Uri Bader

Charles Frances

Karin Melnick

An embedding theorem for automorphism groups of Cartan geometries

Introduction

The main motivation of this paper arises from classical questions about actions of Lie groups preserving a geometric structure: which Lie groups can act on a manifold preserving a given structure, and which cannot? Which algebraic properties of the acting group have strong implications on the geometry of the manifold, such as local homogeneity, or on the dynamics of the action? These questions have been studied extensively by many authors ( [START_REF] Adams | Stuck: The isometry group of a compact Lorentz manifold I[END_REF], [START_REF] Adams | Stuck: The isometry group of a compact Lorentz manifold II[END_REF], [DA], [DAG], [Gr], [Kow], [START_REF] Zeghib | Isometry groups and geodesic foliations of Lorentz manifolds I: Foundations of Lorentz dynamics[END_REF], [START_REF] Zeghib | Isometry groups and geodesic foliations of Lorentz manifolds II: Geometry of analytic Lorentz manifolds with large isometry groups[END_REF], [START_REF] Zimmer | Automorphism groups and fundamental groups of geometric manifolds[END_REF], [START_REF] Zimmer | On the automorphism group of a compact Lorentz manifold and other geometric manifolds[END_REF] among others). These papers primarily prove results in the setting of either rigid geometric structures of algebraic type, which were defined by M. Gromov in [Gr], or of Q-structures of finite type, which were introduced by Cartan.

Our aim in this paper is to develop new tools for studying the kind of questions mentioned above, in the setting of Cartan geometries. This notion of geometric structures was developed by E. Cartan in the first half of the last century. A Cartan geometry infinitesimally models a manifold M on a homogeneous space G/P , where G is a Lie group and P is a closed subgroup. The geometry is flat if and only if M is locally modeled on G/P -that is, M is a (G, G/P )-manifold. The framework of Cartan geometries offers a very satisfactory degree of generality, since it essentially comprises all classical rigid geometric structures, including pseudo-Riemannian metrics, conformal pseudo-Riemannian structures, non-degenerate CR structures, and projective structures.

The main results of this paper are an embedding theorem (theorem 1.2) comparable to Zimmer's well-known embedding theorem for automorphism groups of Q-structures; its algebraic consequences on automorphism groups (theorem 1.3); its consequences for parabolic geometries admitting automorphism groups of high rank (theorem 1.5); and its consequences for geometries additionally equipped with a finite continuous volume form (theorems 1.6 and 1.7). We now discuss in detail these results.

Statement and discussion of main theorem

The Lie algebra of a Lie group G is denoted g. The automorphism group Aut(M, B, ω) consists of the bundle automorphisms of B preserving ω. The diffeomorphisms of M that lift to elements of Aut(M, B, ω) will be denoted Aut M . When the action of G on G/P has finite kernel, a very mild assumption that we will always make in this paper, it can be shown that the projection from Aut(M, B, ω) to Aut M also has finite kernel. The automorphisms of a framing on a manifold form a Lie group (see [Ko] I.3.2), from which it follows that Aut M is a Lie group.

Cartan geometries are closely related, albeit in a nuanced way, to Q-structures of finite type. Kobayashi proved that for Q of finite type, the automorphism group of any Q-structure on any M is a Lie group. For such structures, he constructs a principal subbundle B of the frame bundle of some order, equipped with a framing preserved by all automorphisms of the Q-structure.

His construction thus associates to a Q-structure of finite type a Cartan geometry with the same automorphism group, although the associated Cartan geometry is not canonical. Our theorems on automorphism groups of Cartan geometries thus apply also to automorphism groups of Q-structures of finite type.

For many differential-geometric structures, including pseudo-Riemannian metrics, conformal structures, and, more generally, parabolic structures [CS], there is a canonical associated Cartan geometry modeled on some G/P , and in fact a bijective correspondence between Cartan geometries modeled on G/P and such structures. In this case, the automorphism group of the structure coincides with the automorphism group of the associated (M, B, ω).

The problem of associating a canonical Cartan geometry to a geometric structure is called the equivalence problem (see [Shar]). Many interesting examples can be found in [CS], [CM], [Shar], [Ta]. Some basic examples are given in section 1.3.

Conventions and notations:

The Lie algebra of a Lie group will be denoted by the same symbol printed in the fraktur font; for example, the Lie algebra of G is denoted g.

Throughout the article, M is a connected manifold, and (M, B, ω) is a Cartan geometry modeled on X = G/P . We will make the following global assumptions.

• The pair (G, P ) is such that the G-action on G/P has finite kernel; in practice this assumption is not restrictive.

• The image of P under Ad, denoted Ad g P , is Zariski closed in Aut g.

• For a Lie subgroup H < Aut M , the kernel of Ad : H → Aut h is amenable; note that for H connected, ker Ad = Z(H) is always amenable.

For a subgroup S < H, the group SAd is the Zariski closure of Ad S in Aut h, and Sd is the maximal Zariski closed subgroup of SAd with no proper cocompact algebraic normal subgroups; see section 3 below for details. The main theorem of the paper is the following.

Theorem 1.2. Let H be a Lie subgroup of Aut M . Let S be a subgroup of H preserving a finite measure on M . Then there are

• Š < Ad g P an algebraic subgroup and

• ι : h → g a linear monomorphism
such that ι(h) is Š-invariant, and ι intertwines Š ι(h) and Sd .

Section 4 contains a more detailed statement, in theorem 4.1.

Theorem 1.2 is similar to the embedding theorem of Zimmer [START_REF] Zimmer | On the automorphism group of a compact Lorentz manifold and other geometric manifolds[END_REF], which says that when a connected simple group H acts on a manifold M preserving a finite volume and a Q-structure, with Q algebraic, then the standard representation of Q contains the adjoint representation of H. In particular, Q contains a subgroup locally isomorphic to H. Zimmer's result is proved using algebraic hulls of measurable cocycles and the Borel density theorem. An important first step in the proof is to show that a finite-measure-preserving action of a simple group has discrete stabilizers almost everywhere. In this case, the Lie algebra h linearly embeds in the tangent space at a point, on which Q acts via its standard representation.

Our result here does not require that H be simple, nor be connected, nor preserve a finite measure on M , nor that the action have any discrete stabilizers. The subgroup S < H is not even required to be a Lie subgroup.

The Cartan connection gives linear embeddings of h in g at every point of B. Then measure-theoretic and algebraic arguments, including the Borel density theorem, give the existence of Š < Ad g P restricting to Sd on some image of h. Unfortunately, the theorem gives no information for H discrete.

Of course, theorem 1.2 has nice consequences when H preserves a finite measure (see theorems 1.6, 1.7, 1.8 and 1.9). Only the subgroup S, however, is required to be measure-preserving. When M is compact, for example, the theorem applies to every amenable subgroup S < H. In this way, the theorem yields information on automorphisms of conformal structures, projective structures, and other parabolic structures that do not determine a volume form (see theorems 1.3 and 1.5).

When H is a connected, simple subgroup of Aut M , and H preserves a volume on M and a Q-structure of finite type, then we can obtain the conclusion of Zimmer's theorem from theorem 1.2. On the other hand, Zimmer's embedding theorem deals with arbitrary Q-structures with Q algebraic; in particular, it does not require finite type, so there may not be any associated Cartan geometries. With respect to this comparison, it could be said that we have exchanged the rigidity of simple group actions for rigidity of the structure supporting the action.

Bounds on automorphism groups

Here we deduce from the main theorem coarse information on the possible groups of automorphisms of (M, B, ω) when M is compact.

For a linear group R < GL(R n ), we will define the real rank (resp. the algebraic real rank ), denoted rk R (resp. rk alg R), to be the maximal dimension of a simultaneously R-diagonalizable subgroup of R (resp. of the Zariski closure of a simultaneously R-diagonalizable subgroup of R). The inequality rk R ≤ rk alg R always holds. It can be strict, for example, for

R ∼ = Γ ⋉ R n ,
where Γ is a lattice in a semisimple group G < GL(R n ) with rk G > 0 (see [PR], [Mo]). When R is semisimple, then rk(Ad R) = rk alg (Ad R) is the usual R-rank of R; in fact, for any faithful representation ρ of R, the real rank rk ρ(R) = rk alg ρ(R), and agrees with the usual R-rank of R. For R an abstract semisimple group, we will sometimes write rk R below without reference to a specific representation.

For a Lie group R, we define the nilpotence degree n(R) to be the maximum nilpotence degree of a connected nilpotent subgroup of R.

A rigidity result for parabolic geometries

One class of interesting classical geometries for which the equivalence problem has been solved is that of parabolic geometries (see for example [CS]).

A Cartan geometry (M, B, ω) modeled on X = G/P is said to be parabolic when G is a real semisimple Lie group and P is a parabolic subgroup of G.

The main examples of such geometries follow.

• Conformal pseudo-Riemannian structures.

A conformal pseudo-Riemannian structure of signature (p, q) on a manifold M is the data of a conformal class of metrics

[g] = {e σ g | σ ∈ C ∞ (M )}
where g is a metric of signature (p, q). These structures are in canonical correspondence with Cartan geometries modeled on O(p+1, q +1)/P , where

P ∼ = (R * ×O(p, q)) ⋉ R p+q
is a maximal parabolic subgroup of O(p + 1, q + 1), and the connection ω satisfies certain normalization conditions.

• Nondegenerate CR-structures.

These structures model real hypersurfaces in complex manifolds. A nondegenerate CR-structure of type (p, q) on a (2m + 1)-dimensional manifold M , where p + q = m, is the data of a contact subbundle E ⊂ T M equipped with an almost-complex structure J and a conformal class of Hermitian metrics of type (p, q). Such a structure is equivalent to a unique normal Cartan geometry modeled on SU (p + 1, q + 1)/P , where P is the parabolic subgroup of SU (p + 1, q + 1) stabilizing an isotropic complex line in C p+1,q+1 . Again, the connection ω must satisfy some normalizaiton conditions. The equivalence problem for strictly pseudo-convex CR-structures was first solved by E. Cartan in dimension 3, and in the general case in [CM], [Ta], [CS].

• Projective structures.

A projective structure on an n-dimensional manifold M is a family of smooth curves C : I → M , defined locally by an ODE of the form

c ′′ 1 + P 1 (c ′ ) c ′ 1 = c ′′ 2 + P 2 (c ′ ) c ′ 2 = ... = c ′′ n + P n (c ′ ) c ′ n with P k (x) = Σa k ij x i x j for smooth functions a k ij .
Geodesics of a Riemannian or pseudo-Riemannian metric, for example, satisfy such an equation. To such a projective structure corresponds a unique normalized Cartan geometry modeled on the projective space RP n = P GL(n + 1, R)/P , where P is the stabilizer of a line in R n+1 (see [Shar] ch. 8 for details).

We now come to our main result concerning parabolic geometries. Note that such a geometry (M, B, ω) is modeled on G/P with rk(Ad g P ) = rk G. Theorem 1.3 asserts that when a Lie group H acts by automorphisms of (M, B, ω), then rk alg (Ad H) ≤ rk G. When H is connected, and equality rk(Ad H) = rk G holds, a strong rigidity phenomenon occurs. The universal covers of X and G are denoted X and G, respectively.

Theorem 1.5. Let (M, B, ω) be a compact parabolic Cartan geometry, modeled on X = G/P , where G has finite center, and ω is regular. Let H < Aut M be a Lie subgroup.

1. If rk alg (Ad H) = rk G, then there is a Lie algebra embedding h → g.

If H is connected, and rk(Ad

H) = rk G, then M ∼ = Γ\ X, for some discrete subgroup Γ < G.
The regularity condition on the Cartan connection is very mild. For all parabolic geometries for which the equivalence problem has been solved, regularity is one of the normalization conditions on the Cartan connection. See section 6.1 for the definition of regular connection.

Theorem 1.5 is a wide generalization of previous results obtained in [BN] and [FZ] for conformal actions of connected simple groups on compact pseudo-Riemannian manifolds. For parabolic Cartan geometries modeled on G/P with rk G = 1, existence of a nonproper automorphism group implies that M is equivalent to G/P or the complement of a point in G/P . In conformal Riemannian geometry, this statement is a celebrated theorem of Lelong-Ferrand [LF]; it is proved for general rank-one parabolic Cartan geometries in [F].

Automorphism groups preserving a finite volume

By a finite volume on a manifold M , we mean a continuous volume form ν on M , such that M ν < ∞. This section contains results on actions by automorphisms of a Cartan geometry preserving a finite volume (Theorem 1.6 makes the slightly weaker assumption that the action preserves a finite measure of full support). Isometric actions on pseudo-Riemannian manifolds preserve a volume form, which is always finite if the manifold is compact.

Other interesting examples of such actions are affine actions preserving a finite volume. Note that the invariant volume form need only be C 0 .

Recall that Hd is the discompact radical of HAd (see §3). In [DAG], it is conjectured that any smooth, faithful, volume-preserving action of a simple Lie group on a compact manifold is everywhere locally free. Here, we prove:

Theorem 1.6. Let (M, B, ω) be a Cartan geometry, and let H < Aut M be a Lie subgroup preserving a finite measure of full support in M . Then for almost-every x ∈ M , the stabilizer Lie algebra h x is Hd -invariant. If there are only finitely-many Hd -invariant subalgebras of h, then the H-action is locally free on a dense, open, conull subset of M .

This theorem is proved in [START_REF] Zimmer | Semisimple automorphism groups of G-structures[END_REF] for connected semisimple Lie groups with no compact factor. Our conclusion of local freeness here holds for a wider class of groups, including semidirect products H = Γ ⋉ ψ R, where R is a connected Lie group, Γ is a lattice in a semisimple group L, and ψ is the restriction of a nontrivial representation of L in Aut r.

The next theorem says that when a group of automorphisms of a Cartan geometry preserves a finite volume, then existence of an open orbit implies homogeneity; in this case, the action is everywhere locally free. In general, it is an interesting open question for a geometric manifold when existence of an open locally homogeneous subspace implies local homogeneity everywhere ( [DAG], [Du], [Me]).

Recall that HAd denotes the Zariski closure of Ad H, and Hd its discompact radical.

Theorem 2. The action of H is everywhere locally free.

The stabilizers of points in

H o are lattices Γ ⊂ H o .
Essentially all classical rigid geometric structures can be described both as Cartan geometries and rigid geometric structures of algebraic type in the sense of Gromov. The interplay of these two points of view can be fruitful, as the following corollary shows.

Corollary 1.8. Let (M, B, ω) be a C ω Cartan geometry with M simply connected. Assume that there is a natural associated rigid structure of algebraic type S on M with Aut loc M = Aut loc (M, S). Let ν be a C ω finite volume form on M . Let H < Aut M be a Lie subgroup such that HAd = Hd . If H preserves ν and has a dense orbit in M , then the geometry is homogeneous: there exists H ′ < Aut M acting transitively on M and preserving ν.

Homogeneous reductive geometries

Finally, we record a corollary for those Cartan geometries called reductive. A Cartan geometry (M, B, ω) modeled on G/P is reductive when there is an Ad g P -invariant decomposition g = n ⊕ p, in which the restriction of Ad g P to n is faithful. Note that n is in general a subspace of g, not a subalgebra. Classical examples of reductive geometries are pseudo-Riemannian metrics, for which (g, p) = (so(p, q) ⋉ R n , so(p, q)) or linear connections, for which

(g, p) = (gl(R n ) ⋉ R n , gl(n, R))
Corollary 1.9. Let (M, B, ω) be a reductive Cartan geometry, and g = n⊕p an Ad g P -invariant decomposition. Let H < Aut M be a Lie subgroup such that HAd = Hd . If H acts transitively on M preserving a finite volume, then 1. The group HAd is isomorphic to a subgroup Ȟ < P .

2. There is a linear isomorphism ι : h → n with Ad Ȟ-invariant image, which intertwines the representations of HAd on h and of Ȟ on ι(h).

The identity component

H o is isomorphic to a central extension of a subgroup H ′ < Ȟ.

Automorphisms of a Cartan geometry

Let (M, B, ω) be a Cartan geometry modeled on X = G/P , and H < Aut M a Lie subgroup. The H-action on B is free because it preserves a framing, given by the pullback by ω of any basis of g (see [Ko] I.3.2).

This section introduces several important H-equivariant maps from B to algebraic varieties. These maps will be central in the proof of the main theorem.

The curvature map of a Cartan geometry

The simplest example of a Cartan geometry is the triple (X, G, ω G ), where ω G is the Maurer-Cartan form on G, which evaluates to X on the leftinvariant vector field corresponding to X ∈ g. In addition to fulfilling the definition of a Cartan geometry, the Maurer-Cartan form on G satisfies the structural equation:

dω G (X, Y ) + [ω G (X), ω G (Y )] = 0
for every pair of vector fields X and Y on G.

For a general Cartan geometry (M, B, ω), the 2-form

K(X, Y ) = dω(X, Y ) + [ω(X), ω(Y )]
is not zero, and is called the curvature form of ω. It is a fundamental fact that K vanishes if and only if (M, B, ω) is locally isomorphic to (X, G, ω G ) (see [Shar]). Thus, the manifolds M locally modeled on X, also known as manifolds with (G, X)-structures, correspond exactly to flat Cartan geometries (M, B, ω), modeled on X.

The associated curvature map is

κ : B → Hom(Λ 2 (g/p), g) κ b (σ(X), σ(Y )) = K b (ω -1 b (X), ω -1 b (Y ))
Where X, Y ∈ g, and σ is the projection g → g/p. This map is well-defined because K b vanishes as soon as one of its argument is tangent to a fiber of B → M . The map κ is P -equivariant, where P acts on Hom(Λ 2 (g/p), g) via the adjoint: for any p ∈ Ad P and β ∈ Hom(Λ 2 (g/p), g),

(p.β)(u, v) = p-1 .(β(p.u, p.v))
For any b ∈ B, the Cartan connection induces a natural linear map

ι b : h → g ι b (X) = ω b ( d dt 0 ϕ t X .b)
where ϕ t X is the flow along X for time t. The freeness of the H-action implies that ι b is injective. In general, ι b is not a Lie algebra homomorphism. It nearly is, however, when κ b = 0.

Lemma 2.1. Let (M, B, ω) be a Cartan geometry modeled on G/P . If for some b ∈ B, the curvature κ b vanishes on the image of σ•ι b , then -ι b : h → g is an embedding of Lie algebras.

Proof: Let X, Y ∈ h, viewed as Killing fields on B. Recall that dω(X, Y ) = X.ω(Y ) -Y.ω(X) -ω([X, Y ]) Since X is a Killing field for ω, ω(Y (b)) = ω((ϕ t X ) * b Y (b)) Now (X.ω(Y ))(b) = d dt 0 ω(Y (ϕ t X b)) = d dt 0 ω(Y (ϕ t X b)) -ω((ϕ t X ) * b Y (b)) = ω([X, Y ](b)) Similarly, (Y.ω(X))(b) = ω([Y, X](b)) Then κ b (σ • ι b (X), σ • ι b (Y )) = dω(X(b), Y (b)) + [ω(X(b)), ω(Y (b))] = ω([X, Y ](b)) -ω([Y, X](b)) -ω([X, Y ](b)) + [ω(X(b)), ω(Y (b))] = ω([X, Y ](b)) + [ω(X(b)), ω(Y (b))] = ι b ([X, Y ]) + [ι b (X), ι b (Y )] Therefore, vanishing of κ b implies -ι b ([X, Y ]) = [ι b (X), ι b (Y )] = [-ι b (X), -ι b (Y )]
as desired. ♦

Natural equivariant maps to varieties

To an action of H by automorphisms of a Cartan geometry M modeled on G/P can be associated various equivariant maps to algebraic varieties built from the adjoint representations of H and G.

The maps ι

b : h → g for b ∈ B give a map ι : B → Mon(h, g)
where Mon(h, g) is the variety of all injective linear maps from h to g. The map ι is continuous, since ω is so. Observe that H and P have commuting actions on B, as well as on Mon(h, g), by pre-and post-compositions with the adjoint actions. It will be crucial in the sequel that the latter (Ad H × Ad g P )-action is the restriction of an algebraic action, and that Ad g P is algebraic. To a Lie subgroup H < Aut M , can be associated a real-algebraic space U = Mon(h, g) × Hom(∧ 2 (g/p), g)

where H acts trivially on Hom(∧ 2 (g/p), g), and a continuous map φ : B → U , which is continuous and (H × P )-equivariant.

We denote by P b the algebraic subgroup of Ad g P consisting of those p such that p.ι b (h) = ι b (h). There is a natural algebraic homomorphism The main point in the proof of theorem 1.2 will be precisely to show the existence of b ∈ B such that ρ b : Šb → Sd is a surjection.

ρ b : P b → GL(h) ρ b (p) = ι -1 b • p • ι b Now, if S < H is

Background on measure theory

In this section we present the version of the Borel density theorem that figures in the proof of the main theorem 1.2. Primary references for material here on invariant measures for real-algebraic actions are [Zi3, Chapters 2,3] and [Shal]. We will be considering real-algebraic groups and varieties, and all results will be stated in this setting; in fact, we will use the term "algebraic" to mean real-algebraic below.

For G an algebraic group and V an algebraic variety, an algebraic action of G on V is an action given by an algebraic morphism ϕ : G × V → V . For a locally compact group S and a homomorphism ψ : S → G, where G is an algebraic group, we denote the real points of the Zariski closure of ψ(S) by Sψ or S.

The discompact radical G d of an algebraic group G is the maximal algebraic subgroup of G with no cocompact, algebraic, normal subgroups. It is the minimal algebraic subgroup Q containing all algebraic subgroups with no compact algebraic quotients, which exists by the Noetherian property of algebraic subgroups. By convention, we will put

G d = {e} when G = {e}.
Note that G d is characteristic. See [Shal] for more details. Given ψ : S → G, we denote by Sψ d or Sd the discompact radical of Sψ . The formulations of theorems 1.2 and 1.6 in terms of Hd and Sd apply to a wider class of groups than those for which the Zariski closures HAd or SAd have no compact algebraic quotients. Consider the following group. Let

g =    cos θ -sin θ 0 sin θ cos θ 0 0 0 e λ    ∈ GL(3, R) for some θ = 0 modulo 2π, λ = 0. Let H = g ⋉ R 3 ∼ = Z ⋉ R 3 . Then HAd d ∼ = R *
, but for any S < H, the group SAd is either trivial or has S 1 as a direct factor.

Theorem 3.1 ( [START_REF] Shalom | Invariant measures for algebraic actions, Zariski dense subgroups and Kazhdan's property (T)[END_REF]3.11]). Let ψ : S → Aut V , for S a locally compact group and V an algebraic variety. Assume S preserves a finite measure µ on V . Then µ is supported on the set of Sd -fixed points in V .

Proof: [sketch] By an ergodic decomposition argument, it is enough to assume that the measure µ is S-ergodic. It is well known that S-orbits in V are locally closed [START_REF] Zimmer | Ergodic theory and semisimple groups[END_REF]3.1.1], and it follows that µ is supported on a single S-orbit. We claim that µ is S-invariant. Indeed, Zimmer proved that the stabilizers of probability measures are Zariski closed when V is a quasi-projective variety [START_REF] Zimmer | Ergodic theory and semisimple groups[END_REF]3.2.4], but by a theorem of Chevalley every S-orbit is S-equivariantly quasi-projective. It remains to show that every S-invariant measure is supported on the set of Sd -fixed points. The latter group is generated by algebraic 1-parameter groups, isomorphic to either the additive or the multiplicative group of R. The statement is true for these groups, again by applying an ergodic decomposition and restricting to one orbit. The proof follows. ♦ Corollary 3.2. Let ψ : S → Aut V , for S a locally compact group and V an algebraic variety. Suppose S acts continuously on a topological space M preserving a finite Borel measure µ. Assume φ : M → V is an S-equivariant measurable map. Then φ(x) is fixed by Sd for µ-almost-every x ∈ M .

Proof: Follows from the previous theorem, with the measure φ * (µ). ♦

4 Proof of the main theorem 1.2

This section will be devoted to the proof of theorem 1.2, or more precisely to that of theorem 4.1 below which is a detailed version of it.

We consider (M, B, ω), a fixed Cartan geometry modeled on G/P . Recall the global assumption that Ad g P < Aut g is Zariski closed. The notations are those of section 2.2. Recall that π : B → M is the bundle projection. Proof: Observe first of all that the main point is to prove the surjectivity of ρ b at some point. The two other assertions of the theorem follow by the definitions of Šb and ρ b .

As in section 2.2, consider the real-algebraic variety

U = Mon(h, g) × Hom(∧ 2 (g/p), g)
and the

H × P -equivariant map φ = ι × κ : B → U .
The map φ descends to an H-equivariant map between the spaces of Porbits: φ : M = B/P → V = U/P

The Ad g P -action on U is algebraic, and there is a P -invariant stratification

U = U 0 ∪ • • • ∪ U r
such that, for each j ≥ 0, the stratum U j is Zariski open and dense in ∪ i≥j U i , and each quotient space V i = U i /Ad g P is a smooth algebraic variety (see [Ro] or [Gr] 2.2). Because the H-action on U commutes with Ad g P , it preserves the stratification and acts algebraically on each V i .

Denote by µ i the restriction of µ to M i = φ-1 (V i ). It suffices to prove that ρ b is surjective for µ i -almost-every π(b) ∈ M i , for all i. Because M i is Sinvariant for all i, each measure µ i is S-invariant. Given i such that µ i = 0, corollary 3.2 applied to the S-equivariant map φ :

(M i , µ i ) → V i gives that v = φ(m)

Proofs of bounds on automorphism groups

Recall that Hd denotes the discompact radical of the Zariski closure of Ad H.

A Lie subalgebra of End(V ) will be called algebraic when it is the Lie algebra of a Zariski closed subgroup of GL(V ). Also, throughout this section, M is compact.

Proof of theorem 1.3

To prove (1) of theorem 1.3, choose an R-split subgroup of Ad H with Zariski closure S of dimension rk alg (Ad H). Let S = Ad -1 ( S) < H. The kernel of Ad is amenable by assumption ( §1.1), and S is abelian, so S is amenable. There is a finite S-invariant measure on M , and theorem 4.1 gives an algebraic subgroup Šb < Ad g P such that ρ b ( Šb ) = Sd = S.

The Lie algebra of Šb admits a Levi decomposition into algebraic subalgebras šb = r ⋉ u, where u consists of nilpotent endomorphisms, and r is a reductive Lie algebra (see [START_REF] Morris | Ratner's Theorem on Unipotent Flows[END_REF] ch.4). The homomorphism ρ b induces an epimorphism of algebraic algebras dρ b : šb → s. The image of u consists of elements that are both R-split and nilpotent, so that u ⊂ ker dρ b . The algebra r splits into algebraic subalgebras r = z ⊕ l where l is semisimple, and z is an abelian algebra of C-split endomorphisms. Because s is abelian, l ⊂ ker dρ b . The C-split algebraic subalgebra z decomposes as a sum z = z s ⊕ z e where z s consists of R-split elements, and z e of elements with purely imaginary eigenvalues. Since dρ b is a morphism of real algebraic Lie algebras, it respects this decomposition. Therefore, z e ⊂ ker dρ b , and dρ b (z s ) = s. Then we have rk(Ad g P ) ≥ dim z s ≥ dim s = rk alg (Ad H)

To prove (2), let N be a connected nilpotent subgroup of Ad H of maximal degree, and let N be the Zariski closure. Let S = Ad -1 ( N ) ⊂ H. As above, S is amenable, so theorem 1.2 gives an algebraic subgroup Šb such that ρ b ( Šb ) = Sd . The right-hand side is the discompact radical of N .

Lemma 5.1. Let N be a connected nilpotent group GL(R k ). Let Sd be the discompact radical of the Zariski closure N . Then n(s d ) = n(n).

Proof: Let n = n(n). Any n-tuple (u 1 , . . . , u n ) ∈ n n satisfies [u n , • • • [u 2 , u 1 ] • • • ] = 0
Because this is an algebraic condition, it is satisfied by all n-tuples of elements of n. Therefore n(n) = n(n).

Of course n(s d ) ≤ n(n), so it suffices to show n(s d ) ≥ n(n). Write n = r ⋉ u where u is composed of nilpotents, and r is reductive. The algebra r is reductive and nilpotent, so it is abelian. The adjoint representation of r on u is both nilpotent and reductive, so it is trivial. Therefore r is abelian and centralizes u, so that n(n) = n(u). But u ⊂ sd , so n(n) ≤ n(s d ). ♦ Now as above dρ b is a surjective morphism of algebraic Lie algebras šb → sd . Take an algebraic Levi decomposition šb = r ⋉ u and write r = z ⊕ l as above. Since sd is nilpotent, and l is semisimple, again l ⊂ ker dρ b . The kernel of dρ b in u is ad r-invariant, so there is an ad rinvariant complementary subspace V in u. The subspace z ⊕ V is mapped surjectively on sd by dρ b . Let z ′ be the kernel in z of the adjoint restricted to V , and let z ′′ be a complementary subalgebra in z. Then z = z ′ ⊕ z ′′ . Because elements of z ′′ act via the adjoint on V by nontrivial C-split endomorphisms, none of them is nilpotent. Therefore z ′′ ⊂ ker dρ b . Let u ′ be the Lie algebra generated by V . Then the subalgebra z ′ ⊕ u ′ of šb is nilpotent and maps onto sd via dρ b . By lemma 5.1, we conclude:

n(Ad H) = n(n) = n(s d ) ≤ n(z ′ ⊕ u ′ ) ≤ n(Ad g P ) ♦
5.2 Semi-discrete groups: proof of 1.4

Recall that Γ is a lattice in an algebraic group L < GL(V ) with L = L d , and the group H = Γ ⋉ V acts by automorphisms of the compact Cartan geometry M . Note that L is the Zariski closure of Ad H on V = h, by the Borel density theorem.

Let U = Mon(h, g), and let ι : B → U be the H × P -equivariant map defined in section 2.2. Let Q < L be an amenable, algebraic subgroup with

Q d = Q.
Much as in section 4, we seek a Q-fixed point in U/Ad g P .

The Zariski closure L commutes with Ad g P on U because H does. Let

B ′ = L × Γ B be the L-space of equivalence classes of pairs (h, b) ∈ L × B, where (h, b) ∼ (hγ -1 , γ.b). Define an L × P -equivariant extension of ι ι ′ : B ′ → U [(h, b)] → h.ι(b) Let M ′ = L × Γ M = B ′ /P
Note that M ′ is an M -bundle over L/Γ, which has an H-invariant probability measure ν. The space of probability measures on M ′ projecting to ν is a compact convex L-space, so it contains a fixed point for Q; in other words, there is a finite Q-invariant measure µ on M ′ .

Denote by η the H-equivariant map M → U/Ad g P covered by ι. There is an L-equivariant continuous lift

η ′ : M ′ → U/Ad g P
The L-action on U preserves the stratification as in the proof of 4.1 in section 4, so it acts on the smooth quotient varieties

V i = U i /Ad g P , for each stratum U i ⊂ U . Let µ i be the restriction of η ′ * (µ) to V i . Each µ i is Q-invariant. The corollary 3.2 of the Borel density theorem gives a Q-fixed point v = η ′ ([(h, x)]) = h.η(x)
for some h ∈ L and x ∈ M . Then hQh -1 fixes η(x). In other words, for any b ∈ B lying over x ∈ M , the algebraic subgroup P b < Ad g P , when restricted to ι b (V ), contains a subgroup isomorphic to hQh -1 . Let Šb be the algebraic subgroup ρ -1 b (hQh -1 ), so ρ b is an algebraic epimorphism from Šb onto hQh -1 . Now take Q < L to be an R-split algebraic subgroup of dimension rk L. Then following the proof of 1.3 (1) gives

rk L = dim Q ≤ rk (Ad g P )
Similarly, there is a connected, nilpotent, algebraic Q < L of degree n(L), and following the proof of 1.3 (2) gives n(L) ≤ n(Ad g P ).

6 A rigidity result for parabolic geometries 6.1 Parabolic subalgebras, grading, regular connections

In this section, G is a connected semisimple Lie group. Let a ⊂ g be a maximal R-split abelian subalgebra, and Π ⊂ a * the root system. This set of roots admits a basis Φ = {α 1 , ..., α r } of simple roots, such that any root can be written uniquely as a sum n 1 α 1 + ... + n r α r , where n 1 , ..., n r are integers of the same sign. The simple roots determine a decomposition Π = Π + ∪ {0} ∪ Π - where Π + (resp. Π -) denotes the positive (resp. negative) roots-those for which all the integers n i are positive (resp. negative).

For α ∈ Π, the root space

g α = {Y ∈ g | (ad X)(Y ) = α(X) • Y ∀X ∈ a}
The associated root space decomposition is

g = Σ α∈Π -g α ⊕ m ⊕ a ⊕ Σ α∈Π + g α
The subalgebra m is a compact subalgebra, and a ⊕ m is exactly the centralizer of a in g. We denote by A the connected subgroup of G with Lie algebra a, and by M G the subgroup with Lie algebra m; it is compact if and only if Z(G) is finite.

Let Θ ⊂ Φ, and

Θ -= {α = Σn i α i | α i ∈ Φ and n i ∈ Z ≤0 }
The standard parabolic subalgebra associated to Θ is

p = Σ α∈Θ -g α ⊕ m ⊕ a ⊕ Σ α∈Π + g α
For example, taking Θ = ∅ yields the minimal parabolic subalgebra

p = m ⊕ a ⊕ Σ α∈Π + g α For Θ = Φ, one gets p = g.
Let Θ be a proper subset of Φ. Reordering the elements of Φ if necessary, we can assume Φ \ Θ = {α 1 , ..., α m }. For every j ∈ Z, define

Π j = {Σ r i=1 n i α i | n i ∈ Z ≤0 ∀ i or n i ∈ Z ≥0 ∀ i and Σ m i=1 n i = j}
Then put g j = Σ α∈Π j g α , and g j = Σ i≥j g i .

Let k be the greatest integer such that Π k = ∅. Associated to the parabolic subalgebra p are a grading

g = g -k ⊕ • • • ⊕ g -1 ⊕ g 0 ⊕ g 1 ⊕ • • • ⊕ g k
as well as a filtration

g k ⊂ g k-1 ⊂ ... ⊂ g -k = g with [g i , g j ] ⊂ g i+j . Notice that p = g 0 .

Parabolic geometries

Let G be as above, and P < G a parabolic subgroup with Lie algebra p. Let g i , for i = -k, . . . , k, be the subspaces of the associated grading, as above.

A parabolic geometry is a Cartan geometry (M, B, ω) modeled on X = G/P . As illustrated in section 1.3 of the introduction, many interesting examples of geometric structures have an interpretation as a parabolic geometry.

The Cartan connection ω of a parabolic geometry modeled on X = G/P is regular if the corresponding curvature function satisfies for any i, j < 0

κ(g i , g j ) ⊂ g i+j+1
The regularity assumption on a Cartan connection is natural. Indeed, in every case in which the equivalence problem is solved for parabolic geometries, regularity is one of the normalization conditions on the Cartan connection to make it unique.

6.3 Flatness and completeness: proof of 1.5

Let G be a semisimple Lie group and P a parabolic subgroup as above. Throughout this section, (M, B, ω) is a Cartan geometry modeled on X = G/P with ω regular.

Inside a, define the positive Weyl chamber

a + = {X ∈ a | α(X) > 0 ∀ α ∈ Π + }
Let (a k ) be a sequence of A tending to infinity-that is, leaving every compact subset of A. Fix any norm on g, and write

a k = exp(X k ) = exp(||X k ||u k )
where ||u k || = 1. An an asymptotic direction of (a k ) is any cluster point u ∈ a of the sequence (u k ).

The notion of holonomy sequences will be important in the proof below. Let H < Aut M , for (M, B, ω) a Cartan geometry modeled on G/P , let b ∈ B, and

x = π(b). For h k ∈ H such that h k .x → y ∞ ∈ M , a holonomy sequence associated to (b, h k ), is a sequence (p k ) of P such that h k .b.p -1 k → b ∞ ∈ π -1 (y ∞ )
If (p k ) is a holonomy sequence associated to (b, h k ), then all other holonomy sequences associated to (b, h k ) are (l k p k ) with (l k ) a convergent sequence of P . Also, if b ′ = b.p ′ , then any holonomy sequence associated to (b ′ , h k ) is of the form (p k p ′ ), where (p k ) is a holonomy sequence associated to (b, h k ). These observations lead naturally to the following notion of equivalence.

Definition 6.1. Two sequences (p k ) and (p ′ k ) are equivalent if there are a convergent sequence (l k ) in P and q ∈ P such that p ′ k = l k p k q, for all k ∈ N.

Note that a sequence of P and its conjugate by some p ∈ P are equivalent. Also, two converging sequences of P are equivalent.

Definition 6.2. A holonomy sequence at b ∈ B is a holonomy sequence associated to (b, h k ) for some (h k ) in H. For x ∈ M , a holonomy sequence at x is a holonomy sequence at some b ∈ B for which π(b) = x.

Remark that if (p k ) is a holonomy sequence at x, then any sequence equivalent to (p k ) is also a holonomy sequence at x.

If b ∈ B is such that κ b = 0, then κ b ′ = 0 for every b ′ in the same fiber. In the following, by a slight abuse of language, we will say that the curvature function vanishes at x ∈ M , if κ b = 0 for all b in the fiber of x. We can now prove Proposition 6.3. Let H < Aut M be a Lie subgroup, and S < H.

1. If there is b ∈ B such that Ad A ⊂ Šb , then κ b = 0.
2. Suppose there is x ∈ M such that for any u ∈ a + , there is a holonomy sequence (p k ) ⊂ A at x with asymptotic direction u. Then there is an open neighborhood U of x on which the curvature vanishes.

Proof: We begin with the proof of point (2). Let i ≤ j < 0, and l > i + j. Let α ∈ Π i , β ∈ Π j , and ν ∈ Π l be three roots of g. Choose nonzero vectors v ∈ g α and w ∈ g β . Choose u ∈ a + , such that α(u) + β(u) < ν(u) (Such u always exits, since α + β is in Π i+j , hence cannot equal ν). Finally, choose (p k ) a holonomy sequence at x with u for asymptotic direction. Passing to a subsequence if necessary, write

p k = exp(X k ) with lim k→∞ X k ||X k || = u
By definition of (p k ), there is a sequence (h k ) of H and b ∈ B over x such that h k .x → x ∞ ∈ M , and Let u be in a + and (p k ) a holonomy sequence at x with asymptotic direction u.

b k = h k .b.p -1 k → b ∞ ∈ π -1 (x ∞ ) For all k ∈ N (Ad p k ).κ b (v, w) = κ b k ((Ad p k ).v, (Ad p k ).w) = e α(X k )+β(X k ) κ b k (v, w) Since α(u) + β(u) -ν(u) < 0, e -ν(X k ) (Ad p k ).κ b (v,
Let h k ∈ H and b ∈ π -1 (x) such that b k = h k .b.p -1 k → b ∞ ∈ B Choose an open U ⊂ n -= g -k ⊕ • • • ⊕ g -1 containing 0 n -such that exp b is a diffeomorphism from ω -1 b (U ) onto its image; note that π • exp b (ω -1 b (U )) is open in M . For every k ∈ N, h k . exp b (ω -1 b (U )).p -1 k = exp b k (ω -1 b k ((Ad p k ).U ))
Taking a subsequence if necessary, we assume that u is the only asymptotic direction of (p k ). Since u ∈ a + , the elements Ad p k act by contraction on n -, which implies that (Ad p k ).U → 0 g for the Hausdorff topology. As a consequence, , and it follows from the remark after definition 6.2 that any sequence of A is a holonomy sequence at x. We can now apply proposition 6.3 to obtain an open neighborhood V containing x on which the curvature vanishes. ♦ Corollary 6.5. The Cartan geometry (M, B, ω) is flat.

exp b k (ω -1 b k ((Ad p k ).U )) = h k . exp b (ω -1 b (U )).p -1 k → {b ∞ } Let V = π • exp b (ω -1 b (U )).
Proof: Let N 0 ⊂ M be the subset on which the curvature of ω vanishes. By the previous lemma, N 0 has nonempty interior. If x ∈ ∂N 0 , the topological boundary of N 0 in M , then H.x, the closure of the H-orbit of x, is closed and H-invariant. By lemma 6.4, there are y ∈ H.

x and an open neighborhood V of y such that the curvature vanishes on V . Then y could not have been in ∂N 0 , so ∂N 0 = ∅, which means N 0 = M . ♦ Corollary 6.6. The geometry (M, B, ω) is geometrically isomorphic to Γ\ X.

Proof: By the corollary directly above, the geometry (M, B, ω) is flat-in other words, there is a (G, X)-structure on M . The proof of the completeness of this structure is a straightforward generalization of that in [FZ]. The (G, X)-structure on M defines a developing map δ : M → X, which is a geometric immersion, and a holonomy morphism ρ : Aut M → G satisfying the equivariance relation ρ • δ = δ • ρ (see [Th]). Let π 1 (M ) be the fundamental group of M , which acts by automorphisms on M , and consider Hol(M ) = ρ(π 1 (M )) ⊂ G, the holonomy group of M .

At the infinitesimal level, there is a morphism of Lie algebras dρ e : χ kill ( M ) → g where χ kill ( M ) denotes the Lie algebra of Killing fields on M . Since H acts by automorphisms on M , there is an infinitesimal action of h by Killing fields on M , so that h ֒→ χ kill ( M ). Observe that any Killing field of M lifted from M is centralized by π 1 (M ). Now the morphism dρ e : h → g maps h (resp. a H ) injectively onto a Lie subalgebra h ′ ⊂ g (resp. a ′ H ⊂ h ′ ), which moreover is centralized by Hol(M ). It was established in the proof of lemma 6.4 that a ′ H is included, up to conjugacy, in the sum a⊕m, and projects onto a. Choose u 1 , . . . , u s spanning a ′ H , with projections spanning a. Since those projections are just the Rsplit parts of u 1 , . . . , u s , they are centralized by Hol(M ). Then Hol(M ) centralizes a, so is contained in A × M G . Now Hol(M ) centralizes h ′ , and A acts faithfully on h ′ because its action is conjugated to that of Ad A H on h. It follows that the projection of Hol(M ) on A is trivial, namely Hol(M ) ⊂ M G . The compact group M G is included in K, a maximal compact subgroup of G, which acts transitively on X.

In conclusion, the (G, X)-structure on M can be seen as a (K, X)-structure. These latter structures are known to be complete on closed manifolds (see proposition 3.4.10 of [Th]): the developing map δ is a covering map (The key point is that δ is not changed by viewing the Cartan geometry as a (K, X)-structure rather than a (G, X)-structure). Completeness means M is geometrically isomorphic to X, so Aut M ∼ = G, and M is geometrically isomorphic to Γ\ X, with Γ a discrete subgroup of G acting properly discontinuously on X. ♦ 7 Results about local freeness Assume now that Hd leaves stable only a finite family {0} = F 0 ,. . . ,F s of subalgebras in h. Write Ω = Ω 0 ∪ • • • ∪ Ω s , where Ω i contains all x ∈ Ω such that h x = F i . Let H i be the connected Lie subgroup of H with Lie algebra F i . The action of H i is faithful, so that for i = 0, the interior of Ω i is empty. This follows from the lemma Lemma 7.1. Let {h t } t∈R < Aut M be a 1-parameter subgroup, and U ⊂ M an open subset such that h t U = Id for every t ∈ R. Then h t = Id for every t ∈ R.

Proof: As already noticed, the action of h t on B is free because it preserves a framing. So if suffices to show that h t fixes a point on B for every t ∈ R. Pick x ∈ U and b ∈ B projecting on x. Thanks to the Cartan connection, it is possible to define a developing map D b

x from the space C

1 ([-1, 1], U ) of C 1 curves γ in U with γ(0) = x to the space C 1 ([-1, 1], X) of C 1 curves β in X with β(0) = o,
the class of P in X = G/P . We refer to [Shar] for the construction of the developing map. Now there is a 1-parameter subgroup {p t } < P such that h t .b.p -t = b. It is not difficult to show that

D b

x (h t .γ) = p t .D b x (γ). Because h t U = Id, all developments of curves in U are pointwise fixed by Ad p t . The union of all these developments is a subset of X with nonempty interior, pointwise fixed by p t . As a consequence, p t acts trivially on X for every t ∈ R. Since the kernel of the action of P on X was assumed to be finite, p t = e for every t ∈ R. ♦ Now because Ω is conull for a measure of full support, it is dense in M , and Ω 1 ∪ • • • ∪ Ω s has empty interior. Thus Ω 0 , on which H is locally free, is dense in M . Since the subset of M where the action is locally free is clearly open, theorem 1.6 follows. ♦ 7.2 Open orbit implies homogeneity: proof of 1.7

In this section, we assume that H < Aut M is a Lie subgroup such that HAd , the Zariski closure of Ad H in Aut h, has no algebraic compact quotients-that is, Hd = HAd . The Cartan geometry (M, B, ω) is connected and n-dimensional, equipped with an H-invariant volume form ν of finite total volume.

Recall that σ is the (Ad g P )-equivariant projection from g onto g/p. For every b ∈ B projecting on x ∈ M , there is a natural isomorphism j b : T x M → g/p with j b (u) = σ • ω b (ǔ), where ǔ ∈ T b B projects on u. It is easy to check that this isomorphism is well-defined, and j b.p = (Ad p -1 ) • j b for every p ∈ P and b ∈ B.

By this remark, we see that a vector field on M is nothing else than a map φ : B → g/p satisfying φ(b.p) = (Ad p -1 ).φ(b). In the same way, a continuous n-form ν on the manifold M can be seen as a continuous map ν : B → ∧ n (g/p) * , such that for any n-tuple (u 1 , . . . , u n ) of (g/p) n , ν(b.p)(u 1 , . . . , u n ) = ν(b)((Ad p).u 1 , . . . , (Ad p).u n ) Our hypothesis is that ν(h.b) = ν(b) for all h ∈ H.

Let U be the real algebraic variety Mon(h, g) × ∧ n (g/p) * . The map φ : b → (ι b , ν(b)) is continuous and H × P -equivariant. Here, the action of H on ∧ n (g/p) * is trivial, and for α ∈ ∧ n (g/p) * and p ∈ Ad g P , (p.α)(u 1 , . . . , u n ) = α(p.u 1 , . . . , p.u n ).

  Definition 1.1. Let G be a Lie group and P a closed subgroup. A Cartan geometry (M, B, ω) modeled on (g, P ) is 1. a principal P -bundle π : B → M 2. a g-valued 1-form ω on B satisfying • for all b ∈ B, the restriction ω b : T b B → g is an isomorphism • for all b ∈ B and Y ∈ p, the evaluation ω b ( d dt 0 be tY ) = Y • for all b ∈ B and p ∈ P , the pullback R * p ω = Ad p -1 • ω

  a subgroup, and b ∈ B, define Šb = {p ∈ P b | ρ b (p) ∈ Sd and p.κ b = κ b } The group Šb may not be algebraic in general, but it is if ρ b : Šb → Sd is surjective, as in this case it is the preimage in P b of the algebraic group Sd by the algebraic homomorphism ρ b . The restriction of ρ b to Šb is then an algebraic epimorphism onto Sd .

Theorem 4. 1 .

 1 Let H < Aut M be a Lie subgroup and S < H. Suppose there is an S-invariant measure µ on M . Then there exists Λ ⊂ B such that π(Λ) is µ-conull and, for every b ∈ Λ, ρ b : Šb → Sd is an algebraic epimorphism. In particular, Šb preserves κ b and ρ b intertwines the representation of Šb on ι b (h) and that of Sd on h.

  is an Sd -fixed point for µ i -almost-every m ∈ M i . For such an m, choose b ∈ π -1 (m) ⊂ B, and set u = φ(b). Then Ad g P.u is stabilized by Sd . This means that for any s ∈ Sd , there exists ps ∈ Ad g P , not necessarily unique, such that s.(ι b , κ b ) = ps .(ι b , κ b ), or in other words, ι b •s = ps •ι b , and ps .κ b = κ b . Thus, ps ∈ Šb , and ρ b (p s ) = s. We thus get that ρ b : Šb → Sd is a surjection, and theorem 4.1 is proved. ♦

  w) → 0 Then the component of κ b (v, w) on g ν is zero. Since this is true for any ν ∈ Π l with l > i + j, the component of κ b (v, w) on g i+j+1 is zero, which implies κ b (v, w) = 0 by the regularity condition. Then at any point x ∈ M where the assumptions of point (2) hold, the curvature vanishes. Let b ∈ B and ξ ∈ T b B. Let X ξ be the unique vector field on B having the property that ω b ′ (X ξ (b ′ )) = ω b (ξ) for all b ′ ∈ B. If φ t is the flow generated by X ξ , and if φ 1 is defined, then put exp b (ξ) = φ 1 .b. The map exp b is defined on a neighborhood of 0 b in T b B, and yields a diffeomorphism of a sufficiently small neighborhood of 0 b onto its image.

  For any x ′ ∈ V , the sequence (p k ) is a holonomy sequence at x ′ . The hypotheses of point (2) of the proposition are thus satisfied at any x ′ ∈ V . By what was said above, the curvature vanishes at every point of the open set V . Since a b H ⊂ p, lemma 2.1 ensures that -ι b is a Lie algebra isomorphism between a H and a b H , so a b H is abelian. In fact, for u ∈ a H and v ∈ h (ad ι b (u))(ι b (v)) = -ι b ((ad u).v) In particular, ad a b H acts on ι b (h) as an R-split subalgebra of dimension rk G. Decompose u ∈ a b H into u a + u m . Because a and m are simultaneously diagonalizable on g and u preserves ι b (h), both ad u a and ad u m preserve ι b (h); moreover, the restriction of ad u m is trivial. Then the projection of a b H modulo m is a surjection onto a. Let A b H be the connected subgroup of P with Lie algebra a b H . Any sequence in A b H is a holonomy sequence at the fixed point = π(b). From above, A b H < A × M , and projects onto the first factor. Thus any sequence of A is equivalent to a sequence in A b H

7. 1

 1 Local freeness almost everywhere: proof of 1.6 Here (M, B, ω) is a connected Cartan geometry modeled on G/P , and H < Aut M preserves a finite measure ν on M with full support. Theorem 4.1 yields a subset Ω ⊂ M of full measure such that for every b ∈ B projecting on x ∈ Ω, the image ρ b ( Ȟb ) = Hd . The stabilizer Lie algebra h x has image ι b (h x ) = ι b (h) ∩ p. Since ι b (h) ∩ p is invariant by Ȟb , the Lie algebra h x is invariant by Hd , as desired.

For

  every b ∈ B, we define, as in section 2.2 Ȟb = {p ∈ P b | ρ b (p) ∈ Hd , and p.ν(b) = ν(b)} Following the same proof as for theorem 4.1, we obtain a P -invariant subset Λ ⊂ B projecting on a set of full ν-measure, such that for every b ∈ Λ, ρ b is a surjection from Ȟb onto HAd = Hd . In particular, if b ∈ Λ and h ∈ H, then there is ph ∈ Ȟb , not necessarily unique, such that ι h.b = ph .ι b and ph .ν(b) = ν(b).Observe that if M has an openH-orbit, then it has an open orbit for H o , the identity component of H. Indeed, if H o .x is not open for some x ∈ M , then it has volume zero. But H is a Lie group, so it has countably-many connected components. Thus H.x is the union of countably-many sets of volume zero, so that it also has volume zero, contradicting the assumption that it was open. Let O be an open orbit of H o on M . Since any set of full measure forν intersects O, there is b ∈ Λ projecting on x ∈ O. Denote by O the topological closure of O in M . Pick x ∞ ∈ O and b ∞ ∈ B projecting on x ∞ . There is (h k ) a sequence of H o such that h k .x → x ∞ , and thus a sequence (p k ) of P such that h k .b.p -1 k → b ∞ . Now, the tangent space T x (H o .x) = T x M , so there are X 1 , . . . , X n ∈ h such that (σ • ι b (X 1 ), . . . , σ • ι b (X n )) is a basis of g/p, and thus ν(b)(σ • ι b (X 1 ), . . . , σ • ι b (X n )) = 0.The equivariance property implies that for every p ∈ P ,ν(b.p)(ι b.p (X 1 ), . . . , ι b.p (X n )) = ν(b)(X 1 , . . . , X n )The above equality together with H-invariance of ν implies that there are ǧk ∈ Ȟb such thatν(h k .b.p -1 k )(σ • ι h k .b.p -1 k (X 1 ), . . . , σ • ι h k .b.p -1 k (X n )) = ν(b)(ǧ k .σ(X 1 ), . . . , ǧk .σ(X n )) Since ǧk .ν(b) = ν(b), the right-hand side equals ν(b)(σ • ι b (X 1 ), . . . , σ • ι b (X n )), and the left one tends toν(b ∞ )(σ • ι b∞ (X 1 ), . . . , σ • ι b∞ (X n )). This last term is thus nonzero, which proves that σ • ι b∞ (h) = g/p. It follows that the H o -orbit of x ∞ is open, and that x ∞ ∈ O. Because M is connected, M = O,which proves that H o acts transitively on M . Theorem 1.6 implies that for almost every x ∈ M , the stabilizer subalgebra h x is HAd -invariant. Because H 0 acts transitively, the stabilizer subalgebra is the same at every point of M . Since H acts faithfully, h x = {0} for every x ∈ M . Then M = H o /Γ, where Γ is a discrete subgroup of H o . The n-form ν is left-invariant, so it induces Haar measure on H o /Γ. Since M has finite volume, Γ must be a lattice in H o . Deduction of Corollary 1.8 Let Aut loc (M, ω, ν) (resp. Aut loc (M, S, ν)) be the pseudo-group of local automorphisms of the Cartan geometry (resp. of the structure S) that also preserve ν. The hypotheses of corollary 1.8 imply Aut loc (M, ω, ν) = Aut loc (M, S, ν). The fact that H has a dense orbit ensures that Aut loc (M, ω, ν), hence also Aut loc (M, S, ν), has a dense orbit. Adding an analytic volume form ν to the analytic rigid geometric structure of algebraic type S still yields an analytic rigid geometric structure of algebraic type S ′ = (S, ν). Gromov's open-dense theorem ([Gr] 3.3.A), implies that Aut loc (M, S, ν) has an open dense orbit U . Note that the local Killing fields for this structure also act transitively on U .

We now prove point (1). Since Šb preserves κ b , any sequence (p k ) of A satisfies (Ad p k ).κ b (v, w) = κ b ((Ad p k ).v, (Ad p k ).w) ∀ v, w ∈ g

The proof is then basically the same as the first part of that of point (2). ♦

We can now prove the first point of theorem 1.5. By the hypothesis on rk alg (Ad H), there is an abelian R-split subgroup of Ad H with Zariski closure S of dimension rk G. Let S = Ad -1 ( S) ⊂ H. Since the kernel of Ad on H is amenable by assumption ( §1.1), S is amenable as well. Theorem 4.1 gives b ∈ B such that ρ b maps Šb surjectively on S = Sd .

As in the proof of theorem 1.3 in section 5.1, there is an abelian R-split subalgebra z s ⊂ šb such that dρ b (z s ) = s. Then dim z s = rk G, and there exists a maximal R-split subalgebra a ′ ⊂ p such that ad a ′ = z s . There exists

we can apply point (1) of proposition 6.3. Together with lemma 2.1, it yields that -ι b.p 0 : h → g is an embedding of Lie algebras, proving the first point of the theorem.

We now come to the second part of theorem 1.5. Because the structure on M is analytic and M is simply connected, local Killing fields extend to all of M ([Am], [N], [Gr]). Because M is compact the extended Killing fields are complete. Then Aut (M, S, ν) has an open orbit on M , hence so does Aut (M, ω, ν). The corollary now follows from theorem 1.7. ♦

Homogeneous reductive Cartan geometries

Finally, we prove theorem 1.9. The assumptions here are that the geometry is reductive, and the H-action is transitive and faithful, and preserves a finite volume. By theorem 1.7, the action is everywhere locally free. Now use Theorem 4.1 to obtain b ∈ B such that ρ b is a surjection from Ȟb to HAd . Let Ȟ = Ad -1 g ( Ȟb ) < P . Because H acts locally freely, ι b (h) is transverse to p for all b ∈ B. Let σ be the Ad g P -equivariant projection g to n associated to the decomposition g = n ⊕ p. If Ad p is trivial on ι b (h), then it is trivial on n. Because Ad g P is faithful on n, the kernel of ρ = ρ b • Ad g : Ȟ → HAd is trivial, so ρ is an isomorphism. The map ι = σ • ι b is an isomorphism from h to n, and by theorem 4.1, it intertwines the representations of HAd on h and Ad Ȟ on n. Set H ′ = ρ -1 (Ad H). When H is connected, then H is a central extension of H ′ as claimed in the second point of theorem 1.9. ♦


F Le Campion

G Bourda

P Charlot

J.-F Le Campion

Astrometric suitability of optically-bright ICRF sources for the alignment with the future Gaia celestial reference frame

Keywords:

Astrometric suitability of optically-bright ICRF sources

for the alignment with the future Gaia celestial reference frame

Introduction

The International Celestial Reference System (ICRS) is a kinematical system that assumes that the Universe does not rotate, hence breaking the history of the inertial system materialization from observations attached to the apparent motion of the Sun (Arias et al., 1995). The International Celestial Reference Frame (ICRF) is the realization at radio wavelengths of the ICRS, through very long baseline interferometry (VLBI) measurements of extragalactic radio source positions. It was adopted by the International Astronomical Union (IAU) as the fundamental celestial reference frame during the IAU XXIIIrd General Assembly in Kyoto, Japan, in August 1997. The initial realization was based on the positions measured with VLBI of 212 defining extragalactic radio sources (setting the direction of the ICRF axes). In addition, positions for 396 other sources (divided into 294 candidate and 102 other sources) were included to make the frame denser (Ma et al., 1998). The ICRF was extended at later stages with the positions of another 109 sources commonly referred to as new sources (Fey et al., 2004). Overall, the ICRF currently consists of a set of VLBI coordinates for 717 extragalactic radio sources, most of which show submilliarcsecond accuracy. The accuracy of the individual source positions depends on the number of observations but also on the compactness and positional stability of the sources that show Send offprint requests to: G. Bourda brightness distributions that are often not point-like and timevariable (Ma et al. 1998;Fey and Charlot 2000).

The European space astrometry mission Gaia, to be launched by 2011, will survey about one billion stars in our Galaxy and throughout the Local Group along with 500 000 quasi stellar objects (QSOs), down to an apparent optical magnitude of 20 (Perryman, 2001). Optical positions with Gaia will be determined with an unprecedented accuracy, ranging from a few tens of microarcseconds (µas) at magnitude 15-18 (targeted accuracies are 16 µas at 15 mag and 70 µas at 18 mag) to about 200 µas at magnitude 20 [START_REF] Lindegren | A Giant Step: from Milli-to Micro-arcsecond Astrometry[END_REF]. Unlike Hipparcos, Gaia will permit the construction of an extragalactic frame directly at optical wavelengths, based on the QSOs with the most accurate positions (i.e. the QSOs with magnitude brighter than 18; [START_REF] Mignard | IAU XXV, Joint Discussion 16: The International Celestial Reference System: Maintenance and Future Realization[END_REF]. [START_REF] Mignard | Gaia: A European Space Project[END_REF] demonstrates that the residual spin of the Gaia frame can be determined to 0.5 µas/yr with a "clean sample" of about 10 000 such QSOs. A preliminary Gaia catalog is expected to be available by 2015 with the final version released by 2019.

In the future, aligning the ICRF with the Gaia celestial reference frame will be crucial for ensuring consistency between the measured radio and optical positions. This alignment will be important not only for guaranteeing the proper transition in the case that the fundamental reference frame is moved from the radio domain to the optical domain (as currently anticipated) but also for registering the radio and optical images of any celestial target with the highest accuracy. To align the two frames with the highest accuracy, it is desirable to have several hundred common objects (the more common objects, the more accurate the alignment) with a uniform sky coverage and very accurate radio and optical positions. Obtaining such accurate positions implies that the link sources must have (i) an apparent optical magnitude V brighter than 18, for the highest Gaia astrometric accuracy [START_REF] Mignard | IAU XXV, Joint Discussion 16: The International Celestial Reference System: Maintenance and Future Realization[END_REF], and (ii) no extended VLBI structures, for the highest VLBI astrometric accuracy (Fey and Charlot, 2000).

This paper is aimed at evaluating the suitability of the current individual ICRF extragalactic radio sources for the alignment with the future Gaia frame. First, the ICRF is cross-identified with the Véron-Cetty and Véron (2006) optical catalog of QSOs in order to identify the ICRF sources with a proper optical counterpart for the Gaia link. The astrometric suitability of these sources is then investigated by examining their VLBI brightness distribution and their position accuracy in the ICRF. In this investigation, the structure index defined by Fey and Charlot (2000) is used to identify the sources that have the most compact VLBI brightness distributions. This study results in a sample of 70 ICRF sources, which have appropriate compact structures on VLBI scales and are brighter than magnitude 18. These sources are at present the best candidates for the ICRF-Gaia alignment.

Optical counterparts of ICRF sources

Cross-identification of radio and optical positions

Véron-Cetty and Véron (2006) compiled several optical surveys of active galactic nuclei (AGN) from the literature in order to create a homogeneous, compact, and convenient catalog of 108 080 objects (85 221 quasars, 1 122 BL Lac objects, and 21 737 active galaxies). To identify the ICRF sources that have an optical counterpart, we have cross-identified the positions in this catalog with those in the ICRF. For this cross-identification, it was necessary to adopt an upper limit on the optical-radio position differences above which the corresponding radio and optical sources were considered as different objects.

We tested the impact of the selected upper limit by carrying out the cross-identification of the two catalogs for several values of this parameter, ranging from 1 ′′ to 5 ′′ , the results of which are shown in Table 1. From this table, it is found that there are no major differences whether one sets the upper limit value to 3 ′′ , 4 ′′ or 5 ′′ . On the other hand, significantly fewer sources are found to be common when an upper limit value of 1 ′′ or 2 ′′ is selected. Finally we decided to adopt the upper limit value of 3 ′′ , equivalent to three times the accuracy of the optical positions in the Véron-Cetty and Véron ( 2006) catalog, which is roughly at the 1 ′′ level. Uncertainties in the VLBI positions make a negligible contribution to the position differences since such uncertainties are at the milliarcsecond level.

Based on this cross-identification, we determined that 75 % of the ICRF sources (i.e. 535 sources out of a total of 717 ICRF sources) have an optical counterpart in Véron- Cetty and Véron (2006).This set of sources includes 406 quasars (76 %), 65 BL Lac objects (12 %), and 64 active galaxies (12 %). These results are comparable to those of Souchay et al. (2006) who used the Véron-Cetty and Véron (2003) optical catalog for the crossidentification.

Table 2 presents the distribution of the 535 cross-identified ICRF sources, depending on their categorization in the ICRF as defining, candidate, other or new sources. This table shows that:

-83 % of the ICRF defining sources, 70 % of the ICRF candidate sources, 88 % of the ICRF other sources, and 60 % of (2006;noted VV2006), depending on the source type in this catalog and on the upper limit used for the position differences in the cross-identification. Based on these statistics, there is no evidence for any dependence of the existence of optical counterparts on ICRF source categories. The sample of sources identified here forms the initial pool of ICRF sources from which to identify the best candidates for the Gaia link after consideration of optical magnitudes, source structure, and astrometric accuracy.

Distribution of optical magnitudes

In a second stage, we investigated the distribution of the apparent optical magnitude V for the 535 ICRF sources identified in the previous step. Five of these sources, with undefined optical magnitudes in the Véron-Cetty and Véron (2006) catalog, have not been considered. As shown in Fig. 1, the distribution of optical magnitudes peaks at V between 18 and 19 with the bulk of the sources brighter than V = 20. Overall, there are 485 sources brighter than 20, i.e. detectable by Gaia, while only 45 sources are weaker than 20. When restricting the magnitude limit to 18, the sample is reduced by a factor of 2, leaving only 243 sources. These 243 sources, representing 34 % of the ICRF catalog, would be good candidates for establishing the alignment with the future Gaia celestial reference frame, provided they show limited structure on VLBI scales and have high VLBI positional accuracy.

Astrometric suitability based on observed VLBI structure

The quality of the alignment between the ICRF and the Gaia frame will depend strongly on the astrometric suitability of the sources used for this alignment. To ensure the highest accuracy, the sources with the highest astrometric quality from the above sample should be used.

On the milliarcsecond scale, most of the extragalactic radio sources exhibit spatially extended structures that vary in both time and frequency. As shown by [START_REF] Charlot | [END_REF], such structures may introduce significant errors in the VLBI measurements that deteriorate the source position accuracy. The astrometric suitability of the sources may be estimated from the observed structures through the so-called "structure index", first introduced by Fey and Charlot (1997). The structure index ranges from 1 to 4. Structure index values of 1 or 2 point to the most compact sources with the highest astrometric quality, whereas higher values of the structure index (3 or 4) correspond to sources with extended structures, which are less suitable for astrometry. For the alignment of the ICRF and Gaia frame, only sources with structure index values of 1 or 2, i.e. with the highest astrometric quality, should be used since one wants to determine the link with the highest accuracy.

The structure index distribution for the 485 optically-bright ICRF sources with an apparent optical magnitude V ≤ 20 (see Sect. 2) was investigated and is plotted separately for the sources with V ≤ 18 and for those with V between 18 and 20 in Fig. 2. For these plots, structure indices from the most recent calculation by [START_REF] Charlot | International VLBI Service for Geodesy and Astrometry 2006 General Meeting Proceedings[END_REF] were used, and the distribution is shown for both S and X bands, the two standard frequency bands used in VLBI astrometry. Overall, structure index values are available for 410 sources at X band (205 sources with V ≤ 18 and 205 sources with 18 < V ≤ 20) and 345 sources at S band (169 sources with V ≤ 18 and 176 sources with 18 < V ≤ 20). Interestingly, the sources with magnitude V ≤ 18 tend to have larger structure indices than those with 18 < V ≤ 20, as noticed when comparing the corresponding distributions in Fig. 2. This indicates that the brighter optical sources have statistically more extended radio structures. Based on this analysis, it is also found that only 70 ICRF sources out of those with V ≤ 18 have an excellent or good astrometric suitability (i.e. an X-band structure index value of either 1 or 2) appropriate for determining the Gaia link with the highest accuracy (see list in Table 3). This represents only a small fraction (about 10 %) of the entire ICRF cata- log. The sky distribution of these 70 sources, plotted in Fig. 3, is reasonable but indicates that there is a general lack of sources.

Discussion

This paper is aimed at identifying from the current ICRF extragalactic radio sources the best candidates to establish an accurate alignment with the future Gaia frame. Figure 4 summarizes the successive steps completed to achieve this goal. We first pointed out that 75 % of the ICRF sources have an optical counterpart (535 sources), while less than half of these are potential candidates for the alignment with the future Gaia frame (243 ICRF sources with V ≤ 18, i.e. with the best positions measured by Gaia). And finally, we concluded that only 10 % of the ICRF sources can be identified currently as the best candidates to establish this alignment with the highest accuracy (70 opticallybright ICRF sources with the highest astrometric quality).

To supplement this study, we also investigated the distribution of position accuracy, as available from Ma et al. (1998) and Fey et al. (2004), which is plotted in Fig. 5 for the 485 opticallybright ICRF radio sources with V ≤ 20. From this figure, we can notice that the sources with 18 < V ≤ 20 appear to have higher positional accuracy than those with V ≤ 18. This is especially noticeable when comparing the fraction of sources with position accuracy better than 0.5 mas and those with positional accuracy in the range 0.5-1 mas. This fraction is 71% for the sources with 18 < V ≤ 20 (135 sources out of a total of 190 sources with po- sition accuracy better than 1 mas), whereas it is 55 % for the sources with V ≤ 18 (98 sources out of a total of 177 sources with position accuracy better than 1 mas). This trend is also confirmed when comparing the median positional accuracy, which is 0.60 mas for the sources with V ≤ 18 and 0.47 mas for the sources with 18 < V ≤ 20. Such a finding is consistent with our results on astrometric suitability (Sect. 3), which indicate that the brighter sources (V ≤ 18) have statistically more extended VLBI structures. It is thus likely that the deteriorated position accuracy for the V ≤ 18 sources in Fig. 5 is a result of their having larger structures. A similar correlation between source structure and position accuracy was also reported by Fey and Charlot (2000). In the context of Gaia, this means that the sources that will have the most accurate optical positions measured by Gaia are not statistically those that have the best measured positions in the ICRF. It will thus be crucial in the future to unveil more high-quality optically-bright radio sources suitable for the Gaia link beyond those currently identified from the ICRF and reported in Table 3.

A first opportunity to find such sources may come from the second generation of the ICRF (ICRF-2), currently under development, which will improve the position accuracy of the current ICRF sources and provide VLBI positions of additional sources. This new frame will rely on data acquired by international observing programs organized through the International VLBI Service for Geodesy and Astrometry (Schluter and Behrend, 2007) or other programs such as that developed in the southern hemisphere by the United States Naval Observatory [START_REF] Ojha | [END_REF](Ojha et al., , 2005)). Another option would be to use the Very Long Baseline Array (VLBA) Calibrator Survey (VCS), which comprises milliarcsecond accurate positions for about 3000 additional sources along with images for a large fraction of those sources (Beasley et al., 2002;Fomalont et al., 2003;Petrov et al., 2005Petrov et al., , 2006;;[START_REF] Kovalev | [END_REF]Petrov et al., 2008). Finally, an additional possibility is to search for new VLBI sources, by targeting radio sources weaker than those observed so far in VLBI astrometry (i.e. with flux densities typically below 100 mJy). Such sensitive observations can now be realized owing to recent increases in the VLBI network sensitivity (e.g. recording at 1 Gb/s) and by using a network comprising large antennas like the European VLBI Network (EVN) [START_REF] Charlot | International VLBI Service for Geodesy and Astrometry 2004 General Meeting Proceedings[END_REF]. This identification of new optically-bright VLBI sources for the Gaia frame alignment is the purpose of an observational program initiated in 2007 which focuses on 450 weak radio sources selected from the NVSS (NRAO VLA Sky Survey; [START_REF] Condon | [END_REF] and brighter than the apparent optical magnitude 18. In the initial step, about 90 % of these weak radio sources have been detected during an EVN pilot experiment [START_REF] Bourda | International VLBI Service for Geodesy and Astrometry 2008 General Meeting Proceedings[END_REF], hence drawing good prospects for increasing the pool of VLBI extragalactic radio sources suitable for aligning the ICRF and the future Gaia frame with the highest accuracy. Another issue that might affect the choice of sources for this alignment is the possible variability of the apparent magnitude for the extragalactic sources observed at optical wavelengths. These variations are common and can easily reach one magnitude unit, especially for BL Lac objects that are known to vary on short time scales, as illustrated in Fig. 6. Such variability may affect the selection of candidate sources for the alignment of the ICRF and the Gaia frame if only a limited number of measurements are available for their optical magnitude. For this reason, it is desirable to engage in a long term optical monitoring on these [START_REF] Charlot | Scientific Highlights 2004, Journées 2004 SF2A Proceedings[END_REF]. candidate sources, especially those that have a magnitude near 18, to determine if on average they have an optical magnitude that is brighter or weaker than 18. While the results of this monitoring might impact at some level the list of sources in Table 3, this should not significantly change the percentage of suitable ICRF sources currently available for the Gaia frame alignment.

Finally, it is important to note that obtaining the alignment between the ICRF and the future Gaia frame with the highest accuracy is essential, not only to ensure the consistency between the measured radio and optical positions, but also to pinpoint the relative location of the optical and radio emission in AGNs to a few tens of µas. In fact, the optical and radio emission may not always be spatially coincident at this level of accuracy due to opacity effects in AGN jets. For example, Kovalev et al. (2008) shows that on average the optical-radio core shift is at the level of 100 µas from studying a sample of 29 objects, which is significant considering the expected accuracy of the Gaia catalog and the one foreseen for the ICRF by 2015-2020. It will thus be important to use a large number of objects in aligning the two frames so that positional inconsistencies are averaged out. On the other hand, the differences between the optical and radio positions may provide a direct measurement of such core shifts, which would be of high interest for probing AGN jets properties.

Conclusion

This study focused on the astrometric suitability of the current ICRF extragalactic radio sources for the alignment with the future Gaia frame. We identified 243 candidate sources for this alignment, but only 70 of these (10% of the ICRF) possess the high astrometric quality required to ensure the highest accuracy in the alignment. Accordingly, the current number of VLBI sources for accurately aligning the ICRF and the future Gaia frame is not sufficient. We also showed that the QSOs that will have the most accurate positions measured with Gaia are not those that have the best astrometric positions in the ICRF statistically.

To compensate for the lack of high astrometric quality extragalactic radio sources for this alignment, the VCS catalog, along with the future ICRF-2, will be of high interest. An additional direction would be to identify new appropriate VLBI sources from current deep radio surveys. These solutions mostly concern the northern hemisphere, since the VLBI arrays and observations are concentrated in this part of the world. However, to ensure a homogeneous sky coverage, a major effort is also necessary in the southern hemisphere, most specifically for the declinations below -40 • . To this end, the Asia Pacific Telescope [START_REF] Gulyaev | IAU XXVI, Special Session 5: Astronomy for the Developing World[END_REF] is likely to play a significant role.

Fig. 1 .

 1 Fig. 1. Distribution of the apparent optical magnitude V for the 530 ICRF extragalactic radio sources that have a Véron-Cetty and Véron (2006) optical counterpart.

Fig. 3 .Fig. 4 .

 34 Fig. 3. Sky distribution for the 70 ICRF sources suitable for the alignment with the future Gaia frame.

Fig. 2 .

 2 Fig. 2. Distribution of the X-and S-band structure indices among the optically bright ICRF sources; upper panel: X-band structure index distribution; lower panel: S-band structure index distribution; left-hand side: sources with V ≤ 18; right-hand side: sources with 18 < V ≤ 20.

Fig. 5 .

 5 Fig. 5. Distribution of position accuracy for the 485 ICRF sources with an optical apparent magnitude V brighter than 20.The accuracy is calculated as σ 2 α cos 2 δ + σ 2 δ where σ α and σ δ are the errors in right ascension and declination, as taken from the ICRF catalog. The upper panel shows the positional accuracy for the 243 sources with V ≤ 18, while the lower panel shows the positional accuracy for the 242 sources with 18 < V ≤ 20.

Fig. 6 .

 6 Fig. 6. Variation in the apparent optical magnitude of the BL Lac object Mkn 421 (1101 + 384) from the ICRF, during the year 2002, from optical monitoring at Bordeaux Observatory (Charlot and Le Campion, 2004).

Table 1 .

 1 Number of ICRF extragalactic radio sources crossidentified with the optical catalog of Véron-Cetty and Véron

Table 2 .

 2 Number of ICRF extragalactic radio sources crossidentified with the optical catalog of Véron-Cetty and Véron (2006; noted VV2006), depending on the ICRF source category.

			ICRF catalog
		Defining Candidate	Other	New
		(212)	(294)	(102)	(109)
	VV2006 ∩ ICRF
	Quasars (406)	132	160	70	44
	BL Lac (65)	27	24	11	3
	Active galaxies (64)	16	21	9	18
	Total (535/717)	175/212	205/294 90/102 65/109

Note 1. The upper limit used for the position differences in the crossidentification is 3 ′′ . the ICRF new sources have an optical counterpart in Véron-Cetty and Véron (2006). -33 % of the ICRF sources having an optical counterpart in Véron-Cetty and Véron (2006) are defining sources, 38 % candidate sources, 17 % other sources, and 12 % new sources.

Acknowledgements. The first author is grateful to the CNES (Centre National d'Etudes Spatiales, France) for the post-doctoral position granted at Bordeaux Observatory.

Table 3. List of the 70 ICRF extragalactic radio sources suitable for the alignment with the future Gaia frame. The column labelled 'SI' gives the X-band structure index of the source. The column labelled 'Cat.' details if the source is categorized in the ICRF as defining (d), candidate (c), other (o) or new (n); and the columns labelled 'VV2006 name', 'Type' and 'V' indicate respectively (i) the name of the source in the optical catalog Véron & Véron (2006), (ii) the source type (quasar (Q), BL Lac object (B) or active galaxy (A)), and (iii) its apparent optical magnitude.

ICRF designation

IERS name SI Cat. α δ VV2006 name Type V h min sec


Spiros Kotopoulis

Antje Schommartz

Michiel Postema
email: m.postema@hull.ac.uk

Sonic cracking of blue-green algae

Keywords:

Algae are aquatic organisms classified separately from plants. They are known to cause many hazards to humans and the environment. Algae strands contain nitrogen-producing cells that help them float (heterocysts). It is hypothesized that if the membranes of these cells are disrupted by means of ultrasound, the gas may be released analogous to sonic cracking, causing the strands to sink. This is a desirable ecological effect, because of the resulting suppressed release of toxins into the water.

We subjected small quantities of blue-green algae of the Anabaena sphaerica species to ultrasound of frequencies and pressures in the clinical diagnostic range, and observed the changes in brightness of these solutions over time. Blue green algae were forced to sink at any ultrasonic frequency we studied, supporting our hypothesis that heterocysts release nitrogen under ultrasound insonification in the clinical diagnostic range.

Although the acoustic fields we used to eradicate blue-green algae are perfectly safe in terms of mechanical index, the acoustic pressures surpass the NURC Rules and Procedures by over 35 dB. Therefore, caution should be taken when using these techniques in a surrounding where aquatic or semi-aquatic animals are present.

Introduction

Algae are aquatic, eukaryotic, photosynthetic organisms, ranging in size from single-celled forms to large kelps. Algae are classified separately from plants since they lack true roots, stems, leaves and embryos. Algae are known to cause many health hazards to humans including skin rashes, gastrointestinal, respiratory [START_REF] Turner | Pneumonia associated with contact with cyanobacteria[END_REF], allergic reactions, [START_REF] Stewart | Cyanobacterial lipopolysaccharides and human health: a review[END_REF] and liver cancer [START_REF] Fleming | Blue green algal (cyanobacterial) toxins, surface drinking water, and liver cancer in Florida[END_REF]. In addition, blue-green algae may have implications on aquatic and semi-aquatic animals [START_REF] Bury | The toxicity of cyanobacteria (blue-green algae) to freshwater fish[END_REF]. Eutrophication is the increase in chemical nutrients within the ecosystem, causing blooms of algae and plant life and the subsequent decomposition of blue-green algae by bacteria, an oxygen-consuming process [START_REF] Diego-Mcglone | Eutrophic waters, algal bloom and fish kill in fish farming areas in Bolinao, Pangasinan, Philippines[END_REF]. When billions of such bacterial cells die during a bloom, the water becomes oxygen-depleted, killing off oxygen-dependent organisms [START_REF] Hylland | Interactions between eutrophication and contaminants. IV. Effects on sediment-dwelling organisms[END_REF].

The main factors that influence algae growth are temperature and light [START_REF] Dauta | Growth rate of four freshwater algae in relation to light and temperature[END_REF][START_REF] Allen | Studies on nitrogen-fixing blue-green algae. I. Growth and nitrogen fixation by anabaena cylindrica lemm[END_REF].

At low temperatures and low light conditions, the algae do not photosynthesise and therefore do not bloom. Algae strands contain nitrogen-producing cells that help them float (heterocysts), as demonstrated in Fig. 1. The heterocysts have a diameter between 5 -7 µm [START_REF] Lang | Morphological and ultrastructural changes in vegetative cells and heterocysts of anabaena variabilis grown with fructose[END_REF]. It is hypothesized that if the membranes of these cells are disrupted by means of ultrasound, the gas may be released analogous to sonic cracking [START_REF] Postema | Ultrasound-induced gas release from contrast agent microbubbles[END_REF], causing the strands to sink. At the lake bed, illumination is lower, thus reducing algae multiplication. This is a desirable ecological effect, because of resulting suppressed release of toxins into the water.

There are chemical methods to control certain species of algae, but these have side-effects such as promotion and growth of other species of algae [START_REF] Ferrier | The effects of barley straw (hordeum vulgare) on the growth of freshwater algae[END_REF] whilst also affecting aquatic life in fresh water ponds and lakes. Therefore, ultrasonic algae control has been under investigation [START_REF] Zhang | Ultrasonic frequency effects on the removal of Microcystis aeruginosa[END_REF][START_REF] Ma | Influence of ultrasonic field on microcystins produced by bloom-forming algae[END_REF][START_REF] Zhang | Ultrasonic damages on cyanobacterial photosynthesis[END_REF][START_REF] Hao | Cavitation mechanism in cyanobacterial growth inhibition by ultrasonic irradiation[END_REF][START_REF] Tang | Growth inhibition of the cyanobacterium spirulina (arthrospira) platensis by 1.7 mhz ultrasonic irradiation[END_REF][START_REF] Zhang | Ultrasound-enhanced coagulation for microcystis aeruginosa removal[END_REF]. In [START_REF] Zhang | Ultrasonic frequency effects on the removal of Microcystis aeruginosa[END_REF][START_REF] Ma | Influence of ultrasonic field on microcystins produced by bloom-forming algae[END_REF][START_REF] Zhang | Ultrasonic damages on cyanobacterial photosynthesis[END_REF][START_REF] Hao | Cavitation mechanism in cyanobacterial growth inhibition by ultrasonic irradiation[END_REF][START_REF] Tang | Growth inhibition of the cyanobacterium spirulina (arthrospira) platensis by 1.7 mhz ultrasonic irradiation[END_REF][START_REF] Zhang | Ultrasound-enhanced coagulation for microcystis aeruginosa removal[END_REF], ultrasonic insonification of different species of algae led to a decrease in algae concentrations in the frequency range 20 kHz-1.7 MHz, which is in contrast with [START_REF] Chen | Ultrasound effects on the antioxidative defense systems of porphyridium cruentum[END_REF], where ultrasound was observed to strengthen the cell membranes of red algae. In these studies, the exact acoustic conditions have not been specified other than the frequency and power input. Hence, from the acoustics point of view, they are not repeatable. More importantly, the mechanism causing algae eradication or membrane disruption had not been investigated.

Most commercially available equipment works in the lower ultrasonic range [START_REF] Postema | Onderzoek naar de veilige toepassing van ultrageluid gebruikt om blauwalg te bestrijden[END_REF]. There have been speculations about the physical mechanism behind the algae eradication, specifically about the role of cavitation. In this study, we investigate the effectiveness of ultrasonic insonification in the clinical diagnostic range on Anabaena blue-green algae.

A measure for the safe use of clinical diagnostic ultrasound is the mechanical index (MI) defined by

MI = p - √ f , (1)
where p -is the maximum value of peak negative pressure anywhere in the ultrasound field, measured in water but reduced by an attenuation factor equal to that which would be produced by a medium having an attenuation coefficient of 0.3 dBcm -1 MHz -1 , normalised by 1 MPa, and f is the centre frequency of the ultrasound normalised by 1 MHz [START_REF]Guidelines for the safe use of diagnostic ultrasound equipment[END_REF]. For MI<0.3, the ultrasonic amplitude is considered low. In clinical diagnostics there is a possibility of minor damage to neonatal lung or intestine [START_REF]Guidelines for the safe use of diagnostic ultrasound equipment[END_REF] for 0.3<MI<0.7.

These are considered moderate acoustic amplitudes. For MI>0.7, there is a risk of cavitation if gas cavitation nuclei are present, and there is a theoretical risk of cavitation without the presence of cavitation nuclei [START_REF]Guidelines for the safe use of diagnostic ultrasound equipment[END_REF]. The risk increases with MI values above this threshold [START_REF]Guidelines for the safe use of diagnostic ultrasound equipment[END_REF]. These are considered high acoustic amplitudes. According to the NATO Undersea Research Centre (NURC)

Human Diver and Marine Mammal Risk Mitigation Rules and Procedures [START_REF]Human diver and marine mammal risk mitigation rules and procedures[END_REF], the maximum acoustic pressure to which mammals can be exposed is 708 Pa at frequencies up to 250 kHz. This corresponds to a mechanical index diagnostic range, taking into account both guidelines. The algae used were of the Anabaena species. The Anabaena were obtained from an outside lake and cultured in 2 L of Jaworski's medium [START_REF]Culture Collection of Algae and Protozoa. Media Recipes[END_REF] at room temperature near a South-facing window in an Erlenmeyer flask for 11 days.

The solution was put in a Swirtlock2000 autoclave (Astell Scientific, Kent, UK) at 15-lb pressure for 15 minutes.

Brightness measurements

To measure the time-dependent change in brightness of water containing blue-green algae, the culture was split equally into four 250-mL Perspex beakers: one beaker for each transducer and one control beaker. The transducers were inserted separately in each beaker with the acoustic focus within the sample. Each transducer was turned on for 1 hour. The experimental setup is shown in Fig. 3. A digital photograph of the solution was taken every five minutes using an EOS 350D digital photo camera (Canon Inc, Tokyo, Japan). The lighting and exposure settings were controlled and maintained throughout the insonification. Full manual settings were used:

ISO 100, exposure time 1/50 s, F number 3.50, focal length 18 mm, no flash, centre weighted metering mode, custom white balance, B4,0 shift. The digital photographs were converted to 8-bit grey scale. On the photographs of the insonified solution and control solution, a square area of 160 × 160 pixels was selected, whose average grey scale depth was calculated using MATLAB R (The Mathworks TM , Natick, MA). The change in shade between the first image taken just before insonification and each sequential image after insonification was calculated and graphed for all 18 insonified and control samples. A white sheet was placed behind the beakers to maintain a constant background. The grey scale of the sheet was also measured and used to calibrate the results. In total, 575 measurements were analysed from 122 photographs.

Viability measurements

Every 10 minutes, a 20-µL sample was taken from the insonified solution.

Samples were put on a test slide and observed through a CHA microscope (Olympus Corporation, Tokyo, Japan) with a 10× objective lens (Olympus Corporation, Tokyo, Japan). Digital photographs were captured from the microscope's eye piece using an FE-230 digital camera (Olympus Corporation, Tokyo, Japan). Automatic settings were used with Super Macro mode and a -1.0 exposure adjustment. From these digital images, cell deterioration and chlorophyll damage were determined. To investigate the effect of ultrasound on the viability of the cells, fluorescent light was used. When fluorescent light is projected onto chlorophyll, it is absorbed and re-emitted as a red glow. The red glow denotes that the chlorophyll is still active and can photosynthesize, thus the algae strands are still alive. Fluorescent light was used as the method to check the viability of the chlorophyll, since it is a standardized and accepted method in detecting chlorophyll activity in plants [START_REF] Ventrella | Effect of aggregation state, temperature and phospholipids on photobleaching of photosynthetic pigments in spinach photosystem ii core complexes[END_REF][START_REF] Mateos-Naranjo | Growth and photosynthetic responses to copper stress of an invasive cordgrass, spartina densiflora[END_REF]. Fluorescent light was projected onto these samples for no more than 10 seconds in order to capture the digital image. We may assume that there are no disadvantageous effects of the fluorescent light on the algae [START_REF] Ohl | Detachment and sonoporation of adherent hela-cells by shock wave-induced cavitation[END_REF][START_REF] Kooiman | Ultrasound-triggered local release of lipophilic drugs from a novel polymeric ultrasound contrast agent[END_REF]. The samples were discarded after being exposed to fluorescent light. Three trials of each frequency were performed. One hundred and thirty-one photographs were taken of the microscopic cell structure.

Post-insonification growth measurements

To measure the effect of ultrasound on post-insonification blue-green algae growth, twenty-four 1-mL samples were removed from three solutions that had been subjected to 1 hour of insonifcation with 200-kHz, 1.0-MHz and 2.2-MHz ultrasound, respectively, and put into a culture tray with 96 compartments.

Twenty-four control samples were taken. The culture tray was left in sunlight for 30 days. The grey scale depth was measured for each compartment.

Figure 4 shows the microscopic effect of ultrasound on floating bodies in the algae solution. From 0 minutes to 60 minutes of insonification, no change was seen in the physical structure of the algae for the whole frequency range.

Illuminating the algae with fluorescent light showed that the ultrasound had no effect on the chlorophyll activity for the whole frequency range.

The active chlorophyll shows that the algae strands are still alive and able to photosynthesise after 60 minutes insonification. This indicates that the ultrasound transmitted does not affect the chlorophyll-containing cells themselves, which is desirable since no toxins are released this way.

However, Fig. 5 shows that at all frequencies, for the floating bodies, the insonified samples showed greater brightness than the control samples. For the sunken bodies, all insonified samples showed reduced clarity as compared to the control samples. Thus, the ultrasound has caused the algae to sink. For example, after 60 minutes, the beakers subjected to 200-kHz insonification were 92 ± 12% brighter than the control samples, contrasted by the beaker bottoms, which were 53 ± 27% darker than the control samples.

Clearly, the algae that were floating in the beaker dropped to the bottom at a faster rate than the control sample. This has been attributed to the disruption of the floating bodies by the ultrasound. This is supported by Fig. 6, which

shows that the sunken bodies still have active chlorophyll but contain very few heterocysts.

Figure 7 shows the viability of the culture 30 days after insonification in terms of sample brightness. At 200 kHz, 1.0 MHz, and 2.2 MHz, the samples were 39 ± 14%, 45 ± 17%, and 46 ± 17% brighter than the control samples, respectively. All samples were significantly brighter than the control samples, even at the lower boundaries of the standard deviation. Thus, these results support the hypothesis that the algae that have sunk are less capable of multiplying. Hence, insonification may prevent algae bloom.

Our results can be interpreted as follows. When a heterocyst is subjected to an ultrasound pulse, the bubble encapsulated by it expands, in our frequency range [START_REF] Postema | Ultrasonic bubble in medicine: Influence of the shell[END_REF] during the rarefaction phase of the ultrasound. If the acoustic amplitude is sufficiently high, the encapsulating membrane cannot withhold the bubble from further expanding, resulting in its rupture. This phenomenon is similar to the sonic cracking of micrometer-sized membrane-encapsulated bubbles observed in [START_REF] Postema | Ultrasound-induced gas release from contrast agent microbubbles[END_REF]. Sonic cracking exclusively occurs during the expansion phase of a bubble [START_REF] Postema | Ultrasound-induced encapsulated microbubble phenomena[END_REF].

The resonance frequency f 0 of a an encapsultaed microbubble is given by [START_REF] Postema | Bubble dynamics involved in ultrasonic imaging[END_REF]:

f 0 = 1 2π 3Γ R 0 2 ρ p 0 + 2σ R 0 + 2χ R 0 - 2σ + 6χ R 0 3 ρ , (2)
where p 0 is the ambient pressure, R 0 is the bubble radius, Γ is the polytropic exponent of the gas, ρ is the liquid density, σ is the surface tension, and χ is the elasticity of the encapsulation [START_REF] Postema | Bubble dynamics involved in ultrasonic imaging[END_REF]. Using p 0 =1.013×10 5 Pa, R 0 =3µm, Γ=1.4, ρ is 998 kgm -3 , σ=0.072 Nm -1 , and assuming that the membrane elasticity is similar to that of a lipid encpsulation, χ=0.044 Nm -1 [START_REF] Gorce | Influence of bubble size distribution on[END_REF], we can estimate f 0 ≈1 MHz for Anabaena heterocysts used in our experiments.

Since the greatest change in clarity was seen at this particular frequency, we can safely state that ultrasound insonification close to heterocyst resonance frequency leads to a more effective eradication.

We assume that the correlation between (high)frequency and algae eradication in these studies is related to the ultrasound proximity to heterocyst resonance as well.

According to the NURC Rules and Procedures [START_REF]Human diver and marine mammal risk mitigation rules and procedures[END_REF], the maximum acoustic pressure to which mammals can be exposed is 708 Pa at frequencies up to 250 kHz. The transducers used had acoustic pressures of 40 kPa and 68 kPa at driving frequencies 1.0 MHz and 2.2 MHz, respectively. These pressures surpass the NURC Rules and Procedures by over 35 dB.

Conclusion

At any ultrasonic frequency we studied, blue-green algae were forced to sink. This supports our hypothesis that heterocysts release nitrogen under

58 5 To

 5 investigate the effect of diagnostic ultrasound on algae eradication, three ultrasound transducers were used. A 200-kHz undamped single element transducer containing a PIC155 Piezo crystal (PI Ceramics, Lederhose, Germany), a PA 188 (Precision Acoustics Ltd, Dorchester, UK) 1-MHz undamped single element transducer, and a 2.2-MHz undamped single element transducer containing a Pz37 Piezo crystal (Ferroperm Piezoceramics A/S, Kvistgård, Denmark). The focal distance of the 2.2-MHz transducer was 73 mm. The design of two transducers is shown in Fig. 2. The transducers were subjected to 16-Vpp square pulses at a 11.8-kHz pulse repetition rate transmitted by a V1.0 pulser-receiver (Sonemat, Coventry, UK). Low acoustic amplitudes were used in order to comply with MI<0.3 [20]. The acoustic amplitudes were measured in a separate water tank in the acoustic foci of the transducers with a 0.2-mm needle hydrophone (Precision Acoustics Ltd, Dorchester, UK) connected to a TDS 420A four channel digital oscilloscope (Tektronix Inc, Beaverton, USA). The peak-negative acoustic pressures were 40 kPa for the 1-MHz transducer and 68 kPa for the 2.2-MHz transducer, respectively, i.e., MI 0.1.

19 3 20 4 21 5 22 6 23 7

 1920212223 ultrasound insonification in the clinical diagnostic range. As supported by previous studies, under identical pulse length and pulse repetition, eradication is most effective close to heterocyst resonance, at a driving frequency of roughly 1 MHz. Although the acoustic fields we used to eradicate blue-green algae are safe in terms of mechanical index, the acoustic pressures surpass the NURC Rules and Procedures by over 35 dB. Therefore, caution should be taken when using these techniques in a surrounding where aquatic or semi-aquatic animals are present. Nitrogen-fixating body in Anabaena sphaerica algae. Under fluorescent light the body does not illuminate red, proving there is no chlorophyll in the body. Each frame corresponds to 565 × 565 µm 2 . 18 2 (a) Undamped 2.2-MHz ultrasound transducer with d = 1" diameter and r a = 35 mm acoustic lens. (b) Undamped 200-kHz ultrasound transducer with modification layer h PG = 10 mm. Experimental setup: (a) white paper sheet; (b) control sample; (c) clamp stand; (d) transducer; (e) insonified sample; (f) pulser-receiver. Areas A1 -B2 represent brightness measurement areas, C1 represents the calibration area. Microscopic image sequence showing the effect of 200 kHz -2.5 MHz ultrasound on the floating bodies of algae. Each frame corresponds to 565 × 565 µm 2 . Water brightness as a function of ultrasonic exposure time for samples containing floating algae or sunken algae. Microscopic image sequence of sunken algae Each frame corresponds to 565 × 565 µm 2 . Water clarity 30 days after insonification. 24

Fig. 1 .

 1 Fig. 1. Nitrogen-fixating body in Anabaena sphaerica algae. Under fluorescent light the body does not illuminate red, proving there is no chlorophyll in the body. Each frame corresponds to 565 × 565 µm 2 .

Fig. 2 .

 2 Fig. 2. (a) Undamped 2.2-MHz ultrasound transducer with d = 1" diameter and r a = 35 mm acoustic lens. (b) Undamped 200-kHz ultrasound transducer with modification layer h PG = 10 mm.

Fig. 3 .

 3 Fig. 3. Experimental setup: (a) white paper sheet; (b) control sample; (c) clamp stand; (d) transducer; (e) insonified sample; (f) pulser-receiver. Areas A1 -B2 represent brightness measurement areas, C1 represents the calibration area.

Fig. 4 . 2 .Fig. 5 .

 425 Fig. 4. Microscopic image sequence showing the effect of 200 kHz -2.5 MHz ultrasound on the floating bodies of algae. Each frame corresponds to 565 × 565 µm 2 .

Fig. 6 .Fig. 7 .

 67 Fig. 6. Microscopic image sequence of sunken algae Each frame corresponds to 565 × 565 µm 2 .

Acknowledgments

This work has been supported by EPSRC grant EP/F037025/1 and the HERI Research Pump Priming Fund. We would like to thank John Adams, The University of Hull, for support with the experimental setup.