An embedding theorem for automorphism groups of Cartan geometries

Uri Bader^{∗}, Charles Frances^{†}, and Karin Melnick^{‡}

June 19, 2008

1 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 ( [AS1], [AS2], [DA], [DAG], [Gr], [Kow], [Ze1], [Ze2], [Zi1], [Zi2] 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 give some background and discuss in detail these results.

∗uri.bader@gmail.com

†Charles.Frances@math.u-psud.fr

‡karin.melnick@yale.edu, partially supported by NSF fellowship DMS-855735

1.1 Statement and discussion of main theorem

The Lie algebra of a Lie group G is denoted 𝔤.

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}( \frac{d}{dt} |_{0} be^{tY}) = Y • for all b ∈ B and p ∈ P, the pullback R_{p}^{∗}ω = (Ad p^{−1}) ∘ ω

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 to Q-structures of finite type (See, for example, [Ko]. Note that the term G-structures is more common, but for notational reasons, we use Q). Kobayashi proved that for a linear Lie group 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 associates to a Q-structure of finite type a Cartan geometry with the same automorphism group. 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 S̄^{Ad} is the Zariski closure of Ad S in Aut 𝔥, and S̄_{d} is the maximal Zariski closed subgroup of S̄^{Ad} 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 S̄_{d}.

Section 4 contains a more detailed statement, in theorem 4.1. Theorem 1.2 is similar to the embedding theorem of Zimmer [Zi2], 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 S̄_{d} on some image of 𝔥. Unfortunately, the theorem gives no information for H discrete. An embedding theorem of comparable generality in the setting of rigid geometric structures of algebraic type appears in [Gr] 5.2.A. We find that the value of our embedding theorem in the setting of Cartan geometries lies in the several geometric applications detailed below. 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.

1.2 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.

Theorem 1.3. Let (M, B, ω) be a Cartan geometry modeled on G/P with M compact. Let H be a Lie subgroup of Aut M. Then

1. rk^{alg}(Ad H) ≤ rk(Ad_{𝔤}P).

2. n(Ad H) ≤ n(Ad_{𝔤}P); in particular, if H is connected, then

n(H) ≤ n(P) + 1

The bound in (1) is comparable to a theorem of Zimmer for automorphisms of Q-structures. He proved in [Zi4] that, for M compact, equipped with a Q-structure, where Q < GL(R^{n}) is algebraic, any simple H ⊂ Aut M has rk H ≤ rk Q. Note we assume H is neither simple nor connected.

The bound (1) is tight: if G is a simple noncompact Lie group, and P is a parabolic subgroup of G, then X = G/P is a compact Cartan geometry modeled on itself, on which G acts by automorphisms, and (1) is an equality in this case. The bound in (2) is also tight; an example is found among actions of connected nilpotent groups by isometries of a compact Lorentz manifold. Indeed, in this case, P = O(1, n − 1), the nilpotent subgroups of which are all abelian. The bound given by the theorem in this case is n(H) ≤ 2, which is tight since there exist isometric actions of the Heisenberg group on compact Lorentz manifolds. The bound (2) was first proved for Lorentz-isometric actions by Zimmer in [Zi2]. For parabolic Cartan geometries, we conjecture the sharper bound n(H) ≤ n(P), and we have proved it in the case of conformal pseudo-Riemannian structures ([FM]). The above theorem does not require that H be connected. Nevertheless, when H is a discrete group, it yields no information. The techniques of the proof do yield results for semi-discrete groups:

Theorem 1.4. Let L < GL(V) be an algebraic group with no compact algebraic quotients. Let Γ be a lattice in L, and H = Γ ⋉ V. Suppose that H < Aut M for M a compact Cartan geometry modeled on G/P. Then

1. rk L ≤ rk(Ad_{𝔤}P)

2. n(L) ≤ n(Ad_{𝔤}P)

1.3 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 normalization 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

\frac{c′′_{1} + P_{1}(c′)}{c′_{1}} = \frac{c′′_{2} + P_{2}(c′)}{c′_{2}} = ... = \frac{c′′_{n} + P_{n}(c′)}{c′_{n}}

with P_{k}(x) = Σa_{ij}^{k}x_{i}x_{j} for smooth functions a_{ij}^{k}. 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 X̃ and G̃, 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.

2. If H is connected, and rk(Ad H) = rk G, then M ≅ Γ\X̃, for some discrete subgroup Γ < G̃.

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].

1.4 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 H̄_{d} is the discompact radical of H̄^{Ad} (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 H̄_{d}-invariant. If there are only finitely-many H̄_{d}-invariant subalgebras of 𝔥, then the H-action is locally free on a dense, open, conull subset of M.

This theorem is proved in [Zi5] 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], [Me1]). Recall that H̄^{Ad} denotes the Zariski closure of Ad H, and H̄_{d} its discompact radical.

Theorem 1.7. Let (M, B, ω) be a Cartan geometry, and let H < Aut M be a Lie subgroup. If H preserves a finite volume on M and has an open orbit on M, then

1. The action of H^{o}, the identity component of H, is transitive on M.

2. If H̄^{Ad} = H̄_{d}, the action of H is everywhere locally free.

3. If H̄^{Ad} = H̄_{d}, 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. Here we restrict to real-analytic structures and actions, in order to apply certain results of Gromov.

Corollary 1.8. Let (M, B, ω) be a C^{an} Cartan geometry with M simply connected. Assume that there is a natural associated C^{an} rigid structure 𝒮 of algebraic type on M with Aut_{loc}M = Aut_{loc}(M, 𝒮). Let ν be a C^{an} finite volume form on M. Let H < Aut M be a Lie subgroup such that 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.

1.5 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 H̄^{Ad} = H̄_{d}. If H acts transitively on M preserving a finite volume, then

1. The group H̄^{Ad} 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 H̄^{Ad} on 𝔥 and of Ȟ on ι(𝔥).

3. The identity component H^{o} is isomorphic to a central extension of a subgroup H′ < Ȟ.

2 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.

2.1 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_{b}^{−1}(X), ω_{b}^{−1}(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 p̌ ∈ Ad P and β ∈ Hom(Λ^{2}(𝔤/𝔭), 𝔤),

(p̌.β)(u, v) = p̌^{−1}.(β(p̌.u, p̌.v))

For any b ∈ B, the Cartan connection induces a natural linear map

ι_{b} : 𝔥 → 𝔤
ι_{b}(X) = ω_{b}( \frac{d}{dt}|_{0} φ_{X}^{t}.b)

where φ_{X}^{t} 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_{X}^{t})_{∗b}Y (b))

Now

(X.ω(Y))(b) = \frac{d}{dt}|_{0} ω(Y (φ_{X}^{t}b))
= \frac{d}{dt}|_{0} [ω(Y (φ_{X}^{t}b)) − ω((φ_{X}^{t})_{∗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. ♢

2.2 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 P̌^{b} the algebraic subgroup of Ad_{𝔤}P consisting of those p̌ such that p̌.ι_{b}(𝔥) = ι_{b}(𝔥). There is a natural algebraic homomorphism

ρ_{b} : P̌^{b} → GL(𝔥)
ρ_{b}(p̌) = ι_{b}^{−1} ∘ p̌ ∘ ι_{b}

Now, if S < H is a subgroup, and b ∈ B, define

Š^{b} = {p̌ ∈ P̌^{b} | ρ_{b}(p̌) ∈ S̄_{d} and p̌.κ_{b} = κ_{b}}

The group Š^{b} may not be algebraic in general, but it is if ρ_{b} : Š^{b} → S̄_{d} is surjective, as in this case it is the preimage in P̌^{b} of the algebraic group S̄_{d} by the algebraic homomorphism ρ_{b}. The restriction of ρ_{b} to Š_{b} is then an algebraic epimorphism onto S̄_{d}. The main point in the proof of theorem 1.2 will be precisely to show the existence of b ∈ B such that ρ_{b} : Š_{b} → S̄_{d} is a surjection.

3 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, ch 2, 3], [D], 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 S̄^{ψ} or S̄. 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 S̄_{d}^{ψ} or S̄_{d} the discompact radical of S̄^{ψ}. The formulations of theorems 1.2 and 1.6 in terms of H̄_{d} and S̄_{d} apply to a wider class of groups than those for which the Zariski closures H̄^{Ad} or S̄^{Ad} have no compact algebraic quotients. Consider the following group. Let

g = ( \begin{matrix}cos θ & − sin θ & 0 \\ sin θ & cos θ & 0 \\ 0 & 0 & e^{λ} \end{matrix} ) ∈ GL(3, R)

for some θ ≠ 0 modulo 2π, λ ≠ 0. Let H = ⟨g⟩ ⋉ R^{3} ≅ Z ⋉ R^{3}. Then H̄_{d}^{Ad} ≅ R^{∗}, but for any S < H, the group S̄^{Ad} is either trivial or has S^{1} as a direct factor.

Theorem 3.1 (see [D, 2.6], [Shal, 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 S̄_{d}-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 S̄-orbits in V are locally closed [Zi3, 3.1.1], and it follows that μ is supported on a single S̄-orbit. We claim that μ is S̄-invariant. Indeed, Zimmer proved that the stabilizers of probability measures are Zariski closed when V is a quasi-projective variety [Zi3, 3.2.4], but by a theorem of Chevalley every S̄-orbit is S̄-equivariantly quasi-projective. It remains to show that every S̄-invariant measure is supported on the set of S̄_{d}-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 S̄_{d} 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.

Theorem 4.1. Let H < Aut M be a Lie subgroup and S < H. Suppose there is an S-invariant finite measure μ on M. Then there exists Λ ⊂ B such that π(Λ) is μ-conull and, for every b ∈ Λ,

ρ_{b} : Š^{b} → S̄_{d}

is an algebraic epimorphism. In particular, Š^{b} preserves κ_{b} and ρ_{b} intertwines the representation of Š^{b} on ι_{b}(𝔥) and that of S̄_{d} on 𝔥.

Proof: Observe first of all that the main point is to prove the surjectivity of ρ_{b}. The two other assertions of the theorem follow by the definitions of Š^{b} 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 P - orbits:

ϕ̄: M = B/P → V = U/P

The Ad_{𝔤}P -action on U is algebraic, and there is a Ad_{𝔤}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). Let us call U_{0} = ⋃_{h∈H} h.U′_{0}. This is a Zariski open set, which is in fact a finite union h_{1}.U′_{0} ∪ ... ∪ h_{s}.U′_{0}. Now, each quotient h_{j}.U′_{0}/Ad_{𝔤}P has the structure of a smooth algebraic variety. As a consequence, U_{0}/Ad_{𝔤}P has itself the structure of a smooth algebraic variety, and since the actions of H and Ad_{𝔤}P commute, H acts algebraically on U_{0}/Ad_{𝔤}P. We can then write U as U = U_{0} ∪ F_{0}. Let us apply the previous argument to the smooth part of the Zariski closed set F_{0}. We get a decomposition U = U_{0} ∪ U_{1} ∪ F_{1}, where U_{1} and F_{1} are H × Ad_{𝔤}P -invariant, and U_{1}/Ad_{𝔤}P is a smooth algebraic variety. After finitely many steps, we thus obtain a decomposition:

U = U_{0} ∪ ⋯ ∪ U_{r}

where each U_{i} is H × Ad_{𝔤}P -invariant, and the quotient V_{i} = U_{i}/Ad_{𝔤}P is a smooth algebraic variety, on which H acts algebraically. Denote by μ_{i} the restriction of μ to M_{i} = ϕ̄^{−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 ϕ̄ : (M_{i}, μ_{i}) → V_{i} gives that v = ϕ̄(m) is an S̄_{d}-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 S̄_{d}. This means that for any s ∈ S̄_{d}, there exists p̌_{s} ∈ Ad_{𝔤}P, not necessarily unique, such that s.(ι_{b}, κ_{b}) = p̌_{s}.(ι_{b}, κ_{b}), or in other words, ι_{b}∘s = p̌_{s}∘ι_{b}, and p̌_{s}.κ_{b} = κ_{b}. Thus, p̌_{s} ∈ Š^{b}, and ρ_{b}(p̌_{s}) = s. We thus get that ρ_{b} : Š^{b} → S̄_{d} is a surjection, and theorem 4.1 is proved. ♢

5 Proofs of bounds on automorphism groups

Recall that H̄_{d} 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.

5.1 Proof of theorem 1.3

To prove (1) of theorem 1.3, choose an R-split subgroup of Ad H with Zariski closure S̄ of dimension rk^{alg}(Ad H). Let S = Ad^{−1}(S̄) < H. The kernel of Ad is amenable by assumption (§1.1), and S̄ is abelian, so S is amenable. There is a finite S-invariant measure on M, and theorem 4.1 gives an algebraic subgroup Š^{b} < Ad_{𝔤}P such that ρ_{b}(Š^{b}) = S̄_{d} = S̄. The Lie algebra of Š^{b} admits a Levi decomposition into algebraic subalgebras 𝔰̌^{b} = 𝔯 ⋉ 𝔲, where 𝔲 consists of nilpotent endomorphisms, and 𝔯 is a reductive Lie algebra (see [WM1] ch.4). The homomorphism ρ_{b} induces an epimorphism of algebraic algebras dρ_{b} : 𝔰̌^{b} → 𝔰̄. 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 𝔰̄ 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}) = 𝔰̄. Then we have

rk(Ad_{𝔤}P) ≥ dim 𝔷^{s} ≥ dim 𝔰̄ = rk^{alg}(Ad H)

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

Lemma 5.1. Let N be a connected nilpotent group in GL(R^{k}). Let S̄_{d} be the discompact radical of the Zariski closure N̄. Then n(𝔰̄_{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 𝔫̄. Therefore n(𝔫) = n(𝔫̄). Of course n(𝔰̄_{d}) ≤ n(𝔫̄), so it suffices to show n(𝔰̄_{d}) ≥ n(𝔫̄). Write 𝔫̄ = 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(𝔫̄) = n(𝔲). But 𝔲 ⊂ 𝔰̄_{d}, so n(𝔫̄) ≤ n(𝔰̄_{d}). ♢ Now as above dρ_{b} is a surjective morphism of algebraic Lie algebras 𝔰̌^{b} → 𝔰̄_{d}. Take an algebraic Levi decomposition

𝔰̌^{b} = 𝔯 ⋉ 𝔲

and write 𝔯 = 𝔷 ⊕ 𝔩 as above. Since 𝔰̄_{d} 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 𝔰̄_{d} 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 𝔰̌^{b} is nilpotent and maps onto 𝔰̄_{d} via dρ_{b}. By lemma 5.1, we conclude:

n(Ad H) = n(𝔫) = n(𝔰̄_{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 L-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 h^{−1}Qh fixes η(x). In other words, for any b ∈ B lying over x ∈ M, the algebraic subgroup P̌^{b} < Ad_{𝔤}P, when restricted to ι_{b}(V), contains a subgroup isomorphic to h^{−1}Qh. Let Š^{b} be the algebraic subgroup ρ_{b}^{−1}(h^{−1}Qh), so ρ_{b} is an algebraic epimorphism from Š^{b} onto h^{−1}Qh. 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} ∈ Θ 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_{i=1}^{r}n_{i}α_{i} | n_{i} ∈ Z^{≤0} ∀ i or n_{i} ∈ Z^{≥0} ∀ i and Σ_{i=1}^{m}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}.

6.2 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 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_{k}^{−1} → 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 ⊂ Š^{b}, 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 the same as the direction of 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→∞} \frac{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

b_{k} = h_{k}.b.p_{k}^{−1} → 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, 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. Let u be in 𝔞^{+} and (p_{k}) a holonomy sequence at x with asymptotic direction u. Let h_{k} ∈ H and b ∈ π^{−1}(x) such that

b_{k} = h_{k}.b.p_{k}^{−1} → b_{∞} ∈ B

Choose an open U ⊂ 𝔫^{−} = 𝔤_{−k} ⊕ ⋯ ⊕ 𝔤_{−1} containing 0_{𝔫^{−}} such that exp_{b} is a diffeomorphism from ω_{b}^{−1}(U) onto its image; note that π ∘ exp_{b}(ω_{b}^{−1}(U)) is open in M. For every k ∈ N,

h_{k}. exp_{b}(ω_{b}^{−1}(U)).p_{k}^{−1} = exp_{b_{k}}(ω_{b_{k}}^{−1}((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,

exp_{b_{k}}(ω_{b_{k}}^{−1}((Ad p_{k}).U)) = h_{k}. exp_{b}(ω_{b}^{−1}(U)).p_{k}^{−1} → {b_{∞}}

Let V = π ∘ exp_{b}(ω_{b}^{−1}(U)). 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. We now prove point (1). Since Š^{b} 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 S̄ of dimension rk G. Let S = Ad ^{−1}(S̄) ⊂ 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 Š^{b} surjectively on S̄ = S̄_{d}. As in the proof of theorem 1.3 in section 5.1, there is an abelian R-split subalgebra 𝔷^{s} ⊂ 𝔰̌^{b} such that dρ_{b}(𝔷^{s}) = 𝔰̄. Then dim 𝔷^{s} = rk G, and there exists a maximal R-split subalgebra 𝔞′ ⊂ 𝔭 such that ad 𝔞′ = 𝔷^{s}. There exists p_{o} ∈ P such that (Ad p_{0})(𝔞′) = 𝔞. Since

Ad A < Š^{b.p_{0}} = (Ad p_{0})Š^{b}(Ad p_{0}^{−1})

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. By assumption, there is a connected Lie group A_{H} < H with Lie algebra 𝔞_{H} ⊂ 𝔥, such that Ad A_{H} is R-split, abelian, and has dimension rk G. Moreover, since the kernel of Ad is amenable by assumption, so is A_{H}. We first show Lemma 6.4. Every closed A_{H}-invariant subset of M contains an A_{H}-fixed point x with an open neighborhood on which the curvature vanishes.

Proof: Let F be a closed A_{H}-invariant subset. Apply theorem 4.1 with S = A_{H}. It gives b ∈ π^{−1}(F) such that Š^{b} is mapped surjectively by ρ_{b} on (Ā_{H})_{d}. Exactly as in the proof of the first point, changing b into b.p_{0} if necessary, we obtain Ad A < Š^{b}, and ρ_{b} maps Ad A onto the identity component of (Ā_{H})_{d}. Now (Ā_{H})_{d} acts trivially on 𝔞_{H}, so Ad A acts trivially on 𝔞_{H}^{b} = ι_{b}(𝔞_{H}). Thus 𝔞_{H}^{b} ⊂ 𝔞 ⊕ 𝔪, the centralizer of 𝔞; in particular, 𝔞_{H}^{b} ⊂ 𝔭, which proves that A_{H} fixes x = π(b). Since 𝔞_{H}^{b} ⊂ 𝔭, lemma 2.1 ensures that −ι_{b} is a Lie algebra isomorphism between 𝔞_{H} and 𝔞_{H}^{b}, so 𝔞_{H}^{b} 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_{H}^{b} acts on ι_{b}(𝔥) as an R-split subalgebra of dimension rk G. Decompose u ∈ 𝔞_{H}^{b} 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_{H}^{b} modulo 𝔪 is a surjection onto 𝔞. Let A_{H}^{b} be the connected subgroup of P with Lie algebra 𝔞_{H}^{b}. Any sequence in A_{H}^{b} is a holonomy sequence at the fixed point x = π(b). From above, A_{H}^{b} < A × M, and projects onto the first factor. Thus any sequence of A is equivalent to a sequence in A_{H}^{b}, 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.

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 \overline{H.x}, the closure of the H-orbit of x, is closed and H-invariant. By lemma 6.4, there are y ∈ \overline{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\X̃.

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 : M̃ → X, which is a geometric immersion, and a holonomy morphism ρ : Aut M̃ → 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 M̃, 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}(M̃) → 𝔤

where χ^{kill}(M̃) denotes the Lie algebra of Killing fields on M̃. Since H acts by automorphisms on M, there is an infinitesimal action of 𝔥 by Killing fields on M̃, so that 𝔥 ↪ χ^{kill}(M̃). Observe that any Killing field of M̃ 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 M̃ is geometrically isomorphic to X̃, so Aut M̃ ≅ G̃, and M is geometrically isomorphic to Γ\X̃, with Γ a discrete subgroup of G̃ acting properly discontinuously on X̃. ♢

7 Results about local freeness

7.1 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}(Ȟ^{b}) = H̄_{d}. The stabilizer Lie algebra 𝔥_{x} has image ι_{b}(𝔥_{x}) = ι_{b}(𝔥) ∩ 𝔭. Since ι_{b}(𝔥) ∩ 𝔭 is invariant by Ȟ^{b}, the Lie algebra 𝔥_{x} is invariant by H̄_{d}, as desired. Assume now that H̄_{d} 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_{x}^{b} 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_{x}^{b}(h^{t}.γ) = p^{t}.𝒟_{x}^{b}(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 is a Lie subgroup of Aut M. The Cartan geometry (M, B, ω) is connected and n-dimensional, equipped with an Hinvariant 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 p̌ ∈ Ad_{𝔤}P,

(p̌.α)(u_{1}, …, u_{n}) = α(p̌.u_{1}, …, p̌.u_{n}).

Notice that the H × P action on U is the restriction of an algebraic action of H̄^{Ad} × Ad_{𝔤}P. 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, ϕ(b) is a fixed point for H̄_{d}. Now, H̄_{d} is cocompact in H̄^{Ad} ([Shal, 1.4]). Let K be a compact subset such that H̄^{Ad} = KH̄_{d}. Then we get that for b ∈ Λ, the H̄^{Ad} -orbit of ϕ(b) is compact and in fact equals K.ϕ(b). This means that for every b ∈ Λ, and h ∈ H, there exist p̌_{h} ∈ Ad_{𝔤}P and l_{h} ∈ K satisfying ι_{h.b}(u) = p̌_{h}.ι_{b}(l_{h}.u) for every u ∈ 𝔥, and p̌_{h}.ν(b) = ν(b). Observe that if M has an open H-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 for ν intersects 𝒪, there is b ∈ Λ projecting on x ∈ 𝒪. Denote by \overline{𝒪} the topological closure of 𝒪 in M. Pick x_{∞} ∈ \overline{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_{k}^{−1} → b_{∞}. Let X_{1}, ..., X_{n} ∈ 𝔥. 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:

ν(h_{k}.b.p_{k}^{−1})(σ ∘ ι_{h_{k}.b.p_{k}^{−1}}(X_{1}), …, σ ∘ ι_{h_{k}.b.p_{k}^{−1}}(X_{n}))
= ν(b)(p̌_{h_{k}}.σ ∘ ι_{b}(l_{h_{k}}.X_{1}), …, p̌_{h_{k}}.σ ∘ ι_{b}(l_{h_{k}}.X_{n}))

Looking at a subsequence if necessary, we can assume that l_{h_{k}} tends to l_{∞} ∈ K. Since p̌_{h_{k}}.ν(b) = ν(b), the right-hand side in the previous equality tends to ν(b)(σ ∘ ι_{b}(l_{∞}.X_{1}), …, σ ∘ ι_{b}(l_{∞}.X_{n})), and the left one tends to ν(b_{∞})(σ ∘ ι_{b_{∞}}(X_{1}), …, σ ∘ ι_{b_{∞}}(X_{n})). Now, the tangent space T_{x}(H^{o}.x) = T_{x}M, so there are Y_{1}, …, Y_{n} ∈ 𝔥 such that (σ ∘ ι_{b}(Y_{1}), …, σ ∘ ι_{b}(Y_{n})) is a basis of 𝔤/𝔭, and thus ν(b)(σ ∘ ι_{b}(Y_{1}), …, σ ∘ ι_{b}(Y_{n})) ≠ 0. Choosing X_{1} = l_{∞}^{−1}.Y_{1}, ..., X_{n} = l_{∞}^{−1}.Y_{n}, we get that ν(b_{∞})(σ ∘ι_{b_{∞}}(X_{1}), …, σ ∘ι_{b_{∞}}(X_{n})) ≠ 0, and 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. Assume now that H̄^{Ad} = H̄_{d}. Theorem 1.6 implies that for almost every x ∈ M, the stabilizer subalgebra 𝔥_{x} is H̄^{Ad} -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. 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. ♢

7.3 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 Ȟ^{b} to H̄^{Ad} . Let Ȟ = Ad_{𝔤}^{−1}(Ȟ^{b}) < 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 from 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 → H̄^{Ad} 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 H̄^{Ad} 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. ♢

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