕ n (x), then χ ∈ Γ * X . As in the proof of the proposition this follows from Proposition 2.3 of [Sc98], since the subgroup of T-valued coboundaries of (Γ, X, ν) is closed in the group of cocycles endowed with the topology of convergence in measure.
In order to show that each element of Γ * X has finite order, we observe that, using [JoSc87], (Γ, X, ν) has no non atomic Z-factor up to orbit equivalence. Hence, for every χ ∈ Γ * X and some n ∈ N * one has χ n = 1. Since Γ is finitely generated, Γ * X is a closed subgroup of a torus. Therefore Γ * X is finite.
Conversely, if µ does not satisfy (SG), then for some c of modulus 1 and a sequence (ϕ n ) in L 2 (X) with ϕ n = 1, we have lim n P ϕ n − cϕ n 2 = 0. As in the proof of the proposition, we can use the condition that ρ 0 does not contain weakly Id Γ to get that P − I is invertible on L 2 0 (X) and obtain that lim n |ϕ n | − 1 2 = 0. Then, writing ϕ n (x) = |ϕ n (x)| e iαn(x) , we get that for a subsequence (n k ) of integers, lim k e i(αn k (x)−αn k (γx)) = c.

Examples
4.1. Random walk in random scenery . As an example corresponding to Corollary 3.11, we consider a group Γ, a probability measure µ on Γ such that A := supp(µ) is symmetric and (supp(µ)) 2 generates Γ as a group. We denote by (Σ n (ω), ω ∈ A Z ) the left random walk on Γ defined by µ and we consider the visits of Σ n (ω) to a random scenery on Γ.
Such a random scenery is defined by a finite set C ⊂ R d , a probability measure η on C with supp(η) = C and v∈C η(v) v = 0. To each γ ∈ Γ, one associates a random variable x γ with values in C. The variables x γ are assumed to be i.i.d. with law η.
The scenery defines a point x = (x γ ) γ∈Γ of the Bernoulli scheme X = C Γ endowed with the product measure µ ⊗Γ and Γ acts on C Γ by left translations: if x = (x γ , γ ∈ Γ), then ax = (x aγ , γ ∈ Γ). If we define f (x) = x e ∈ C, the cumulated scenery is given by S n (ω, x) = n k=1 f (a k (ω)...a 1 (ω)x). One can give the following interpretation: the random walker collects the quantity x γ when visiting the site γ and his "cumulated gain" at time n along the path defined by ω is S n (ω, x).
We consider the Markov operator P on X associated with µ and its restriction Π 0 to L 2 0 (X). It is well known (see [BeHaVa08], Ex E45, p. 394) that the action of Γ on L 2 0 (X) decomposes as a direct sum of tensor products of the regular representation in ℓ 2 (Γ). A typical summand is ⊗ k 1 ℓ 2 (Γ) and if r 0 (µ) is the spectral radius of the convolution operator by µ in ℓ 2 (Γ), we have r(Π 0 ) = sup k≥1 (r 0 (µ)) k = r 0 (µ).
Assume that Γ is non amenable. Then we have r 0 (µ) < 1 (see [Ke59]), hence property (SG) is satisfied. If we assume that C ⊂ R d is not supported on a coset of a closed subgroup of R d , the hypothesis and the conclusions 2 and 3 of Corollary 3.11 are valid. Hence, with the above notations, it follows: Proposition 4.1. Let Γ be a non amenable group, µ a probability measure on Γ such that supp(µ) is symmetric and (supp(µ)) 2 generates Γ, Σ n (ω) the corresponding random walk on Γ. We assume that Γ is endowed with an R d -valued random scenery with law η, that C = supp(η) is finite with v∈C vη(v) = 0, and supp(η) is not contained in a coset of a closed subgroup of R d . We denote by S n (ω, x) the accumulated scenery along the random walk and byσ the transformation on Then, the convergence of 1 √ n S n (ω, x) to a non degenerate normal law is valid. If d ≤ 2,σ is ergodic and (S n ) is recurrent with respect to µ × ν × ℓ. If d ≥ 3, µ ⊗Z × ν-a.e., lim n S n (ω, x) = +∞.
Here, due to the strong transience properties of Γ, S n (ω, x) behaves qualitatively like a sum of i.i.d. random variables. Let us consider Γ = Z m , for m large.

4.2.
Random walks on extensions of tori. Now we present a special case where Condition (AP) can be checked.
Since x → {x} is bounded and {x} dν(x) = 0, the proposition is a direct consequence of Proposition 3.9, Theorem 3.6 and the following lemma.
Lemma 4.4. Let µ be a probability measure on Sp(2d, Z) and let Γ be the subgroup generated by supp(µ). For λ ∈ R 2d let P λ be the operator on L 2 (T 2d ) defined by Then, if Γ acts Q-irreducibly on R 2d and Γ is not virtually abelian, we have r(P λ ) < 1, for λ ∈ R 2d \ {0}. In particular (SG) and (AP) are valid.
Proof. We will use as an auxiliary tool the Heisenberg group H 2d+1 and its automorphism group Sp(2d, R) ⋉ R 2d . The group H 2d+1 has a one dimensional center C isomorphic to R and a lattice ∆ such that ∆ ∩ C is isomorphic to Z, and ∆/∆ ∩ C is isomorphic to Z 2d .
We denote byμ the push-forward of µ by the map a →â. We denote byΓ the group generated by supp(μ) and we consider the convolution action ofμ on L 2 (X).
Since lim n r n = 0, in order to show that r(P λ ) < 1, i.e., P n λ < 1 for some n > 0, it suffices to find u ′ ∈ Z 2d such that c λ − 2πu ′ < 1. This is possible at least for a multiple of λ: one can find k ∈ N, k = 0, and u ′ ∈ Z 2d such that kλ − 2πu ′ < c −1 . Now, if r(P λ ) = 1, one has also from Corollary 3.10 that, for any k ∈ Z, r(P kλ ) = 1. From above this is impossible; hence r(P λ ) < 1.

4.3.
Random walks on coverings. Let G be a Lie group, H a closed subgroup such that G/H has a G-invariant measure m. If µ is a probability measure on G, we consider the random walk on G/H defined by µ, and the corresponding skew productσ on G Z × G/H endowed with the measure µ ⊗Z × m. Then one can ask for the ergodicity of such a skew product and its stochastic properties. If H is a normal subgroup of another group L ⊂ G such that G/L is compact, G/H is fibred over G/L and one can use harmonic analysis on G/L and H/L.
A special case of Proposition 4.5 below corresponds to the abelian coverings of compact Riemann surfaces of genus g ≥ 2. In this case, H is a subgroup ∆ ′ of a cocompact lattice ∆ in SL(2, R) and G/∆ ′ can be seen as the unit tangent bundle of the covering.
Proposition 4.5. Let G be a simple non compact real Lie group of real rank 1, µ a symmetric probability measure with finite support A ⊂ G such that the closed subgroup G µ generated by A is non amenable. Let ∆ be a co-compact lattice in G, ∆ ′ a normal subgroup such that ∆/∆ ′ = Z d , m the Haar measure on G/∆ ′ .
Proof. Since ∆ ′ is normal in ∆, the group Λ = ∆/∆ ′ ∼ Z d acts by right translations on G/∆ ′ and this action of Λ commutes with the left action of G.
The G-space G/∆ ′ can be written as X × Λ where X ⊂ G/∆ ′ is a Borel relatively compact fundamental domain of Λ in G/∆ ′ . We will denote by y the projection of y ∈ G/∆ ′ on X identified with G/∆, by m the Haar measure on G/∆, and by (g, x) → g.x the natural action of g ∈ G on an element x of the fundamental domain X Let z(y) be the Λ-valued Borel function on G/∆ ′ defined by y = yz(y). Then the G-action on X × Λ can be written as g(x, t) = (g.x, t + z(gx)) where the group Λ = Z d is written additively.
For g ∈ G and x ∈ X, writing Z(g, x) := z(gx), we obtain a cocycle: Actually the cocycle relation is valid in restriction to Γ.
Since G is simple and G µ is non amenable, we know ( [FuSh99], Theorem 6.11) that the convolution operator Π 0 on X = G/∆ defined by µ has a spectral radius r(Π 0 ) < 1 on L 2 0 (X). On the other hand, if for any a ∈ supp(µ), x ∈ X, we write c a (x) = z(ax) = Z(a, x) andã(x, t) = (a.x, t + c a (x)), we are in the situation of Section 3.
In order to verify this, we observe that, since X is relatively compact and supp(µ) is finite, the functions c a (x) are uniformly bounded. Furthermore, the cocycle relation for Z(g, x) gives for any g ∈ G, x ∈ X: Z(g −1 , x) + Z(g, g −1 .x) = 0; hence (Z(g −1 , x) + Z(g, x)) dm(x) = 0. Since µ is symmetric, we have the centering condition: c a (x) dm(x) dµ(a) = 0.
Since G µ is non amenable and G is simple, the result will follow from Theorem C, part 2 of [Sh00], if we can show that ρ λ does not contain weakly the representation Id G . By definition, ρ λ is the induced representation to G of the representation λ ∆ of ∆ defined by the character λ. Clearly, if λ = 0, λ ∆ does not contain weakly Id ∆ . Since G/∆ has a finite G-invariant measure, it follows from Proposition 1.11b, p. 113 of [Ma91] that ρ λ does not contain weakly Id G . 4.4. Random walks on motion groups. Let G be the motion group SU(d) ⋉ C d , d ≥ 2. Write X = SU(d), ν for the Haar measure on X, V = C d . We identify a vector in V with the corresponding translation in G and we write G = X V . Let Γ ⊂ SU(d) be a dense subgroup with property (SG) and A a finite generating set of Γ. As mentioned is Section 2, such groups exists if d ≥ 2. To each a ∈ A we associateã ∈ G withã = a τ a , where τ a ∈ V . We consider a probability measure µ on A with supp(µ) = A and we denote byμ its push-forward onÃ := {ã, a ∈ A}.
In contrast to the above examples the main role here will be played byΓ, the subgroup of G generated byÃ. Let us consider the convolutionsμ n , n ∈ N, on G and the natural affine action of G on V .
Proposition 4.6. Assume that Γ ⊂ SU(d) is such that the natural representation of Γ in L 2 0 (SU(d)) does not contain weakly Id Γ and the affine action ofÃ on V has no fixed point. Then there exists c > 0 such that for any continuous function f with compact support on G, lim nμ n (f ) n d = c(ν ⊗ ℓ)(f ). In particular, for any f, f ′ continuous non negative functions on G with compact support, we have: Furthermore the convolution equationμ * f = f on G, with f ∈ L ∞ (ν ⊗ ℓ), has only constant solutions.
Proof. We will use the results of Section 3; the link with Section 3 is as follows. The mapsã on X × V are defined here as left multiplication on G = XV by aτ a : where x −1 (τ a ) is the vector obtained from τ a by the linear action of x.
Hence the action of A on X is by left multiplication on the group SU(d) and c a (x) = x −1 (τ a ). The centering condition c a (x) dµ(a) dν(x) = 0 is valid here, since it reduces to x −1 (τ a ) dν(x) = 0, which is a consequence of the fact that this integral is the barycenter of the sphere SU(d)τ a of center 0 and radius τ a , hence is equal to 0.
Then the action ofΓ ⊂ G on X × V is by left multiplication on G = XV . This action is part of the action of G on itself by left translation.
Let us fix some notations. For x ∈ X, v ∈ V , with the above notations, x(v) corresponds to the element xvx −1 of G. We observe that, if g = x g τ g and h = x h τ h , then where the action of G on X is given by (g, x) → x g x: It follows that forγ ∈Γ, c(γ, x) as defined in Section 3 is equal to x −1 (τγ) and is the restriction toΓ × X of the cocycle on G × X given by c(g, x) = x −1 (τ g ).
We show now that the closure H ofΓ is equal to G. We observe that H ∩ V is a normal subgroup of H and the action by conjugation of G on V reduces to the linear action of G.
Since Γ is dense in SU(d) and W = H ∩ V is Γ-invariant, W is a closed SU(d)invariant subgroup of V . Hence W = {0} or V .
Suppose we are in the first case. Then the projection H → SU(d) is injective. In particularΓ is isomorphic to Γ. In connection with Section 3, we may observe that c(γ, x) = x −1 (τγ) defines also a cocycle on Γ × X since τγ depends only on γ; hence c(γ, x) = c(γ, x).
We will use the following lemma.
Lemma 4.7. Assume H is a closed subgroup of G = SU(d) ⋉ C d , d ≥ 2, such that H ∩ C d = {0} and the projection of H on SU(d) is dense. Then H is conjugate to SU(d).
Proof. Let π be the projection of G onto SU(d). Observe that π(H) is a Lie subgroup of SU(d) isomorphic to H. Also π(H) contains a finitely generated countable subgroup ∆ which is dense in π(H), hence in SU(d). Then ∆ is non amenable since otherwise, using [Ti72], ∆ would have a polycyclic subgroup ∆ 0 with finite index. Then the closure of ∆ 0 would be solvable and equal to SU(d), which is impossible since d ≥ 2.
Let H 0 be the connected component of identity in H and observe that π(H 0 ) is normal in π(H). It follows that the Lie algebra of π(H 0 ) is invariant under the adjoint action of π(H), hence invariant under the action of its closure SU(d). Then, using the exponential map, we see that π(H 0 ) is a normal Lie subgroup of SU(d).
Since SU(d) is a simple Lie group, we get π(H 0 ) = {e} or π(H 0 ) = SU(d). In the first case, H would be a discrete subgroup of G, hence amenable like G. This imply that π −1 (∆) ⊂ H would be amenable. Hence ∆ itself would be amenable which is a contradiction. Hence π(H) = SU(d) and π is an isomorphism of H onto SU(d). In particular H is compact and its affine action on V has a fixed point τ ∈ V . Hence τ −1 Hτ = SU(d).
The existence of a fixed point for the affine action of H on V , as shown in the lemma, contradicts the hypothesis onÃ, hence W = V . Since the projection of H on SU(d) is dense, we get H = G. Now we are going to apply Theorem 3.6, part 1b). For this we have to verify (AP). If (AP) is not valid, there exists (λ, θ) ∈ V × R and d(x) with |d(x)| = 1, such that for any a ∈ A: e i λ,c(a,x) = e iθ d(xa)/d(x).
As observed above, c(a, x) extends to G as the cocycle c(g, x) which is equal to x −1 (τ g ) on (g, x) = (x g τ g , x). Then we have e iθ = e i λ,c(a,x) d(x)/d(a.x), and the right hand side is the restriction toÃ × X of the cocycle c λ (g, x) = e i λ,c(g,x) d(x)/d(x g x) on G × X.
This cocycle takes values e iθ onÃ, hence its values are also independent of x on the groupΓ. SinceΓ is dense in G and c λ is measurable on G × X, using the L 2 continuity of the translation, it is also independent of x on G, hence it defines a character on G.
This means that the function on G defined by ψ(xv) = e −i λ,v d(x) is invariant by left translation by any elementγ ∈Γ. SinceΓ is dense in G, hence ergodic on G, ψ is constant, i.e. λ = 0, d = 1. It follows that (AP) is valid. Hence the result.
Since (AP) is valid, the last assertion is a consequence of 3) in Theorem 3.6.
There exists various possibilities for the geometry of the subgroupΓ inside G, as the following proposition shows.
Proposition 4.8. With the above notations, assume that the finite set A ⊂ SU(d) generates a dense subgroup Γ and the affine action ofÂ on V has no fixed point. Then 1) If Γ has property (T), thenΓ ∩ V is dense in V .
Proof. 1) We show using arguments as in the proof of Proposition 4.6 thatΓ is dense in G. We observe thatΓ∩V is a normal subgroup ofΓ and the action by conjugation of G on V reduces to the linear action of SU(d).
Since Γ is dense in SU(d) andΓ ∩ V is Γ-invariant, its closure W is a closed SU(d)invariant subgroup of V . Hence W = {0} or V .
This contradicts the hypothesis onÃ, henceΓ ∩ V = {0} is not valid. Thereforẽ Γ ∩ V is dense in V . The fact thatΓ is dense in G follows, but was already proved in Proposition 4.6.
2) We denote by π the natural projection of G onto SU(d), and we observe that π(ã) = a for any a ∈ A; hence π(Γ) = Γ. Since Γ is free it follows that the restriction of π toΓ is an isomorphism ofΓ onto Γ.
In particular, since π(V ) = {0} we haveΓ ∩ V = {0} andΓ is free. The density of Γ in G has been shown in the proof of Proposition 4.6.

Questions
1) In the situation of random walks in random scenery (example 4.1), with Γ = Z m , m ≥ 3, is the local limit theorem for S n (ω, x) ∈ V valid ?
2) In the situation of motion groups (example 4.4), for d ≥ 2, ifΓ ⊂ SU(d) ⋉ C d and Γ ⊂ SU(d) is dense, is the local limit theorem for S n (ω, x) ∈ V still valid ?
What can be said about the equidistribution of the orbits ofΓ on V ?
What are the bounded solutions of the equationμ * f = f , f ∈ L ∞ (ν ⊗ ℓ), on G.
3) In the above considerations the maps a ∈ A are chosen with probability µ(a) which does not depend on x and the product space is endowed with the product measure P = µ ⊗N * . One can extends this framework by choosing the maps a ∈ A according to a weight µ(x, a) depending on x ∈ X and consider the corresponding Markovian model. One can also replace the shift invariant measure P by a Gibbs measure.
A question is then the validity of the results obtained above in these more general situations.