R Émi Boutonnet
email: rboutonnet@ucsd.edu

Adrian Ioana
email: aioana@ucsd.edu

Alireza Salehi Golsefidy
email: asalehigolsefidy@ucsd.edu

LOCAL SPECTRAL GAP IN SIMPLE LIE GROUPS AND APPLICATIONS

. We present several applications to the Banach-Ruziewicz problem, orbit equivalence rigidity, continuous and monotone expanders, and bounded random walks on G. In particular, we prove that, up to a multiplicative constant, the Haar measure is the unique Γ-invariant finitely additive measure defined on all bounded measurable subsets of G.

A.I. was partially supported by NSF Grant DMS #1161047, NSF Career Grant DMS #1253402, and a Sloan Foundation Fellowship.

A.S.G. was partially supported by NSF Grant DMS #1303121, and a Sloan Foundation Fellowship.

Introduction

1.1. Background and motivation. Spectral gap for probability measure preserving actions is a fundamental notion in mathematics with a wide range of applications. The goal of this paper is to introduce and study a notion of spectral gap for general measure preserving actions.

We begin our discussion by recalling the following:

Definition 1.1. A measure preserving action Γ (X, µ) of a countable group Γ on a standard probability space (X, µ) is said to have spectral gap if there exist S ⊂ Γ finite and κ > 0 such that

F 2 κ g∈S g • F -F 2 for any F ∈ L 2 (X, µ) with X F dµ = 0.
Here, g •F denotes the function given by (g •F )(x) = F (g -1 x), for every g ∈ Γ and x ∈ X. To justify the terminology, consider the self-adjoint averaging operator P S (ξ) = 1 2|S| g∈S (g • F + g -1 • F ). Then the constant function 1 is an eigenfunction of P S with eigenvalue 1, and the existence of κ > 0 as above is equivalent to the presence of a gap right below 1 in the spectrum of P S .

Let G be a compact Lie group and denote by m G its Haar measure. An important question, which has been investigated intensively over the last three decades, is whether the left translation action Γ (G, m G ), associated to a countable dense subgroup Γ < G, has spectral gap. Interest in this question first arose in the early 1980s, in connection with Ruziewicz's problem for the n-sphere S n (also known as the Banach-Ruziewicz problem). The latter asks if the Lebesgue measure on S n is the unique finitely additive, rotation-invariant measure defined on all Lebesgue measurable subsets. For n = 1, Banach used the amenability of SO(2) (as a discrete group) to show that the answer is negative [START_REF] Banach | Sur le problème de la mesure[END_REF]. For n 2, however, the problem remained open for a long time. First, it was realized that the existence of a countable dense subgroup of SO(n+1) with the spectral gap property implies an affirmative answer [START_REF] Del Junco | Counterexamples in ergodic theory and number theory[END_REF][START_REF] Rosenblatt | Uniqueness of invariant means for measure-preserving transformations[END_REF]. By using Kazhdan's property (T), Margulis [START_REF] Margulis | Some remarks on invariant means[END_REF] and Sullivan [START_REF] Sullivan | For n > 3 there is only one finitely additive rotationally invariant measure on the n-sphere on all Lebesgue measurable sets[END_REF] then obtained an affirmative answer for every n 4. The remaining cases n = 2, 3 were finally settled in the affirmative by Drinfeld [Dr84] via the construction of a countable dense subgroup of SU (2) with the spectral gap property. An optimal such construction was achieved soon after by Lubotzky, Phillips, and Sarnak [START_REF] Lubotzky | Hecke operators and distributing points on the sphere I[END_REF][START_REF] Lubotzky | Hecke operators and distributing points on S 2 II[END_REF] (see [START_REF] Oh | The Ruziewicz problem and distributing points on homogeneous spaces of a compact Lie group[END_REF] for a generalization to compact simple Lie groups not locally isomorphic to SO(3)). For all of this, see the excellent survey [START_REF] Lubotzky | Discrete Groups, Expanding Graphs and Invariant Measures[END_REF]. Later on, a new robust method for proving the spectral gap property for subgroups of SU (2) was developed by Gamburd, Jakobson, and Sarnak [START_REF] Gamburd | Spectra of elements in the group ring of SU (2)[END_REF]. It is worth pointing out that in all of these results, the subgroups involved are generated by matrices with algebraic entries.

In 2006, a breakthrough was made by Bourgain and Gamburd who established the spectral gap property for any dense subgroup of SU (2) generated by matrices with algebraic entries [START_REF] Bourgain | On the spectral gap for finitely-generated subgroups of SU (2)[END_REF]. Their approach followed two earlier major works: the authors' work on expansion for Cayley graphs of SL 2 (F p ) [START_REF] Bourgain | Gamburd: Uniform expansion bounds for Cayley graphs of SL2(Fp)[END_REF], and Helfgott's product theorem for subsets of SL 2 (F p ) [START_REF] Helfgott | Growth and generation in SL2(Z/pZ)[END_REF]. Subsequently, Bourgain and Gamburd established the spectral gap property for dense subgroups of G = SU (d) generated by matrices with algebraic entries, for any d 2 [START_REF] Bourgain | Gamburd: A spectral gap theorem in SU (d)[END_REF]. Recently, this was generalized further by Benoist and de Saxcé to cover arbitrary connected compact simple Lie groups G [START_REF] Benoist | Saxcè: A spectral gap theorem in simple Lie groups[END_REF].

If G is a compact group and Γ is a countable dense subgroup with the spectral gap property, then the Haar measure m G is the unique finitely additive Γ-invariant measure defined on all measurable subsets of G. One of the main motivations for this paper is to formulate and prove analogues of the results from [BG06, [START_REF] Bourgain | Gamburd: A spectral gap theorem in SU (d)[END_REF][START_REF] Benoist | Saxcè: A spectral gap theorem in simple Lie groups[END_REF] that apply to general simple Lie groups G. By analogy with the compact case, it would thus be desirable to find a notion of spectral gap for infinite measure preserving actions, which in the case of left translation actions on locally compact groups G, implies a uniqueness property for its left Haar measures as finitely additive measures. 1.2. Local spectral gap. As we explain in Corollary D, the following new notion of spectral gap satisfies the desired property.

Definition 1.2. Let Γ (X, µ) be a measure preserving action of a countable group Γ on a standard measure space (X, µ). We say that Γ (X, µ) has local spectral gap with respect to a measurable set B ⊂ X of finite measure if there exist S ⊂ Γ finite and κ > 0 such that Remark 1.3. We continue with a few remarks on this definition:

(1) Although the action Γ (X, µ) is not required to be ergodic, this is automatic if the action has local spectral gap with respect to a set B such that ∪ g∈Γ g • B is co-null in X.

(2) If (X, µ) is a probability space and B = X, then local spectral gap coincides with spectral gap. Assume that (X, µ) is an infinite measure space with X being a locally compact space and µ a Radon measure. Then the notion of local spectral gap aims to capture the intuitive idea that functions on X which are locally almost Γ-invariant, must be locally almost constant (see also Proposition 2.2). This is different from the "global" notion of spectral gap requiring that there is no sequence of unit almost Γ-invariant functions in L 2 (X, µ). Indeed, since any sequence of unit almost Γ-invariant functions in L 2 (X, µ) converges weakly to 0 on compact subsets of X, the latter reflects only the dynamics of the action at infinity.

(3) The notion of local spectral gap appears implicitly in Margulis' positive resolution of the Banach-Ruziewicz problem for R n (n 3). More precisely, with the above terminology, he first shows the existence of a subgroup Γ < R n SO(n) such that the action Γ (R n , λ n ) has local spectral gap, and then concludes that the Lebesgue measure λ n is indeed the unique finitely additive isometry-invariant measure defined on all bounded measurable subsets of R n [START_REF] Margulis | Finitely-additive invariant measures on Euclidian spaces[END_REF]. (4) While local spectral gap might depend on the choice of B, the following independence result can be easily shown: assume that B 1 , B 2 are measurable subsets of X such that B 1 ⊂ K •B 2 and B 2 ⊂ K • B 1 , for some finite set K ⊂ Γ. Then local spectral gap with respect to B 1 is equivalent to local spectral gap with respect to B 2 (see Proposition 2.3).

Notation. Let G be locally compact second countable group and H < G be a closed subgroup.

Here and after, we assume that the locally compact topology on G is Hausdorff. We denote by m G a fixed left Haar measure of G. We also denote by m G/H a fixed quasi-invariant Borel regular measure on G/H which is "nice", in the sense that it arises from a rho-function for the pair (G, H) (see [START_REF] Bekka | Kazhdan's property (T)[END_REF]Theorem B.1.4.]).

The following is our main result.

Theorem A (local spectral gap). Let G be a connected simple Lie group. Denote by g the Lie algebra of G and by Ad : G → GL(g) its adjoint representation. Let Γ < G be a dense subgroup. Assume that there is a basis B of g such that the matrix of Ad(g) in the basis B has algebraic entries, for any g ∈ Γ. Let B ⊂ G be a measurable set with compact closure and non-empty interior.

Then the left translation action Γ (G, m G ) has local spectral gap with respect to B.

In the case G is compact, Theorem A recovers the main results of [START_REF] Bourgain | On the spectral gap for finitely-generated subgroups of SU (2)[END_REF][START_REF] Bourgain | Gamburd: A spectral gap theorem in SU (d)[END_REF][START_REF] Benoist | Saxcè: A spectral gap theorem in simple Lie groups[END_REF]. On the other hand, if G is not compact, Theorem A reveals an entirely new type of phenomenon for locally compact groups.

Remark 1.4. The assumption on Γ < G is in particular satisfied if G = SL n (R), for some n 2, and Γ is a dense subgroup of G such that every matrix g ∈ Γ has algebraic entries.

Remark 1.5. In view of Remark 1.3 (4), the conclusion of Theorem A does not depend on the choice of the set B. Indeed, if B 1 ⊂ G has compact closure and B 2 ⊂ G has non-empty interior, then there exists a finite set K ⊂ Γ such that

B 1 ⊂ K • B 2 .
Theorem A is a consequence of our main technical result proving a restricted spectral gap estimate in the spirit of Bourgain and Yehudayoff's pioneering work [START_REF] Bourgain | Expanders and dimensional expansion[END_REF][START_REF] Bourgain | Yehudayoff: Expansion in SL2(R) and monotone expanders[END_REF].

Theorem B (restricted spectral gap). Assume that Γ < G are as in Theorem A. Let B ⊂ G be a measurable set with compact closure. Let U be a neighborhood of the identity element in G.

Then there exist a finite set T ⊂ Γ ∩ U and a finite dimensional subspace V ⊂ L 2 (B) such that the probability measure µ = 1 2|T | g∈T (δ g + δ g -1 ) satisfies µ * F 2 < 1 2 F 2 , for every F ∈ L 2 (B) V .

Note that unlike Theorem A, this result is new even in the case of compact groups, where it leads to some unexpected consequences (see Remark 1.7).

Theorem B concerns the left regular representation of G. The proof of Theorem B moreover shows that for any 0 < r < 1 there exists a finite set T ⊂ Γ ∩ U such that the conclusion holds with r in place of 1 2 . As a consequence, it follows that a more general statement, addressing all quasi-regular representations of G, holds true.

Corollary C. Assume that Γ < G are as in Theorem A. Let H < G be a closed subgroup and denote by π : G → U(L 2 (G/H, m G/H )) be the associated quasi-regular unitary representation. Let B ⊂ G/H be a measurable set with compact closure. Let U be a neighborhood of the identity in G.

Then there exist a finite set T ⊂ Γ ∩ U and a finite dimensional subspace V ⊂ L 2 (B) such that the probability measure µ = 1 2|T | g∈T (δ g +δ g -1 ) satisfies π(µ)(F ) 2 < 1 2 F 2 , for any F ∈ L 2 (B) V .

Here, for a probability measure µ, we denote by π(µ) the averaging operator g∈G µ({g})π(g).

Corollary C generalizes [BY11, Theorem 5] which deals with the case when G is SL 2 (R) and H is the subgroup of upper triangular matrices. Then G/H can be identified with the real projective line, P 1 (R). The proof of [START_REF] Bourgain | Yehudayoff: Expansion in SL2(R) and monotone expanders[END_REF]Theorem 5] is specific to this situation, as it relies on the fact that the action of SL 2 (R) on P 1 (R) is 2-transitive to show a certain mixing property. Corollary C provides an alternative approach to the mixing property in this case. Corollary C is new in all other cases with G non-compact, including the simplest one when G = SL 2 (R) and H is trivial.

Remark 1.6. In the case G has trivial center, the proof of Theorem B yields a more quantitative statement (see Theorem 6.7). To explain this, identify G with a subgroup of GL n (R), for some n, and endow it with the metric induced by the Hilbert-Schmidt norm . 2 . For ε > 0, denote B ε (1) = {g ∈ G| g -1 2 < ε}.

Then the proof of Theorem B shows that there is a constant C > 1 (depending on Γ) such that for any small enough ε > 0, there exist a finite set T ⊂ Γ ∩ B ε (1) which freely generates a group, and a finite dimensional subspace V ⊂ L 2 (B) such that denoting µ = 1 2|T | g∈T (δ g + δ g -1 ) we have

• |T | < 1 ε C , and • µ * F 2 < ε F 2 , for every F ∈ L 2 (B) V .
Remark 1.7. Theorem B (and its quantitative version) sheds some new light on the spectra of averaging operators on compact groups. In order to briefly recall known results along these lines, assume for simplicity that G = SU (2). Then the irreducible representations of G can be listed as π n : G → U(H n ), where dim(H n ) = n + 1, for every n 0, and by the Peter-Weyl theorem we have that L 2 (G) = n 0 H ⊕(n+1) n . Let T ⊂ G be a finite set which freely generates a subgroup, consider the probability measure µ = 1 2|T | g∈T (δ g + δ g -1 ), and denote by P µ the operator F → µ * F . Then P µ is self-adjoint and since P µ 1, its spectrum is contained in [-1, 1]. Moreover, since P µ can be identified with n 0 π n (µ) ⊕n+1 , it is also diagonalizable.

The asymptotic distribution of the eigenvalues of P µ has been studied in [START_REF] Lubotzky | Hecke operators and distributing points on the sphere I[END_REF][START_REF] Gamburd | Spectra of elements in the group ring of SU (2)[END_REF], where it is shown that most of them lie in the interval -

√ 2|T |-1 |T | , √ 2|T |-1 |T |
. More precisely, if d n denotes the number of eigenvalues of π n (µ) that lie outside this interval (so-called "exceptional" eigenvalues), then dn n → 0 (see [LPS86, Theorem 1.1]). Assume from now on that the elements of T have algebraic entries. Then the following sharper estimate holds: dn n 1 log n , for large n (see [GJS99, Theorem 1.3]). A remarkable fact, discovered by Lubotzky, Phillips and Sarnak is that for certain sets T , the operator P µ has no exceptional eigenvalues, i.e. d n = 0, for all n 1 (see [START_REF] Lubotzky | Hecke operators and distributing points on the sphere I[END_REF][START_REF] Lubotzky | Hecke operators and distributing points on S 2 II[END_REF]). As already mentioned above, the more recent work [START_REF] Bourgain | On the spectral gap for finitely-generated subgroups of SU (2)[END_REF] implies that P µ has a spectral gap, i.e. the supremum κ µ of the spectrum of P µ but 1 satisfies κ µ < 1. However, besides these facts, not much is known about the exceptional eigenvalues of P µ . In particular, to the best of our knowledge, it is unknown whether κ µ is ever an eigenvalue of P µ .

Theorem B implies that k µ can be an isolated eigenvalue of P µ , and thus P µ can have a second spectral gap. Moreover, it shows that operators of the form P µ may have arbitrarily many gaps at the top of their spectrum. To make this precise, let ε > 0 small enough, and let T ⊂ Γ ∩ B ε (1) and µ as given by Remark 1.6. Then P µ has only finitely many eigenvalues outside the interval [-ε, ε]. On the other hand, since T ⊂ B ε (1), the number of eigenvalues of P µ belonging to the interval ( 1 2 , 1) gets arbitrarily large, as ε → 0. In fact, it is easy to see that this number is 1 ε 2 . In combination with [LPS86, Theorem 1.1] the following picture emerges: the spectrum of P µ contains

• the whole interval - √ 2|T |-1 |T | , √ 2|T |-1 |T |
• only finitely many points, all of which are isolated eigenvalues, outside -1

|T | 1 C , 1 |T | 1 C . • |T | 2 C points in the interval ( 1 2 , 1).
1.3. Applications. We now turn to discussing several applications of our main results.

1.3.1. The Banach-Ruziewicz problem. The original Banach-Ruziewicz problem asks whether the Lebesgue measure on S n (resp. R n ) is the unique rotation-invariant (resp. isometry-invariant) finitely additive measure defined on all bounded Lebesgue measurable sets. This problem is an illustration of a general question: let Γ be a locally compact group acting isometrically on a locally compact metric space X with an invariant Radon measure µ. Is µ the unique Γ-invariant finitely additive measure defined on all µ-measurable subsets of X with compact closure? Here and after, uniqueness is of course meant up to a multiplicative constant.

If the space X is compact and the group Γ is countable discrete, then a positive answer to this question is closely connected to the spectral gap of the action. The connection stems from the well-known fact that the action Γ (X, µ) has spectral gap if and only if integration against µ is the unique Γ-invariant mean on L ∞ (X, µ) (see [START_REF] Rosenblatt | Uniqueness of invariant means for measure-preserving transformations[END_REF][START_REF] Schmidt | Amenability, Kazhdan's property T, strong ergodicity and invariant means for ergodic groupactions[END_REF]). On the other hand, if µ is unique among invariant finitely additive measures, then integration against µ is unique among invariant means. The converse of this statement is also true for certain classes of actions, including left translation actions on compact groups (see Remark 7.2).

In Section 7, we generalize these results to the case when X is locally compact. Assume that every orbit Γ • x is dense in X, and denote by L ∞ c (X, µ) the algebra of L ∞ -functions with compact support. Firstly, we prove that the action Γ (X, µ) has local spectral gap with respect to a measurable set with compact closure and non-empty interior if and only if integration against µ is the unique Γ-invariant positive linear functional on L ∞ c (X, µ) (see Theorem 7.6). This result is partially inspired by Margulis' work [START_REF] Margulis | Finitely-additive invariant measures on Euclidian spaces[END_REF], which we follow closely in the proof of the only if assertion. Secondly, in the case of left translation actions Γ (G, m G ) on locally compact groups G, we show that the Haar measure m G is unique among invariant finitely additive measures if and only if integration against m G is unique among invariant positive linear functionals on L ∞ c (G, m G ) (see Theorem 7.1). Altogether, by combining these two results we derive the following: Theorem D. Let G be a locally compact second countable group and Γ < G be a countable dense subgroup. Denote by C(G) the family of measurable subsets A ⊂ G with compact closure.

Then the following conditions are equivalent: Note that in order to treat arbitrary locally compact groups, we use the structure theory of locally compact groups [START_REF] Montgomery | Topological transformation groups[END_REF] as well as Breuillard and Gelander's topological Tits alternative [START_REF] Breuillard | A topological Tits alternative[END_REF].

(1) If ν : C(G) → [0, ∞) is a Γ-invariant,
As an immediate consequence of Theorems A and D we deduce the following uniqueness characterization of Haar measures on simple Lie groups, in the spirit of the Banach-Ruziewicz problem:

Corollary E. Assume that Γ < G are as in Theorem A.

Then, up to a multiplicative constant, the Haar measure m G of G is the unique finitely additive Γ-invariant measure defined on C(G).

1.3.2. Orbit equivalence rigidity. Next, we apply our results to the theory of orbit equivalence of actions. This area has flourished in the last 15 years, with many new exciting developments (see the surveys [START_REF] Popa | Deformation and rigidity for group actions and von Neumann algebras[END_REF][START_REF] Furman | A survey of Measured Group Theory, Geometry, Rigidity, and Group Actions[END_REF][START_REF] Gaboriau | Orbit equivalence and measured group theory[END_REF]). To recall the notion of orbit equivalence, consider two ergodic measure preserving actions Γ (X, µ) and Λ (Y, ν) of countable groups Γ, Λ on standard measure spaces (X, µ), (Y, ν). The actions are called orbit equivalent if there exists a measure class preserving Borel isomorphism θ :

X → Y such that θ(Γ • x) = Λ • θ(x), for µ-almost every x ∈ X.
The simplest instance of when the actions are orbit equivalent is when they are conjugate, i.e. there exists a measure class preserving Borel isomorphism θ : X → Y and a group isomorphism δ : Γ → Λ such that θ(g • x) = δ(g) • θ(x), for all g ∈ Γ and µ-almost every x ∈ X.

In general, however, orbit equivalence is a much weaker notion of equivalence than conjugacy. This is best illustrated by the striking theorem of Ornstein-Weiss and Connes-Feldman-Weiss showing that if the groups Γ, Λ are both infinite amenable and the measure spaces (X, µ), (Y, ν) are either both finite or both infinite, then the actions are orbit equivalent (see [START_REF] Ornstein | Ergodic theory of amenable groups. I. The Rokhlin lemma[END_REF][START_REF] Connes | An amenable equivalence relations is generated by a single transformation[END_REF]). In sharp contrast, there exist "rigid" situations when for certain classes of actions of non-amenable groups one can deduce conjugacy from orbit equivalence.

It was recently discovered in [START_REF] Ioana | Orbit equivalence and Borel reducibility rigidity for profinite actions with spectral gap[END_REF] that such a rigidity phenomenon occurs for left translation actions on compact groups in the presence of spectral gap. More precisely, let Γ < G and Λ < H be countable dense subgroups of compact connected Lie groups with trivial centers. Assuming that Γ (G, m G ) has spectral gap, it follows from [Io13, Corollary 6.3] that the actions Γ (G, m G ) and Λ (H, m H ) are orbit equivalent if and only if they are conjugate. Most recently, this result has been generalized to the case when G and H are arbitrary, not necessarily compact, connected Lie groups with trivial centers (see [Io14, Theorems A and 4.1]). The only difference is that in the locally compact setting, the spectral gap assumption has to be replaced with the assumption that the action Γ (G, m G ) is strongly ergodic.

To recall the latter notion, let Γ (X, µ) be an ergodic measure preserving action. Then, loosely speaking, strong ergodicity requires that any sequence of asymptotically invariant subsets of X must be asymptotically trivial. In order to make this precise, since the measure µ can be infinite, we first choose a probability measure µ 0 on X with the same null sets as µ. The action is said to be strongly ergodic if any sequence {A n } of measurable subsets of X satisfying µ 0 (g [START_REF] Schmidt | Asymptotically invariant sequences and an action of SL(2; Z) on the 2-sphere[END_REF]. It is easy to see that this definition does not depend on the choice of µ 0 .

•A n ∆ A n ) → 0, for all g ∈ Γ, must satisfy µ 0 (A n )(1 -µ 0 (A n )) → 0 [CW80,
For translation actions on compact groups, strong ergodicity is implied by the spectral gap property, which is now known to hold in considerably large generality by [START_REF] Bourgain | On the spectral gap for finitely-generated subgroups of SU (2)[END_REF][START_REF] Bourgain | Gamburd: A spectral gap theorem in SU (d)[END_REF][START_REF] Benoist | Saxcè: A spectral gap theorem in simple Lie groups[END_REF]. On the other hand, in the case of translation actions on locally compact non-compact groups, strong ergodicity seems much harder to work with, and so far could only be checked in two rather specific situations (see [START_REF] Ioana | Strong ergodicity, property (T), and orbit equivalence rigidity for translation actions[END_REF]Propositions G and H]).

Nevertheless, strong ergodicity is implied by local spectral gap, for arbitrary ergodic measure preserving actions. Moreover, for translation actions on locally compact groups, we prove that local spectral gap and strong ergodicity are equivalent (see Theorem 7.1). This generalizes [AE10, Theorem 4], which dealt with the compact case. Consequently, all actions covered by Theorem A are strongly ergodic, which in combination with [START_REF] Ioana | Strong ergodicity, property (T), and orbit equivalence rigidity for translation actions[END_REF] allows us to conclude the following: Corollary F. Assume that Γ < G are as in Theorem A. Suppose that G has trivial center. Let H be any connected Lie group with trivial center and Λ < H be any countable dense subgroup. Remark 1.9. In the context of Corollary F, assume moreover that Γ is a free group. Since the left translation action Γ (G, m G ) is strongly ergodic, it is not amenable in the sense of [START_REF] Zimmer | Amenable ergodic group actions and an application to Poisson boundaries of random walks[END_REF]. Then [HV12, Theorem A] (which builds on [START_REF] Ozawa | On a class of II1 factors with at most one Cartan subalgebra[END_REF][START_REF] Popa | Unique Cartan decomposition for II1 factors arising from arbitrary actions of free groups[END_REF]) implies that L ∞ (G) is the unique Cartan subalgebra of the L ∞ (G) Γ, up to unitary conjugacy. In combination with Corollary F, we deduce that the crossed product von Neumann algebras L ∞ (G) Γ and L ∞ (H) Λ are isomorphic if and only if there is a topological isomorphism δ : G → H such that δ(Γ) = Λ. 1.3.3. Continuous and monotone expanders. Our main results also lead to a general construction of continuous and monotone expanders, extending the main result of [START_REF] Bourgain | Yehudayoff: Expansion in SL2(R) and monotone expanders[END_REF]. Expander graphs are infinite families of highly connected sparse finite graphs. It is sometimes desirable to find expander graphs within certain classes of graphs. A finite graph is called monotone if it is defined by monotone functions. This means that the vertex set of the graph can be identified with [n] = {1, 2, ..., n} in such a way that there exist partially defined monotone maps

ϕ i : [n] → [n], 1 i d, such that two vertices a, b are connected iff b = ϕ i (a), for some i.
Bourgain and Yehudayoff recently found the first explicit construction of constant degree monotone expander graphs [START_REF] Bourgain | Expanders and dimensional expansion[END_REF][START_REF] Bourgain | Yehudayoff: Expansion in SL2(R) and monotone expanders[END_REF]. Their approach is to first build a continuous monotone expander and then discretize it to obtain monotone expanders. In their terminology, a continuous expander consists of a family of smooth partially defined maps ϕ i : B → B, 1 i d, where B is a compact subset of a manifold endowed with a finite measure A → |A|, such that the following holds: there is κ > 0 such that for every measurable set

A ⊂ B with |A| |B| 2 , we have | ∪ d i=1 ϕ i (A)| (1 + κ)|A|.
As a consequence of Theorem B, we obtain the following result.

Corollary G. Assume that Γ < G are as in Theorem A. Let H < G be a closed subgroup and B ⊂ G/H be a measurable set with compact closure and non-empty interior. For a measurable subset A ⊂ G/H, denote |A| := m G/H (A).

Then there exists a finite set S ⊂ Γ finite for which there is a constant κ > 0 such that for any measurable set A ⊂ B with |A| |B| 2 we have

| ∪ g∈S g • A ∩ B| (1 + κ)|A|.
Moreover, if B is open and connected, and ε > 0 is given, then S ⊂ Γ can be taken inside B ε (1).

Assume that G is equal to SL 2 (R), H is the subgroup of upper triangular matrices, identify G/H with the real projective line P 1 (R) = R ∪ {∞}, and let B = [0, 1]. With this notation, [BY11,

Theorem 4] provides a finite set S ⊂ SL 2 (Q) which satisfies the conclusion of Corollary G. Moreover, S can be taken close enough to the identity so that the restriction g of every g ∈ S to B ∩ g -1 B is monotonically increasing. Therefore, the family {g} g∈S is a continuous monotone expander.

Corollary G generalizes [BY11, Theorem 4] by showing the existence of such a set S inside any dense subgroup of G generated by algebraic elements. Note that, as opposed to [START_REF] Bourgain | Yehudayoff: Expansion in SL2(R) and monotone expanders[END_REF], our construction of S is not explicit. On the other hand, unlike [START_REF] Bourgain | Yehudayoff: Expansion in SL2(R) and monotone expanders[END_REF], our construction does not rely on the strong Tits alternative from [START_REF] Breuillard | A strong Tits alternative[END_REF].

1.3.4. Spectral gap for delayed bounded random walks. Our last application concerns random walks on Lie groups that are bounded and delayed, in a sense made precise below. Let G be a connected simple Lie group and S ⊂ G be a finite symmetric set. Denote k = |S| and enumerate S = {g 1 , ..., g k }. Let B ⊂ G be a measurable set which is bounded (i.e. has compact closure). We define a random walk on B as follows: a given point x ∈ B moves with probability 1 k to each of the points h 1 x, h 2 x, ..., h k x, where h i = g i , if g i x ∈ B, and

h i = e, if g i x / ∈ B.
In other words, with probability 1 k , x either moves to g i x or stays put, depending on whether g i x belongs to B or not. The associated transition operator P S : L 2 (B) → L 2 (B) is given by

P S (F ) = 1 k k i=1 1 B∩g i B g i • F + 1 B\g i B F , for every F ∈ L 2 (B).
Then P S is symmetric, P S 1, and P S (1 B ) = 1 B , where 1 B denotes the characteristic function of B. Theorem B allows us to deduce the existence of many sets S such that P S has a spectral gap.

Corollary H. Assume that Γ < G are as in Theorem A.

Then there exists a finite symmetric set S ⊂ Γ such that the operator P S : L 2 (B) → L 2 (B) satisfies

P S |L 2 (B) C1 B < 1.
When G is compact and B = G, this result is a consequence of [START_REF] Bourgain | On the spectral gap for finitely-generated subgroups of SU (2)[END_REF][START_REF] Bourgain | Gamburd: A spectral gap theorem in SU (d)[END_REF][START_REF] Benoist | Saxcè: A spectral gap theorem in simple Lie groups[END_REF]. Corollary H is new in all other cases, including the case when G is compact and B is a proper subset.

1.4. On the proof of restricted spectral gap. Our approach to proving restricted spectral gap is a combination of general results from [START_REF] De Saxcé | A product theorem in simple Lie groups[END_REF][START_REF] Benoist | Saxcè: A spectral gap theorem in simple Lie groups[END_REF], refinements of techniques from [BG10,SGV11], and ideas from [START_REF] Bourgain | Yehudayoff: Expansion in SL2(R) and monotone expanders[END_REF] on how to treat non-compact situations. It relies on the remarkable strategy invented by Bourgain and Gamburd [START_REF] Bourgain | Gamburd: Uniform expansion bounds for Cayley graphs of SL2(Fp)[END_REF][START_REF] Bourgain | On the spectral gap for finitely-generated subgroups of SU (2)[END_REF] to prove spectral gap in the compact setting.

To briefly recall this strategy, consider a symmetric probability measure µ on a compact group G, for which we want to establish the spectral gap property. A first step is to show that the convolution powers of µ become "flat" rather quickly. Then one uses a mixing inequality to deduce spectral gap for the corresponding operator P µ : L 2 (G) → L 2 (G) given by P µ (F ) = µ * F .

Flattening. The term flat roughly means that after "discretizing" the group G, the measure has a small 2-norm, compared to the scale at which we discretize G. In [START_REF] Benoist | Saxcè: A spectral gap theorem in simple Lie groups[END_REF], Benoist and de Saxcé proved a general flattening lemma for connected compact simple Lie groups G. They showed that if a measure ν on G is not already flat and does not concentrate on any proper closed subgroup of G, then its convolution square ν * ν will be significantly flatter. A repeated application of this result shows that a measure on G with small mass on closed subgroups will flatten rather quickly.

Escaping subgroups. Thus, in order to show that P µ has spectral gap, it is necessary to show that, quite quickly, convolution powers of µ have small mass on closed subgroups. To guarantee this, one needs (due to the currently available techniques) to impose a diophantine assumption on the support of µ. Typically, one assumes that µ is supported on finitely many elements with algebraic entries (when viewed as matrices via the adjoint representation).

Mixing inequality. The concluding part, deducing spectral gap out of flatness of some small power of µ, relies on a mixing inequality. If G is a finite group, the mixing inequality bounds the norm of the operator P µ in terms of the 2-norm of µ (see [START_REF] Babai | Product Growth and Mixing in Finite Groups[END_REF] and [START_REF] Tao | Expansion in finite simple groups of Lie type[END_REF]Proposition 1.3.7]). This step relies on the representation theory of the ambient group G. Specifically, one usually uses the idea, due to Sarnak and Xue [START_REF] Sarnak | Bounds for multiplicities of automorphic representations[END_REF], of exploiting "high multiplicity" of eigenvalues.

In the non-compact setting, we will prove restricted spectral gap with a similar strategy. Recall, however, that our aim is somewhat different from the spectral gap property for compact groups. Indeed, we are given a connected simple Lie group G, a dense subgroup Γ < G and an open ball B ⊂ G. Our goal is to produce a measure µ supported on Γ and on an arbitrarily small neighborhood of 1, such that the averaging operator P µ : L 2 (B) → L 2 (G) has norm less than 1 2 (after discarding a finite dimensional subspace V ⊂ L 2 (B)). Let us emphasize the main differences that occur in the proof.

Firstly, we show that the mixing inequality still holds in our setting, leading to a result that might be of independent interest (see Theorem 5.1). Our proof is inspired by the "geometric approach" introduced in [BY11, BG10], but here we address a far greater level of generality. Also, our proof is elementary, in that it only relies on basic results from the representation theory of G, and essentially self-contained. Using this inequality, we reduce to the task of producing a measure µ with support contained in Γ and arbitrarily close to 1, whose convolution powers flatten rather quickly.

The flattening lemma of [START_REF] Benoist | Saxcè: A spectral gap theorem in simple Lie groups[END_REF] relies on two main tools: a product theorem due to de Saxcé [dS14], and the non-commutative Balog-Szemerédi-Gowers Lemma due to Tao [START_REF] Tao | Product set estimates for non-commutative groups[END_REF]. It turns out that these two tools actually hold for general (not necessarily compact) connected simple Lie groups. So, by reproducing the proof of [BdS14, Lemma 2.5], we get a similar flattening lemma in the locally compact setting (see Corollary 4.2). An important aspect is that our lemma only applies to measures whose support is controlled (relative to the scale at which we discretize G).

Next, refining techniques from [SGV11, Section 3] we construct a measure µ, supported on Γ and on an arbitrarily small neighborhood of 1, that will escape proper subgroups quickly when taking convolution powers (Propositions 3.2 and 3.9). Therefore, we are almost in position to apply the flattening lemma to some convolution powers of µ. However, we need to make sure that these convolution powers still have a controlled support. This amounts to bounding the speed of escape of subgroups in terms of the size of the support of µ. A priori, the measure µ that we construct does not admit such a nice bound. As in [START_REF] Bourgain | Yehudayoff: Expansion in SL2(R) and monotone expanders[END_REF], an application of the pigeonhole principle allows us to construct a new measure µ with an improved bound. Then µ satisfies all the required assumptions to ensure that it will become flat quickly enough. Finally, our mixing inequality will allow us to show restricted spectral gap for this new measure µ .

We will provide more quantitative statements of the main steps of the proof in Section 2.5. 1.5. Organization of the paper. Besides the introduction, this paper has seven other sections and an appendix. In Section 2, we establish some basic properties of local spectral gap, explain how Theorem A follows from Theorem B, and provide a detailed outline of the proof of Theorem B. Sections 3, 4, and 5 are each devoted to one of the three main parts of the proof of Theorem B. In Section 6 we conclude the proof of Theorem B and derive Corollary C. In Sections 7 and 8, we prove Theorem D and Corollaries G, H, respectively. Finally, the Appendix deals with the proof of Lemma 4.1. 1.6. Acknowledgements. We are grateful to Cyril Houdayer, Hee Oh and Peter Sarnak for helpful comments.

Preliminaries

2.1. Terminology. We begin by introducing various terminology concerning analysis on groups. Let G be a locally compact second countable (l.c.s.c.) group and fix a left Haar measure m G .

Given a measurable set A ⊂ G and a measurable function f : G → C, we denote

|A| := m G (A), G f (x) dx := G f dm G , and f p,A := 1 A f p = A |f (x)| p dx 1 p .
We denote by M(G) the family of Borel probability measures on G. Let f, g : G → C be measurable functions and µ, ν ∈ M(G). Then the convolution functions f * g, µ * f : G → C and probability measure µ * ν are defined (when the integrals make sense) by the formulae

(f * g)(x) = G f (y)g(y -1 x) dy, (µ * f )(x) = G f (y -1 x) dµ(y) and G F d(µ * ν) = G G F (xy) dµ(x)dν(y)
for any continuous F : G → C. We will often use the following inequalities

f * g 2 f 1 g 2 , f * g ∞ f 2 g 2 and µ * f 2 f 2 .
Further, we denote by f : G → C the function given by f (x) = f (x -1 ). Similarly, μ is the Borel probability measure given by G F dμ = G F dµ, for any continuous F : G → R. We say that µ is symmetric if μ = µ. For n 1, we denote by µ * n the n-fold convolution product of µ with itself. We also denote by supp(µ) the support of µ. If µ and ν have finite support, then μ({x}) = µ({x -1 }) and (µ * ν)({x}) = y∈G µ({y})ν({y -1 x}), for any x, y ∈ G.

If G is unimodular, we denote by λ, ρ : G → U(L 2 (G)) the left and right regular representations of G given by λ g (f

)(x) = f (g -1 x), ρ g (f )(x) = f (xg), for every f ∈ L 2 (G) and any g, x ∈ G. Notice that λ g (f ) = δ g * f and ρ g (f ) = f * δ g -1
, where δ g denotes the Dirac measure at g ∈ G.

Next, we establish a useful result that we will need later on.

Lemma 2.1. Let µ be a symmetric Borel probability measure on G and n 1.

Then (1) µ * n * f 2 µ * f 2n 2 , for every f ∈ L 2 (G) with f 2 = 1. (2) µ * n (A) 2 µ * (2n) (A -1 A), for every measurable set A ⊂ G. Proof. (1) Since µ is symmetric, we have µ * m * f 2 2 = µ * m * f, µ * m * f = µ * 2m * f, f µ * 2m * f 2 , for any m 0. By induction, it follows that µ * 2 m * f 2 µ * f 2 m , for all m 0. Choose m 0 such that 2 m n < 2 m+1 . Then µ * n * f 2 µ * 2 m+1 * f 2 µ * f 2 m+1 µ * f 2n , as claimed.
(2) Indeed, we have

µ * n (A) 2 = µ * n (A -1 )µ * n (A) µ * (2n) (A -1 A).
2.2. Basic properties of local spectral gap. We continue with several elementary properties of local spectral gap, starting with an easy, but useful, equivalent formulation of local spectral gap.

Proposition 2.2. Let Γ (X, µ) be a measure preserving action of a countable group Γ, and B ⊂ X a measurable set of finite measure. Then Γ (X, µ) has local spectral gap with respect to B if and only if there exist a finite set F ⊂ Γ and a constant κ > 0 such that the following holds:

ξ - 1 µ(B) B ξ dµ 2,B κ g∈F g • ξ -ξ 2,B for any ξ ∈ L 2 (G).
Proof. The if implication is clear. To prove the only if implication, suppose that Γ (X, µ) has local spectral gap with respect to B. Then there are a finite set F ⊂ Γ and κ > 0 such that η 2,B κ g∈F g • η -η 2,B , for any η ∈ L 2 (X) with B η dµ = 0. We may assume that e ∈ F . Let ξ ∈ L 2 (X) and put α = 1 µ(B) B ξ dµ. Let C = ∪ g∈F g -1 B and define η = ξ -α1 C ∈ L 2 (G). Then B η dµ = 0 and g • η -η 2,B = g • ξ -ξ 2,B , for all g ∈ F . The conclusion now follows.

Proposition 2.3. Let Γ (X, µ) be a measure preserving action of a countable group Γ, and B 1 , B 2 ⊂ X measurable sets of finite measure. Assume there is a finite set K ⊂ Γ such that B 1 ⊂ ∪ h∈K hB 2 and B 2 ⊂ ∪ h∈K hB 1 .

Then Γ (X, µ) has local spectral gap with respect to B 1 if and only if it does with respect to B 2 .

Proof. Assume that local spectral gap holds with respect to B 1 , but not B 2 . Let ξ n ∈ L 2 (X) be a sequence satisfying ξ n 2,B 2 = 1, B 2 ξ n dµ = 0, for all n, and g • ξ n -ξ n 2,B 2 → 0, for every g ∈ Γ.

If g ∈ Γ, then we have

g • ξ n -ξ n 2,B 1 h∈K g • ξ n -ξ n 2,hB 2 = h∈K (h -1 g) • ξ n -h -1 • ξ n 2,B 2 h∈K (h -1 g) • ξ n -ξ n 2,B 2 + h -1 • ξ n -ξ n 2,B 2 .
This implies that g • ξ n -ξ n 2,B 1 → 0, for every g ∈ Γ. Since we have local spectral gap with respect to B 1 , Proposition 2.2 provides scalars α n ∈ C such that ξ n -α n 2,B 1 → 0. By reasoning as above, it follows that ξ n -α n 2,B 2 → 0. Since B 2 ξ n dµ = 0, for all n, we get that α n → 0. Hence, ξ n 2,B 2 → 0, which gives the desired contradiction.

Next, we establish that local spectral gap passes to direct product actions.

Proposition 2.4. For i ∈ {1, 2}, let Γ i (X i , µ i ) be a measure preserving action which has local spectral gap with respect to a measurable set B i ⊂ X i of finite measure.

Then the product action

Γ 1 × Γ 2 (X 1 × X 2 , µ 1 × µ 2 ) has local spectral gap with respect to B 1 × B 2 .
Proof. By Lemma 2.2, for i ∈ {1, 2}, we can find F i ⊂ Γ i finite and κ i > 0 such that

(2.1) ξ - 1 µ i (B i ) B i ξ dµ i 2 2,B i κ i g∈F i g • ξ -ξ 2 2,B i for any ξ ∈ L 2 (X i ). Denote (X, µ) = (X 1 × X 2 , µ 1 × µ 2 ) and B = B 1 × B 2 . Let ξ ∈ L 2 (X, µ) and put α = 1 µ(B) B ξ dµ. For y ∈ X 2 , define ξ y (x) = ξ(x, y) and f (y) = 1 µ 1 (B 1 ) B 1 ξ y dµ 1 . Then it is easy to see that f ∈ L 2 (X 2 ) and g • f -f 2 2,B 2 1 µ 1 (B 1 ) g • ξ -ξ 2 2,B 1 ×B 2 , for all g ∈ Γ 2 . Since 1 µ 2 (B 2 ) B 2 f dµ 2 =
α, by using the last inequality and applying (2.1) to f we get that

(2.2) f -α 2 2,B 2 κ 2 g∈F 2 g • f -f 2 2,B 2 κ 2 µ 1 (B 1 ) g∈F 2 g • ξ -ξ 2 2,B 1 ×B 2
On the other hand, by applying (2.1) to ξ y we get that ξ

y -f (y) 2 2,B 1 κ 1 g∈F 1 g • ξ y -ξ y 2 2,B 1 . By integrating over y ∈ B 2 , we derive that (2.3) B |ξ(x, y) -f (y)| 2 dµ(x, y) κ 1 g∈F 1 g • ξ -ξ 2 2,B 1 ×B 2 .
It is now clear that the combination of (2.2) and (2.3) implies the conclusion.

Finally, we record a result asserting that local spectral gap passes through certain quotients. Since its proof is very similar to that of Corollary C, we leave its details to the reader.

Proposition 2.5. Let G be a l.c.s.c. group, H < G a closed subgroup, and Γ < G a countable dense subgroup. Assume that G/H admits a G-invariant Borel regular measure m G/H . Suppose that the left translation action Γ (G, m G ) has local spectral gap.

Then the left translation action Γ (G/H, m G/H ) has local spectral gap.

Deduction of Theorem

A from Theorem B. The aim of this subsection is to show that Theorem B implies Theorem A. This relies on the following result.

Proposition 2.6. Let G be a l.c.s.c. group, Γ < G a countable dense subgroup, and B ⊂ G a measurable set with non-empty interior and compact closure. Assume that there exists a constant c > 0 satisfying the following property: for any neighborhood U of the identity, there are a finite set S ⊂ Γ ∩ U and a finite dimensional vector space

V ⊂ L 2 (G) such that for all ξ ∈ L 2 (B) V we have max g∈S g • ξ -ξ 2 c ξ 2 .
Then the left translation action Γ (G, m G ) has local spectral gap with respect to B.

Proof. Assume by contradiction that the conclusion is false. Then there is a sequence

ξ n ∈ L 2 (G) satisfying ξ n 2,B = 1, B ξ n dµ = 0, for all n, and lim n→∞ g • ξ n -ξ n 2,B = 0, for all g ∈ Γ.
If C ⊂ G is a compact set, then C can be covered with finitely many of the sets {gB} g∈Γ . It follows that sup n ξ n 2,C < ∞ and lim

n→∞ g • ξ n -ξ n 2,C = 0, for all g ∈ Γ. Since G is second countable, we can find a subsequence {ξ n k } of {ξ n } and ξ ∈ L 2 loc (G) (i.e. a locally L 2 -integrable function) such that 1 C ξ n k → 1 C ξ,
weakly, for every compact set C ⊂ G. But then ξ must be Γ-invariant, and hence constant by ergodicity. Since ξ n has mean zero on B, for all n, we derive that ξ = 0, almost everywhere. This argument implies that 1 C ξ n → 0, weakly, for any compact set C ⊂ G.

Let lim

n be a bounded linear functional on ∞ (N) which extends the limit. Then

ν(C) = lim n 1 C ξ n 2 2
defines a Γ-invariant finitely additive measure on bounded Haar measurable subsets of G. Since ν(B) = 0, we get that ν = 0. Since B has non-empty interior, by using finite additivity, we can find two open sets B 1 B 2 ⊂ B such that ν(B 1 ) = 0, ν(B 2 \ B 1 ) (c 2 ν(B 1 ))/4, and there exists a closed intermediate subset

B 1 ⊂ F ⊂ B 2 .
By local compactness, we can find an intermediate open set B 0 between B 1 and B 2 and a neighborhood U of the identity small enough so that B 1 ⊂ gB 0 ⊂ B 2 , for g ∈ U . Put p = 1 B 0 . As the sequence {pξ n } converges weakly to 0 and is supported on B, the following claim contradicts our assumption on c.

Claim. For all g ∈ Γ ∩ U , we have lim

n g • (pξ n ) -(pξ n ) 2 c 2 lim n pξ n 2 . Indeed, for g ∈ Γ ∩ U we can estimate lim n g • (pξ n ) -(pξ n ) 2 = lim n (g • p)(g • ξ n ) -(pξ n ) 2 lim n (g • p)ξ n -(pξ n ) 2 + lim n (g • p)(g • ξ n ) -(g • p)ξ n 2 = lim n (g • p -p)ξ n 2 + lim n g • ξ n -ξ n 2,gB 0 But by the above, lim n g • ξ n -ξ n 2,gB 0 = 0. Moreover, since B 1 ⊂ gB 0 ⊂ B 2 , we get that lim n (g • p -p)ξ n 2 2 ν(B 2 \ B 1 ) c 2 4 ν(B 1 ) c 2 4 ν(B r (1)) = c 2 4 lim n pξ n 2 2
Proof of Theorem A. Assume that Theorem B holds and let us explain how Theorem A follows.

Let S ⊂ Γ be a finite set and denote µ

= 1 |S| g∈S δ g . If ξ ∈ L 2 (G), then g∈S g • ξ -ξ 2 2 = 2|S| ξ 2 2 -µ * ξ, ξ .
Thus, if we have that µ * ξ < 1 2 ξ 2 , then max g∈S g • ξ -ξ 2 > ξ 2 . By combining Theorem B and Proposition 2.6, we conclude that Γ (G, m G ) has local spectral gap.

2.4. Reduction to groups with trivial center. Next, we will argue that in order to prove Theorem B, we may reduce to the case when G has trivial center.

Assume that Theorem B holds for connected simple Lie groups with trivial center. Let G be a connected simple Lie group, B ⊂ G a measurable set with compact closure and non-empty interior, and c > 0. Let π : G → GL(g) be the adjoint representation of G. Put G 0 = π(G) and Γ 0 = π(Γ).

Since π has discrete kernel, we can find a small enough compact neighborhood of the identity C ⊂ G and

ε 0 > 0 such that π is 1-1 on ∪ g∈Bε 0 (1) gC. Let K ⊂ G be a finite set such that B ⊂ ∪ h∈K Ch. Write B as a disjoint union B = h∈K C h , where C h is a subset of Ch, for every h ∈ K.
Since G 0 has trivial center, the conclusion of Theorem B holds for (G 0 , Γ 0 , π(C)) by our assumption.

It is then easy to see that Theorem B also holds for (G, Γ, C).

Thus, given ε > 0, there are a finite set T ⊂ Γ∩B ε (1) and a finite dimensional subspace

W ⊂ L 2 (C) such that µ := 1 2|T | g∈T (δ g + δ g -1 ) satisfies µ * F 2 < c |K| F 2 , for every F ∈ L 2 (C) W. Let V ⊂ L 2 (B) be the linear span of {1 C h ρ h (W )|h ∈ K}, where {ρ g } g∈G denotes the right regular representation of G. Let F ∈ L 2 (B) V . If h ∈ K, then ρ -1 h (1 C h F ) ∈ L 2 (C) W , and therefore µ * (1 C h F ) 2 = µ * (ρ -1 h (1 C h F )) 2 < c |K| ρ -1 h (1 C h F ) 2 = c |K| 1 C h F 2 . Since F = h∈K 1 C h F , we deduce that µ * F 2 < c F 2 . Since c > 0 is arbitrary, and V ⊂ L 2 (B)
is finite dimensional, this implies that G satisfies the conclusion of Theorem B.

2.5. Outline of the proof of Theorem B. The previous subsection allows us to work hereafter with connected simple groups with trivial center. Note, however, that although our results will be stated only for groups with trivial center, they have analogues for general connected simple groups.

In order to outline the proof of Theorem B, let us introduce some more notation. Let G be a connected simple Lie group with trivial center, and Γ < G be a dense subgroup. Suppose there is a basis B of g such that the matrix of Ad(g) in the basis B has algebraic entries, for any g ∈ Γ.

Let n be the dimension of G, g its Lie algebra, and Ad: G → GL(g) its adjoint representation. We identify G ∼ = Ad(G), g ∼ = R n via the basis B, and GL(g

) ∼ = GL n (R) ⊂ M n (R).
In particular, in this identification we have that Γ < GL n ( Q).

For α = (α i,j ) n i,j=1 ∈ M n (R), we denote by α 2 = ( n i,j=1 |α i,j | 2 ) 1/2 its Hilbert-Schmidt norm. We endow G with the metric given by (g, h) → Ad(g) -Ad(h) 2 . Abusing notation, we write g -h 2 := Ad(g) -Ad(h) 2 and g 2 := Ad(g) 2 . Note that

gh -gk 2 g 2 h -k 2 , for all g, h, k ∈ G.
For x ∈ G and δ > 0, we denote B δ (x) := {y ∈ G| x-y 2 δ}. For δ > 0, we let

A (δ) = ∪ x∈A B δ (x)
be the δ-neighborhood of A ⊂ G, and denote

P δ := 1 B δ (1) |B δ (1)| ∈ L 1 (G) +,1 .
As explained at the end of the Introduction, the proof of Theorem B splits into three parts, dealt with in the following three sections.

• In Section 3 we produce measures with small support that Escape subgroups quickly.

There are two steps for this. First, we produce for all ε > 0 a finite set S ⊂ Γ ∩ B ε (1) and constants d, C > 0 such that for δ > 0 small enough, the measure µ S := 1 2|S| g∈S (δ g + δ g -1 ) satisfies µ * n S (H (δ) ) δ d , for all proper closed subgroups H and n ≈ C log 1 δ . This step is obtained by combining Propositions 3.2 and 3.9. The set S that we obtain freely generates a free group.

One can of course get a better constant C by modifying accordingly the value of d. But for a fixed d, the value of C depends on ε. Namely, it could happen that C → ∞ as ε → 0. As explained in the introduction, we want to control the speed of escape in terms of ε. So the second step is to upgrade the set S to a set T , also contained in B ε (1), such that the following holds (Theorem 3.1).

There are constants d 1 , d 2 > 0 not depending on T such that the probability measure µ T satisfies

µ * n T (H (δ) ) δ d 1 , for all δ > 0 small enough, all proper closed subgroups H and n ≈ d 2 log 1 δ log 1 ε .
This improvement is obtained using the pigeonhole principle and the freeness of the elements of S.

• In Section 4 we extend the 2 -flattening lemma from [START_REF] Benoist | Saxcè: A spectral gap theorem in simple Lie groups[END_REF].

Our generalization of the flattening lemma [BdS14, Lemma 2.5] to the locally compact setting does not require much additional effort. However, it only applies for measures with controlled support. But we anticipated this issue in part 1 above, by controlling the speed of escape in terms of ε.

Indeed, we want to apply the flattening lemma to the measure

µ * n T , with n ≈ d 2 log 1 δ log 1 ε . Now, the support of µ * n T is contained in B δ -β (1)
, with β > 0 arbitrarily small. Since the "controlled support" condition that we require is soft enough, we are in position to apply our flattening Lemma 4.1. Thus, our main result (Corollary 4.2) shows that the measure µ T produced in Section 3 will flatten rather quickly: given α > 0, we have µ * n T * P δ 2 δ -α , for δ small enough and n ∼ log 1 δ . • In Section 5 we prove a Mixing inequality and combine it with the above to conclude.

More precisely, we show that if µ T is the measure produced in Section 3, then the convolution operator F ∈ L 2 (B) → (µ T * F ) ∈ L 2 (G) has norm less than 1/2, when restricted to the orthogonal complement of a finite dimensional subspace V ⊂ L 2 (B). The first observation is that this flexibility of discarding a finite dimensional subspace V when trying to bound the norm of µ T * F 2 , allows us to restrict our study to functions F that live at a "small scale". Namely, it will be enough to consider functions F that do not change much when "discretizing" the group with high accuracy. This reduction is achieved via a Littlewood-Paley type decomposition (Theorem 6.2 and Corollary 6.6). Then we are left to show a mixing inequality (Theorem 5.1). This is inspired by [BG10, Lemma 10.35], and should be thought of as an analogue of the well-known mixing inequality for finite groups (see e.g. [Ta15, Proposition 1.3.7]), after discretizing the group. We will then be able to conclude restricted spectral gap by combining this inequality with the flattening obtained in Section 4.

Escape from subgroups

The goal of this section is to prove the following: Theorem 3.1 (escape from subgroups). Let G be a connected simple Lie group with trivial center, and Ad: G → GL(g) be its adjoint representation. Let Γ < G be a countable dense subgroup. Assume that there there is a basis B of g such that the matrix of Ad(g) in the basis B has algebraic entries, for every g ∈ Γ.

Then there are constants d 1 , d 2 > 0 depending on Γ only such that the following holds.

Given ε 1 > 0, we can find 0 < ε < ε 1 and a finite set T ⊂ Γ∩B ε (1) which freely generates a subgroup of Γ such that for any small enough δ > 0, the probability measure

µ = 1 2|T | g∈T (δ g + δ g -1 ) satisfies µ * 2n (H (δ) ) δ d 1 , where n = d 2 log 1 δ log 1 ε ,
for any proper closed connected subgroup H < G.

3.1. Ping-pong. The first ingredient in the proof of Theorem 3.1 is a proposition which, roughly speaking, asserts the existence of representations ρ i : Γ → GL(V i ), i ∈ I, and M 2 such that

• the intersection of Γ with any proper closed subgroup of G stabilizes a line in some V i , and

• we can find a set S ⊂ Γ of simultaneous "ping-pong players" for all the ρ i 's in any given neighborhood of the identity in G such that |S| = M .

Proposition 3.2. Let G be a connected simple real Lie group with trivial center. Let Γ < G be a finitely generated dense subgroup. Assume that there there is a basis B of the Lie algebra g of G such that the matrix of Ad(g) in the basis B has algebraic entries, for every g ∈ Γ.

Then there exist finitely many vector spaces V i , i ∈ I, defined over local fields K i , representations ρ i : Γ → GL(V i ), and an integer M 2 such that the following properties hold true:

(1) For any proper closed subgroup

H < G such that Γ ∩ H is non-discrete, there exist i ∈ I and [v] ∈ P(V i ) such that ρ i (g)([v]) = [v], for all g ∈ Γ ∩ H.
(2) For any η > 0, there is a finite set S ⊂ Γ satisfying |S| = M and S ⊂ B η (1) such that for all i ∈ I and every g ∈ S := S ∪ S -1 , we can find two sets

K (i) g ⊂ U (i)
g ⊂ P(V i ) such that the following conditions hold: (a) For every g ∈ S we have ρ i (g)(U

(i) g ) ⊂ K (i) g . (b) Every line [v] ∈ P(V i ) is contained in at least two of the sets {U (i) g } g∈ S . (c) For every g 1 , g 2 ∈ S we have K (i) g 1 ⊂ U (i) g 2 , unless g 1 g 2 = 1. (d) For every g 1 , g 2 ∈ S we have K (i) g 1 ∩ K (i) g 2 = ∅, unless g 1 = g 2 .
Before proving Proposition 3.2, let us record a simple observation that will be used later.

Lemma 3.3. [START_REF] Salehi Golsefidy | Varjú: Expansion in perfect groups[END_REF] In the setting from Proposition 3.2, let i ∈ I and v ∈ V i \ {0}. Let g = g n g n-1 ...g 1 be a reduced word on length n in S. Assume that

ρ i (g)([v]) = [v] and let 1 j < n. (1) If ρ i (g j g j-1 ...g 1 )([v]) ∈ U g j+1 , then g j+1 , ..., g n are uniquely determined by v. (2) If ρ i (g j g j-1 ...g 1 )([v]) / ∈ U g j+1 , then ρ i (g l g l-1 ...g 1 )([v]) / ∈ U g l+1 , for all 1 l j.
Proof. For simplicity, denote ρ = ρ i and

K g = K (i) g , U g = U (i) g , for all g ∈ S. Assume that ρ(g j g j-1 ...g 1 )([v]) ∈ U g j+1 . Since ρ(g j+1 )(U g j+1 ) ⊂ K g j+1 , we get ρ(g j+1 g j ...g 1 )([v]) ∈ K g j+1 . Since g j+1 = g -1
j+2 , we have that K g j+1 ⊂ U g j+2 , and hence ρ(g j+1 g j ...g 1 )(

[v]) ∈ U g j+2 . Using induction it follows that ρ(g p g p-1 ...g 1 )([v]) ∈ K gp , for all j + 1 p n. Thus, [v] = ρ(g n ...g 1 )([v]) ∈ K gn .
Since the sets {K g } g∈ S are mutually disjoint, g n is therefore determined by v. Further, we have that ρ

(g -1 n )([v]) = ρ(g n-1 ...g 1 ) ∈ K g n-1 . Since ρ(g -1 n )([v]
) is determined by v, we deduce that g n-1 is also determined by v. The first assertion now follows by induction. Since the beginning of the proof implies the second assertion, the proof is complete.

The rest of this subsection is devoted to proving Proposition 3.2. The proof is very similar to the the proof of [SGV11, Proposition 21]. Consider a connected simple real Lie group G with trivial center, together with a finitely generated dense subgroup Γ as in the statement of Proposition 3.2. By identifying G with Ad(G), we can assume that G is the connected component of a real algebraic group G = Ad(G)

Z ⊂ GL(g). By our assumptions on Γ we can find a number field k with an embedding k ⊂ R and a basis of

g such that Γ ⊂ GL d (k) ⊂ GL d (R) ∼ = GL(g). Since Γ is Zariski dense in G,
(k) ⊂ g(k) is invariant under H(k), but not under G(k).
Next, define the finite set of representations ρ i , i ∈ I, to be the collection of all (non-trivial) irreducible subrepresentations of the representations of G on m j=1 g, m < d = dim G. These are algebraic representations, defined over a finite extension k of k. We will show that Proposition 3.2 holds when we view these representations ρ i as defined over appropriate places K i of k . Note that no matter how we choose the places K i , we still get representations of Γ ⊂ G(k) which satisfy item (1) of the proposition, by the above paragraph. Let us now choose the places K i for which we will be able to prove that item (2) of the proposition also holds true.

Lemma 3.4. Use the above notation. Then for every i ∈ I, there are a local field K i and a sequence (h n ) n in Γ which converges to 1 in the real topology such that (ρ i (h n )) n goes to infinity in the K i -topology.

Proof. Fix i ∈ I. First we claim that there is a sequence (h n ) n ⊂ Γ which converges to 1 in the real topology such that the elements ρ i (h n ) are pairwise distinct.

To prove the claim, we view the representations ρ i as representations over C by fixing an embedding k ⊂ C. This way, it makes sense to talk about ρ i (G). Note that the image ρ i (B 1 (1)) of the unit ball of G is connected. Since the representation ρ i is non-trivial and Γ is dense in G, there exists a sequence (h n ) n ⊂ Γ ∩ B 1 (1) such that ρ i (h n ) is non-trivial and converges to 1 in the complex topology. In particular, after passing to a subsequence of h n if necessary, we get that the elements ρ i (h n ) are distinct. As (h n ) n is a bounded sequence and ρ i (h n ) converges to 1, we deduce that h n converges to 1 as well, proving our claim.

Next, denote by R the ring generated by the coefficients of the elements Ad(g), g ∈ Γ. Since Γ is finitely generated, R ⊂ k is a finitely generated subring. The discrete diagonal embedding of k in its adèle group gives a discrete embedding of R in a product of finitely many places K ν , ν ∈ S, of k .

From this we obtain a discrete embedding

ρ i (G(R)) → Π ν∈S ρ i (G(K ν )). In particular, ρ i (Γ) is discrete inside Π ν∈S ρ i (G(K ν )). Therefore there exists a field K i := K ν such that the infinite set {ρ i (h n )} is unbounded as a subset of ρ i (G(K ν )).

Below, we denote by Γ

1 the set of sequences (h n ) ⊂ Γ which converge to 1 in the real topology. For i ∈ I, we view ρ i : G → GL(V i ) as a representation over K i , and equip GL(V i ) with the operator norm • i corresponding to the absolute value on K i . We denote by A i the set of cluster points in the K i -topology of sequences of the form A key fact that we will use is that since G is simple, we have that ρ i (g) has determinant 1 for all g ∈ G, and in particular for all g ∈ Γ. Hence, if

(ρ i (h n )/ ρ i (h n ) i ) n , where (h n ) ∈ Γ 1 . Finally,
(h n ) n ⊂ Γ and (ρ i (h n )) n is unbounded in the K i -topology, then the normalized sequence (ρ i (h n )/ ρ i (h n ) K i ) n has a non-invertible cluster point. So by our choice of K i , we have r i < d i := dim(V i ).
Let us mention the following stability result for the sets A i .

Lemma 3.5. If b and b belong to A i and bb = 0, then some scalar multiple of bb belongs to A i .

In particular rk(bb ) ≥ r i .

Proof.

If b = lim n ρ i (g n )/ ρ i (g n ) i and b = lim n ρ i (h n )/ ρ i (h n ) i , with (g n ) n , (h n ) n ∈ Γ 1 , then the product sequence (g n h n ) n ⊂ Γ converges to 1 in the real topology. Moreover, lim n ρ i (g n h n ) ρ i (g n ) i ρ i (h n ) i = bb , so that lim n ρ i (g n h n ) i ρ i (g n ) i ρ i (h n ) i = bb i . Therefore lim n ρ i (g n h n ) ρ i (g n h n ) i = bb bb i .
Now, we turn to the construction of the set S from Proposition 3.1. The following lemma will produce the first element of S. In the context of Lemma 3.6, it will be g n with n large enough, depending on η. The other elements of S will arise as appropriate conjugates of this first element.

Lemma 3.6. There exists a sequence (g n ) n ∈ Γ 1 such that for all i ∈ I

(1) lim n ρ i (gn)

ρ i (gn) i = b i for some b i with rk(b i ) = r i ; (2) Range(b i ) ∩ Ker(b i ) = {0}.
Proof. We proceed in three steps.

Step 1. There exists a sequence (h n ) n ∈ Γ 1 which satisfies (1) above for all i ∈ I.

We proceed by induction. Enumerate the set

I = {1, • • • , |I|}. Assume that (k n ) n ∈ Γ 1 satisfies (1)
for all indices i < i 0 , for some 1 i 0 < |I|. Taking a subsequence if necessary, we can assume that the sequence (

ρ i 0 (k n )/ ρ i 0 (k n ) i 0 ) converges to some element b i 0 ∈ End(V i ).
Since the rank of b i 0 could be greater than r i 0 , we also consider a sequence

(k n ) n ∈ Γ 1 such that (ρ i 0 (k n )/ ρ i 0 (k n ) K i 0
) n converges to some b i 0 with rank r i 0 . Taking a subsequence we can assume

(ρ i (k n )/ ρ i (k n ) i
) n converges to some element b i for all i < i 0 , with possibly rk(b i ) > r i .

We will prove the existence of an element g ∈ Γ such that the sequence (gk n g -1 k n ) n satisfies (1) for all i i 0 . Note that no matter how we choose g, this sequence is inside Γ 1 . In fact it suffices to find g ∈ Γ such that ρ i (g -1 )b i ρ i (g)b i = 0, for all i i 0 . Indeed, then Lemma 3.5 implies that the sequence (ρ i (gk

n g -1 k n )/ ρ i (gk n g -1 k n ) i ) n converges to some non-zero multiple of ρ i (g -1 )b i ρ i (g)b i , which has rank at most equal to min(rk(b i ), rk(b i )) = r i .
For each i i 0 , the set

X i = {g ∈ G(K i ) | Range(ρ i (g)b i ) Ker(b i )} is a Zariski open set in G which is non-empty because ρ i is irreducible. Therefore Γ (∩ i i 0 X i ) is nonempty. This proves Step 1.
Step 2. There exists a sequence (g n ) n ∈ Γ 1 such that (1) is true for any i ∈ I and the corresponding elements b i satisfy b 2 i = 0. Consider a sequence (h n ) as in Step 1, and denote by b i the corresponding elements. We will find an element g ∈ Γ such that the sequence of elements g n := gh n g -1 h n does what we want. As above, for any i, the set

X i of elements g ∈ G(K i ) such that Range(ρ i (g)b i ) Ker(b i ) is a non-empty Zariski-open set. So is the set Y i of g ∈ G(K i ) such that Range(ρ i (g -1 )b i ) Ker(b i ), for all i |I|. Take g ∈ Γ (∩ i |I| (X i ∩ Y i )) so that the element a i := ρ i (g)b i ρ i (g -1 )b i is non-zero.
Then for all i, the sequence (ρ i (g n )/ ρ i (g n ) i ) n converges to some nonzero multiple b i of a i . We claim that a 2 i = 0. Indeed, Lemma 3.5 implies that the rank of a i is equal to r i = rk(ρ i (g)b i ). This means that the range of a i is equal to the range of ρ i (g)b i . Since g ∈ X i , it follows that b i a i is non-zero. Using again Lemma 3.5, we get that the rank of b i a i is equal to r i = rk(b i ). This means that the range of b i a i is equal to the range of b i . But since g ∈ Y i we see that b i ρ i (g -1 )b i a i = 0. This shows that a 2 i = 0.

Step 3. The sequence from Step 2 satisfies the conclusion of the lemma.

We just need to check that for all i, any element b ∈ A i with rank r i and such that b 2 = 0 satisfies Range(b)∩Ker(b) = {0}. Indeed, if b 2 = 0 then Lemma 3.5 implies that some multiple of b 2 belongs to A i . Hence rk(b 2 ) = r i = rk(b). This precisely means that Range(b) ∩ Ker(b) = {0}.

Before actually proving Proposition 3.2, let us give two easy lemmas.

Lemma 3.7. Given a local field K, consider a sequence of invertible elements

(g n ) n ⊂ GL d (K), such that lim n g n g n = b and lim n g -1 n g -1 n = b , for some non-invertible elements b, b ∈ M d (K). Then bb = 0, so that Range(b ) ⊂ Ker(b).
Proof. Note that bb is a scalar matrix, being the limit of the sequence (1/ g n g -1 n

) n . Since it is non-invertible, it must be 0.

Lemma 3.8. Let ρ : G(K) → GL(W ρ ) be an irreducible algebraic representation over a local field K. Let V + 1 , V - 1 , V + 2 , V - 2 ⊆ W ρ be non-zero, proper subspaces such that V + 1 ∩ V - 1 = V + 2 ∩ V - 2 = {0} and V + 1 ⊆ V - 2 and V + 2 ⊆ V - 1 .
For M 1, denote by X M ⊆ G(K) M the set of M -tuples (h 1 , . . . , h M ) satisfying the following two conditions.

(1) For 1 s = t M , we have ρ(h

s )V + 1 ρ(h t )(V - 1 ∪ V - 2 ) and ρ(h s )V + 2 ρ(h t )(V - 1 ∪ V - 2 );
(2) For any subset S ⊂ {1, . . . , M } and any choice of Let us prove by induction over M that B M is a non-empty Zariski open set. For M = 1, the condition (2) is empty, so this is clearly true. Assuming the result for M , let us check it for M + 1. Consider the finite collection of all vector spaces of the form

V s ∈ {V - 1 , V - 2 }, s ∈ S, we have dim(∩ s∈S ρ(h s )V s ) max(0, dim(W ρ ) -|S|).
E α = ∩ s∈S ρ(h s )V s ⊂ W ρ , where S ⊂ {1, . . . , M } and V s ∈ {V - 1 , V - 2 }, for all s ∈ S. Then B M +1 is equal to (h 1 , . . . , h M +1 ) (h 1 , . . . , h M ) ∈ B M and E α ρ(h M +1 )(V - 1 ∪ V - 2
) for all α with E α = ∅ . Using this, it can be easily seen that B M +1 is a finite intersection of non-empty Zariski open sets. Therefore, B M +1 is Zariski open, as well as non-empty by the Zariski connectedness of G.

Proof of Proposition 3.2. Consider the representations ρ i over local fields K i , i ∈ I, defined above. For i ∈ I, we consider the representation ρ i : g → ρ i (g -1 ) t . Note that by the definition of r i , we clearly have r i = r i .

Applying Lemma 3.6 to the set of representations {ρ i } i∈I ∪{ρ i } i∈I , we obtain a sequence (g n ) n ⊂ Γ which converges to 1 in the real topology, and elements b i , b i ∈ End(V i ) such that for all i ∈ I,

• lim n ρ i (gn) ρ i (gn) K i = b i and lim n ρ i (g -1 n ) ρ i (g -1 n ) K i = b i , • rk(b i ) = rk(b i ) = r i , and • Range(b i ) ∩ Ker(b i ) = Range(b i ) ∩ Ker(b i ) = {0}.
By Lemma 3.7, we can add the following property to the above list:

• Range(b i ) ⊂ Ker(b i ) and Range(b i ) ⊂ Ker(b i ).
Now, for i ∈ I, the sets

V + i,1 = Range(b i ), V - i,1 = Ker(b i ), V + i,2 = Range(b i ) and V - i,2 = Ker(b i ) satisfy the hypothesis of Lemma 3.8. Put M := max i (dim(ρ i )) + 1. For i ∈ I, denote by X i the non-empty Zariski open subset of G(K i ) M
given by Lemma 3.8 applied to these sets. Pick an

M -tuple (h 1 , • • • , h M ) ∈ Γ (∩ i X i ).

Before going further, let us mention that ρ

i (h s )V + i,1 = Range(h s b i h -1 s ), ρ i (h s )V - i,1 = Ker(h s b i h -1 s ), whereas ρ i (h s )V + i,2 = Range(h s b i h -1 s ), ρ i (h s )V - i,2 = Ker(h s b i h -1 s ).
Then by the definition of X i , for every i ∈ I and 1 s = t N , we have

ρ i (h s )V + i,1 ρ i (h t )V - i,2 . This means that (h t b i h -1 t ).(h s b i h -1 s ) = 0. But both (h s b i h -1 s ) and (h t b i h -1 t )
belong to A i and have rank r i . Thus, their product has rank equal to r i by Lemma 3.5. From this we deduce that

ρ i (h s )V + i,1 ∩ρ i (h t )V - i,2 = {0}. Similarly, ρ i (h s )V + i,1 ∩ρ i (h t )V - i,1 = {0} and ρ(h s )V + i,2 ∩ρ(h t )(V - i,1 ∪V - i,2 ) = {0}.
Using the above properties, for every i ∈ I and 1 s N , we can find compact neighborhoods

K i,s , K i,s ⊂ P(V i ) of P(ρ i (h s )V + i,1 ) and P(ρ i (h s )V + i,2
) respectively, and open sets U i,s , U i,s ⊂ P(V i ) which are complements of neighborhoods of P(ρ i (h s )V - i,1 ) and P(ρ i (h s )V - i,2 ), respectively, such that:

• K i,s ⊂ U i,s and K i,s ⊂ U i,s for all s; • K i,s ∩ U i,s = ∅ = K i,s ∩ U i,s ; • For all s = t, K i,s ⊂ U i,t ∩ U i,t and K i,s ⊂ U i,t ∩ U i,t ;
• For any x ∈ P(V i ), we can find at least two indices s for which x ∈ U i,s or x ∈ U i,s .

The last fact is due to property (2) from Lemma 3.8, which implies that for any set S ⊂ {1, • • • , M } with |S| = M -1 and any choice of V s ∈ {V - i,1 , V - i,2 }, s ∈ S, we have ∩ s∈S ρ(h s )V s = {0}. Finally, given η > 0, we can find n large enough so that for all i ∈ I and all s, we have

h s g n h -1 s (U i,s ) ⊂ K i,s , h s g -1 n h -1 s (U i,s ) ⊂ K i,s and h s g n h -1 s , h s g -1 n h -1 s ∈ B η (1). We define S to be the set of elements {h s g n h -1 s |1 s M }. If g = h s g n h -1 s ∈ S, define K (i) g = K i,s and
U (i) g = U i,s , and if g = h s g -1 n h -1 s ∈ S -1 , define K (i) g = K i,s and
U (i) g = U i,s
. These sets are easily seen to satisfy the desired properties.

3.2.

From subgroups to neighborhoods of subgroups. The goal of this section is to prove the following proposition, which roughly says that algebraic points with small logarithmic height cannot be very close to a proper algebraic subgroup. Our method is fairly similar to [SGV11, Proposition 16] (see also [START_REF] Benoist | Saxcè: A spectral gap theorem in simple Lie groups[END_REF]Proposition 3.11] or [Va10, Proposition 4]).

Proposition 3.9. Let G be a connected simple Lie group and T ⊂ G a finite subset. Assume that there there is a basis B of the Lie algebra g of G such that the matrix of Ad(g) in the basis B has algebraic entries, for every g ∈ T .

Then there exists a constant C > 0 (depending on T ) such that for every integer n 1 and any non-discrete proper closed subgroup H < G, we can find a proper closed subgroup H < G such that

W n (T ) ∩ H (e -Cn ) ⊆ H , where W n (T ) = {g 1 g 2 ...g n | g 1 , g 2 , ..., g n ∈ T ∪ T -1 }.
Notation. In this subsection, we use the notation O X (a) to denote a positive quantity bounded by Ca, for some constant C > 0 depending only on X. We also use the notation a X b to mean the existence of some constant C > 0 depending only on X such that a Cb. Lemma 3.10. Let X ⊆ M n (R) be a finite subset. Suppose the R-span A of X is an R-algebra, and V := R n is a simple A-module. Then there exists c 0 > 0 such that for every l ∈ V * and v ∈ V

max x∈X |l(xv)| c 0 l 2 v 2 . Proof. Let H X (l, v) := max x∈X |l(xv)|.
We need to show that the infimum of H X (l, v) on the pair of unit vectors is positive. Suppose the contrary. So by the continuity of H X : V * × V → R, there are unit vectors l 0 and v 0 such that H X (l 0 , v 0 ) = 0. This implies that for any a ∈ A we have l 0 (av 0 ) = 0. Hence the A-module generated by v 0 is a proper subspace which contradicts the simplicity of V .

Lemma 3.11. Let G be a simple Lie group and T ⊆ G be a finite symmetric set such that Γ = T is a dense subgroup of G. Suppose that the matrix of Ad(g) with respect to a basis B of the Lie algebra g of G has algebraic entries, for every g ∈ T . Then there exists C 1 > 0 such that the following holds:

If n 1 is an integer, then for any proper non-discrete closed subgroup H of G, there are non-zero

vectors v ∈ g ⊗ R C and l ∈ g * ⊗ R C such that l(Ad(γ)(v)) = 0, for any γ ∈ W n (T ) ∩ H (e -C 1 n ) . Proof. Since Γ is a dense subgroup of G, the R-span A of Ad(Γ) in End R (g) is equal to the R-span of Ad(G). Denote by d the dimension of G. It is easy to see that the R-span of W d 2 (T ) is equal to A.
Hence by Lemma 3.10, there exists c 0 > 0 such that for any l ∈ g * and v ∈ g we have (3.1) max

γ∈W d 2 (T ) |l(Ad(γ)(v))| c 0 l 2 v 2 ,
as the adjoint representation is irreducible.

Let H < G be a proper non-discrete closed subgroup and fix n ≥ 1. Let v ∈ g, l ∈ g * such that

(1) v 2 = 1 and l 2 = 1.

(2) v ∈ h and h ⊆ ker l where h := Lie(H) is the Lie algebra of H.

By using (3.1) and rescaling v, we find γ 0 ∈ W d 2 (T ), v H ∈ g, and l H ∈ g * such that

(1)

l H 2 = 1, v H 2 1/c 0 . (2) l H (Ad(γ 0 )(v H )) = 1. (3) for any h ∈ H, l H (Ad(h)(v H )) = 0.
By the hypothesis, g has a basis

B := {v 1 , . . . , v d } such that v * i (Ad(γ)(v j )) ∈ Q, for
* i (Ad(γ)(v j )) ∈ O k (S), for any γ ∈ Γ. For any g ∈ G, let η g (x, y) ∈ R[x 1 , . . . , x d , y 1 , . . . , y d ] be the polynomial η g ([l] B * , [v] B ) := l(Ad(g)(v)), where [l] B * (resp. [v] B
) is the vector of coordinates of l in the basis B * (resp. B). It is clear that η g is a degree 2 polynomial in 2d variables. Fix a constant C 1 > 0 large enough, depending only on T (we will be more specific later). Now, suppose that the following system of polynomial equations do not have a common solution over C

η γ (x, y) = 0 for any γ ∈ W ≤n (T ) ∩ H (e -C 1 n ) , η γ 0 (x, y) -1 = 0.
We notice that the coefficients of η γ are in O k (S). We view O k (S) as a discrete subring of p∈V k (∞)∪S k p , where V k (∞) is the set of Archimedean places of k. It is clear that the Snorm (the maximum norm in p∈V k (∞)∪S k p ) of the coefficients of η γ for γ ∈ W n (T ) is at most e O T (n) . Then by the effective Nullstellensatz [MW83, Theorem IV] there are polynomials q γ (x, y), q γ 0 (x, y) ∈ O k [x 1 , . . . , x d , y 1 , . . . , y d ] and a ∈ O k such that (1)

γ∈W n (T )∩H (e -C 1 n ) q γ (x, y)η γ (x, y) + q γ 0 (x, y)η γ 0 (x, y) = a.

(2) deg q γ , deg q γ 0 d,deg k 1.

(3) The S-norms of the coefficients of q γ and q γ 0 are at most e O T (n) . (4) The S-norm of a is at most e O T (n) , and it is non-zero.

Since a ∈ O k is non-zero, we have 1

|N k/Q (a)| = p∈V k (∞) |a| p (min p∈V k (∞) |a| p ) a deg k-1 S . Thus (3.2) min p∈V k (∞) |a| p e -O T (n) . Suppose p 0 ∈ V k (∞)
is the place which gives us the embedding of Ad(Γ) into End R (g).

So by the properties of l H and v H mentioned above we have that

|η γ (l H , v H )| p 0 e -C 1 n/2 , |q γ (l H , v H )| p 0 e O T (n) ,
for any γ ∈ W n (T ) ∩ H (e -C 1 n ) . Hence we have

| γ∈W n (T )∩H (e -C 1 n ) q γ (l H , v H )η γ (l H , v H ) + q γ 0 (l H , v H )η γ 0 (l H , v H )| p 0 e O T (n)-C 1 n/2 e -C 1 n/4
if we chose C 1 large enough. But if we chose C 1 perhaps even larger (but still depending only on T ) this contradicts (3.2).

Proof of Proposition 3.9. Let G be the Zariski-closure of Ad(G) in GL(g). By Lemma 3.11, there exists a constant C 1 > 0 such that for any n and any non-discrete proper closed subgroup H of G there is a variety X (depending on H and n) of G whose dimension is strictly less than dim G such that Ad(W n (T ) ∩ H (e -C 1 n ) ) ⊆ X. Using the generalized Bezout theorem it was proved in [EMO05, Proposition 3.2] that there is N (X) 1 such that W N (X) (A) ⊆ X whenever A generates a Zariski-dense subgroup of G. Moreover, by the proof of [EMO05, Proposition 3.2], N (X) is bounded above by some bound depending on the number of irreducible components of X and the maximal degree of an irreducible component of X. Since X is the intersection of G with a hyperplane, we conclude that N := sup X N (X) < ∞. This number N only depends on T .

Next, we show that there exists C > 0 (depending only on T ) such that for all multiple n of N ,

(3.3) W N W n/N (T )) ∩ H (e -Cn ) ⊆ W n (T ) ∩ H (e -C 1 n ) .
This -coupled with the above paragraph -implies that for all multiple n of N and all proper closed subgroup H of G, the set W n/N (T ) ∩ H (e -Cn ) is contained in a proper algebraic subgroup of G.

For any γ i ∈ W n/N (T ) ∩ H (e -Cn ) , there are h i ∈ H such that Ad(γ i ) -Ad(h i ) 2 e -Cn and Ad(γ i ) 2 e O T (n) . Hence, Ad(h i ) 2 e O T (n) and

Ad(γ 1 • • • γ N ) -Ad(h 1 • • • h N ) 2 = N -1 i=0 (Ad(γ 1 • • • γ N -i h N -i+1 • • • h N ) -Ad(γ 1 • • • γ N -i-1 h N -i • • • h N )) 2 N -1 i=0 ( N -i-1 j=1 Ad(γ j ) 2 )( N j=N -i+1 Ad h j 2 )( Ad γ N -i -Ad h N -i 2 ) e O T (n)-Cn e -C 1 n if C
T 1, which implies (3.3).

3.3. Proof of Theorem 3.1. By [BrG02, Corollary 2.5], Γ contains a finitely generated subgroup which is dense in G. Thus, we may assume that Γ is finitely generated. Let ρ i : Γ → GL(V i ), i ∈ I, be the representations and M 2 be the integer given by Theorem 3.2.

By a result of Kazhdan and Margulis (see [Ra72, Theorem 8.16]), there is a neighborhood U of the identity in G such that for any discrete subgroup Σ < G, Σ ∩ U is contained in a connected nilpotent subgroup of G. Let U 0 ⊂ U be an open set such that U contains the closure of U -1 0 U 0 . Throughout the proof, we fix two constants κ > 1 and η > 0 (depending on G only) such that (a) B R (1) can be covered by at most R κ of the sets {gU 0 } g∈G , whenever R > 2,

(b) B R (1) can be covered by at most R r κ balls in G of radius r 2 , whenever R > 2r > 0, (c) x -1 2 x κ 2 , for every x ∈ G, and (d) (1 + η) (3κ+4)κ < 2M -1 2M -2 1 13 .
Let S ⊂ Γ be a set satisfying Theorem 3.2 such that S = S ∪ S -1 ⊂ B η (1) and |S| = M . For i ∈ I, let

K (i) g ⊂ U (i)
g (g ∈ S) be the subsets of V i provided by Theorem 3.2. The usual ping-pong lemma implies that S freely generates a subgroup of Γ, which we denote by S . Let |g| S be the length of an element g ∈ S with respect to S. We denote by W n (S) the set of elements of length n, and by W n (S) the set of elements of length at most n.

Let

1 be an integer and put ε = (1 + η) -. In part 1 of the proof, we construct a finite set T ⊂ Γ ∩ B ε (1). Our construction is inspired by the proof of [BY11, Lemma 3]. In the rest of the proof (parts 2-4), we provide constants d 1 , d 2 > 0 and show that T satisfies the conclusion of Theorem 3.1, whenever is large enough. This will clearly imply Theorem 3.1.

Y = {w = s 1 s 2 ...s |s 1 = a, s = b, s 2 , ..., s -1 ∈ S, s i+1 = s -1 i , for all 1 i < }. Let Z = {w 3 |w ∈ Y }. Since | S| = 2M , we get that |Z| = |Y | (2M -1) -3 . Since S ⊂ B η (1), it follows that W n (S) ⊂ B (1+η) n (1)
, for all n 1. Since Z ⊂ W 3 (S), we get that Z ⊂ B (1+η) 3 (1).

By using (b), Z can be covered by at most

(1+η) 3 ε (1+η) 3κ κ = (1 + η) (3κ+4)κ balls of radius ε 2(1+η) 3κ .
From this we deduce that there is g 0 ∈ Z such that

(3.4) |B ε (1+η) 3κ (g 0 ) ∩ Z| |Z| (1 + η) (3κ+4)κ (2M -1) -3 (1 + η) (3κ+4)κ We define T = g -1 0 (B ε (1+η) 3κ (g 0 ) ∩ Z) \ {1} and T = T ∪ T -1 . Then |T | (2M -1) -3 (1+η) (3κ+4)κ -1. Since by inequality (d) we have that 2M -1 (1+η) (3κ+4)κ > (2M -2) 1 13 (2M -1)
12 13 , we get that

(3.5) |T | [(2M -2) 1 13 (2M -1) 12 13 ] (2M -1) 4 [(2M -2) 1 13 (2M -1)
12 13 ] -5 , for all 1.

If g ∈ T , then g 0 g -g 0 2 ε (1+η) 3κ . Since g 0 2 (1 + η) 3 , we get g -1 0 2 g 0 κ 2

(1 + η) 3κ .

Altogether, it follows that g -1 2 g -1 0 2 g 0 g -g 0 2 ε, for all g ∈ T . Hence T ⊂ B ε (1). We end part (1) of the proof by recording a useful property of T .

Claim 1. If g ∈ W n (T ), then n |g| S 6n . Thus, T freely generates a free subgroup of Γ.

Proof. It is enough to show that n |g| S 3n , for all g ∈ W n (Z) and n 1. Let g = g εn n ...g ε 1 1 , where n 1 and g 1 , ..., g n ∈ Z, ε 1 , ..., ε n ∈ {±1} are such that g

ε i+1 i+1 g ε i i = 1, for all 1 i n -1. Let w 1 , ..., w n ∈ Y such that g 1 = w 3 1 , ..., g n = w 3 n . Then g = w 2εn n (w εn n w ε n-1 n-1 )w ε n-1 n-1 (w ε n-1 n-1 w ε n-2 n-2 )...(w ε 2 2 w ε 1 1 )w 2ε 1 1 . Since w ε i+1
i+1 w ε i i = 1 and w ε i w ε i is already reduced, after making all the possible cancellations, the middle w ε i i from g ε i i = w ε i i w ε i i w ε i i will not be affected. This implies the conclusion.

Part 2: bounding the number of returns.

We continue by showing that the number of elements g ∈ W n (T ) which fix a given line [v] ∈ P(V i ), for some i ∈ I, is bounded above by |W n (T )| 1-c 0 , for a constant c 0 > 0.

Claim 2. There exist n 0 1, 0 1 and c 0 > 0 such that given i ∈ I and v ∈ V i , we have

|{g ∈ W n (T )|ρ i (g)([v]) = [v]}| |W n (T )| 1-c 0 ,
for all 0 and n n 0 .

Proof of Claim 2. Let i ∈ I and v ∈ V i . For simplicity, we denote ρ = ρ i and

K g = K (i) g , U g = U (i) g , for g ∈ S. Fix n 1 and define A = {g ∈ W n (T )|ρ(g)([v]) = [v]}. Denote N = n
2 . In order to estimate |A|, we partition A into two subsets according to the reduced form of g. Let g ∈ A and g = k p k p-1 ...k 1 be its reduced form with respect to S, where p = |g| S and k 1 , ..., k p ∈ S. By Claim 1 we get that 2N n p 6n 12N + 6. We define

B = {g ∈ A|ρ(k N k N -1 ...k 1 )([v]) ∈ U k N +1 } and C = A \ B.
We proceed by estimating |B| and |C| separately. ε n-1 n-1 ...g ε 1 1 , where g 1 , ..., g n ∈ T , ε 1 , ..., ε n ∈ {±1} and g

ε j j = g ε j+1
j+1 , for all 1 j < n. Let w 0 ∈ Y such that g 0 = w 3 0 and w 1 , ..., w n ∈ Y \ {w 0 } such that g 1 = w -3 0 w 3 1 , ..., g n = w -3 0 w 3 n . Then the reduced form can be written as g = h n w εn n h n-1 ...h 1 w ε 1 1 h 0 , where h i ∈ Γ satisfies |h i | S 5 , and the factor w ε i i corresponds to the middle w ε i i from w 3ε i i . We claim that g εq q is uniquely determined, for any q such that n q 11n 12 + 1. More precisely, we will show by induction that ε q , w q and h q are uniquely determined, for all q with n q 11n 12 + 1.

First, if q = n, we have that either

h n = w -3 0 w n , if ε n = 1, or h n = w -1 n , if ε n = -1. Since |h n | S 4 and p -N n -N n 2 > 4
, it follows that w n and ε n are determined. Specifically, there are two cases: (1) k p ...k p-+1 = w -1 0 or (2) k p ...k p-+1 = w -1 n . In case (1) ε n = 1, w n = k p-3 ...k p-4 +1 , and h n = w -3 0 w n , while in case (2)

ε n = -1, w n = k -1 p-+1 ...k -1 p and h n = w -1 n .
Assume that ε n , ..., ε q+1 , w n , ..., w q+1 , h n , ..., h q+1 are determined, for some q with n q 11n 12 + 1. Since |h n w εn n ...h q+1 w ε q+1 q+1 | S 6(n -q) and p -N n 2 6(n -q + 1) = 6(n -q) + 6 , we deduce that the first 6 letters from the left in the reduced word h q w εq q ...h 1 w ε 1 1 h 0 with respect to S are determined. Note that h q ∈ {w q+1 w -3 0 w q , w q+1 w -1 q }, if ε q+1 = 1, and h q ∈ {w -1 q+1 w q , w -1 q+1 w 3 0 w -1 q }, if ε q+1 = -1. Since ε q+1 , w q+1 are determined and |h q w εq q | S 6 , it follows easily that ε q , w q and h q are determined. This finishes the proof of our assertion.

Therefore, if q = 11n 12 +2, then g εn n , ..., g εq q ∈ T are uniquely determined for every g = g εn n ...

g ε 1 1 ∈ B. Since g ε 1 1 , ..., g ε q-1 q-1 can each take at most 2|T | values, we get that |B| (2|T |) q-1 (2|T |) 11n 12 +1 . Claim 4. |C| [4(2M -2) ] n 6 (2|T |) n+2-n 6 , for all n 1. Proof of Claim 4. Assume that g = k p k p-1 ...k 1 ∈ C. Then the second part of Lemma 3.3 implies that ρ(k j k j-1 ...k 1 )([v]) / ∈ U k j+1
, for all 1 j N . Below we will use this fact as follows. Suppose that k 1 , ..., k j are already determined, for some 1 j N . Since ρ(k j k j-1 ...k 1 )([v]) belongs to at least 2 of the sets {U g } g∈ S and | S| = 2M , we derive that k j+1 ∈ S can take at most 2M -2 values. Now, since g ∈ W n (T ), we can write g = g εn n g

ε n-1 n-1 ...g ε 1 1 , where g 1 , ..., g n ∈ T , ε 1 , ..., ε n ∈ {±1} and g

ε j j = g ε j+1
j+1 , for all 1 j < n. Let w 1 , ..., w n ∈ Y \ {w 0 } such that g 1 = w -3 0 w 3 1 , ..., g n = w -3 0 w 3 n . Let q with 1 q n 12 -1 and assume that g ε 1 1 , ..., g εq q are already determined. In other words, assume that w 1 , ..., w q and ε 1 , ..., ε q are determined. Our goal is to estimate the number of possible values of g ε q+1 q+1 ∈ T . Depending on the values of ε q , ε q+1 ∈ {±1} we are in one of four cases. We assume that ε q = ε q+1 = 1, since the estimates in the other three cases are entirely similar. In this case, we have g = g εn n ...g ε q+2 q+2 (w -3 0 w 2 q+1 )(w q+1 w -1 0 )(w -2 0 w 3 q )g ε q-1 q-1 ...g ε 1 1 . Let j such that we have (w -2 0 w 3 q )g ε q-1 q-1 ...g ε 1 1 = k j k j-1 ...k 1 . Then j and k 1 , ..., k j are determined. Note that j 6q . Write w 0 = r 1 ...r , w q+1 = s 1 ...s , where r 1 , ..., r , s 1 , ..., r ∈ S. Notice that |w q+1 w -1 0 | S is even and 2

|w q+1 w -1 0 | S 2 -2. Let 1 -1 such that |w q+1 w -1 0 | S = 2 . Assume that 1
-1 is determined. Then s +1 = r +1 , ..., s = r , hence s +1 , ..., s are determined. Since w q+1 w -1 0 = s 1 ...s r -1 ...r -1 1 , we get that k j+1 = r -1 1 , ..., k j+ = r -1 and k j+ +1 = s , ..., k j+2 = s 1 . Hence k 1 , ..., k j+ are determined. As j + 2 6 q + 2( -1)

< N we get ρ(k j+ ...k 1 )([v]) / ∈ U k j+ +1 .
The beginning of the proof implies that k j+ +1 and hence s can take at most 2M -2 values. Moreover, if k j+ +1 , ..., k j+ +p are determined, for some 1 p -1, then since ρ(k j+ +p ...k 1 )([v]) / ∈ U k j+ +p+1 , we deduce that k j+ +p+1 and therefore s -p can take at most 2M -2 values. It follows that there are at most (2M -2) possibilities for s 1 , ..., s .

We derive that in the case ε q = ε q+1 = 1, the total number of possible values of w q+1 is at most

-1 =1 (2M -2) (2M -2)
. By adapting the above argument, it follows that the number of possible values of w q+1 is at most (2M -2) in the other three cases as well. Altogether, we get that if g ε 1 1 , ..., g εq q are determined and 1 q n 12 -1, then g ε q+1

q+1 can take at most 4(2M -2) values. Let q = n 12 . Thus, if g ε 1 1 is determined, then g ε 2 2 ...g εq q can take at most [4(2M -2) ] q-1 values.

11n 12 +1 + [4(2M -2) ] n 12 (2|T |) n+2-n 12 [(2M -2) 1 12 (2M -1) 11 12 ] (n+24)( +2)
for all n > 12 and every 1. Equations (3.5) and (3.6) together imply that there exist n 0 1, c 0 > 0 and l 0 1 such that |A| |T | (1-c 0 )n , for all n n 0 and every

0 . Since |W n (T )| = 2|T |(2|T | -1) n-1 > |T | n ,
∈ Γ|ρ i (g)([v]) = [v]}).
Claim 5. There exist n 1 1 and c > 0 such that for every i ∈ I, v ∈ V i and 0 , we have that

µ * n ({g ∈ Γ|ρ i (g)([v]) = [v]}) |T | -cn , for all n n 1 . Proof of Claim 5. Denote ρ = ρ i and A = {g ∈ Γ|ρ(g)([v]) = [v]}.
Let n 10n 0 , where n 0 is as in Part 2. For every k 1, fix g k ∈ W k (T ). Then µ * n ({g}) = µ * n ({g k }), for all g ∈ W k (T ). Since µ * n is supported on words of length at most n in T , we get

µ * n (A) = n k=0 µ * n (A ∩ W k (T )) = n k=0 |A ∩ W k (T )|µ * n ({g k }).
Let us now majorize each of the terms involved. First, by Kesten's theorem [START_REF] Kesten | Symmetric random walks on groups[END_REF] we have that

µ * n ({g}) 2|T | -1 |T | n
, for all g ∈ Γ.

Moreover, we deduce from Part 2 that for n 10n 0 , we have that

|A ∩ W k (T )| |W k (T )| 1-c 0 (2|T | -1) -c
1 k< n 10 (2|T |) k 2|T | -1 |T | n + (2|T | -1) -c 0 n 10 n 10 k n |W k (T )|µ * n ({g k }) 1 k< n 10 (2|T |) k 2|T | -1 |T | n + (2|T | -1) -c 0 n 10 (2|T |) n 10 2|T | -1 |T | n + (2|T | -1) -c 0 n 10 .
The conclusion of Claim 5 is now immediate.

Part 4: end of the proof.

We are now ready to conclude the proof. Let n 1 , c, 0 be given as above and let C be the constant given by Proposition 3.9. Since M 2, we get that (2M -1)

12 13 3 12 13 > e. By using (3.5), and after taking a larger 0 , we may assume that |T | e and that (4n + 1)2 (κ+1)n e n 4 , for any n 1 and 0 .

Claim 6. Let δ > 0 be small enough and n be an integer such that log(1+η)

7C log 1 δ log 1 ε n log(1+η) 6C log 1 δ log 1 ε . Then µ * n (H (δ) ) δ min{c, 1 4 } 7C
, for every proper closed connected subgroup H < G.

Proof of Claim 6. Fix n as in the claim and let H < G be a proper closed connected subgroup. Thanks to Proposition 3.9, we can find a proper closed subgroup H < G such that W 6n (S) ∩ H (e -C6n ) ⊂ H .

Let g ∈ supp(µ * n ) ∩ H (δ) . Then g ∈ W n (T ) and since T ⊂ W 6 (S), we deduce that g ∈ W 6n (S).

Since ε = (1 + η) -, hence = log 1 ε log(1+η) , the assumption on n implies that δ e -6n C . By using the previous paragraph, we derive that g ∈ H . Since µ is supported on T , we also have g ∈ T . Denoting Γ 0 = T ∩ H , we therefore get that

(3.7) µ * n (H (δ) ) µ * n (Γ 0 ).
We continue by treating two separate cases:

Case 1. Γ 0 is non-discrete in G.
In this case, since Γ 0 ⊂ Γ ∩ H , we get that Γ ∩ H is non-discrete. Proposition 3.2 implies the existence of i ∈ I and

[v] ∈ P(V i ) such that ρ i (g)([v]) = [v]
, for all g ∈ Γ ∩ H . Since |T | e , by combining 3.7 with Part 3, we get that for δ > 0 small enough so that n n 1 ,

(3.8) µ * n (H (δ) ) µ * n (Γ ∩ H ) µ * n ({g ∈ Γ|ρ i (g)([v]) = [v]}) |T | -cn e -cn .
Since n

log (1+η) log 1 δ 7C log 1 ε , we get n log 1 δ 7C
. This implies that e -cn δ c 7C , proving the claim.

Case 2. Γ 0 is discrete in G.

In this case, by the definition of U , we have that Γ 1 := Γ 0 ∩U is a nilpotent group. Since Γ 1 < T and T is a free group, Γ 1 must be a cyclic group. As a consequence, we have that

|Γ 0 ∩ U ∩ supp(µ * 2n )| |Γ 1 ∩ supp(µ * 2n )| = |Γ 1 ∩ W 2n (T )| 4n + 1.
Next, if N = (1 + ε) κn , then by (a) we can find g 1 , ..., g N ∈ G such that B (1+ε

) n (1) ⊂ ∪ N i=1 g i U 0 . Since supp(µ * n ) ⊂ B (1+ε) n (1), we thus get that Γ 0 ∩supp(µ * n ) ⊂ ∪ N i=1 (Γ 0 ∩g i U 0 ∩supp(µ * n )). Recall that U -1 0 U 0 ⊂ U . So, if 1 i N and x, y ∈ Γ 0 ∩g i U 0 ∩supp(µ * n ), then x -1 y ∈ Γ 0 ∩U ∩supp(µ * 2n ). This implies that |Γ 0 ∩ g i U 0 ∩ supp(µ * n )| |Γ 0 ∩ U ∩ supp(µ * 2n )| 4n + 1, for every 1 i N . Altogether, we get that |Γ 0 ∩ supp(µ * n )| (4n + 1)N (4n + 1)(1 + ε) κn .
In combination with Kesten's theorem, we derive that

µ * n (Γ 0 ) (4n + 1)(1 + ε) κn 2|T | -1 |T | n .
Since |T | e , we get

√ 2|T |-1 |T | 2 √ |T | 2e -2 . Since (4n + 1)(1 + ε) κn 2 n (4n + 1)2 (κ+1)n e n 4
, by using 3.7 we conclude that (3.9)

µ * n (H (δ) ) µ * n (Γ 0 ) e -n 4 . Since n log(1+η) 7C log 1 δ log 1 ε = log 1 δ 7C
, by combining 3.8 and 3.9 we get that

µ * n (H (δ) ) e -min{c, 1 4 }n e -min{c, 1 4 } 7C log 1 δ = δ min{c, 1 4 } 7C , then either ν * P δ 2 δ -α , or ν * ν * P δ 2 δ γ ν * P δ 2 .
We first claim that there is a constant C > 1 depending only on G such that the following holds. Let ρ ∈ (0, 1), R > 2, a, b, x ∈ B R (1) and h, k ∈ G such that x -1 a -h 2 ρ and x -1 b -k 2 ρ. Then b -1 a -k -1 h 2 R C ρ. Indeed, the claim follows since there is a constant c > 1 depending only on G such that y -1 2 ( y 2 + 1) c , for any y ∈ G, and we have that

b -1 a -k -1 h 2 = (x -1 b) -1 (x -1 a) -k -1 h 2 x -1 2 a 2 (x -1 b) -1 -k -1 2 + k -1 2 x -1 a -h 2 x -1 2 a 2 (x -1 b) -1 2 k -1 2 x -1 b -k 2 + k -1 2 x -1 a -h 2 .
Let k 1 be the smallest integer such that δ kγ P δ 2 δ -α , for any δ > 0 small enough. Let ε > 0 small enough such that

2 k d 2 4 log 1 ε < min{β, γ d 1 C } ε
. Let µ be a Borel probability measure on G which is supported on B ε (1) and satisfies the hypothesis. The proof relies on the following:

Claim. If δ > 0 is small enough and n is an integer such that d 2 log 1 δ 4 log 1 ε n min{β, γ d 1 C } log 1 δ ε , then the measure ν = µ * n satisfies conditions (a) and (b). Proof of the claim. Since supp(ν) ⊂ B (1+ε) n (1) and (1 + ε) n [(1 + ε) 1 ε ] β log 1
δ < e β log 1 δ = δ -β , we get that ν satisfies (a). To verify (b), let ρ δ and H < G be a proper closed connected subgroup.

We may assume that ρ δ

4γ d 1 , because otherwise δ -γ ρ d 1 4 > 1 and (b) is trivially satisfied. Let m = d 2 log 1 ρ 1 2 log 1 ε = d 2 log 1 ρ 2 log 1 ε
. Then m 2n and the hypothesis implies that

µ * 2m (H ρ ( 1 2 ) ) ρ d 1 2 . For x ∈ G, denote A x = xH (ρ) ∩ supp(µ * m ). Since µ * 2n = µ * (2n-m) * µ * m , we have (4.1) ν * ν(H (ρ) ) = µ * 2n (H (ρ) ) sup x∈supp(µ * (2n-m) ) µ * m (xH (ρ) ) = sup x∈supp(µ * (2n-m) ) µ * m (A x ).
Further, since µ * m is symmetric, Lemma 2.1 implies that

(4.2) µ * m (A x ) µ * 2m (A -1 x A x ) 1 2 , for any x ∈ G. Let x ∈ supp(µ * (2n-m) ) and a, b ∈ A x . Since supp(µ * k ) ⊂ B (1+ε) k (1), for any k 1, we have that a, b, x ∈ B (1+ε) 2n (1)
. By the definition of A x , we can find h, k ∈ H such that x -1 a -h 2 ρ and

x -1 b -k 2 ρ. The earlier claim implies that b -1 a -k -1 h 2 (1 + ε) 2Cn ρ. Since n < γ d 1 C log 1 δ ε , we get that (1 + ε) 2Cn < e 2Cnε < δ -2γ d 1 ρ -1 2 . Thus b -1 a -k -1 h 2 ρ 1 2 . Since k -1 h ∈ H and a, b ∈ A x are arbitrary, we deduce that A -1 x A x ⊂ H (ρ 1
2 ) . By combining (4.1) and (4.2) we therefore derive that

ν * ν(H (ρ) ) µ * 2m (H (ρ 1 2 ) ) 1 2 ρ d 1 4 ,
which finishes the proof of the claim.

Let δ > 0 and put n 0 = d 2 log 1 δ 4 log 1 ε and n 1 = 2 k n 0 . We claim that µ * n 1 * P δ 2 δ -α , for any small enough δ > 0. Once this claim is proven, the conclusion follows for c 0 = 2 k-2 d 2 since

n 1 2 k-2 d 2 log 1 δ log 1 ε and µ * n * P δ 2 = µ * (n-n 1 ) * (µ * n 1 * P δ ) 2 µ * n 1 * P δ 2 , for any n n 1 .
Assume by contradiction that the claim is false and let 0 i k. Then 2 i n 0 n 1 and therefore

µ * 2 i n 0 * P δ 2 µ * n 1 * P δ 2 > δ -α . On the other hand, d 2 log 1 δ 4 log 1 ε 2 i n 0 min{β, γ d 1 C } log 1 δ ε .
The claim implies that µ 2 i n 0 satisfies conditions (a) and (b). As µ * 2 i n 0 * P δ 2 > δ -α we must have µ * 2 i+1 n 0 * P δ 2 δ γ µ * 2 i n 0 * P δ 2 , for every 0 i k.

By combining these inequalities we deduce that µ * n 1 * P δ 2 δ kγ µ * n 0 * P δ 2 δ kγ P δ 2 δ -α , which is a contradiction.

A mixing inequality

The goal of this section is to prove an analogue for simple Lie groups of the well-known mixing inequality for quasirandom finite groups (see [START_REF] Tao | Expansion in finite simple groups of Lie type[END_REF]Proposition 1.3.7]). In the next section, we will combine this mixing inequality with Corollary 4.2 and a Littlewood-Paley decomposition on simple Lie groups to deduce Theorem B.

Theorem 5.1 (mixing inequality). Let G be a connected simple Lie group with trivial center. Denote by d the dimension of G, and let B ⊂ G be a measurable set with compact closure.

Then there exist constants a, b, κ > 0 such that for every F ∈ L 2 (B) with F 2 = 1, we have

f * F 16d 2 a P δ * F 2 + bδ κ ,
for all f ∈ L 2 (G) with f 2 = 1 and all 0 < δ < 1. This result and its proof are inspired by [BG10, Lemma 10.35], which dealt with the case G = SU (d), for d 2. In particular, we borrow from [START_REF] Bourgain | Gamburd: A spectral gap theorem in SU (d)[END_REF] the idea of reducing to functions F that satisfy an additional "symmetry", i.e. are eigenvectors for a maximal torus of G. This reduction is crucial, as it will allow us to exploit certain cancellations appearing in the integrals.

Turning to the proof of Theorem 5.1, we start with a classical lemma which can be easily deduced from [RS88, Section 2]. We denote by C 1 c (G) the space of compactly supported C 1 -functions on G. Lemma 5.2. Let G and H be two Lie groups of dimensions n and m. Assume that n m.

Consider an analytic function

φ : G → H such that the derivative dφ x : g → h has rank m, at almost every point x ∈ G. Let ψ ∈ C 1 c (G) and denote by µ = φ * (ψ • dm G ) the push-forward measure of the measure ψ • dm G on G through φ.
Then µ is absolutely continuous with respect to m H , and the Radon-Nykodym derivative ρ : H → R is L 1 -Hölder: there exist α > 0 and C > 0 such that

H |ρ(g -1 h) -ρ(h)|dh C g -1 α
2 , for every g ∈ H.

By applying this lemma, we obtain the following:

Lemma 5.3. Let G be a connected simple Lie group and H < G be a connected compact Lie subgroup of dimension 1.

Define π : G × H 2 → G by letting π(g, t 1 , t 2 ) = t 1 gt -1 1 t 2 g -1 t -1 2 , for all g ∈ G, t 1 , t 2 ∈ H. Let ψ ∈ C 1 c (G) and define ν = π * ((ψ • dm G ) × m H × m H ).
Then ν * n is absolutely continuous with respect to m G , and the corresponding Radon-Nykodym derivative is L 1 -Hölder, for every integer n dim(G).

Proof. Let n 1. Then ν * n = π (n) * ((ψ • dm G ) n × m 2n H ), where π (n) : G n × H 2n → G is defined as π (n) (g 1 , ..., g n , t 1 , ..., t 2n ) = n i=1 π(g i , t 2i-1 , t 2i ) = n i=1 (t 2i-1 g i t -1 2i-1 t 2i g -1 i t -1 2i ).
By Lemma 5.2, we only have to check that the derivative of the analytic function π (n) has rank d, at almost every point, as soon as n d := dim(G).

Fix n d. Let g and h be the Lie algebras of G and H, respectively. Let Ad: G → GL(g) be the adjoint representation of G. Since dim(H) = 1 and H is connected, there is b ∈ g such that

h = {ub | u ∈ R} and H = {exp(ub) | u ∈ R}.
Let X n be the set of (g 1 , ..., g n , t 1 , ..., t 2n ) ∈ G n × H 2n such that the following set spans g:

{Ad( i-1 j=1 π(g j , t 2j-1 , t 2j ))(b) -Ad(( i-1 j=1 π(g j , t 2j-1 , t 2j ))t 2i-1 g i t -1 2i-1 )(b) | 1 i n}. Claim 1. rk(d(π (n) ) x ) = d, for every x ∈ X n .
Proof of Claim 1. Take x = (g 1 , ..., g n , t 1 , ..., t 2n ) ∈ X n . Proving the claim amounts to showing that the map πn :

y → π (n) (y)π (n) (x) -1 is such that d(π n ) x has rank d.
For all 1 i n, define a map ϕ i : R → G by the formula

ϕ i (u) = πn (g 1 , ..., g n , t 1 , ..., t 2i-2 , exp(ub)t 2i-1 , t 2i • • • , t 2n )
The derivative ϕ i (0) ∈ g belongs to the range of the derivative d(π n ) x , while an easy computation gives that

ϕ i (0) = Ad( i-1 j=1 π(g j , t 2j-1 , t 2j ))(b) -Ad(( i-1 j=1 π(g j , t 2j-1 , t 2j ))t 2i-1 g i t -1 2i-1 )(b).
Since x ∈ X n , the set {ϕ i (0) | 1 i n} spans g, and d(π n ) x is therefore onto.

Claim 2. X n is a nonempty Zariski open subset of G n × H 2n , for every n d.

Proof of Claim 2. Since X n is clearly Zariski open, for every n 1, it remains to argue that X n is nonempty, whenever n d. Since G is simple, g is the only non-trivial Ad(G)-invariant subspace of g. Thus, the span of {Ad(g)(b) -Ad(h)(b)|g, h ∈ G} is equal to g. Equivalently, we derive that the span of {Ad(g)(b) -b|g ∈ G} is also equal to g. We can therefore find g 1 , ..., g d ∈ G such that {Ad(g i )(b) -b|1 i d} spans g. Define g d+1 = .... = g n = t 1 = ... = t 2n = 1. Then it is clear that (g 1 , ..., g n , t 1 , ..., t 2n ) ∈ X n , which shows that X n is nonempty, as claimed.

Finally, if n d, then Claim 2 implies that X n a co-null subset of G n × H 2n .

We are now ready to prove Theorem 5.1.

Proof of Theorem 5.1.

Let F ∈ L 2 (B) and f ∈ L 2 (G) with f 2 = 1. Since F * F is supported on BB -1 , we have f * F 2 2 = f * F, f * F = f * f, F * F f * f 2,BB -1 F * F 2 . Since f * f ∞ 1, we get that f * f 2,BB -1 |BB -1 | 1/2
. Moreover, for every g ∈ G, we have that F * F (g) = G F (g -1 x)F (x) dx = λ g (F ), F . By putting these facts together, we get that

f * F 16d 2 |BB -1 | 4d ( G | λ g (F ), F | 2 dg) 4d .
Thus, the conclusion reduces to proving the following: ( * ) there exist constants a, b, κ > 0 such that for every F ∈ L 2 (B) with F 2 = 1, we have that

G | λ g (F ), F | 2 dg 4d a P δ * F 2 + bδ κ , for all 0 < δ < 1.
To this end, we fix a compact connected Lie subgroup H of G with dimension 1. Below, we denote by x, y, z, g elements of G and by t, t 1 , t 2 elements of H. Writing dx (respectively, dt) will refer to integration against the Haar measure of G (respectively, H).

Let B ⊂ G be an open set with compact closure which contains B -1 B. Let ψ ∈ C 1 c (G) be a non- negative function which is equal to 1 on B. Define π : G×H 2 → G by π(x, t 1 , t 2 ) = t 1 xt -1 1 t 2 x -1 t -1 2 , for all x ∈ G and t 1 , t 2 ∈ H. Let ν = π * ((ψ • dm G ) × m 2
H ). Lemma 5.3 implies that ν * d is absolutely continuous with respect to m G and the corresponding Radon-Nykodym derivative ρ is L 1 -Hölder. In other words, there exist κ > 0 and C > 0 such that

G |ρ(g -1 h) -ρ(h)|dh C g -1 2κ 2 , ∀g ∈ G.
For x ∈ G, we define an operator

R x : L 2 (G) → L 2 (G) by the formula (R x f )(z) = H f (ztxt -1 ) dt, f ∈ L 2 (G), z ∈ G. Claim 1. For every f ∈ L 2 (G), we have B R x (f ) 2 2 dx f * ν 2 f 2 . Proof of Claim 1. Let f ∈ L 2 (G). Since (R * x R x f )(z) = H 2 f (zt 1 x -1 t -1 1 t 2 xt -1 2 ) dt 1 dt 2 , the claim follows from the following calculation B R x (f ) 2 2 dx G R x (f ) 2 2 ψ(x) dx = G R * x R x (f ), f ψ(x) dx = G G×H 2 f (zt 1 x -1 t -1 1 t 2 xt -1 2 ) ψ(x) dx dt 1 dt 2 f (z) dz = G (f * ν)(z)f (z) dz f * ν 2 f 2 .
Next, using that ρ is L 1 -Hölder, we deduce the following claim:

Claim 2. There is c > 0 such that

P δ * f * ρ -f * ρ 2 cδ κ f 2 , for all f ∈ L 2 (B) and 0 < δ < 1.
Proof of Claim 2. Take f ∈ L 2 (B) and δ > 0. Note that for x ∈ G, we have

(P δ * f * ρ -f * ρ)(x) = 1 |B δ | B δ (1)×B f (z)(ρ(z -1 y -1 x) -ρ(z -1 x)) dy dz.
Using the Cauchy-Schwarz inequality and L 1 -Hölder condition for ρ, we get that

P δ * f * ρ -f * ρ 2 2 is at most equal to 1 |B δ | 2 G B δ (1)×B |f (z)| 2 |ρ(z -1 y -1 x) -ρ(z -1 x)| dy dz B δ (1)×B |ρ(z -1 y -1 x) -ρ(z -1 x)| dy dz dx 2 ρ 1 |B δ | G×B δ (1)×B |f (z)| 2 |ρ(z -1 y -1 x) -ρ(z -1 x)| dx dy dz 2 ρ 1 |B δ | B |f (z)| 2 G×B δ (1) |ρ(z -1 y -1 zx) -ρ(x)| dx dy dz 2C ρ 1 B |f (z)| 2 sup y∈B δ (1) z -1 yz -1 2κ 2 dz c 2 δ 2κ f 2 2 ,
for some constant c > 0 independent of f and δ.

Let F ∈ L 2 (B) with F 2 = 1. The proof of ( * ) splits into two cases.

Case 1. We first prove assertion ( * ) in the following case: there is a character η : H → T such that for all t ∈ H, F (xt) = η(t)F (x), for almost every x ∈ G.

Then for almost every (x, y, t) ∈ G 2 × H we have that F (xt)F (yt) = F (x)F (y). By using this fact we get that

G | λ g (F ), F | 2 dg = G 3 F (g -1 x)F (x) F (g -1 y)F (y) dg dx dy = G 3 H F (g -1 xt)F (x) F (g -1 yt)F (y) dg dx dy dt
Using left invariance of the Haar measure and unimodularity on the g and y variables, we get

G | λ g (F ), F | 2 dg = G 3 H F (gt -1 y -1 xt)F (x) F (g)F (y) dg dx dy dt = G 3 H F (gt -1 y -1 t)F (x) F (g)F (xy) dg dx dy dt = G ( F * F )(y) R y -1 F, F dy = G ( F * F )(y) F, R y F dy Since F * F ∞ 1 and F * F ∈ L 2 ( B), we conclude from Claim 1 and Lemma 2.1 that G | λ g (F ), F | 2 dg B R x (F ) 2 dx | B| 1/2 B R x (F ) 2 2 dx 1/2 | B| 1 2 F * ν 1 2 2 | B| 1 2 ν 1 4 F * ν * d 1 4d 2 = | B| 1 2 ν 1 4 F * ρ 1 4d
2 . On the other hand, Claim 2 yields

F * ρ 2 P δ * F * ρ 2 + F * ρ -P δ * F * ρ 2 ρ 1 P δ * F 2 + cδ κ .
Thus, if we let a = | B| 2d ν d ρ 1 and b = c| B| 2d ν d , the desired inequality ( * ) follows in Case 1. Moreover, notice the crucial fact that a and b are independent of the character η.

Case 2. We now prove ( * ) for an arbitrary function F ∈ L 2 (B) with F 2 = 1.

Consider the unitary representation H σ L 2 (G) corresponding to the right multiplication action H G. Since H is compact and abelian, we can decompose

L 2 (G) = η∈Char(H) H η ,
where H η denotes the eigenspace of σ corresponding to a character η : H → T.

Thus, we can decompose F = η F η , where F η (xt) = η(t)F (x), for almost every x ∈ G, t ∈ H. Note that the functions F η do not necessarily belong to L 2 (B). However, F η belongs to the closure of the linear span of σ(H)F , and therefore to L 2 (BH), for every η.

By applying Case 1 with BH instead of B, and using homogeneity, we get that there exist constants a, b, κ > 0 (independent of F ) such that for all η ∈ Char(H) we have

( G | λ g (F η ), F η | 2 dg) 4d a P δ * F η 2 F η 16d-1 2 + bδ κ F η 16d 2 .
Since all the norms on R 2 are equivalent, we find a , b > 0 (only depending on a, b, d) such that for all η ∈ Char(H) we have that

( G | λ g (F η ), F η | 2 dg) 1/2 a P δ * F η 1/8d 2 F η 2-1/8d 2 + b δ κ/8d F η 2 2 .
But since λ g (F η ) ∈ H η and P δ * F η ∈ H η for all η, by using the triangle inequality for . 2 and Hölder's inequality we get that

G | λ g (F ), F | 2 dg 1/2 = G η λ g (F η ), F η 2 dg 1/2 η G | λ g (F η ), F η | 2 dg 1/2 η a P δ * F η 1/8d 2 F η 2-1/8d 2 + b δ κ/8d a η P δ * F η 2 2 1/16d η F η 2 2 1-1/16d + b δ κ/8d = a P δ * F 1/8d 2 + b δ κ/8d .
Using again the equivalence of norms in R 2 and modifying the values of a and b if necessary, the conclusion follows. We define bounded linear operators

∆ i : L 2 (G) → L 2 (G), i 0, as follows ∆ 0 (F ) = P 1/2 * F ∆ i (F ) = P 2 -(i+1) * F -P 2 -i * F, for all i 1.
Remark 6.1. The decomposition F = i 0 ∆ i (F ) is analogous to the classical Littlewood-Paley decomposition on R n , in the following sense. For any i 0, the function ∆ i (F ) "lives" at scale 2 -i : it is essentially constant at scales 2 -i and essentially has mean zero on balls of radius 2 -i .

We now prove that the operators ∆ i , i 0, yield an almost orthogonal decomposition of L 2 (G). This will allow us to reduce to functions living at an arbitrary small scale in the proof of restricted spectral gap Theorem B. Theorem 6.2. There exists a constant C > 0 such that for all F ∈ L 2 (G) and any µ ∈ M(G) with supp(µ) ⊂ B 1 (1), we have that

(1) i 0 ∆ i (F ) 2 2 C F 2 2 . (2) µ * F 2 2 C i 0 µ * ∆ i (F ) 2 2 . (3) i 0 2 i/2 P 2 -2i * ∆ i (F ) -∆ i (F ) 2 2 C i 0 ∆ i (F ) 2 2 . (4) i 0 2 i/2 P 2 -i/2 * ∆ i (F ) 2 2 C i 0 ∆ i (F ) 2 2 .
The first ingredient of the proof of Theorem 6.2 is the following lemma. This lemma and its proof are a variation of [KS71, Lemma 11] due to Knapp and Stein.

Lemma 6.3 (Cotlar-Stein). Consider a Hilbert space H and bounded operators T i : H → H, i 0.

Assume that there exists ϕ : Z → R + with Φ := n∈Z ϕ(n) < ∞ such that for all i, j 0, we have

T * j T i 1/2 ϕ(j -i) and T i T * j 1/2 ϕ(i -j). For k 0, denote Φ k := |n| k ϕ(n).
Then for all ξ ∈ H and all k 0 we have

i,j: |i-j| k | T i ξ, T j ξ | Φ k Φ ξ 2 .
Proof. Fix ξ ∈ H and k 0. For every i, j 0, we choose a scalar α i,j in such a way that | T i ξ, T j ξ | = α i,j T i ξ, T j ξ , and that α i,j = 0 whenever | T i ξ, T j ξ | = 0. Then for all N 0, the operator R N := 0 i,j N : |i-j| k α i,j T * j T i is self-adjoint. In order to prove the lemma, it is sufficient to check that the operator norm of R N is at most Φ k Φ, for all N 0. Take N 0. Since R N is self-adjoint, R N p = R p N , for all integers p 1. This leads to the estimate:

R N p 0 i 1 ,j 1 ,...,ip,jp N : |i l -j l | k, ∀1 l p T * j 1 T i 1 T * j 2 T i 2 • • • T * jp T ip .
Since the general term of this sum is bounded by the following two quantities

T * j 1 T i 1 T * j 2 T i 2 • • • T * jp T ip T * j 1 T i 1 T * j 2 T i 2 • • • T * jp T ip and T * j 1 T i 1 T * j 2 T i 2 • • • T * jp T ip T * j 1 T i 1 T * j 2 • • • T i p-1 T * jp T ip , we get that R N p 0 i 1 ,j 1 ,...,ip,jp N : |i l -j l | k, ∀1 l p ( T * j 1 T * j 1 T i 1 T i 1 T * j 2 • • • T * jp T ip T ip ) 1/2
N ( max

0 i N T i )( max 1 j N 1 i N : |i-j| k T * j T i 1/2 ) p ( max 0 i N 1 j N T i T * j 1/2 ) p-1 N ( max 0 i N T i )Φ p k Φ p-1 .
Since p 1 is arbitrary, we indeed get that R N Φ k Φ.

Remark 6.4. The case k = 0 of Lemma 6.3 recovers the classical Cotlar-Stein lemma (see [START_REF] Stein | Harmonic Analysis: Real-variable Methods, Orthogonality and Oscillatory Integrals[END_REF] Chapter VII]) which asserts that, under the same assumptions as above, the sum i 0 T i converges in the strong operator topology. Lemma 6.3 also implies that the sum i 0 T i ξ 2 is finite, for all ξ ∈ H. Later on, we will use the following inequalities, which follow easily from Lemma 6.3 (6.1)

i 0 T i ξ 2 Φ 2 ξ 2 , (6.2) i 0 T i ξ 2 2 k i 0 T i ξ 2 + Φ k Φ ξ 2 , for all k 0.
In order to prove Theorem 6.2 we will also need the following lemma which allows us to quantify the "orthogonality" between the operators ∆ i , i 0. For a Borel probability measure µ ∈ M(G) we denote by T µ : L 2 (G) → L 2 (G) the contractive operator given by T µ (F ) = µ * F . Lemma 6.5. There exists a constant C 0 > 0 such that for any Borel probability measure µ ∈ M(G) with supp(µ) ⊂ B 1 (1) we have that

(P δ 1 -P δ 2 ) * µ * P δ 3 1 C 0 δ 2 δ 3
, for all 0 < δ 1 δ 2 δ 3 < 1, and

∆ * j T * µ T µ ∆ i C 0 2 |i-j| and T µ ∆ i ∆ * j T * µ C 0 2 |i-j| , for all i, j 0.
Proof. Denote by B 1 , B 2 and B 3 the balls centered at 1 with respective radii δ 1 , δ 2 and δ 3 . Note that (P δ 1 -P δ 2 ) * µ * P δ 3 1 G (P δ 1 -P δ 2 ) * δ x * P δ 3 1 dµ(x). So it suffices to prove that lemma for Dirac measures µ = δ x , with x ∈ B 1 (1). Fix x ∈ B 1 (1). Then for all y ∈ G, we have

|(P δ 1 -P δ 2 ) * δ x * P δ 3 (y)| P δ 1 -P δ 2 1 /|B 3 | 2/|B 3 |.
Let us now bound the measure of the support of (P δ 1 -P δ 2 ) * δ x * P δ 3 . One easily checks that this support is contained in

B 2 xB 3 ∩ B 2 x(G \ B 3 ).
Firstly, if y ∈ B 2 xB 3 , we write y = axb, where a ∈ B 2 , b ∈ B 3 . Then we have that y 2 8 and

x -1 y -1 2 x -1 y -b 2 + δ 3 x -1 2 y -xb 2 + δ 3 x -1 2 xb 2 δ 2 + δ 3 C 1 δ 2 + δ 3 , where C 1 > 0 is independent of x ∈ B 1 (1). Secondly, if y ∈ B 2 x(G \ B 3 ), we write y = a xb , where a ∈ B 2 and b / ∈ B 3 . Then we see that xb 2 a -1 2 y 2 8 a -1 2 and x -1 y -1 2 b -1 2 -x -1 y -b 2 δ 3 -x -1 2 y -xb 2 δ 3 -x -1 2 xb 2 δ 2 δ 3 -C 2 δ 2 , where C 2 > 0 is independent of x ∈ B 1 (1).
Therefore, the support of (P δ 1 -

P δ 2 ) * δ x * P δ 3 is contained in x(B δ 3 +C 1 δ 2 \ B δ 3 -C 2 δ 2 ). Altogether, we get that (P δ 1 -P δ 2 ) * δ x * P δ 3 1 2 |B δ 3 +C 1 δ 2 \ B δ 3 -C 2 δ 2 | |B δ 3 | ,
which implies the first inequality. By using the fact that x -1 -1 2 x -1 2 x -1 2 and arguing similarly to the above, it follows that the quantities ( Pδ 1 -Pδ 2 ) * µ * P δ 3 1 and (P δ 1 -P δ 2 ) * µ * Pδ 3 1 are bounded above by C 0 δ 2 /δ 3 , for some possibly larger constant C 0 > 0. Since f * g 2 f 1 g 2 , for all f ∈ L 1 (G), g ∈ L 2 (G), these estimates imply the rest of the asserted inequalities.

Proof of Theorem 6.2. Let C 0 > 0 be the constant provided by Lemma 6.5 and define ϕ :

Z → R + by letting ϕ(n) = C 1/2 0
2 |n|/2 . Then Lemma 6.5 gives that for any finitely supported probability measure on G with supp(µ) ⊂ B 1 (1), the operators

T i := T µ ∆ i on L 2 (G) satisfy T * j T i 1/2 ϕ(j -i) and T i T * j 1/2
ϕ(i -j), for all i, j 0.

Let Φ and Φ k be as defined in Lemma 6.3 and take k large enough so that Φ k Φ < 1. Let F ∈ L 2 (G). Since lim δ→0 P δ * F -F 2 = 0, we get that F = i 0 ∆ i (F ). By combining this fact with equations (6.1) and (6.2), we derive that

(6.3) 1 Φ 2 i 0 ∆ i (F ) 2 2 F 2 2 k 1 -Φ k Φ i 0 ∆ i (F ) 2 2
Similarly, for all µ ∈ Prob(G) with supp(µ) ⊂ B 1 (1), we have

µ * F 2 2 k 1 -Φ k Φ i 0 µ * ∆ i (F ) 2 2 ,
Further, Lemma 6.5 implies that for all i 0, we have

P 2 -2i * ∆ i (F ) -∆ i (F ) 2 4C 0 2 i F 2 and P 2 -i/2 * ∆ i (F ) 2 C 0 2 i/2 F 2 . Therefore, i 0 2 i/2 P 2 -2i * ∆ i (F ) -∆ i (F ) 2 2 2(4C 0 ) 2 F 2 2 and i 0 2 i/2 P 2 -i/2 * ∆ i (F ) 2 2 √ 2 √ 2 -1 C 2 0 F 2 2 .
It is now clear that the conclusion of Theorem 6.2 holds for C > 0 large enough (but still independent of µ and F ).

6.2. Reduction to functions living at a small scale. We continue with a consequence of Theorem 6.2 that will allow us to reduce the problem of proving restricted spectral gap to functions that live at an arbitrarily small scale δ > 0.

Corollary 6.6. Let C > 0 be the constant provided by Theorem 6.2. Let 0 < r < 1. Let B ∈ C(G) and µ ∈ M(G) be a Borel probability measure with supp(µ) ⊂ B 1 (1). Assume that for any finite dimensional subspace

V ⊂ L 2 (B), there is F ∈ L 2 (B) V such that µ * F 2 > r F 2 . Let B ⊂ G
be an open set with compact closure which contains the closure of B.

Then for every δ 0 > 0, there exist F ∈ L 2 ( B) and 0 < δ < δ 0 such that

(1) µ * F 2 > r F 2 /(2C). (2) P δ * F -F 2 < δ 1/16 F 2 . ( 3
) P δ 1/4 * F 2 < δ 1/16 F 2 .
Proof. Let δ 0 > 0. Choose N 1 such that 2 -N < δ 0 /2 and 2 -N/4 < r 2 /(16C 3 ). Since B has compact closure, the operator L 2 (B)

F → P δ * F ∈ L 2 (G) is compact, for any δ > 0. Hence, the operator L 2 (B) F → ∆ i (F ) ∈ L 2 (G) is compact, for all i 0. The hypothesis implies that we can find F 0 ∈ L 2 (B) such that µ * F 0 2 > r F 0 2 and N -1 i=0 ∆ i * F 0 2 2 < r 2 F 0 2 2 /(2C).
By using Theorem 6.2 (2), we derive that i 0 µ *

∆ i (F 0 ) 2 2 µ * F 0 2 2 /C > r 2 F 0 2 2 /C. Since N -1 i=0 µ * ∆ i (F 0 ) 2 < r 2 F 0 2 2 /(2C), we get that i N µ * ∆ i (F 0 ) 2 2 > r 2 F 0 2 2 /(2C).
In combination with Theorem 6.2 (1) we deduce that

i N µ * ∆ i (F 0 ) 2 2 > r 2 ( i N ∆ i (F 0 ) 2 2 )/(2C 2 ), or equivalently (6.4) i N ( ∆ i (F 0 ) 2 2 -µ * ∆ i (F 0 ) 2 2 ) 1 - r 2 2C 2 i N ∆ i (F 0 ) 2 2 . Since i 0 ∆ i (F 0 ) 2 2 F 0 2 2 /C by Theorem 6.2 (2) and N -1 i=0 ∆ i * F 0 2 2 < F 0 2 2 /(2C), we get that i 0 ∆ i (F 0 ) 2 2 < 2( i N ∆ i (F 0 ) 2 2 )
. By combining this inequality with Theorem 6.2 (3) and using that 2 -N/4 < r 2 /(16C 3 ) we deduce that i N

2 i/4 P 2 -2i * ∆ i (F 0 ) -∆ i (F 0 ) 2 2 2 -N/4 i 0 2 i/2 P 2 -2i * ∆ i (F 0 ) -∆ i (F 0 ) 2 2 (6.5) 2 -N/4 C i 0 ∆ i (F 0 ) 2 2 < 2 -N/4+1 C i N ∆ i (F 0 ) 2 2 r 2 8C 2 i N ∆ i (F 0 ) 2 2 .
Similarly, by using Theorem 6.2 (4), we get that (6.6)

i N 2 i/4 P 2-i/2 * ∆ i (F 0 ) 2 2 < r 2 8C 2 i N ∆ i (F 0 ) 2 2 .
By combining equations (6.4), (6.5), and (6.6) we can find i N such that

( ∆ i (F 0 ) 2 2 -µ * ∆ i (F 0 ) 2 2 )+2 i/4 P 2 -2i * ∆ i (F 0 )-∆ i (F 0 ) 2 2 +2 i/4 P 2-i/2 * ∆ i (F 0 ) 2 2 < (1- r 2 4C 2 ) ∆ i (F 0 ) 2 2 .
Let F := ∆ i (F 0 ) and δ := 2 -2i . Then δ 2 -2N < δ 0 . Moreover, the above inequality implies that µ * F 2 2 > r 2 F 2 2 /(4C 2 ), P δ * F -F 2 2 < δ 1/8 F 2 2 , and P δ 1/4 * F 2 < δ 1/8 F 2 2 . Finally, notice that since F 0 ∈ L 2 (B), the support of F is contained in B 2 -i+1 (1)B ⊂ B δ 0 (1)B and hence in B, if δ 0 > 0 is small enough. 6.3. Proof of Theorem B. Next, we prove the following "quantitative restricted spectral gap" theorem for all measures with small support that escape subgroups at a controlled speed. It is clear that this result in combination with Theorem 3.1 immediately implies Theorem B. Theorem 6.7. Let G be a connected simple Lie group with trivial center and B ⊂ G a measurable set with compact closure. Let d 1 , d 2 > 0 be given.

Then there exist c > 0 and ε 2 > 0 such that the following holds true. Let 0 < ε < ε 2 and µ ∈ M(G) be a Borel probability on G with supp(µ) ⊂ B ε (1). Assume that for all δ > 0 small enough, we have that for any proper connected closed subgroup H < G,

µ * 2n (H (δ) ) δ d 1 , where n = d 2 log 1 δ log 1 ε .
Then there exists a finite dimensional subspace V ⊂ L 2 (B) such that µ * F 2 < ε c F 2 , for every We start by quantifying how small ε > 0 should be. First, Theorem 5.1 provides constants a, b, κ > 0 such that for any F ∈ L 2 ( B) with F 2 = 1 we have (6.7) f * F 16d 2 a P δ * F 2 + bδ κ , for all f ∈ L 2 (G) with f 2 = 1 and all 0 < δ < 1.

F ∈ L 2 (B) V .
Put q = min{ 1 16 , κ/4} and let C > 0 be the constant provided in Theorem 6.2. Choose • 0 < α < q 16d and denote by c 0 and ε 0 the corresponding constants given by Corollary 4.2. • c > 0 such that 2c 0 c < min{ 1 16 , q 16d -α}. • 0 < ε < ε 0 small enough so that 2c 0 (c

+ log 2C log 1 ε ) < min{ 1 16 , q 16d -α}.
Next, take a probability measure µ on G supported on B ε (1) such that for all δ > 0 small enough,

we have µ * 2n (H (δ) ) δ d 1 , where n = d 2 log 1 δ log 1 ε
, for any proper connected closed subgroup H < G.

By Corollary 4.2, there exists δ 0 > 0 such that for all 0 < δ < δ 0 , we have that (6.8)

µ * n * P δ 2 δ -α , for all n c 0 log 1 δ log 1 ε .
Taking δ 0 smaller if necessary, we can assume that δ

2c 0 (c+ log 2C log 1 ε ) > (a + b) 1 16d δ q 16d -α + δ 1 16
, for δ < δ 0 . Now, assume by contradiction that the measure µ does not satisfy the conclusion of the theorem. Then by Corollary 6.6, there exists F ∈ L 2 ( B) with F 2 = 1 and 0 < δ < δ 0 such that (1) µ * F 2 > ε c /(2C).

(2) P δ * F -F 2 < δ 1/16 .

(3) P . Since µ is symmetric, by using Lemma 2.1 we derive that

ε c 2C 2n µ * F 2n µ * n * F 2 µ * n * P δ * F 2 + P δ * F -F 2
On the other hand, by combining (6.7) and (6.8) we get that

µ * n * P δ * F 2 a P δ 1/4 * F 2 + δ κ/4 1 16d µ * n * P δ 2
(aδ

1 16 + bδ κ/4 ) 1 16d δ -α (a + b) 1 16d δ q 16d -α .
By putting the last two inequalities together we get that ε c 2C 2n (a + b)

1 16d δ q 16d -α + δ 1 16 . Since ε c 2C 2n ε c 2C 2c 0 log 1 δ log 1 ε = δ 2c 0 (c+ log 2C log 1 ε )
, this contradicts the choice of δ > 0. , for all x, g ∈ G.

Put

B 1 = B 1 (1) • B ⊂ G/H and B 2 = B 1 (1) -1 • B 1 ⊂ G/H. Let p : G → G/H be the canonical projection. Let B1 , B2
⊂ G be open sets with compact closures such that B i ⊂ p( Bi ) for i = 1, 2. Replacing Bi by Bi • K for some compact set K ⊂ H with non empty interior, we may also assume that H 1 Bi (xh)dh is bounded away from 0 uniformly in x ∈ Bi . Then using (6.9) there exists β > 0 such that F 2 /β F • p 2, Bi β F 2 for all F ∈ L 2 (B i ), for both i = 1 and i = 2.

Fix ε ∈ (0, 1) small enough so that | ρ(x) ρ(gx) -1| 1 4 , for all g ∈ B ε (1) and x ∈ B1 . Take r > 0 such that 2rβ 4 < 1/16. By Theorem 6.7 there exist a finite dimensional space V ⊂ L 2 ( B2 ) and a finite set T ⊂ Γ such that the measure µ

= 1 2|T | g∈T (δ g + δ g -1 ) satisfies supp(µ) ⊂ B ε (1) and µ * F 2 < r F 2 , for all F ∈ L 2 ( B2 ) V .
Take a sequence of functions F n ∈ L 2 (B) which converges weakly to 0 and such that F n 2 = 1 for all n. To prove the corollary, it is enough to show that eventually π(µ)(F n ) 2 < 1 2 .

First, remark that by our choice of ε, we have

|1 -ρ(gx) ρ(x) | 1 4 ρ(gx)
ρ(x) for all g ∈ supp(µ) and x ∈ B1 . Thus, Equation (6.10) gives for all F ∈ L 2 (B):

π(µ)(F ) -µ * F 2 = 1 2|T | g∈T ∪T -1 ( ρ(g •) ρ -1)F (g -1 •) 2 1 4 1 2|T | g∈T ∪T -1 ρ(g •) ρ F (g -1 •) 2 = 1 4 F 2 .
Therefore, for all n we have (6.11)

π(µ)(F n ) 2 µ * F n 2 + 1 4 .
So we are left to bound µ * F n 2 by 1 4 for all n large enough. Since µ * F n is supported on B 1 , by the definition of β, we have that

µ * F n 2 2 β 2 (µ * F n ) • p 2 2, B1 = β 2 µ * (F n • p) 2 2, B1 = β 2 µ * (F n • p)1 B1 , µ * (1 B2 .(F n • p)) β 2 µ * (F n • p) 2, B1 µ * (1 B2 .(F n • p)) 2
The second line above comes from the fact that 1 B1 1 g• B2 for all g ∈ supp(µ) ⊂ B 1 (1). Using the same fact we moreover see that µ *

(F n • p) 2, B1 F n • p 2, B2 β F n 2 = β. In summary, µ * F n 2 2 β 3 µ * (1 B2 .(F n • p)) 2 .
Using (6.9) one easily checks that the sequence (1 B2 .(F n • p)) n ⊂ L 2 ( B2 ) goes weakly to 0. Hence, we deduce from the restricted spectral gap assumption on µ that for n large enough,

µ * (1 B2 .(F n • p)) 2 < 2r 1 B2 .(F n • p) 2 = 2r F n • p 2, B2 2rβ.
Altogether, we get that for n large enough:

µ * F n 2 2 < 2rβ 4 1 16 .
Combining this with (6.11), we indeed get that π(µ)(F n ) 2 < 1 2 for n large enough.

The Banach-Ruziewicz problem

This section is devoted to the proof of Theorem D. Moreover, we will show the following:

Theorem 7.1. Let G be a l.c.s.c. group and Γ < G a countable dense subgroup.

Then the following four conditions are equivalent:

(1) If ν : C(G) → [0, ∞) is a Γ-invariant, finitely additive measure, then there exists α 0 such that ν(A) = α|A|, for all G is free, it follows that G is Γ-paradoxical (see Definition 7.9). The proof of [Lu94, Theorem 2.1.17] implies that any subsets B, C ⊂ G with nonempty interior are equidecomposable. Further, the proof of [Lu94, Proposition 2.1.12] gives that any finitely additive Γ-invariant measure ν : C(G) → [0, ∞) is absolutely continuous with respect to m G . It follows readily that (1) holds. Theorem 7.1 is therefore contained in the literature when G is compact. Our contribution is to show that it holds for arbitrary locally compact groups.

A ∈ C(G). (2) If Φ : L ∞ c (G, m G ) → C is a Γ-invariant, positive linear functional, then there exists α 0 such that Φ(f ) = α G f dm G , for all f ∈ L ∞ c (G, m G ). (
Turning to locally compact groups G, the non-trivial implications, which we will address below, are (2) =⇒ (3), (2) =⇒ (1), and (4) =⇒ (3). 7.1. Local spectral gap and uniqueness of invariant means. In order to prove implication (2) =⇒ (3) from Theorem 7.1, we give an equivalent formulation of local spectral gap in terms of uniqueness of invariant linear functionals (see Theorem 7.6). This generalizes a well-known result for probability measure preserving actions. Let Γ (X, µ) be a probability measure preserving action of a countable group Γ. Then integration against µ defines a Γ-invariant mean (i.e. a unital positive linear functional) on L ∞ (X, µ). In the early 1980's, it was realized that whether this is the unique Γ-invariant mean on L ∞ (X, µ) is equivalent to the spectral gap of the action. More precisely, the following was shown:

Theorem 7.3. [Ro81,Sc81] Let Γ
(X, µ) be an ergodic measure preserving action of a countable group Γ on a probability space (X, µ). Consider the following conditions:

(1) If Φ : L ∞ (X, µ) → C is a Γ-invariant mean, then Φ(f ) = X f dµ, for all f ∈ L ∞ (X, µ).
(2) There does not exist a sequence {A n } of measurable subsets of X such that µ(A n ) > 0, for all n, lim n→∞ µ(A n ) = 0, and lim n→∞ µ(gA n ∆A n )/µ(A n ) = 0, for all g ∈ Γ.

(3) If a sequence ϕ n ∈ L 1 (X, µ) of positive functions satisfies X ϕ n dµ = 1, for all n, and lim

n g • ϕ n -ϕ n 1 = 0, for all g ∈ Γ, then lim n ϕ n -1 1 = 0. Then µ(A n ∩ A) k i=1 µ(g -1 i A n ∩ B) and we get that lim sup n→∞ µ(A n ∩ A)/µ(A n ∩ B) k. Since | 1 µ(An∩B) An f dµ| f ∞ µ(An∩A)
µ(An∩B) , it follows that Φ(f ) is well-defined. Further, we have

| An g • f dµ - An f dµ| g -1 An∆An |f | dµ = (g -1 An∆An)∩A |f | dµ f ∞ µ((g -1 A n ∆A n ) ∩ A).
Since A ∈ C B (X), we have lim

n µ((gA n ∆A n ) ∩ A)/µ(A n ∩ B) = 0.
In combination with the above, this implies that Φ(g • f ) = Φ(f ). Therefore, Φ is Γ-invariant. Since Φ(1 B ) = 1 and condition (1) is assumed true, we get that Φ(f ) = 1 µ(B) X f dµ, for all f ∈ L ∞ B (X, µ). We are now ready to derive a contradiction. Firstly, assume that = lim n µ(A n ∩ B) = 0. In this case, after passing to a subsequence, we may assume that 0 < µ(A n ∩ B) < 3 -n µ(B), for all n 1. Let C = B \ (∪ n 1 A n ). It follows that µ(C) > 0 and hence Φ(1 C ) = µ(C)/µ(B) > 0. On the other hand, it is clear from the definition of Φ that Φ(1 C ) = 0. This gives a contradiction.

Secondly, assume that 0 < < µ(B).

If C ∈ C B (X), then lim n µ(A n ∩ C) = Φ(1 C ) = ( µ(C))/µ(B). Denoting A m,n = A m ∩ A n , we further get that (7.1) lim m lim n µ(A m,n ∩ C) = lim m µ(A m ∩ C) µ(B) = 2 µ(C) µ(B) 2 .
In particular, lim This implies that 2 /µ(B) ∈ S. Since 2 /µ(B) < , this contradicts the minimality of .

(2) =⇒ (3). The proof relies on a variation of Namioka's trick. Suppose (2) is true. By contradiction, assume that there is a sequence of positive functions ϕ n ∈ L 1 (X, µ) satisfying B ϕ n dµ = µ(B), for all n, and lim

n g • ϕ n -ϕ n 1,B = 0, for all g ∈ Γ, such that ϕ n -1 1,B → 0.
After passing to a subsequence, assume that there is δ > 0 such that ϕ n -1 1,B δ, for all n 1.

Let c ∈ (0, δ 2µ(B) ) and denote δ

0 := δ-2cµ(B) 2 > 0. Fix n 1. We define ψ n : X → R by letting ψ n (x) = ϕ n (x) -1. Since B ψ n dµ = 0 and c1 {ψn c} + ψ n 1 {0 ψn<c} c1 {ψn 0} cϕ n , we have δ B |ψ n | dµ = 2 B ψ n 1 {ψn 0} dµ = 2 B ψ n 1 {ψn c} dµ + 2 B ψ n 1 {0 ψn<c} dµ = 2 B (ψ n -c)1 {ψn c} dµ + 2 B (c1 {ψn c} + ψ n 1 {0 ψn<c} ) dµ 2 B (ψ n -c)1 {ψn c} dµ + 2c B ϕ n dµ = 2 B (ϕ n -(1 + c))1 {ϕn (1+c)} dµ + 2cµ(B).
It follows that B (ϕ n -(1 + c))1 {ϕn (1+c)} dµ δ 0 . For t 0, we put A t,n = {x ∈ X|ϕ n (x) t}. By combining this inequality with Fubini's theorem we get that

(7.2) ∞ 1+c µ(A t,n ∩ B) dt = B ∞ 1+c 1 {ϕn t} dt dµ = B (ϕ n -(1 + c))1 {ϕn (1+c)} dµ δ 0 .
Next, let g ∈ Γ. By using a similar calculation to the above we get that

∞ 0 µ((gA t,n ∆A t,n ) ∩ B) dt = B ∞ 0 |1 gAt,n (x) -1 At,n (x)| dt dµ(x) = B ∞ 0 |1 {ϕn(g -1 x) t} -1 {ϕn(x) t} | dt dµ(x) = B |ϕ n (g -1 x) -ϕ n (x)| dµ(x) = g • ϕ n -ϕ n 1,B .
Now, fix a finite set F ⊂ Γ and ε > 0. We claim that there is a measurable set A ⊂ X satisfying µ(A ∩ B) ∈ (0, µ(B) 1+c ], and µ((gA∆A) ∩ B)/µ(A ∩ B) < ε, for all g ∈ F . To this end, note that since g • ϕ n -ϕ n 1,B → 0, for all g ∈ Γ, we can find n 1 such that g∈F g • ϕ n -ϕ n 1,B < εδ 0 . By combining 7.2 with the last displayed identity it follows that

ε ∞ 1+c µ(A t,n ∩ B) dt εδ 0 > ∞ 0 ( g∈F µ((gA t,n ∆A t,n ) ∩ B)) dt.
From this we deduce that there is t

1 + c such that g∈F µ((gA t,n ∆A t,n ) ∩ B) < εµ(A t,n ∩ B). Also, tµ(A t,n ) B ϕ n dµ = µ(B), hence µ(A t,n ) µ(B) t µ(B)
1+c . Thus, A = A t,n verifies the claim. By using the claim we can construct a sequence {A n } of measurable subsets of X which satisfy that µ(A n ∩ B) ∈ (0, µ(B) 1+c ], for all n, and lim n µ((gA n ∆A n ) ∩ B)/µ(A n ∩ B) = 0, for all g ∈ Γ. Since

(2) is assumed true, we would get that lim n µ(A n ∩ B) = µ(B), which is a contradiction.

(3) =⇒ (1). The proof follows the proof of [Ma82, Theorem 3] and relies on Day's convexity trick. Suppose that (3) holds and let Φ : L ∞ B (X, µ) → C be a Γ-invariant, positive linear functional. If Φ(1 B ) = 0, then we clearly have that Φ ≡ 0. Thus, we may assume that Φ(1 B ) > 0. After replacing Φ with µ(B)

Φ(1 B ) Φ, we may further assume that Φ(1 B ) = µ(B). Let f 0 ∈ L ∞ B (X, µ). We will show that Φ(f 0 ) = X f 0 dµ. To this end, we denote by C the support of f 0 and fix ε > 0

. Let F ⊂ Γ be a finite set. Define Y = C ∪ B ∪ (∪ g∈F g -1 B). Since µ(Y ) < ∞, we can find a net of positive functions ϕ i ∈ L 1 (Y ) such that (7.3) lim i Y ϕ i f dµ = Φ(f ), for all f ∈ L ∞ (Y ).
Then for all g ∈ F and every f ∈ L ∞ (B) we have that f, g

-1 • f ∈ L ∞ (Y ) and hence (7.4) lim i B (g • ϕ i -ϕ i )f dµ = lim i Y ϕ i (g -1 • f -f ) dµ = Φ(g -1 • f -f ) = 0. Denote by A the set of positive functions ϕ ∈ L 1 (Y ) such that | Y ϕf 0 dµ -Φ(f 0 )| < ε and B ϕ dµ = Φ(1 B
). Then 7.3 and 7.4 imply that 0 = (0) g∈F belongs to the weak closure of

A F := {((g • ϕ -ϕ) |B ) g∈F |ϕ ∈ A} ⊂ L 1 (B) |F | .
Since A F is a convex set, its weak and norm closures concide. Thus, 0 belongs to the norm closure of A F , for every finite set F ⊂ Γ. It follows that there is a sequence ϕ n ∈ A such that lim

n→∞ g • ϕ n -ϕ n 1,B = 0, for all g ∈ Γ. Since B ϕ n dµ = Φ(1 B ) = µ(B), for all n, condition (3)
gives that lim n→∞ ϕ n -1 1,B = 0. Further, we get that lim n→∞ ϕ n -1 1,gB = 0, for all g ∈ Γ.

Since Y ∈ C B (X), we deduce that lim

n→∞ ϕ n -1 1,Y = 0. Since | Y ϕ n f 0 dµ -Φ(f 0 )| < ε, for all n, we get that | Y f 0 dµ -Φ(f 0 )| ε.
As ε > 0 is arbitrary, we conclude that Φ(f 0 ) = X f 0 dµ.

(3) =⇒ (4). Assume that (3) holds and that (4) is false. Then we can find a sequence ξ n ∈ L 2 (X) such that B ξ n dµ = 0 and ξ n 2,B = µ(B), for all n, and lim n→∞ g • ξ n -ξ n 2,B = 0, for all g ∈ Γ. Moreover, we may assume that ξ n is real-valued, for all n. (4) =⇒ (3). The proof of this implication is easy and so we leave it to the reader.

Define ϕ

n := |ξ n | 2 ∈ L 1 (X). Then B ϕ n dµ = µ(B)
(2) =⇒ (5). To prove this final implication, we need a lemma whose proof we leave to the reader. Lemma 7.7. Let (X, µ) be a standard measure space. Let ν be a Borel probability measure on X which is equivalent to µ. Let {A n } be a sequence of measurable subsets of X. Then lim n→∞ ν(A n ) = 0 if and only if lim n→∞ µ(A n ∩ B) = 0, for every measurable set B ⊂ X with µ(B) < ∞. Now, let ν be a Borel probability measure on X which is equivalent to µ. Let {A n } be a sequence of measurable subsets of X such that lim Proposition 7.8 is proved by adapting an argument due to Tarski, who used the Banach-Tarski paradox to show that any SO(n + 1)-invariant, finitely additive measure defined on the Lebesgue subsets of S n , n 2, is absolutely continuous with respect to the Lebesgue measure (see [Lu94, Proposition 2.2.12]). As such, we need to recall the notion of equidecomposability:

Proof. Since G is a l.c.s.c. group, we can find a left-invariant compatible metric d on G which is proper, in the sense that the closed ball B r (x) := {y ∈ G|d(x, y) r} is compact, for every r > 0 and x ∈ G (see e.g. [START_REF] De Cornulier | de la Harpe: Metric geometry of locally compact groups, book in progress[END_REF]). We denote B r = B r (e), for all r > 0.

The idea of the proof is to show that there exists a bounded set D ⊂ G which has non-empty interior and is paradoxical. By Lemma 7.10 we are in one of two cases.

Case 1. Γ contains a non-abelian free subgroup Γ 0 which is not discrete in G.

We claim that D := B 1 is paradoxical. Assume by contradiction that B 1 is not paradoxical. Then by a theorem of Tarski (see [START_REF] Wagon | The Banach-Tarski paradox[END_REF]Corollary 9.2]) we can find a Γ-invariant, finitely additive measure ϕ : P(G) → [0, ∞] such that ϕ(B 1 ) = 1. Note that if r > 0, then we can find g 1 , ..., g n ∈ Γ such that B r ⊂ ∪ n i=1 g i B 1 and B 1 ⊂ ∪ n i=1 g i B r . From this we deduce that 0 < ϕ(B r ) < ∞, for all r > 0. We define ρ : (0, ∞) → (0, ∞) by letting ρ(r) = ϕ(B r ). Then ρ is an increasing function and hence has at most countably many points of discontinuity.

Let s > 0 be a point at which ρ is continuous. Next, we define Φ : P(G) → [0, 1] by letting Φ(A) = ϕ(A ∩ B s )/ϕ(B s ), for every A ∈ P(G). Then Φ is a finitely additive measure, Φ(G) = 1, and we have the following To see this, let g ∈ G and A ∈ P(G). Then we have that g

-1 B s \ B s ⊂ B s \ B s and that B s \ g -1 B s ⊂ B s \ B s
, where s = s + d(g, e) and s = max{s -d(g, e), 0}. Since ρ is continuous at s, we get that lim

g→e ϕ(g -1 B s \ B s ) = 0 and lim g→e ϕ(B s \ g -1 B s ) = 0. Moreover, if g ∈ Γ, then we have that |ϕ(gA ∩ B s ) -ϕ(A ∩ B s )| = |ϕ(A ∩ g -1 B s ) -ϕ(A ∩ B s )| ϕ(g -1 B s \ B s ) + ϕ(B s \ g -1 B s ).
The combination of these two facts clearly implies 7.5. Now, let a and b be two free generators of Γ 0 . Let g n ∈ Γ 0 \{e} be a sequence such that lim n→∞ g n = e.

Note that for every g ∈ Γ 0 \ {e}, at least one of the pairs {g, aga -1 } and {g, bgb -1 } generates a copy of F 2 . Therefore, after passing to a subsequence and after eventually swapping a and b, we may assume that g n and h n = ag n a -1 generate a copy of F 2 , for all n. Note that lim

n→∞ g n = lim n→∞ h n = e.
We define X = n 1 X n , where X n = G, to be the disjoint union of infinitely many copies of G. Let c and d be two free generators of F 2 . Let lim n be a bounded linear functional on ∞ (N) which extends the limit. We consider an action F 2 X given by c • x = g n x and d • x = h n x, for all n 1 and x ∈ X n . Finally, we define Ψ : P(X) → [0, 1] by Ψ(A) = lim n Φ(A ∩ X n ), for all A ∈ P(X). Then Ψ is a finitely additive measure and Ψ(X) = 1. Moreover, Ψ is F 2 -invariant. Indeed, let A ∈ P(X) and write A = n 1 A n , where A n ⊂ X n . Since lim n→∞ g n = e, equation 7.5 implies that

Ψ(c • A) = lim n Φ(g n A n ) = lim n Φ(A n ) = Ψ(A).
Similarly, if follows that Ψ(d • A) = Ψ(A). Altogether, Ψ : P(X) → [0, 1] is an F 2 -invariant, finitely additive measure with Ψ(X) = 1. However, since the action F 2 X is free, this contradicts the non-amenability of F 2 . Thus, we conclude that B 1 is paradoxical.

Case 2. There exists a compact subgroup K < G such that Γ 0 := Γ ∩ K is non-amenable.

Let D ∈ C(G) be a K-invariant set with non-empty interior. Since the left multiplication action Γ 0 D is free and Γ 0 is non-amenable, we get that D is Γ 0 -paradoxical and hence Γ-paradoxical.

Altogether, we have shown the existence of a subset D ⊂ G which has non-empty interior and is paradoxical. Let B, C ⊂ G be two bounded subsets with non-empty interior.

Since Γ < G is dense, we can find integers p, q 1 such that B pD and D qC. Since D is paradoxical, we have 2 n D D, for all integers n 1. By combining these facts, we get that qB qpD 2 qp D D qC. The cancellation law implies that B C. Similarly, we get that C B, and [Lu94, Proposition 2.1.2.] implies that B ∼ C, as desired.

Proof of Proposition 7.8. Let ν : C(G) → [0, ∞) be a Γ-invariant, finitely-additive measure and B ∈ C(G) such that |B| = 0. We will show that ν(B) = 0. To this end, let ε > 0.

Let C ∈ C(G) be a set with non-empty interior such that ν(C) < ε. By Lemma 7.11 we have B C, hence we can find a subset D ⊂ C such that B ∼ D. We can therefore decompose B = ∪ k i=1 B i and

D = ∪ k i=1 D i such that B i ∩ B j = D i ∩ D j = ∅,
for all 1 i < j k, and there exist g 1 , ..., g k ∈ Γ such that g i B i = D i , for all 1 i k. Since |B| = 0, we get |D i | = |B i | = 0, for all 1 i k.

In particular, B i , D i are measurable sets with compact closures, for all 1

i k. Since ν is Γ-invariant, we deduce that ν(B) = k i=1 ν(B i ) = k i=1 ν(D i ) = ν(D)
. Therefore, since D ⊂ C, we conclude that ν(B) ν(C) < ε showing that ν(B) = 0 because ε > 0 was arbitrary. (3) =⇒ (4). This follows from the implication (4) =⇒ (5) from Theorem 7.6. After passing to a subsequence, we may assume that the limit = lim n→∞ |A n ∩ B| exists. By using the strong ergodicity assumption and Lemma 7.7 it is easy to see that if > 0, then = |B|. Thus, in order to derive the conclusion, it suffices to prove that = 0 leads to a contradiction. We will achieve this by adapting the "averaging" argument from [AJN07, Lemma 14]. We continue with the following claim:

Claim. If A ⊂ G is a measurable set and p 1 is an integer, then

C p |(Ag -1 1 ∪ ... ∪ Ag -1 p ) ∩ B| dg 1 ...dg p = |C| p B 1 -1 - |A ∩ xC|) |C| p dx.
Proof of the claim. Let q 1 be an integer and put β q = C q |Ag -1 1 ∩ ... ∩ Ag -1 q ∩ B| dg 1 ...dg q . By using Fubini's theorem we have that

β q = C q B
1 Ag -1 1 (x)...1 Ag -1 q (x) dx dg 1 ...dg q = B C q 1 x -1 A (g 1 )...1 x -1 A (g q ) dg 1 ...dg q dx = B |x -1 A ∩ C| q dx = B |A ∩ xC| q dx. Now, the inclusion-exclusion principle gives that the left side of the claimed identity is equal to α p = p q=1 (-1) q-1 p q β q |C| p-q . In combination with the above, the claim follows. As one may observe, the claim deals with right translates while the group that we consider is not necessarily unimodular. We will then have to consider the constant τ = sup{∆(g) , g ∈ C}, where ∆ is the modular function on G. This constant is finite since C has compact closure.

Since lim t→0

(1 -t)

1 -(1 -t n ) pn 1 -(1 -t n ) α[ 1 tn ] > 1 - 1 2 α .
By combining this inequality with the claim above, we find g n,1 , g n,2 , ..., g n,pn ∈ C such that Ãn := ∪ pn i=1 A n g -1 n,i satisfies

|gC ∩ A n | = |C ∩ g -1 A n | = |C ∩ (g -1 A n ∩ B)| |C ∩ A n | + |(g -1 A n ∩ B) \ A n |,
dividing by |A n | and using (8.1) yields that ν(gC) ν(C). Since the same argument shows that ν(C) = ν(g -1 (gC)) ν(gC), the assertion follows.

Next, we claim that ν extends to a Γ-invariant finitely additive measure ν : C(G) → [0, ∞). Let C ∈ C(G). Since Γ is dense in G, we can find a measurable partition {C i } k i=1 of C and g 1 , ..., g p ∈ Γ such that g i C i ⊂ B, for every 1 i p. We define ν(C) = p i=1 ν(g i C i ). To see that ν is well-defined, consider another measurable partition {D j } q j=1 of C and h 1 , ..., h q ∈ Γ such that h j D j ⊂ B, for every 1 j q. Then for all i, j, we have that g i (C i ∩ D j ), h j (C i ∩ D j ) ⊂ B, thus ν(g i (C i ∩ D j )) = ν(h j (C i ∩ D j )). Using this fact we derive that

p i=1 ν(g i C i ) = p i=1 q j=1 ν(g i (C i ∩ D j )) = q j=1 p i=1 ν(h j (C i ∩ D j )) = q j=1 ν(h j D j ),
showing that ν is well-defined. It is now clear that ν is finitely additive and extends ν. In combination with (8.1) it follows that lim n g • η n -η n 2 2 2|gB \ B|.

Let ε > 0 such that sup g∈Bε(1) |gB \ B| < α 128 . By applying Theorem 6.7 we can find a probability measure µ supported on Γ ∩ B ε (1) and a finite dimensional subspace V ⊂ L 2 (B) such that we have µ * F 2 < 8.2. Proof of Corollary H. Let S = {g 1 , ..., g k } be a finite symmetric subset of G. Recall that the operator P S : L 2 (B) → L 2 (B) is given by P S (F ) = 1 k k i=1 1 B∩g i B g i • F + 1 B\g i B F . We start by giving a useful formula for P S (F ), F . Since 1 B\g i B F = F -1 B∩g i B F , we get that

P S (F ), F = F 2 2 - 1 k k i=1 F 2 2,B∩g i B -1 B g i • F, 1 g i B F . Since g i • F -F 2 2,B∩g i B = F 2 2,B∩g -1 i B + F 2 2,B∩g i B -2 1 B g i • F, 1 g i B F
, for all i, and S is symmetric, we deduce that (8.3) P S (F ), F = F 2 2 -

1 2k k i=1 g i • F -F 2 2,B∩g i B .
Since g • F -F 2,B∩gB 2 F 2 , this calculation implies that P S is symmetric and P S 1. Moreover, if there is 1 i k such that B ∩ g i B = ∅, then P S (F ), F (-1 + 2 k ) F 2 2 , and therefore the spectrum of P S satisfies σ(P S ) ⊂ [-1 + 2 k , 1]. Assume that the conclusion of Corollary H is false. Thus, the restriction of P S to L 2 (B) C1 B has norm 1, for every finite symmetric set S ⊂ Γ. Fix g ∈ Γ such that gB ∩ B = ∅. It follows that 1 ∈ σ(P S |L 2 (B) C1 B ), for every finite symmetric set S ⊂ Γ which contains g. Using (8.3) we conclude that there is a sequence F n ∈ L 2 (B) C1 B such that F n 2 = 1, for all n, and g • F n -F n 2,B∩gB → 0, for every g ∈ Γ.

We claim that F n → 0, weakly. Indeed, any weak limit point F of {F n } satisfies F (g -1 x) = F (x), for all g ∈ Γ and almost every x ∈ B ∩ gB. It is clear that F can be extended to a Γ-invariant function F ∈ L 2 loc (G). Since Γ < G is dense, F and therefore F must be a constant function. Since F has zero integral, we get that F ≡ 0, which proves the claim.

  Basic properties of local spectral gap 2.3. Deduction of Theorem A from Theorem B 2.4. Reduction to groups with trivial center 2.5. Outline of the proof of Theorem B 3. Escape from subgroups 3.1. Ping-pong 3.2. From subgroups to neighborhoods of subgroups 3.3. Proof of Theorem 3.1 4. 2 -flattening 5. A mixing inequality R.B. was partially supported by NSF Grant DMS #1161047, NSF Career Grant DMS #1253402 and ANR Grant NEUMANN.

F|F | 2 dµ 1 2

 1 2,B κ g∈S g • F -F 2,B for any F ∈ L 2 (X, µ) with B F dµ = 0.Here, F 2,B := B denotes the L 2 -norm of the restriction of F to B.

  finitely additive measure, then there exists α 0 such that ν(A) = α m G (A), for all A ∈ C(G).(2) The left translation action Γ (G, m G ) has local spectral gap.

  we put r i = min b∈A i rk(b), where rk(b) is the rank of b.

  Then X M is a nonempty Zariski-open set. Proof. Denote by A M (resp. B M ) the set of M -tuples satisfying condition (1) (resp. (2)). Then A M is clearly a finite intersection of Zariski open sets, which are non-empty by irreducibility of ρ.

  any γ ∈ Γ and 1 i, j d, where B * := {v * 1 , . . . , v * d } is the dual basis. Since Γ is finitely generated, there are a number field k and a finite set of places S of k such that v

Part 1 :

 1 construction of the set T . Let a, b ∈ S with a = b and define

Claim 3 .

 3 |B| (2|T |) 11n 12 +1 , for all n > 12. Proof of Claim 3. Assume that g = k p k p-1 ...k 1 ∈ B. Then the first part of Lemma 3.3 implies that k N +1 , ..., k p-1 , k p are uniquely determined by v. Now, since g ∈ W n (T ), we can write g = g εn n g

  the conclusion of Claim 2 follows. Part 3: bounding the probability of return. Define µ = 1 2|T | g∈T (δ g + δ g -1 ). By using Part 2 and following closely the proof of [Va10, Proposition 9] (see also [SGV11, Proposition 7]) we next estimate µ * n ({g

  0 n 10 |W k (T )| for all k n/10. When k < n/10, we use the brutal bound |A ∩ W k (T )| |W k (T )| (2|T |) k . Altogether, we get µ * n (A)

6.

  Proofs of Theorem B and Corollary C 6.1. A Littlewood-Paley decomposition on Lie groups. Let G be a connected simple Lie group with trivial center. In order to prove Theorem B, we next introduce a Littlewood-Paley decomposition on G. This is analogous to the Littlewood-Paley decomposition on G = SU (d) defined by Bourgain and Gamburd in [BG10, Section 10]. As before, we endow G with the . 2 metric and denote by C(G) the family of measurable subsets of G with compact closure.

  Theorem 6.7 also implies the quantitative version of Theorem B referred to in Remarks 1.6 and 1.7. Proof. Let B ⊂ G and d 1 , d 2 > 0 be as in the statement of the theorem. Let B ⊂ G be an open set with compact closure which contains the closure of B. Denote d = dim(G).

  m,n ∩ B) = 2 /µ(B). Let g ∈ Γ. Since > 0, the assumptions on {A n } imply that lim n µ((gA n ∆A n ) ∩ C) = 0. Since g Am,n ∆A m,n ⊂ (gA m ∆A m ) ∪ (gA n ∆A n ), we get that lim m lim n µ((gA m,n ∩ A m,n ) ∩ C) = 0. It follows that there is a sequence of the form Ãk = A m(k),n(k) such that lim k µ( Ãk ∩ B) = 2 /µ(B) and lim k µ((g Ãk ∆ Ãk ) ∩ C) = 0, for all C ∈ C B (X) and g ∈ Γ.

  and lim n→∞ g • ϕ n -ϕ n 1,B = 0, for all g ∈ Γ. By using condition (3) we deduce that limn→∞ ϕ n -1 1,B = lim n→∞ |ξ n | 2 -1 1,B = 0. Since ξ n is real-valued, itfollows that there exists a sequence {A n } of measurable subsets of X such that lim n→∞ ξ n -(1 An -1 X\An ) 2,B = 0. By using the defining properties of {ξ n } we get that lim n→∞ µ((gA n ∆A n ) ∩ B) = 0 and lim n→∞ µ(A n ∩ B) = µ(B)/2. This contradicts condition (2), which, as shown above, is equivalent to (3).

  n→∞ ν(gA n ∆A n ) = 0, for all g ∈ Γ. Lemma 7.7 implies that lim n→∞ µ((gA n ∆A n ) ∩ B) = 0, for all g ∈ Γ. Since condition (2) holds, we get lim n→∞ µ(A n ∩ B) = µ(B). Using the almost Γ-invariance of {A n }, it follows that lim n→∞ µ(A n ∩ C) = µ(C) and hence that lim n→∞ µ(A c n ∩ C) = 0, for all C ∈ F B (X). Since the action Γ (X, µ) is ergodic, we have that µ(X \(∪ g∈Γ gV )) = 0. By combining the last two facts we deduce that lim n→∞ µ(A c n ∩B) = 0, for every measurable set B ⊂ X with µ(B) < ∞. Applying Lemma 7.7 again yields that lim n→∞ ν(A c n ) = 0. This implies that the asymptotically invariant sequence {A n } is trivial. 7.2. Absolute continuity of invariant finitely additive measures. Towards proving implication (2) =⇒ (1) from Theorem 7.1 we first establish that any Γ-invariant finitely additive measure on C(G) is absolutely continuous with respect to m G . Proposition 7.8. Let G be a l.c.s.c. group and Γ < G be a countable dense subgroup such that Γ ∩ G 0 is non-amenable, for any open subgroup G 0 < G. Let ν : C(G) → [0, ∞) be a Γ-invariant, finitely additive measure. If B ∈ C(G) and |B| = 0, then ν(B) = 0.

7. 3 .

 3 Proof of Theorem 7.1. We are now ready to prove Theorem 7.1. Let S ⊂ L ∞ c (G, m G ) be the set of functions of the form n i=1 c i 1 A i , where c 1 , ..., c n ∈ C and A 1 , ..., A n ∈ C(G).(1) =⇒ (2). Assume that (1) holds true. Let Φ :L ∞ c (G, m G ) → C be a Γ-invariant, positive linear functional. Then ν : C(G) → [0, ∞) given by ν(A) = Φ(1 A ) is a Γ-invariant, finitely additive measure. Since (1) holds, we can find α 0 such that ν(A) = α|A|, for all A ∈ C(G). Then we clearly have that Φ(f ) = α G f dm G , for all f ∈ S. Let f ∈ L ∞ c (G, m G ) be a real-valued function. Then -f ∞ 1 B f f ∞ 1 B, where B denotes the support of f . Since Φ is a positive, we get that -α|B| f ∞ Φ(f ) α|B| f ∞ . Moreover, we may find a sequence {f n } of real-valued functions which belong to S and whose support is included in B such that lim n→∞ f -f n ∞ = 0. By applying the above inequality to f -f n , it follows that Φ(f ) = lim n→∞ Φ(f n ) = lim n→∞ α G f n dm G = α G f dm G . This proves condition (2).

( 2 )

 2 =⇒ (1). By using Proposition 7.8, Φ : S → C given by Φ(f ) = n i=1 c i ν(A i ), for every f = n i=1 c i 1 A i ∈ S, is a well-defined Γ-invariant positive linear functional. Moreover, if B is the support of f , then |Φ(f )| ν(B) f ∞ . By arguing as above, we get that Φ extends to a Γ-invariant positive linear functional Φ :L ∞ c (G, m G ) → C.By applying (2) to Φ, the conclusion follows.(2) ⇐⇒ (3). Let B ∈ C(G) be a set with non-empty interior. ThenL ∞ B (G, m G ) = L ∞ c (G, m G) by Remark 7.5 and the conclusion follows from Theorem 7.6.

( 4 )

 4 =⇒ (3). Assume that Γ (G, m G ) is strongly ergodic and let ν be a Borel probability measure on G which is equivalent to m G . Let B ∈ C(G) be an open set with compact closure.Let {A n } be a sequence of measurable subsets of G such that |A n ∩ B| > 0, for all n, and limn→∞ |(gA n ∆A n ) ∩ B|/|A n ∩ B| = 0, for all g ∈ Γ. Since L ∞ B (G, m G ) = L ∞ c (G, m G ),Theorem 7.6 guarantees that in order to prove local spectral gap with respect to B, it suffices to show that lim n→∞ |A n ∩ B| = |B|.

  Now, we let C = B -1 B and D = BC. We claim that sup n |A n ∩ D|/|A n ∩ B| < ∞. Indeed, since D ∈ C(G) and ∪ g∈Γ gB = G, we can find h 1 , ..., h p ∈ Γ such that D ⊂ ∪ p i=1 h i B. Then we have that|A n ∩ D| p i=1 |h -1 i A n ∩ B| p|A n ∩ B| + p i=1 |(h -1 i A n ∆A n ) ∩ B|and the claim follows. Since = 0, the claim implies that lim n→∞ |A n ∩ D| = 0 and we can find κ ∈ (0, 1) such that (7.6) |C| |A n ∩ D| κ |C| |A n ∩ B| , for all n.

  |B| < | Ãn ∩ B|, for all n N . Moreover, we also have the upper bound (7.9)| Ãn ∩ B| pn i=1 |A n g -1 i ∩ B| pn i=1 |A n ∩ Bg i |τ τ p n |A n ∩ D| |B| 2, for all n N .

Finally, let

  g ∈ Γ and K ∈ C(G). We claim that lim n→∞ |(g Ãn ∆ Ãn ) ∩ K| = 0. To this end, note that g Ãn ∆ Ãn ⊂ ∪ pn i=1 (gA n ∆A n )g -1 n,i . Denote L = KC and notice that Kg n,i ⊂ L. Also, since B ⊂ D, we get that p n |C|/|A n ∩ B|. By combining all of these facts we get that|(g Ãn ∆ Ãn ) ∩ K| pn i=1 |(gA n ∆A n )g -1 n,i ∩ K| p n τ |(gA n ∆A n ) ∩ L| τ |C| |(gA n ∆A n ) ∩ L| |A n ∩ B| .Since L ∈ C(G), we have that lim n→∞ |(gA n ∆A n ) ∩ L|/|A n ∩ B| = 0. This proves our claim. Let µ be a Borel probability measure on G which is equivalent to m G . By combining the claim and Lemma 7.7 we get that lim n→ µ(g Ãn ∆ Ãn ) = 0, for all g ∈ Γ. Since the action Γ (G, m G ) is strongly ergodic, we conclude that lim n→∞ µ( Ãn )(1 -µ( Ãn )) = 0. By applying Lemma 7.7 again, we get that lim n→∞ | Ãn ∩ B|(|B| -| Ãn ∩ B|) = 0. This however contradicts the inequalities (7.8) and(7.9). We have therefore shown that the case = 0 leads to a contradiction, as desired.8. Proofs of Corollaries G and H8.1. Proof of Corollary G. We only treat the case when H is trivial. The general case can then be deduced in a similar fashion as in the proof of Corollary C. Assume that the conclusion is false.Then we can find a sequence of non-negligible measurable setsA n ⊂ B such that |A n | |B| 2 and (8.1) |(gA n ∩ B) \ A n | |A n | → 0, for every g ∈ Γ.Our first goal is to show that the sets A n are equidistributed, in the following sense: (8.2) lim n |C ∩ A n | |A n | = |C| |B| , for every C ∈ C(B). To this end, we denote by C(B) the collection of all measurable subsets of B, and define a finitely additive measure ν : C(B) → [0, 1] by letting ν(C) = lim n |C∩An| |An| , for every C ∈ C(B). Let us first show that ν(gC) = ν(C), whenever C ∈ C(B) and g ∈ Γ satisfy gC ⊂ B. Since

  Since the action ΓG has local spectral gap and ν(B) = 1, Theorem D implies that ν(C) = |C| |B| , for every C ∈ C(G), thus proving (8.2).Note that α := inf n |A n | > 0. Otherwise, after replacing {A n } n with a subsequence, we may assumethat n |A n | < B. But then C = B \ (∪ n A n ) would satisfy ν(C) = ν(C) = 0, while |C| > 0. Define η n = 1 An -|An| |B| 1 B ∈ L 2 (B). Further, let g ∈ Γ, and note that if A ⊂ B is a subset, then gA \ A ⊂ ((gA ∩ B) \ A) ∪ (gB \ B). Using this, for every n we get thatg • η n -η n 2 1 gAn -1 An 2 + 1 gB -1 B 2 = 2|gA n \ A n | + 2|gB \ B| 2|(gA n ∩ B) \ A n )| + 2|gB \ B| + 2|gB \ B|.

F 2 Moreover, since η n 2 2=

 22 , for every F ∈ L 2 (B) V . By using the previous paragraph we havelim n µ * η n -η n 2 sup |A n |(1 -|An| |B| ) and |A n | |B| 2 , we deduce that √ α 2 < η n 2 < |B|.Using this, we conclude that lim n µ * η n 2 > n η n 2 . On the other hand, (8.2) implies that η n → 0, weakly in L 2 (B). This gives a contradiction, and finishes the proof of the main assertion.Let us prove the moreover assertion. Assuming that this assertion is false, we can find a sequence of non-negligible measurable sets A n ⊂ B such that |A n | |B| 2 and (8.1) holds for all g ∈ Γ ∩ B ε (1). Let B 0 ⊂ B be a non-negligible measurable set with B 0 B -10 ⊂ B ε (1). Put A n := A n ∩ B 0 and assume that lim n |A n | |An| > 0. We claim that lim n |A n | = |B 0 |. Indeed, it is easy to see that lim n |gA n ∩B 0 \A n | |A n | = 0, for every g ∈ Γ ∩ B ε (1). Since gB 0 ∩ B 0 = ∅ forces g ∈ B ε(1), the last identity holds for every g ∈ Γ, and the first part of the proof implies the claim.Finally, choose a neighborhood B 1 of the identity such thatB 1 B -1 1 ⊂ B ε (1). Denote by C the set of x ∈ B such that lim n |An∩B 1 x| |An| > 0. By the claim, if x ∈ C, then lim n |A n ∩B 1 x∩B| = |B 1 x∩B| > 0. Since B isopen, it is easy to check that C is both an open and closed subset. This contradicts the connectedness of B.

  Next, we define a finitely additive measure ν :C(B) → [0, 1] by letting ν(C) = lim n F n 2 2,C . It is easy to check that ν(gC) = ν(C), whenever C ∈ C(B) and g ∈ Γ satisfy gC ⊂ B. By repeating the reasoning from the proof of Corollary G it follows that ν(C) = |C| |B| , for every C ∈ C(B). Since g • F n -F n 2 g • F n -F n 2,B∩gB + F n 2,B\g -1 B + F n 2,B\gB ,we get that lim n g • F n -F n 2 2 |gB\B| |B| , and the proof of Corollary G gives a contradiction.

  Then the left translation actions Γ (G, m G ) and Λ (H, m H ) are orbit equivalent if and only if there is a topological isomorphism δ : G → H such that δ(Γ) = Λ.

	Remark 1.8. If Γ < G is a countable subgroup that contains Γ, then Theorem A implies that the
	action Γ	(G, m

G ) has local spectral gap. Hence, Corollary F remains valid if Γ is replaced by Γ.

  we see that G is in fact defined over k. Now, note that in order to check item (1) of Proposition 3.2 for a subgroup H < G, it suffices to check it for the closure of Γ ∩ H (in the real topology). This shows that we only have to deal with proper closed subgroups H < G which are non-discrete in G and such that Γ ∩ H is dense in H. But if H is such a subgroup, then its Zariski closure H ⊂ G is a proper algebraic subgroup which is defined over k, because Γ ∩ H ⊂ GL d (k) is a Zariski dense subgroup of it. Hence the Lie algebra h ⊂ g of H is a non-trivial proper subspace of g defined over k which is globally invariant under H but not under G, because G is simple. Altogether, we find that the line in dim h j=1 g(k) corresponding to the subspace h

  This finishes the proof of Claim 4.End of proof of Claim 2. By combining Claims 3 and 4, and using that |T | (2M -1) -1 , we get

	(3.6)	|A| (2|T |)
		As
	g ε 1 1 , g	ε q+1

q+1 , ..., g εn n can each take at most 2|T | values, we get that |C| [4(2M -2) ] n 12 (2|T |) n+2-n 12 .

  6.4. Proof of Corollary C. Let Γ, G, H and B ⊂ G/H be as in the statement of Corollary C. Recall that the measure m G/H on G/H arises from a rho-function for the pair (G, H) (see [BdHV08, Theorem B.1.4.]). Thus, there exists a continuous function ρ : G → R *Of course, equality 6.9 holds more generally for any function f ∈ L 1 (G) with compact support. The measure m G/H is not necessarily G-invariant, but the function ρ allows to determine the translates of g • m G/H (see[START_REF] Bekka | Kazhdan's property (T)[END_REF] Lemma B.1.3]):

	(6.10)	d(g • m G/H ) dm G/H	(xH) =	ρ(gx) ρ(x)
				+ such that
	(6.9)	f (x)ρ(x)dx =
	G	G/H H

f (xh)dh dm G/H (xH), for all f ∈ C c (G).

  3) The translation action Γ (G, m G ) has local spectral gap with respect to a measurable set B ⊂ G with compact closure and non-empty interior. (4) The translation action Γ (G, m G ) is strongly ergodic.

	Remark 7.2. Suppose that G is compact. It is clear that (1) =⇒ (2). Further, it is well-known that
	(2) ⇐⇒ (3) and (3) =⇒ (4) (see Theorem 7.3). The implication (4) ⇐⇒ (3) was established recently
	in [AE10, Theorem 4]. Let us also explain why (2) =⇒ (1). Note that if (2) holds, then (3) does as
	well, hence Γ is non-amenable. Since the action Γ

  Then we have that B ⊂ xC and hence |A n ∩ B| |A n ∩ xC|. Defining t n = |A n ∩ B|/|C| we see that t n |A n ∩ D|/|C| ε. By (7.6) we get p n

	1 t = 1 e , we can find ε ∈ (0, 1) such that
	(7.7)	(1 -t) [ 1 t ] <	1 2	, for all t ∈ (0, ε].
	Since lim
					α tn	α[ 1 tn ],
	where α = κ|B| 2τ |C| . By using (7.7) we deduce that
	1 -1 -	|C| |A n ∩ xC|	pn

n→∞ |A n ∩ D| = 0, we can find N such that |A n ∩ D|/|C| ε, for all n N . Fix n N and let p n := |B| 2τ |C| |C| |An∩D| . Also, let x ∈ B.

which proves Claim 6.

Finally, put d 1 = min{c, 1 4 } 7C and d 2 = log(1+η) 12C . Then Claim 6 implies that d 1 , d 2 > 0 satisfy the conclusion of Theorem 3.1.

2 -flattening

A key step in Bourgain and Gamburd's remarkable strategy [START_REF] Bourgain | Gamburd: Uniform expansion bounds for Cayley graphs of SL2(Fp)[END_REF] for proving spectral gap is the so-called 2 -flattening lemma. In [START_REF] Bourgain | On the spectral gap for finitely-generated subgroups of SU (2)[END_REF] and [START_REF] Bourgain | Gamburd: A spectral gap theorem in SU (d)[END_REF], Bourgain and Gamburd established a flattening lemma for probability measures on SU (2) and SU (d), d 2, respectively. Bourgain and Yehudayoff then proved a flattening lemma for probability measures on SL 2 (R) whose support is large but "controlled" [START_REF] Bourgain | Yehudayoff: Expansion in SL2(R) and monotone expanders[END_REF]. All of these results rely on product theorems for the respective Lie groups. In an important recent development, de Saxcé obtained a product theorem for arbitrary connected simple Lie groups [dS14]. This allowed Benoist and de Saxcé [START_REF] Benoist | Saxcè: A spectral gap theorem in simple Lie groups[END_REF] to extend the flattening lemmas of [START_REF] Bourgain | On the spectral gap for finitely-generated subgroups of SU (2)[END_REF][START_REF] Bourgain | Gamburd: A spectral gap theorem in SU (d)[END_REF] to any compact connected simple Lie group.

In this section, we first note that the product theorem of [dS14] allows to derive a flattening lemma in the spirit of [START_REF] Bourgain | Yehudayoff: Expansion in SL2(R) and monotone expanders[END_REF]Lemma 4.1] for arbitrary connected simple Lie groups.

Lemma 4.1 ( 2 -flattening, [START_REF] Benoist | Saxcè: A spectral gap theorem in simple Lie groups[END_REF]). Let G be a connected simple Lie group with trivial center. Given α, κ > 0, there exist β, γ > 0 such that the following holds for any δ > 0 small enough.

Suppose that µ is a symmetric Borel probability measure on G such that (1) supp(µ) ⊂ B δ -β (1), (2) µ * P δ 2 δ -α , and (3) (µ * µ)(H (ρ) ) δ -γ ρ κ , for all ρ δ and any proper closed connected subgroup H < G.

Then µ * µ * P δ 2 δ γ µ * P δ 2 .

Lemma 4.1 follows by adapting the proof of [START_REF] Benoist | Saxcè: A spectral gap theorem in simple Lie groups[END_REF]Lemma 2.5] in order to deal with non-compact Lie groups G and measures µ with large controlled support (in the sense of (1)). Nevertheless, for completeness, we include the details of proof in the Appendix.

For now, we assume this lemma, and continue towards proving our main results. More precisely, by applying Lemma 4.1 repeatedly we obtain: Corollary 4.2. Let G be a connected simple Lie group with trivial center, and d 1 , d 2 > 0 be given. Then for every α > 0, there exist ε 0 > 0 and c 0 > 0 such the following holds.

Let 0 < ε < ε 0 and µ be a Borel probability measure on G such that supp(µ) ⊂ B ε (1). Assume that for any δ > 0 small enough we have µ * 2n (H (δ) ) δ d 1 , for any proper connected closed subgroup

Then for any δ > 0 small enough we have µ * n * P δ 2 δ -α , for any integer n c 0

Proof. Let α > 0. By Lemma 4.1 there are β, γ > 0 such that for any δ > 0 small enough the following holds: if ν is a symmetric Borel probability measure on G which satisfies (a) supp(ν) ⊂ B δ -β (1), and

4 , for all ρ δ and any proper closed connected subgroup H < G, (4) The action Γ (X, µ) has spectral gap. (5) The action Γ (X, µ) is strongly ergodic.

Then conditions (1)-( 4) are equivalent and they all imply condition (5).

The equivalence of (1) and ( 2) is due to Rosenblatt [START_REF] Rosenblatt | Uniqueness of invariant means for measure-preserving transformations[END_REF]Theorem 1.4]. The equivalence of (1), (3) and (4), and the fact that (1) implies (5) are due to Schmidt [Sc81, Propositions 2.2 and 2.3] (see also [START_REF] Furman | Sharp ergodic theorems for group actions and strong ergodicity[END_REF]Section 5], where a gap from [START_REF] Schmidt | Amenability, Kazhdan's property T, strong ergodicity and invariant means for ergodic groupactions[END_REF] is fixed).

The main goal of this section is to generalize Theorem 7.3 to arbitrary measure preserving actions.

In order to state our result, we need to introduce some notation:

Notation 7.4. Let Γ (X, µ) be a measure preserving action of a countable group Γ on a standard measure space (X, µ). Let B ⊂ X be a measurable set. We denote by

• C B (X) the family of measurable subsets C ⊂ X for which we can find g 1 , ..., g n ∈ Γ such that C ⊂ ∪ n i=1 g i B, almost everywhere, and by

(X, µ) be an ergodic measure preserving action of a countable group Γ on a standard measure space (X, µ). Let B ⊂ X be a measurable set with 0 < µ(B) < ∞. Consider the following conditions:

(2) If a sequence {A n } of measurable subsets of X satisfies that µ(A n ∩ B) > 0, for all n, and lim

(3) If a sequence ϕ n ∈ L 1 (X, µ) of positive functions satisfies B ϕ n dµ = µ(B), for all n, and lim

(4) The action Γ (X, µ) has local spectral gap with respect to B. (5) The action Γ (X, µ) is strongly ergodic.

Then conditions (1)-(4) are equivalent and they all imply condition (5).

Theorem 7.6 is motivated in part by Margulis' proof of [START_REF] Margulis | Finitely-additive invariant measures on Euclidian spaces[END_REF]Theorem 3]. Note that in the case µ(X) = 1 and B = X, it recovers Theorem 7.3.

Proof. Let lim n be a bounded linear functional on ∞ (N) which extends the usual limit.

(1) =⇒ (2). Assume that (1) holds and (2) is false. Let S be the set of L 0 for which there exists a sequence {A n } of measurable subsets of X such that µ(A n ∩ B) > 0, for all n, lim n µ((gA n ∆A n ) ∩ B)/µ(A n ∩ B) = 0, for all g ∈ Γ, and lim n µ(A n ∩ B) = L. It is easy to see that S ⊂ [0, ∞) is a non-empty closed set. We denote by the minimum of S and by {A n } the corresponding sequence of measurable subsets of X. Since (2) is false, < µ(B).

We define a positive linear functional Φ :

We claim that Φ is well-defined and Γ-invariant. To this end, let f ∈ L ∞ B (X, µ) and g ∈ Γ. Denote by A the support of f . Since A ∈ C B (X), we can find g 1 , ..., g k ∈ Γ with A ⊂ ∪ k i=1 g i B.

Definition 7.9. Let Γ X be an action of a group Γ on a set X. Denote by P(X) the power set of X and let B, C ∈ P(X). We say that B is Γ-equidecomposable to C if we can decompose

for all 1 i < j k, and there exist g 1 , ..., g k ∈ Γ such that g i B i = C i , for all 1 i k. In this case, we write B ∼ C. Also, we write B C if B ∼ C 0 , for some subset C 0 ⊂ C. More generally, if m and n are positive integers, we say that mB is equidecomposable to nC (and write mB ≺ nC) if we can decompose B in m ways and use translations with elements of Γ to rebuild n copies of C. Note that the following cancellation law holds: if mB ∼ mC, then B ∼ C (see [START_REF] Lubotzky | Discrete Groups, Expanding Graphs and Invariant Measures[END_REF]page 13]). With this terminology, B is said to be paradoxical if B ∼ 2B.

The proof of Proposition 7.8 relies on the following two lemmas, the first of which is due to Breuillard and Gelander.

Lemma 7.10. [START_REF] Breuillard | A topological Tits alternative[END_REF] Let G be a l.c.s.c group and Γ < G be a countable dense subgroup. Assume that Γ ∩ G 0 is non-amenable, for every open subgroup G 0 < G.

Then at least one of the following two conditions holds:

(1) Γ contains a non-abelian free subgroup which is not discrete in G.

(2) There exists a compact subgroup K < G such that Γ ∩ K is non-amenable.

Proof. Let G 0 be the connected component of the identity in G. Then G/G 0 is a totally disconnected lcsc group and therefore admits an open compact subgroup L. Since G 0 := L•G 0 is an open subgroup of G, it is enough to prove the conclusion of the lemma for the inclusion Γ ∩ G 0 < G 0 . Thus, after replacing G by G 0 , we may assume that G is a compact extension of a connected group.

By the structure theory of locally compact groups [MZ55, Theorem 4.6], we can find a compact normal subgroup K < G such that H := G/K is a real Lie group. By replacing G with an open subgroup, we may assume that H is connected. Denote by p : G → H the quotient homomorphism. Let h be the Lie algebra of H and consider the adjoint homomorphism q :=Ad: H → GL(h). Since H is connected, the kernel of q is equal to the center Z(H) of H.

By applying the topological Tits alternative [BG04, Theorem 1.3] to the inclusion of q(p(Γ)) into GL(h) ∼ = GL dim(H) (R) (endowed with the standard Euclidean topology), we are in one of two cases: (i) q(p(Γ)) contains a free dense subgroup, or (ii) q(p(Γ)) contains an open solvable subgroup Σ.

In case (i), there is a free subgroup

In case (ii), we have that q -1 (Σ) is an open subgroup of p(Γ). Since H is connected and p(Γ) < H is dense, it follows that q -1 (Σ) = p(Γ). Since the kernel of q is equal to Z(H), we get that p(Γ)/(q -1 (Σ) ∩ Z(H)) ∼ = Σ. Given that Σ is solvable and Z(H) is abelian, we conclude that p(Γ) is amenable. Since Γ is non-amenable, we derive that Γ ∩ K is non-amenable, which proves (2).

Lemma 7.11. Let G be a lcsc group and Γ < G be a countable dense subgroup such that Γ ∩ G 0 is non-amenable, for every open subgroup G 0 < G.

Then any sets B, C with compact closures and non-empty interiors are Γ-equidecomposable. Let G be a connected simple Lie group with trivial center and denote by g its Lie algebra. The goal of this appendix is to prove Lemma 4.1. To this end, by relying on results from [START_REF] De Saxcé | A product theorem in simple Lie groups[END_REF][START_REF] Tao | Product set estimates for non-commutative groups[END_REF] and following closely the proof of [BdS14, Lemma 2.5], we first prove the following 2 -flattening lemma.

In order to do so, it will be more convenient to work with an invariant metric on G, rather than the . 2 -metric used in the rest of the paper. We therefore fix an Euclidean structure on g, and endow G with the corresponding left-invariant Riemannian metric, denoted by d.

For further reference, we note that there is a constant C > 1 such that

This is clear if x belongs to a small enough neighborhood V of the identity. On the other hand, if x ∈ V , then one can easily prove such an inequality by using the KAK decomposition of G (see [LMR00, Section 3] for details).

Let δ > 0. We denote by B(x, δ) = {y ∈ G|d(x, y) < δ} the open ball of radius δ centered at x ∈ G, and by A [δ] = ∪ x∈A B(x, δ) the δ-neighborhood of a set A ⊂ G, both with respect to the metric d.

For a probability measure µ on G, we denote µ δ = µ * Q δ , where

Lemma A.1 ( 2 -flattening, [START_REF] Benoist | Saxcè: A spectral gap theorem in simple Lie groups[END_REF]). Let G be a connected simple Lie group with trivial center. Given α, κ > 0, there exists ε > 0 such that the following holds for any δ > 0 small enough:

Suppose that µ is a symmetric Borel probability measure on G such that

(1) supp(µ) ⊂ B(1, ε log 1 δ ), (2) µ δ 2 δ -α , and (3) (µ * µ)(H [ρ] ) δ -ε ρ κ , for all ρ δ and an proper closed connected subgroup H < G.

Notation. We use the notation O(ε) to denote a positive quantity which is bounded by Cε, for some constant C > 0 depending only on G. We also use the notation φ ψ for functions φ, ψ : (0, ∞) → (0, ∞) to mean the existence of a constant C depending only on G such that φ(δ) Cψ(δ), for any small enough δ > 0. If φ ψ and ψ φ, we write φ ψ.

A.1. Ingredients of the proof of Lemma A.1. The proof of Lemma A.1 relies on Bourgain and Gamburd's strategy [START_REF] Bourgain | Gamburd: Uniform expansion bounds for Cayley graphs of SL2(Fp)[END_REF][START_REF] Bourgain | On the spectral gap for finitely-generated subgroups of SU (2)[END_REF]. In order to implement this strategy, we use de Saxcé's product theorem [dS14, Theorem 3.9]. Recall that if A is a subset of G and δ > 0, then N (A, δ) denotes the least number of open balls of radius δ needed to cover A.

Theorem A.2 (product theorem, [dS14]). Let G be a simple Lie group of dimension d. Then there exists a neighborhood U of the identity in G such that the following holds.

Given α ∈ (0, d) and κ > 0, there exist ε 0 = ε 0 (α, κ) and τ = τ (α, κ) > 0 such that, for any δ > 0 small enough, if A ⊂ U is a set satisfying

(1) N (A, δ) δ -d+α-ε 0 , (2) N (A, ρ) ρ -κ δ ε 0 , for all ρ δ, and

(3) N (AAA, δ) δ -ε 0 N (A, δ), then A is contained in a neighborhood of size δ τ of a proper closed connected subgroup of G.

To prove Lemma A.1, we will also need Tao's non-commutative Balog-Szemerédi-Gowers Lemma [Ta06, Theorem 6.10]. If A, B are subsets of a metric group G and δ > 0, then the δ-multiplicative

The following inequality will be used in the proof of Lemma A.1

, which together prove the desired inequality.

Recall that a subset H ⊂ G is called a K-approximative subgroup, for some K > 1, if it is symmetric and there is a symmetric set X ⊂ HH of cardinality at most K such that HH ⊂ XH.

Theorem A.3 (non-commutative Balog-Szemerédi-Gowers lemma, [START_REF] Tao | Product set estimates for non-commutative groups[END_REF]). Let G be a Lie group endowed with a left-invariant Riemannian metric. Then there exist constants c > 0 and R > 0 such that the following holds for any δ ∈ (0, 1) and K 2.

Suppose that A, B are non-empty subsets of G contained in B(1, 1) such that

Then there is a K c -approximate subgroup H of G and elements x, y ∈ G such that

A final ingredient in the proof of Lemma A.1 is an approximation of the measure µ δ by dyadic level sets [START_REF] Lindenstrauss | Saxcé: Hausdorff dimension and subgroups of SU (2)[END_REF]. A family of sets {A i } i∈I is called essentially disjoint if there is a constant C such that the intersection of more than C distinct sets A i is empty.

Lemma A.4. [START_REF] Lindenstrauss | Saxcé: Hausdorff dimension and subgroups of SU (2)[END_REF] Let µ be a Borel probability measure on G. Let δ ∈ (0, 1) and C be a maximal δ-separated subset of G.

). Then we have the following

(1) at most O(1) log 1 δ of the sets A i are non-empty, (2) A i is an essentially disjoint union of balls of radius δ, for all i 0, (3)

Proof. These assertions follow from [LdS13, Lemma 4.4], but for completeness, we include a proof.

Since and similarly µ 2δ (x) µ 3δ (y). Assuming µ δ (y) > 0, let x ∈ C with y ∈ B(x, δ). Then µ 2δ (x) > 0, hence x ∈ C i and y ∈ A i , for some i 0. This implies that µ δ (y)

On the other hand, if y ∈ A i , for i > 0, then there is x ∈ C such that y ∈ B(x, δ) and µ 2δ (x) > 2 i-1 . Hence, 2 i 1 A i (y) = 2 i < 2µ 2δ (x) µ 3δ (y). Since the balls {B(x, δ)} x∈C are essentially disjoint, y belongs to O(1) of the sets {A i } i>0 and it follows that i>0 2 i 1 A i (y) µ 3δ (y). This proves (3).

Lemma A.5. Let a > 0 and µ be a Borel probability measure on G.

, thus proving the first inequality. To prove the second inequality we may assume a > 1. Then for any x ∈ G we have

Thus, µ aδ 2 µ δ 2 . Since µ aδ 2 µ δ 2 by the above, we are done.

A.2. Proof of Lemma A.1. Assume that µ δ * µ δ 2 > δ ε µ δ 2 , for some ε > 0. Following closely the proof of [BdS14, Lemma 2.5], we will reach a contradiction for any ε small enough.

Let U be the neighborhood of 1 ∈ G provided by Theorem A.2, and let 0 < r < 1 with B(1, r) ⊂ U .

Let R be the constant given by Theorem A.3, and {A i } 0 i log 1 δ be the sets given by Lemma A.4. Let C > 1 the constant appearing in inequality (A.1). By using (A.1) one easily checks that (A.2) d(x -1 , y -1 ) Ce 2Cd(x,1) (e Cd(x,y) -1), for any x, y ∈ G. ε) , A i can be covered by at most δ -O(ε) sets of diameter at most δ 3Cε 2 . Since A i is a union of balls of radius δ, and δ δ 3Cε 2 , for δ small enough, we can decompose

, where each set A i,k is the union of some of the balls of radius δ that make up A i , and has diameter at most δ 3Cε . Moreover, by (A.2) the diameter of A -1 i,k is at most Ce 2C(ε log 1 δ +3δ) (e Cδ 3Cε -1) δ Cε . Thus, for δ small enough, A i,k has diameter at most min{1, r} andA -1 i,k has diameter at most 1, for all k. Before continuing, let us also note that

Since the sum on the right contains δ -O(ε) terms, we can find 0 i, j log 1 δ and 1 k, l δ -O(ε) such that if we denote A i = A i,k and A j = A j,l , then

This implies that

. Since A i is an essentially disjoint union of balls of radius δ, we have |A i | δ d N (A i , δ). Altogether, we deduce that

.

Since A i and A j -1 have diameters at most 1, we can find g, h ∈ B(1, ε log 1 δ + 3δ + 1) such that gA i ⊂ B(1, 1) and A j h ⊂ B(1, 1). On the other hand, combining the last inequality with A.3 yields

By applying Theorem A.3 to gA i and A j h, we deduce the existence of a δ -O(ε) -approximate subgroup H ⊂ B(1, R) and elements z, t ∈ B(1, R) such that N (H, δ) δ -O(ε) N (gA i , δ)

1 2 , N (zH ∩gA i , δ) δ O(ε) N (gA i , δ) and N (Ht∩A j h, δ) δ O(ε) N (A j h, δ). Let v = g -1 z and w = th -1 . By using (A.3) we further get that

The next claim allows us to replace H with its 4δ-neighborhood:

Claim. H := H [4δ] satisfies the following:

(1)

Proof of the claim.

(1) Recall that there is a subset

) by (A.7), we get that vH intersects at least δ O(ε) N (A i , δ) of the balls {B(x, δ)} x∈C i . Thus, v H ∩ A i = (vH) [4δ] ∩ A i contains at least δ O(ε) N (A i , δ) of the balls {B(x, 3δ)} x∈C i . Since the balls {B(x, δ)} x∈C i and hence the balls {B(x, 3δ)} x∈C i are essentially disjoint, v H ∩A i must contain at least δ O(ε) N (A i , δ) disjoint balls from the collection {B(x, 3δ)} x∈C i .

On the other hand, for every x ∈ C i we have that µ 2δ (x) > 2 i-1 and hence

(2) Let X ⊂ G be a set of cardinality δ -O(ε) such that HH ⊂ HX. Since H ⊂ B(1, R) and R is an absolute constant, by using (A.2) it follows that H H ⊂ (HH) Let us show that for ε > 0 small enough, the set H2 ∩ U satisfies the assumptions of Theorem A.2. Firstly, using (2) above in combination with (A.5) and (A.6) we get that

Secondly, since A i has diameter at most r, we get that

In combination with (1) from the claim, it follows that

δ -ε ρ κ . By using the fact that µ and Q δ are symmetric, we get that

Since this holds for any ball of radius ρ that intersects U , in combination with (A.9) it gives that (A.10) N ( H2 ∩ U, ρ) δ O(ε) ρ -κ , for all 1 ρ δ.

Thirdly, let C ⊂ H be a set of diameter at most r and fix x ∈ C. Since B(1, r) ⊂ U , we get that

Equations (A.8), (A.10), (A.11) together guarantee that we are in position to apply Theorem A.2 to H2 ∩ U . Thus, there is a proper closed connected subgroup

On the other hand, by using (A.9) and reasoning similarly to the above we conclude that

Since the hypothesis implies that µ * µ(L [δ τ +O(1)δ] ) δ -ε (δ τ +O(1)δ) κ , it is now clear that choosing ε > 0 small enough yields a contradiction. To finish the proof, assume that µ additionally satisfies µ * P δ 2 δ -α , and (µ * µ)(H (ρ) ) δ -γ ρ κ , for all ρ δ and any proper closed connected subgroup H < G, for some γ > 0.

By using (a), (c) and Lemma A.5 we get that

δ -(γ+κO(β)) ρ κ , for all ρ δ and any proper closed connected subgroup H < G.

If β, γ > 0 are chosen small enough, then Theorem A.1 implies that µ δ * µ δ 2 < δ ε µ δ 2 . Moreover, if β, γ are small enough, then by combining this inequality with (a) and (b) we derive that

This concludes the proof of Lemma 4.1.