PROPER AFFINE ACTIONS AND GEODESIC FLOWS OF HYPERBOLIC SURFACES

W. GOLDMAN, F. LABOURIE, AND G. MARGULIS

Abstract. Let Γ_{0} ⊂ O(2, 1) be a Schottky group, and let Σ = H^{2}/Γ_{0} be the corresponding hyperbolic surface. Let C(Σ) denote the space of geodesic currents on Σ. The cohomology group H^{1}(Γ_{0}, V) parametrizes equivalence classes of affine deformations Γ_{u} of Γ_{0} acting on an irreducible representation V of O(2, 1). We define a continuous biaffine map

C(Σ) × H^{1}(Γ_{0}, V) ⟶^{Ψ} ℝ

which is linear on the vector space H^{1}(Γ_{0}, V). An affine deformation Γ_{u} acts properly if and only if Ψ(μ, [u]) ≠ 0 for all μ ∈ C(Σ). Consequently the set of proper affine actions whose linear part is a Schottky group identifies with a bundle of open convex cones in H^{1}(Γ_{0}, V) over the Teichmüller space of Σ.

Contents

Introduction 2 Notation and Terminology 5 1. Affine geometry 6 2. Flat bundles associated to affine deformations 10 3. Sections and subbundles 12 4. Hyperbolic geometry 14 5. Labourie’s diffusion of Margulis’s invariant 20 6. Properness 25 References 28

Date: March 2, 2006. 1991 Mathematics Subject Classification. 57M05 (Low-dimensional topology),

20H10 (Fuchsian groups and their generalizations).

Key words and phrases. proper action, hyperbolic surface, Schottky group, geodesic flow, geodesic current, flat vector bundle, flat affine bundle, homogeneous bundle, flat Lorentzian metric, affine connection.

Goldman gratefully acknowledge partial support from National Science Foundation grant DMS-0103889. Margulis gratefully acknowledge partial support from National Science Foundation grant DMS-0244406. Labourie thanks l’Institut Universitaire de France.

Introduction

It is well known that every discrete group of Euclidean isometries of ℝ^{n} contains a free abelian subgroup of finite index. In [31], Milnor asked if every discrete subgroup of affine transformations of ℝ^{n} must contain a polycyclic subgroup of finite index. Furthermore he showed that this question is equivalent to whether a discrete subgroup of Aff(ℝ^{n}) which acts properly must be amenable. By Tits [36], this is equivalent to the existence of a proper affine action of a nonabelian free group. Margulis subsequently showed [28, 29] that proper affine actions of nonabelian free groups do indeed exist. The present paper completes the classification of proper affine actions on ℝ^{3} for a large class of discrete groups.

If Γ ⊂ Aff(ℝ^{n}) is a discrete subgroup which acts properly on ℝ^{n}, then a subgroup of finite index will act freely. In this case the quotient ℝ^{n}/Γ is a complete affine manifold M with fundamental group π_{1}(M) ≅ Γ. When n = 3, Fried-Goldman [19] classified all the amenable such Γ (including the case when M is compact). Furthermore the linear holonomy homomorphism

π_{1}(M) ⟶^{𝕃} GL(ℝ^{3})

embeds π_{1}(M) onto a discrete subgroup Γ_{0} ⊂ GL(ℝ^{3}), which is conjugate to a subgroup of O(2, 1)^{0} (Fried-Goldman [19]). Mess [30] proved the hyperbolic surface

Σ = Γ_{0}\O(2, 1)^{0}

is noncompact. Goldman-Margulis [22] and Labourie [27] gave alternate proofs of Mess’s theorem, using ideas which evolved to the present work. Labourie’s result applies to affine deformations of Fuchsian actions in all dimensions.

Thus the classification of proper 3-dimensional affine actions reduces to affine deformations of free discrete subgroups Γ_{0} ⊂ O(2, 1)^{0}. An affine deformation of Γ_{0} is a group Γ of affine transformations whose linear part equals Γ_{0}, that is, a subgroup Γ ⊂ Isom^{0}(𝔼^{2,1}) such that the restriction of 𝕃 to Γ is an isomorphism Γ ⟶ Γ_{0}.

Equivalence classes of affine deformations of Γ_{0} ⊂ O(2, 1)^{0} are parametrized by the cohomology group H^{1}(Γ_{0}, V), where V is the linear holonomy representation. Given a cocycle u ∈ Z^{1}(Γ_{0}, V), we denote the corresponding affine deformation by Γ_{u}. Drumm [12, 13, 15, 11] showed that Mess’s necessary condition of non-compactness is sufficient: every noncocompact discrete subgroup of O(2, 1)^{0} admits proper affine deformations. In particular he found an open subset of H^{1}(Γ_{0}, V) parametrizing proper affine deformations [17].

This paper gives a criterion for the properness of an affine deformation Γ_{u} in terms of the parameter [u] ∈ H^{1}(Γ_{0}, V).

With no extra work, we take V to be representation of Γ_{0} obtained by composition of the discrete embedding Γ_{0} ⊂ SL(2, ℝ) with any irreducible representation of SL(2, ℝ). Such an action is called Fuchsian in Labourie [27]. Proper actions occur only when V has dimension 4k + 3 (see Abels [1], Abels-Margulis-Soifer [2, 3] and Labourie [27]). In those dimensions, examples of proper affine deformations of Fuchsian actions can be constructed using methods of Abels-Margulis-Soifer [2, 3].

Theorem. Suppose that Γ_{0} contains no parabolic elements. Then the equivalence classes of proper affine deformations of Γ form an open convex cone in H^{1}(Γ_{0}, V).

The main technique in this paper involves a generalization of the invariant constructed by Margulis [28, 29] and extended to higher dimensions by Labourie [27]. For an affine deformation Γ_{u}, a class function Γ_{0} ⟶^{α_{u}} ℝ exists which satisfies the following properties:

• α_{[u]}(γ^{n}) = |n|α_{[u]}(γ) (when r is odd);

• α_{[u]}(γ) = 0 ⟺ γ fixes a point in E.

• The function α_{[u]} depends linearly on [u];

• The map

H^{1}(Γ_{0}, V) ⟶ ℝ^{Γ_{0}}
[u] ⟼ α_{[u]}

is injective ([18]).

• Suppose dim V = 3. If Γ_{[u]} acts properly, then |α_{[u]}(γ)| is the Lorentzian length of the unique closed geodesic in E/Γ_{[u]} corresponding to γ.

• (Opposite Sign Lemma) Suppose Γ acts properly. Either α_{[u]}(γ) >

0 for all γ ∈ Γ_{0} or α_{[u]}(γ) < 0 for all γ ∈ Γ_{0}. This paper provides a more conceptual proof of the Opposite Sign Lemma by extending Margulis’s invariant to a continuous function Ψ_{u}on a connected space C(Σ) of probability measures. If

α_{[u]}(γ_{1}) < 0 < α_{[u]}(γ_{2}),

then an element μ ∈ C(Σ) “between” γ_{1} and γ_{2} exists, satisfying Ψ_{u}(μ) = 0.

Associated to the linear group Ω_{0} is a complete hyperbolic surface Σ = Γ_{0}\H^{2}. A geodesic current [6] is a Borel probability measure on the unit tangent bundle UΣ of Σ invariant under the geodesic flow ϕ. We modify α_{[u]} by dividing it by the length of the corresponding closed geodesic in the hyperbolic surface. This modification extends to a continuous function Ψ_{[u]} on the compact space C(Σ) of geodesic currents on Σ.

Here is a brief description of the construction, which first appeared in Labourie [27]. For brevity we only describe this for the case dim V = 3. Corresponding to the affine deformation Γ_{[u]} is a flat affine bundle 𝔼 over UΣ, whose associated flat vector bundle has a parallel Lorentzian structure 𝔹. Let ξ be the vector field on UΣ generating the geodesic flow. For any section s of 𝔼, its covariant derivative ∇_{ξ}(s) is a section of the flat vector bundle 𝕍 associated to 𝔼. Let ν denote the section of 𝕍 which associates to a point x ∈ UΣ the unit-spacelike vector corresponding to the geodesic associated to x. Let μ ∈ C(Σ) be a geodesic current. Then Ψ_{[u]}(μ) is defined by:

∫_{UΣ} 𝔹(∇_{ξ}(s), ν) dμ.

Nontrivial elements γ ∈ Γ_{0} correspond to periodic orbits c_{γ} for ϕ. The period of c_{γ} equals the length ℓ(γ) of the corresponding closed geodesic in Σ. Denote the subset of C(Σ) consisting of measures supported on periodic orbits by C_{per}(Σ). Denote by μ_{γ} the geodesic current corresponding to c_{γ}. Because α(γ^{n}) = |n|α(γ) and ℓ(γ^{n}) = |n|ℓ(γ), the ratio α(γ)/ℓ(γ) depends only on μ_{γ}.

Theorem. Let Γ_{u} denote an affine deformation of Γ_{0}.

• The function

C_{per}(Σ) ⟶ ℝ
μ_{γ} ⟼ \frac{α(γ)}{ℓ(γ)}

extends to a continuous function

C(Σ) ⟶^{Ψ_{[u]}} ℝ.

• Γ_{u} acts properly if and only if Ψ_{[u]}(μ) ≠ 0 for all μ ∈ C(Σ).

Since C(Σ) is connected, either Ψ_{[u]}(μ) > 0 for all μ ∈ C(Σ) or Ψ_{[u]}(μ) < 0 for all μ ∈ C(Σ). Compactness of C(Σ) implies that Margulis’s invariant grows linearly with the length of γ:

Corollary. For any u ∈ Z^{1}(Γ_{0}, V), there exists C > 0 such that

|α_{[u]}(γ)| ≤ Cℓ(γ)

for all γ ∈ Γ_{u}.

Margulis [28, 29] originally used the invariant α to detect properness: if Γ acts properly, then all the α(γ) are all positive, or all of them are negative. (Since conjugation by a homothety scales the invariant:

α_{λu} = λα_{u}

uniform positivity and uniform negativity are equivalent.) Goldman [20, 22] conjectured this necessary condition is sufficient: that is, properness is equivalent to the positivity (or negativity) of Margulis’s invariant. Jones [23] proved this when Σ is a three-holed sphere. In this case the cone corresponding to proper actions is defined by the inequalities corresponding to the three boundary components γ_{1}, γ_{2}, γ_{3} ⊂ ∂Σ:

{[u] ∈H^{1}(Γ_{0}, V) | α_{[u]}(γ_{i}) > 0 for i = 1, 2, 3}
⋃ {[u] ∈ H^{1}(Γ_{0}, V) | α_{[u]}(γ_{i}) < 0 for i = 1, 2, 3}

However, when Σ is a one-holed torus, Charette [9] recently showed that the positivity of α_{[u]} requires infinitely many inequalities, suggesting the original conjecture is generally false.

We thank Virginie Charette, David Fried, Todd Drumm, and Jonathan Rosenberg for helpful conversations. We are grateful to the hospitality of the Centre International de Recherches Mathématiques in Luminy where this paper was initiated. We are also grateful to Gena Naskov for several corrections.

Notation and Terminology

Denote the tangent bundle of a smooth manifold M by T M. For a given Riemannian metric on M, the unit tangent bundle UM consists of unit vectors in T M. For any subset S ⊂ M, denote the induced bundle over S by US. Denote the (real) hyperbolic plane by H^{2}.

Let 𝕍 be a vector bundle over a manifold M. Denote by R^{0}(𝕍) the vector space of sections of 𝕍. Let ∇ be a connection on 𝕍. If s ∈ R^{0}(𝕍) is a section and X is a vector field on M, then

∇_{X}(s) ∈ R^{0}(𝕍).

denotes covariant derivative of s with respect to X.

An affine bundle 𝔼 over M is a bundle of affine spaces over M. The vector bundle 𝕍 associated to 𝔼 is the vector bundle whose fiber 𝕍_{x} over x ∈ M is the vector bundle associated to the fiber 𝔼_{x}. Denote the affine space of sections of E by S(𝔼). If s_{1}, s_{2} ∈ S(𝔼) are two sections of E, the difference s_{1} − s_{2} is the section of 𝕍, whose value at x ∈ M is the translation s_{1}(x) − s_{2}(x) of 𝔼_{x}.

Denote the convex set of Borel probability measures on a topological space X by P(X). Denote the cohomology class of a cocycle z by [z]. A flow is an action of the additive group ℝ of real numbers. The transformations in a flow ϕ are denoted ϕ_{t}, for t ∈ ℝ.

1. Affine geometry

This section collects general properties on affine spaces, affine transformations affine deformations of linear group actions. We are primarily interested in affine deformations of linear actions factoring through the irreducible 2r + 1-dimensional real representation V_{r} of

G_{0} ≅ PSL(2, ℝ),

where r is a positive integer.

1.1. Affine spaces and their automorphisms. Let V be a real vector space. Denote its group of linear automorphisms by GL(V).

An affine space E (modelled on V) is a space equipped with a simply transitive action of V. Call V the vector space underlying E, and refer to its elements as translations. Translation τ_{v} by a vector v ∈ V is denoted by addition:

E ⟶^{τ_{v}} E
x ⟼ x + v.

Let x, y ∈ E. Denote the unique vector v ∈ V such that τ_{v}(x) = y by subtraction:

v = y − x.

Let E be an affine space with associated vector space V. Choosing an arbitrary point O ∈ E (the origin) identifies E with V via the map

V ⟶ E
v ⟼ O + v.

An affine automorphism of E is the composition of a linear mapping (using the above identification of E with V) and a translation, that is,

E ⟶^{g} E
O + v ⟼ O + 𝕃(g)(v) + u(g)

which we simply write as

v ⟼ 𝕃(g)(v) + u(g).

The affine automorphisms of E form a group Aff(E), and (𝕃, u) defines an isomorphism of Aff(E) with the semidirect product V ⋊ GL(V). The linear mapping 𝕃(g) ∈ GL(V) is the linear part of the affine transformation g, and

Aff(E) ⟶^{𝕃} GL(V)

is a homomorphism. The vector u(g) ∈ V is the translational part of g. The mapping

Aff(E) ⟶^{u} V

satisfies a cocycle identity:

(1.1)  u(γ_{1}γ_{2}) = u(γ_{1}) + 𝕃(γ_{1})u(γ_{2})

for γ_{1}, γ_{2} ∈ Aff(E).

1.2. Affine deformations of linear actions. Let Γ_{0} ⊂ GL(V) be a group of linear automorphisms of a vector space V. Denote the corresponding Γ_{0}-module by V as well.

An affine deformation of Γ_{0} is a representation

Γ_{0} ⟶^{ρ} Aff(E)

such that 𝕃 ∘ ρ is the inclusion Γ_{0} ↪ GL(V). We confuse ρ with its image Γ := ρ(Γ_{0}), which we also refer to as an affine deformation of Γ_{0}. Note that ρ embeds Γ_{0} as the subgroup Γ of GL(V). In terms of the semidirect product decomposition

Aff(E) ≅ V ⋊ GL(V)

an affine deformation is the graph ρ = ρ_{u} (with image denoted Γ = Γ_{u}) of a cocycle

Γ_{0} ⟶^{u} V

that is, a map satisfying the cocycle identity (1.1). Write

γ = ρ(γ_{0}) = (u(γ_{0}), γ_{0}) ∈ V ⋊ Γ_{0}

for the corresponding affine transformation:

x ⟼^{γ} γ_{0}x + u(γ_{0}).

Cocycles form a vector space Z^{1}(Γ_{0}, V). Cocycles u_{1}, u_{2} ∈ Z^{1}(Γ_{0}, V) are cohomologous if their difference u_{1} − u_{2} is a coboundary, a cocycle of the form

Γ_{0} ⟶^{δv_{0}} V
γ ⟼ v_{0} − γv_{0}.

Cohomology classes of cocycles form a vector space H^{1}(Γ_{0}, V). Affine deformations ρ_{u_{1}}, ρ_{u_{2}} are conjugate by translation by v_{0} if and only if

u_{1} − u_{2} = δv_{0}.

Thus H^{1}(Γ_{0}, V) parametrizes translational conjugacy classes of affine deformations of Γ_{0} ⊂ GL(V)

An important affine deformation of Γ_{0} is the trivial affine deformation: When u = 0, the affine deformation Γ_{u} equals Γ_{0} itself.

1.3. Margulis’s invariant of affine deformations. Consider the case that G_{0} = PSL(2, ℝ) and 𝕃 is an irreducible representation of G_{0}. For every positive integer r, let 𝕃_{r} denote the irreducible representation of G_{0} on the 2r-symmetric power V_{r} of the standard representation of SL(2, ℝ) on ℝ^{2}. The dimension of V_{r} equals 2r+1. The central element −𝕀 ∈ SL(2, ℝ) acts by (−1)^{2r} = 1, so this representation of SL(2, ℝ)) defines a representation of PSL(2, ℝ) = SL(2, ℝ)/{±𝕀}.

Furthermore the G_{0}-invariant nondegenerate skew-symmetric bilinear form on ℝ^{2} induces a nondegenerate symmetric bilinear form B on V_{r}, which we normalize in the following paragraph.

An element γ ∈ G_{0} is hyperbolic if it corresponds to an element γ̃ of SL(2, ℝ) with distinct real eigenvalues. Necessarily these eigenvalues are reciprocals λ, λ^{−1}, which we can uniquely specify by requiring |λ| < 1. Furthermore we choose eigenvectors v_{+}, v_{−} ∈ ℝ^{2} such that:

• γ̃(v_{+}) = λv_{+};

• γ̃(v_{−}) = λ^{−1}v_{−};

• The ordered basis {v_{−}, v_{+}} is positively oriented. Then the action 𝕃_{r} has eigenvalues λ^{2j}, for

j = −r, 1 − r, ⋯ − 1, 0, 1, …, r − 1, r,

where the symmetric product

v_{−}^{r−j}v_{+}^{r+j} ∈ V_{r}

is an eigenvector with eigenvalue λ^{2j}. In particular γ fixes the vector

x^{0}(γ) := cv_{−}^{r}v_{+}^{r},

where the scalar c is chosen so that

B(x^{0}(γ), x^{0}(γ)) = 1.

Call x^{0}(γ) the neutral vector of γ.

The subspaces

V^{−}(γ) := ∑_{j=1}^{r} ℝ(v_{−}^{r+j}v_{+}^{r−j})

V^{+}(γ) := ∑_{j=1}^{r} ℝ(v_{−}^{r−j}v_{+}^{r+j})

are γ-invariant and V enjoys a γ-invariant B-orthogonal direct sum decomposition

V = V^{−}(γ) ⊕ ℝ(x^{0}(γ)) ⊕ V^{+}(γ).

For any norm ∥ ∥ on V, there exists C, k > 0 such that

∥γ^{n}(v)∥ ≤ Ce^{−kn}∥v∥for v ∈ V^{+}(γ)
(1.2)  ∥γ^{−n}(v)∥ ≤ Ce^{−kn}∥v∥for v ∈ V^{−}(γ).

Furthermore

x^{0}(γ^{n}) = |n|x^{0}(γ)

if n ∈ ℤ, n ≠ 0, and

V^{±}(γ^{n}) = \begin{cases}V^{±}(γ) & if n > 0 \\ V^{∓}(γ) & if n < 0 \end{cases} .

For example, consider the hyperbolic one-parameter subgroup A_{0} comprising a(t) where

a(t) : \begin{cases}v_{+} & ⟼ e^{t/2}v_{+} \\ v_{−} & ⟼ e^{−t/2}v_{−} \end{cases}

In that case the action on V_{1} corresponds to the one-parameter group of isometries of ℝ^{2,1} defined by

a(t) ⟼ [\begin{matrix}cosh(t) & 0 & sinh(t) \\ 0 & 1 & 0 \\ sinh(t) & 0 & cosh(t) \end{matrix}]

with neutral vector

v_{0} = [\begin{matrix}0 \\ 1 \\ 0 \end{matrix}] .

Next suppose that g ∈ Aff(E) is an affine transformation whose linear part γ = 𝕃(γ) is hyperbolic. Then there exists a unique affine line l = l_{g} ⊂ E which is g-invariant. The line l is parallel to the neutral vector x^{0}(γ). The restriction of g to l_{g} is a translation by

B( gx − x , x^{0}(γ) ) x^{0}(γ)

where x^{0}(γ) is chosen so that B(x^{0}(γ), x^{0}(γ)) = 1.

Suppose that Γ_{0} ⊂ G_{0} be a Schottky group, that is, a nonabelian discrete subgroup containing only hyperbolic elements. Such a discrete subgroup is a free group of rank at least two. The adjoint representation of G_{0} defines an isomorphism of G_{0} with the identity component O(2, 1)^{0} of the orthogonal group of the 3-dimensional Lorentzian vector space ℝ^{2,1}.

Let u ∈ Z^{1}(Γ_{0}, V) be a cocycle defining an affine deformation ρ_{u} of Γ_{0}. In [28, 29], Margulis constructed the invariant

Γ_{0} ⟶^{α_{u}} ℝ
γ ⟼ B(u(γ), x^{0}(γ)).

This well-defined class function on Γ_{0} satisfies

α_{u}(γ^{n}) = |n|α_{u}(γ)

(when r is odd) and depends only the cohomology class [u] ∈ H^{1}(Γ_{0}, V). Furthermore α_{u}(γ) = 0 if and only if ρ_{u}(γ) fixes a point in E. Two affine deformations of a given Γ_{0} are conjugate if and only if they have the same Margulis invariant (Drumm-Goldman [18]). An affine deformation γ_{u} of Γ_{0} is radiant if it satisfies any of the following equivalent conditions:

• Γ_{u} fixes a point;

• Γ_{u} is conjugate to Γ_{0};

• The cohomology class [u] ∈ H^{1}(Γ_{0}, V) is zero;

• The Margulis invariant α_{u} is identically zero. For further discussion see [9, 10, 17, 20, 22, 27].)

The centralizer of Γ_{0} in the general linear group GL(ℝ^{3}) consists of homotheties

v ⟼^{h_{λ}} λv

where λ ≠ 0. The homothety h_{λ} conjugates an affine deformation (ρ_{0}, u) to (ρ_{0}, λu). Thus conjugacy classes of non-radiant affine deformations are parametrized by the projective space ℙ(H^{1}(Γ_{0}, V)). Our main result is that conjugacy classes of proper actions comprise a convex domain in ℙ(H^{1}(Γ_{0}, V)).

2. Flat bundles associated to affine deformations

We define two fiber bundles over UΣ. The first bundle 𝕍 is a vector bundle associated to the original group Γ_{0}. The second bundle 𝔼 is an affine bundle associated to the affine deformation Γ. The vector bundle underlying 𝔼 is 𝕍. The vector bundle 𝕍 has a flat linear connection and the affine bundle 𝔼 has a flat affine connection, each denoted ∇. (For the theory of connections on affine bundles, see Kobayashi-Nomizu [25].) The vector field Ξ tangent to the flow Φ defined above identifies with the unique vector field on the total space of 𝔼 which is ∇-horizontal and covers the vector field ξ defining the geodesic flow on UΣ.

2.1. Semidirect products and homogeneous affine bundles. Consider a Lie group G_{0}, a vector space V, and a linear representation G_{0} ⟶^{𝕃} GL(V). Let G be the corresponding semidirect product V ⋊ G_{0}. Multiplication in G is defined by

(2.1)  (v_{1}, g_{1}) · (v_{2}, g_{2}) := (v_{1} + 𝕃(g_{1})v_{2}, g_{1}g_{2}).

Projection G ⟶^{Π} G_{0} defines a trivial bundle with fiber V over G_{0}. It is equivariant with respect to the action of G on the total space by left-multiplication and the action of G on the base obtained from leftmultiplication on G_{0} and the homomorphism 𝕃. Since 𝕃 is a homomorphism, (2.1) implies equivariance of Π.

Consider Π as a (trivial) affine bundle by disregarding the origin in V. This affine structure is G-invariant: For (2.1) implies that the action of γ_{1} = (v_{1}, g_{1}) on the total space covers the action of g_{1} on the base. On the fibers the action is affine with linear part g_{1} = 𝕃(γ_{1}) and translational part v_{1} = u(γ_{1}). Denote this G-homogeneous affine bundle over G_{0} by 𝔼̃.

Consider Π also as a (trivial) vector bundle. By (2.1), this structure is G_{0}-invariant. Via 𝕃, this G_{0}-homogeneous vector bundle becomes a G-homogeneous vector bundle 𝕍̃.

This G-homogeneous vector bundle underlies 𝔼̃: Let (v′_{2} − v_{2}, 1) be the translation taking (v_{2}, g_{2}) to (v′_{2}, g_{2}). Then (2.1) implies that γ_{1} = (v_{1}, g_{1}) acts on (v′_{2} − v_{2}, g_{2}) by 𝕃:

((v_{1} + g_{1}(v′_{2})) − (v_{1} + g_{1}(v_{2})), g_{1}g_{2}) = (𝕃(g_{1})(v′_{2} − v_{2}), g_{1}g_{2}).

In our examples, 𝕃 preserves a a bilinear form B on V. The G_{0}- invariant bilinear form V ×V ⟶^{B} ℝ defines a bilinear pairing 𝕍̃×𝕍̃ ⟶^{𝔹} ℝ of vector bundles.

2.2. Homogeneous connections. The G-homogeneous affine bundle 𝔼̃ and the G-homogeneous vector bundle 𝕍̃ admit flat connections (each denoted ˜∇) as follows. To specify ˜∇, it suffices to define the covariant derivative of a section s̃ over a smooth path g(t) in the base. For either 𝔼̃ = G or 𝕍̃ = G, a section is determined by a smooth path v(t) ∈ V:

ℝ ⟶^{s̃} V ⋊ G_{0} = G
t ⟼ (v(t), g(t)).

Define

\frac{D}{dt}s̃(t) := \frac{d}{dt}v(t) ∈ V.

If now s̃ is a section of 𝔼̃ or 𝕍̃, and X is a tangent vector field, define

˜∇_{X}(s̃) = \frac{D}{dt}s̃(g(t))

where g(t) is any smooth path with

g′(t) = X(g(t)).

The resulting covariant differentiation operators define connections on 𝕍 and 𝔼 which are invariant under the respective G-actions.

2.3. Flatness. For each v ∈ V,

G_{0} ⟶^{s̃_{v}} V ⋊ G_{0} = G
g ⟼ (v, g)

defines a section whose image is the coset {v} × G_{0} ⊂ G. Clearly these sections are parallel with respect to ˜∇. Since the sections s̃_{v} foliate G, the connections are flat.

If 𝕃 preserves a bilinear form B on V, the bilinear pairing 𝔹 on 𝕍̃ is parallel with respect to ˜∇.

3. Sections and subbundles

Now we describe the sections and subbundles of the homogeneous bundles over G_{0} ≅ PSL(2, ℝ) associated to the irreducible 2r + 1dimensional representation V_{r}.

3.1. The flow on the affine bundle. Right-multiplication by a(t) on G defines a flow Φ̃_{t} on 𝔼̃. Since A_{0} ⊂ G_{0}, this flow covers the flow ϕ_{t} on G_{0} defined by right-multiplication by a(t) on G_{0}. That is, the diagram

𝔼̃ ⟶^{Φ̃_{t}}  𝔼̃
Π̃↓  ↓Π̃
UH^{2} ⟶_{ϕ̃_{t}} UH^{2}

commutes. The vector field Φ̃ on 𝔼̃ generating this flow covers the vector field ϕ̃ generating ϕ_{t}.

Furthermore s̃_{v}(ga(−t)) = s̃(v)a(−t) implies s̃_{v} ∘ ϕ̃_{t} = Φ̃_{t} ∘ s̃_{v}, whence Ξ̃ is the ˜∇-horizontal lift of ξ̃. The flow Φ̃_{t} commutes with the action of G. Thus Φ̃_{t} is a flow on

the flat G-homogeneous affine bundle 𝔼̃ covering ϕ_{t}.

Right-multiplication by a(t) on G also defines a flow on the flat Ghomogeneous vector bundle 𝕍̃ covering ϕ̃_{t}. Identifying 𝕍̃ as the vector bundle underlying 𝔼̃, the ℝ-action is just the linearization DΦ̃_{t} of the action Φ̃_{t}:

𝕍̃ ⟶^{DΦ̃_{t}}  𝕍̃
↓  ↓
UH^{2} ⟶_{ϕ̃_{t}} UH^{2}

3.2. The neutral section. The G-action and the flow DΦ̃_{t} on 𝕍̃ preserve a section ν̃ ∈ 𝕍̃ defined as follows. The one-parameter subgroup A_{0} fixes the neutral vector v_{0} ∈ V. Let ν̃ denote the section of 𝕍̃ defined by:

UH^{2} ≈ G_{0} ⟶^{ν̃} V ⋊ G_{0} ≈ 𝕍̃
g ⟼ (𝕃(g)v_{0}, g).

In terms of the group operation on G, this section is given by rightmultiplication by v_{0} ∈ V ⊂ G acting on g ∈ G_{0} ⊂ G. Since 𝕃 is a homomorphism,

ν̃(hg) = hν̃(g),

so ν̃ defines a G-invariant section of 𝕍̃.

Although ν̃ is not parallel in every direction, it is parallel along the flow Φ̃_{t}:

Lemma 3.1. ˜∇ξ̃(ν̃) = 0.

Proof. Let g ∈ G_{0}. Then

˜∇ξ̃(ν̃)(g) = \frac{D}{dt}|_{t=0}ν̃(ϕ̃_{t}(g)) = \frac{D}{dt}|_{t=0}ga(−t)v_{0} = 0

since a(−t)v_{0} = v_{0} is constant. □

The section ν̃ is the diffuse analogue of the neutral eigenvector x^{0}(γ) of a hyperbolic element γ ∈ O(2, 1)^{0} discussed in §1.3. Another approach to the flat connection ∇ and the neutral section ν is given in Labourie [27].

3.3. Stable and unstable subbundles. Let V^{0} ⊂ V denote the line of vectors fixed by A_{0}, that is the line spanned by v_{0}. The eigenvalues of a(t) acting on V are the 2r + 1 distinct positive real numbers

e^{r}, e^{r−2}, …, 1, …, e^{2−r}, e^{−r}

and the eigenspace decomposition of V is invariant under a(t). Let V^{+} denote the sum of eigenspaces for eigenvalues > 1 and V^{−} denote the sum of eigenspaces for eigenvalues < 1. The corresponding decomposition

V = V^{−} ⊕ V^{0} ⊕ V^{+}

is a(t)-invariant and defines a (left)G-invariant decomposition of the vector bundle 𝕍̃ into subspaces which are invariant under DΦ̃_{t}.

4. Hyperbolic geometry

Let ℝ^{2,1} be the 3-dimensional real vector space with inner product

B(u, v) := u_{1}v_{1} + u_{2}v_{2} − u_{3}v_{3}

and G_{0} = O(2, 1)^{0} the identity component of its group of isometries. G_{0} consists of linear isometries of ℝ^{2,1} which preserve both an orientation of ℝ^{2,1} and a time-orientation (or future) of ℝ^{2,1}, that is, a connected component of the open light-cone

{v ∈ ℝ^{2,1} | B(v, v) < 0}.

Then

G_{0} = O(2, 1)^{0} ≅ PSL(2, ℝ) ≅ Isom^{0}(H^{2})

where H^{2} denotes the real hyperbolic plane. The representation ℝ^{2,1} is isomorphic to the irreducible representation V_{2} defined in §1.3.

4.1. The hyperboloid model. We define H^{2} as follows. We work in the irreducible representation ℝ^{2,1} = V_{1} (isomorphic to the adjoint representation of G_{0}). Let B denote the invariant bilinear form (isomorphic to the Killing form). The two-sheeted hyperboloid

{v ∈ ℝ^{2,1} | B(v, v) = −1}

has two connected components. If v_{1} lies in this hyperboloid, then its connected component equals

(4.1)  {v ∈ ℝ^{2,1} | B(v, v) = −1, B(v, v_{1}) < 0}.

For clarity, we fix a timelike vector v_{1}, for example

v_{1} = [\begin{matrix}0 \\ 0 \\ 1 \end{matrix}] ∈ ℝ^{2,1},

and define H^{2} as the connected component (4.1).

The Lorentzian metric defined by B restricts to a Riemannian metric of constant curvature −1 on H^{2}. The identity component G_{0} of O(2, 1)^{0} is the group Isom^{0}(𝔼^{2,1}) of orientation-preserving isometries of H^{2}. The stabilizer of v_{1}

K_{0} := PO(2).

is the maximal compact subgroup of G_{0}. Evaluation at v_{1}

G_{0}/K_{0} ⟶ ℝ^{2,1}
gK_{0} ⟼ gv_{1}

identifies H^{2} with the homogeneous space G_{0}/K_{0}.

The action of G_{0} canonically lifts to a simply transitive (left-) action on the unit tangent bundle UH^{2}. The unit spacelike vector

(4.2)  v̇_{1} := [\begin{matrix}0 \\ 1 \\ 0 \end{matrix}] ∈ ℝ^{2,1}

defines a unit tangent vector to H^{2} at v_{1}, which we denote:

(v_{1}, v̇_{1}) ∈ UH^{2}.

Evaluation at (v_{1}, v̇_{1})

G_{0} ⟶^{E} UH^{2}
g_{0} ⟼ g_{0}(v_{1}, v̇_{1})

G_{0}-equivariantly identifies G_{0} with the unit tangent bundle UH^{2}. Here G_{0} acts on itself by left-multiplication. The action on UH^{2} is the action induced by isometries of H^{2}. Under this identification, K_{0} corresponds to the fiber of UH^{2} above v_{1}.

4.2. The geodesic flow. The Cartan subgroup

A_{0} := PO(1, 1).

is the one-parameter subgroup comprising

a(t) = [\begin{matrix}1 & 0 & 0 \\ 0 & cosh(t) & sinh(t) \\ 0 & sinh(t) & cosh(t) \end{matrix}]

for t ∈ ℝ. Under this identification,

a(t)v_{1} ∈ H^{2}

describes the geodesic through v_{1} tangent to v̇_{1}.

Right-multiplication by a(t) on G_{0} identifies with the geodesic flow ϕ̃_{t} on UH^{2}. First note that ϕ̃_{t} on UH^{2} commutes with the action of G_{0} on UH^{2}, which corresponds to the action on G_{0} on itself by leftmultiplications. The ℝ-action on G_{0} corresponding to ϕ̃_{t} commutes with left-multiplications on G_{0}. Thus this ℝ-action on G_{0} must be right-multiplication by a one-parameter subgroup. At the basepoint (v_{1}, v̇_{1}), the geodesic flow corresponds to by right-multiplication by A_{0}. Hence this one-parameter subgroup must be A_{0}. Therefore rightmultiplication by A_{0} on G_{0} induces the geodesic flow ϕ̃_{t} on UH^{2}.

Denote the vector field corresponding to the geodesic flow by ξ̃:

ξ̃(x) := \frac{d}{dt}|_{t=0}ϕ̃_{t}(x).

4.3. The convex core. Since Γ_{0} is a Schottky group, the complete hyperbolic surface is convex cocompact. That is, there exists a compact geodesically convex subsurface core(Σ) such that the inclusion

core(Σ) ⊂ Σ

is a homotopy equivalence. core(Σ) is called the convex core of Σ, and is bounded by closed geodesics ∂_{i}(Σ) for i = 1, …, k. Equivalently the discrete subgroup Γ_{0} is finitely geneerated and contains only hyperbolic elements.

Another criterion for convex cocompactness involves the ends e_{i}(Σ) of Σ. Each component e_{i}(Σ) of the closure of Σ \ core(Σ) is diffeomorphic to a product

e_{i}(Σ) ⟶^{≈} ∂_{i}(Σ) × [0, ∞).

The corresponding end of UΣ is diffeomorphic to the product

e_{i}(UΣ) ⟶^{≈} ∂_{i}(Σ) × S^{1} × [0, ∞).

The corresponding Cartesian projections

e_{i}(Σ) ⟶ ∂_{i}(Σ)

and the identity map on core(Σ) extend to a deformation retraction

Σ ⟶^{Π_{core(Σ)}} core(Σ).

We use the following standard compactness results (see Kapovich [24], §4.17 (pp.98–102) or Canary-Epstein-Green [8] for discussion and proof):

Lemma 4.1. Let Σ = H^{2}/Γ_{0} be the hyperbolic surface where Γ_{0} is convex cocompact and let R ≥ 0. Then

K_{R} := {y ∈ Σ | d(y, core(Σ)) ≤ R}

is a compact geodesically convex subsurface, upon which the restriction of Π_{core(Σ)} is a homotopy equivalence.

Lemma 4.2. Let Σ be as above, UΣ its unit tangent bundle and ϕ_{t} the geodesic flow on UΣ. Then the space of ϕ-invariant Borel probability measures on UΣ is a convex compact space C(Σ) with respect to the weak ⋆-topology.

Proof. The subbundle Ucore(Σ) comprising unit vectors over points in core(Σ) is compact.

Any geodesic which exits core(Σ) into an end e_{i}(Σ) remains in the same end e_{i}(Σ). Therefore every recurrent geodesic ray eventually lies in core(Σ). By the Poincaré recurrence theorem, every ϕ-invariant probability measure on UΣ is supported over core(Σ). Indeed, the union of all recurrent orbits of ϕ is a closed ϕ-invariant subset U_{rec}Σ ⊂ UΣ which we now show is compact.

On the unit tangent bundle UH^{2}, every ϕ orbit tends to a unique ideal point on S_{∞}^{1} = ∂H^{2}; let

UH^{2} ⟶^{η} S_{∞}^{1}

denote the corresponding map. Then the ϕ-orbit of x̃ ∈ UH^{2} defines a geodesic which is recurrent in the forward direction if and only if η(x̃) ∈ Λ, where Λ ⊂ S_{∞}^{1} is the limit set of Γ_{0}. The geodesic is recurrent in the backward direction if η(−x̃) ∈ Λ. Then every recurrent orbit of the geodesic flow lies in the compact set U_{rec}Σ consisting of vectors x ∈ Ucore(Σ) with

η(x̃), η(−x̃) ∈ Λ.

Since U_{rec}Σ is compact, P(U_{rec}Σ) is a compact convex set, and the closed subset of ϕ-invariant probability measures on U_{rec}Σ is also a compact convex set. Since every ϕ-invariant probability measure on UΣ is supported on U_{rec}Σ, the assertion follows. □

4.4. Bundles over UΣ. Let Γ ⊂ G be an affine deformation of a discrete subgroup Γ_{0} ⊂ G_{0}. Since Γ is a discrete subgroup of G, the quotient 𝔼 := 𝔼̃/Γ is an affine bundle over UΣ = UH^{2}/Γ_{0} and inherits a flat connection ∇ from the flat connection ˜∇ on 𝔼̃. Furthermore the flow Φ̃_{t} on 𝔼̃ descends to a flow Φ on 𝔼 which is the horizontal lift of the flow ϕ on UΣ.

The vector bundle 𝕍 underlying 𝔼 is the quotient

𝕍 := 𝕍̃/Γ = 𝕍̃/Γ_{0}

and inherits a flat linear connection ∇ from the flat linear connection ˜∇ on 𝕍̃. The flow DΦ̃_{t} on 𝕍̃ covering ϕ̃_{t}, the neutral section ν̃, and the stable-unstable splitting all descend to a flow DΦ, a section ν and a splitting

(4.3)  𝕍 = 𝕍^{−} ⊕ 𝕍^{0} ⊕ 𝕍^{+}

of the flat vector bundle 𝕍 over UΣ. In particular

(Π_{core(Σ)})^{∗}𝕍_{core(Σ)} ≅ 𝕍

since Π_{core(Σ)} is a homotopy equivalence.

We choose a Euclidean metric g on 𝕍 with the following properties: • The neutral section ν is bounded with respect to g;

• The flat linear connection ∇ is bounded with respect to g;

• The flow Dϕ exponentially expands the subbundle V^{+} and ex-

ponentially contracts V^{−}. One may construct such a metric by choosing any Euclidean inner product on the vector space V and extending it to a Euclidean metric on the vector bundle

𝕍̃ ≈ G = G_{0} × V

which is invariant under left-multiplications in G_{0}. This Euclidean metric satisfies the following hyperbolicity condition: Constants C, k > 0 exist so that

∥D(ϕ_{t})v∥ ≤ Ce^{kt}∥v∥ as t ⟶ −∞,

for v ∈ 𝕍^{+}, and

∥D(ϕ_{t})v∥ ≤ Ce^{−kt}∥v∥ as t ⟶ +∞.

for v ∈ 𝕍^{−}, which follow immediately from (1.2).

4.5. Proper Γ-actions and proper ℝ-actions. Let X be a locally compact Hausdorff space with homeomorphism group Homeo(X). Suppose that H is a closed subgroup of of Homeo(X) with respect to the compact-open topology. Recall that H acts properly if the mapping

H × X ⟶ X × X
(g, x) ⟼ (gx, x)

is a proper mapping. That is, for every pair of compact subsets K_{1}, K_{2} ⊂ X, the set

{g ∈ H | gK_{1} ∩ K_{2} ≠ ∅}

is compact. The usual notion of a properly discontinuous action is a proper action where H is given the discrete topology. In this case, the quotient X/H is a Hausdorff space. If H acts freely, then the quotient mapping X ⟶ X/H is a covering space. For background on proper actions, see Bourbaki [7], Koszul [26] and Palais [32].

The question of properness of the Γ-action is equivalent to that of properness of an action of ℝ.

Proposition 4.3. An affine deformation Γ acts properly on E if and only if A_{0} acts properly by right-multiplication on G/Γ.

The following lemma is a key tool in our argument. (A related statement is proved in Benoist [5], Lemma 3.1.1. For a proof in a different context see Rieffel [33, 34].)

Lemma 4.4. Let X be a locally compact Hausdorff space. Let A and B be commuting groups of homeomorphisms of X, each of which acts properly on X. Then the following conditions are equivalent:

(1) A acts properly on X/B;

(2) B acts properly on X/A;

(3) A × B acts properly on X.

Proof. We prove (3) ⟹ (1). Suppose A × B acts properly on X but A does not act properly on X/B. Then a B-invariant subset H ⊂ X exists such that:

• H/B ⊂ X/B is compact;

• The subset

A_{H/B} := {a ∈ A | a(H/B) ∩ (H/B) ≠ ∅}

is not compact. We claim a compact subset K ⊂ X exists satisfying B · K = H.

Denote the quotient mapping by

X ⟶^{Π_{B}} X/B.

For each x ∈ Π_{B}^{−1}(H/B), choose a precompact open neighborhood U(x) ⊂ X. The images Π_{B}(U(x)) define an open covering of H/B. Choose a finite subset x_{1}, …, x_{l} such that the open sets Π_{B}(U(x_{i})) cover H/B. Then the union

K′ := ⋃_{i=1}^{l} \overline{U(x_{i})}

is a compact subset of X such that Π_{B}(K′) ⊃ H/B. Taking K = K′ ∩ Π_{B}^{−1}(H/B) proves the claim.

Since A acts properly on X, the subset

(A × B)_{K} := {(a, b) ∈ A × B | aK ∩ bK ≠ ∅}

is compact. But A_{H/B} is the image of the compact set (A × B)_{K} under Cartesian projection A × B ⟶ A and is compact, a contradiction.

We prove that (1) ⟹ (3). Suppose that A acts properly on X/B and K ⊂ X is compact. Then Cartesian projection A × B ⟶ A maps

(4.4)  (A × B)_{K} ⟶ A_{(B·K)/B}.

A acts properly on X/B implies A_{(B·K)/B} is a compact subset of A. But (4.4) is a proper map, so (A × B)_{K} is compact, as desired.

Thus (3) ⟺ (1). The proof (3) ⟺ (2) is similar.

□

Proof of Proposition 4.3. Apply Lemma 4.4 with X = G and A to be the action of A_{0} ≅ ℝ by right-multiplication and B to be the action of Γ by left-multiplication. The lemma implies that Γ acts properly on E = G/G_{0} if and only if G_{0} acts properly on Γ\G.

Apply the Cartan decomposition G_{0} = K_{0}A_{0}K_{0}. Since K_{0} is compact, the action of G_{0} on E is proper if and only if its restriction to A_{0} is proper. □

5. Labourie’s diffusion of Margulis’s invariant

In [27], Labourie defined a function UΣ ⟶^{F_{u}} ℝ, corresponding to the invariant α_{u} defined by Margulis [28, 29]. Margulis’s invariant α = α_{u} is an ℝ-valued class function on Γ_{0} whose value on γ ∈ Γ_{0} equals

B( ρ_{u}(γ)O − O , x^{0}(γ) )

where x^{0}(γ) ∈ V is the neutral vector of γ (see §1.3 and the references listed there).

Now the origin O ∈ E will be replaced by a section s of 𝔼, the vector x^{0}(γ) ∈ V will be replaced by the neutral section ν of 𝕍, and the linear action of Γ_{0} will be replaced by the geodesic flow ϕ_{t} on UΣ.

Let s be a section of 𝔼. Its covariant derivative with respect to ξ is a section 𝕍.∇_{ξ}(s) ∈ R^{0}(𝕍). Pairing with ν ∈ R^{0}(𝕍) via

𝕍 × 𝕍 ⟶^{𝔹} ℝ

produces a function UΣ ⟶^{F_{u,s}} ℝ:

F_{u,s} := 𝔹(∇_{ξ}(s), ν).

(Compare Labourie [27].)

5.1. The invariant is well-defined. Henceforth choose a Riemannian metric g on 𝕍 such that B and ν are bounded with respect to g. Let S(𝔼) denote the space of continuous sections s of 𝔼, differentiable along ξ, such that ∇_{ξ}(s) is bounded with respect to g. In particular F_{u,s} is bounded and measurable.

Lemma 5.1. Let s ∈ S(𝔼) and μ be a ϕ-invariant Borel probability measure on UΣ. Then

∫_{UΣ} F_{u,s} dμ

is independent of s.

Proof. Let s_{1}, s_{2} ∈ S(𝔼) be sections. Then s = s_{1} − s_{2} is a section of 𝕍 and

∇_{ξ}(s_{1}) − ∇_{ξ}(s_{2}) = ∇_{ξ}(s).

Therefore

∫_{UΣ} F_{u,s_{1}}dμ − ∫_{UΣ} F_{u,s_{2}}dμ = ∫_{UΣ} 𝔹(∇_{ξ}(s), ν)dμ
= ∫_{UΣ} ξ𝔹(s, ν)dμ − ∫_{UΣ} 𝔹(s, ∇_{ξ}(ν))dμ.

The first term vanishes since μ is ϕ-invariant:

∫_{UΣ} ξ𝔹(s, ν)dμ = ∫_{UΣ} \frac{d}{dt}(ϕ_{t})^{∗}(𝔹(s, ν))dμ

= \frac{d}{dt} ∫_{UΣ}(ϕ_{t})^{∗}(𝔹(s, ν))dμ

= \frac{d}{dt} ∫_{UΣ} 𝔹(s, ν)d((ϕ_{t})_{∗}μ) = 0.

The second term vanishes since ∇_{ξ}(ν) = 0 (Lemma 3.1). □

Thus

Ψ_{[u]}(μ) := ∫_{UΣ} F_{u,s}dμ

is a well-defined function C(Σ) ⟶^{Ψ_{[u]}} ℝ whose continuity follows from the definition of the weak topology on P(UΣ).

5.2. Periodic orbits. Suppose that

t ⟼ Γ_{0}ga(t)

defines a periodic orbit of the geodesic flow ϕ_{t}. Suppose that γ is the corresponding element of Γ_{0}. Then

Γ_{0}ga(t + T) = Γ_{0}γga(t) = Γ_{0}ga(t)

where T = ℓ(γ) is the period of the orbit.

The following notation is useful. If x_{0} ∈ UΣ and T > 0, let ϕ_{[0,T]}^{x_{0}} denote the map

[0, T] ⟶^{ϕ_{[0,T]}^{x_{0}}} UΣ
(5.1)  t ⟼ ϕ_{t}(x_{0}).

Let T = ℓ(γ) denote the period of the periodic orbit corresponding to γ and μ_{[0,T]} denote Lebesgue measure on [0, T]. Then

(5.2)  μ_{γ} := \frac{1}{T} (ϕ_{[0,T]}^{x_{0}})_{∗}(μ_{[0,T]})

defines the geodesic current associated to the periodic orbit γ.

Proposition 5.2 (Labourie [27], Proposition 4.2). Let γ ∈ Γ_{0} be hyperbolic and let μ_{γ} ∈ C_{per}(Σ) be the corresponding geodesic current. Then

(5.3)  α(γ) = ℓ(γ) ∫_{UΣ} F_{u,s} dμ_{γ}.

Proof. Let x_{0} ∈ UΣ is a point on the periodic orbit, and consider the map (5.1) and the geodesic current (5.2).

Choose a section s ∈ S(𝔼). The pullback of s to the covering space UH^{2} ≈ G_{0} is the graph of a map

ṽ : G_{0} ⟶ V

satisfying

ṽ ∘ γ = ρ(γ)ṽ.

Then

(5.4)  ∫_{UΣ} F_{u,s} dμ_{γ} = \frac{1}{T} ∫_{0}^{T} F_{u,s}(ϕ_{t}(x_{0}))dt,

which we evaluate by passing to the covering space UH^{2} ≈ G_{0}. Lift the basepoint x_{0} to x̃_{0} ∈ UH^{2}. Then ϕ_{[0,T]}^{x_{0}} lifts to the map

[0, T] ⟶^{ϕ̃x̃_{0} ; [0,T]} UH^{2}
t ⟼ ϕ̃_{t}(x̃_{0}).

Let g ∈ G_{0} correspond via E to x̃_{0}. Then the periodic orbit lifts to the trajectory

ℝ ⟶ G_{0}
t ⟼ ga(−t).

Since the periodic orbit corresponds to the deck transformation γ (which acts by left-multiplication on G_{0}),

γg = ga(−T)

which implies

(5.5)  γ = ga(−T)g^{−1}.

Evaluate ∇_{ξ}s and ν along the trajectory

ϕ̃_{t}x̃_{0} ⟷ ga(−t)

by lifting to the covering space and computing in G. Lift s to

G_{0} ⟶^{s̃} V ⋊ G_{0} = G
g ⟼ (ṽ(g), g)

PROPER AFFINE ACTIONS 23

where ṽ : G_{0} ⟶ V satisfies

ṽ(γg) = ρ(γ)ṽ(g).

Then

(∇_{ξ}s̃)(ϕ_{t}x̃_{0}) = \frac{D}{dt}ṽ(ga(−t)).

In semidirect product coordinates, the lift ν̃ is defined by the map

G_{0} ⟶^{ν̃} V
g ⟼ 𝕃(g)v_{0}.

Lemma 5.3. ν̃(ga(−t)) = x^{0}(γ) for all t ∈ ℝ. Proof.

ν̃(ga(−t)) = 𝕃(ga(−t)) v_{0}
= 𝕃(ga(−t)) x^{0}(a(−T))
= x^{0}((ga(−t))a(−T)(ga(−t))^{−1})
= x^{0}((ga(−T)g^{−1}) = x^{0}(γ) (by (5.5)).
□

We evaluate the integrand in (5.4):

F_{u,s}(ϕ_{t}x_{0}) = 𝔹(∇_{ξ}s, ν)(ϕ_{t}x_{0})

= B(∇_{ξ}s̃(ga(−t)), ν̃(ga(−t))

= B(\frac{D}{dt}ṽ(ga(−t)), x^{0}(γ))

= \frac{d}{dt}B(ṽ(ga(−t)), x^{0}(γ))

and

∫_{0}^{T} F_{u,s}(ϕ_{t}x_{0})dt = B(ṽ(ga(−T)), x^{0}(γ)) − B(ṽ(g), x^{0}(γ))
= B(ṽ(γg) − ṽ(g), x^{0}(γ))
= B(ρ(γ)ṽ(g) − ṽ(g), x^{0}(γ))
= α(γ)

as claimed. □

5.3. Nonproper deformations. Now we prove that 0 ∉ Ψ_{[u]}(C(Σ)) implies Γ_{u} acts properly.

Proposition 5.4. Suppose that Φ defines a non-proper action. Then there exists a ϕ-invariant Borel probability measure μ on UΣ such that Ψ_{[u]}(μ) = 0.

Proof. By Proposition 4.3, we may assume that Φ defines a non-proper action on 𝔼. There exist compact subsets K_{1}, K_{2} ⊂ 𝔼 for which the set of t ∈ ℝ such that

Φ_{t}(K_{1}) ∩ K_{2} ≠ ∅

is noncompact. Thus there exists an unbounded sequence t_{n} ∈ ℝ, and sequences P_{n} ∈ K_{1} and Q_{n} ∈ K_{2} such that

Φ_{t_{n}}P_{n} = Q_{n}.

Passing to subsequences, we may assume that t_{n} ↗ +∞ and

(5.6)  lim_{n→∞} P_{n} = P_{∞},  lim_{n→∞} Q_{n} = Q_{∞}.

for some P_{∞}, Q_{∞} ∈ 𝔼. For n = 1, …, ∞, the images

p_{n} = Π(P_{n}), q_{n} = Π(Q_{n})

are points in UΣ such that

(5.7)  lim_{n→∞} p_{n} = p_{∞},  lim_{n→∞} q_{n} = q_{∞}

and ϕ_{t_{n}}(p_{n}) = q_{n}. Choose R > 0 such that

d(p_{∞}, Ucore(Σ)) < R,
d(q_{∞}, Ucore(Σ)) < R.

Passing to a subsequence we may assume that all p_{n}, q_{n} lie in the compact set UK_{R}. Since K_{R} is geodesically convex, the curves

{ϕ_{t}(p_{n}) | 0 ≤ t ≤ t_{n}}

also lie in UK_{R}(Lemma 4.1).

Choose a section s ∈ S(𝔼). For n = 1, …, ∞, write

P_{n} = s(p_{n}) + a_{n}ν + p_{n}^{+} + p_{n}^{−}
(5.8)  Q_{n} = s(q_{n}) + b_{n}ν + q_{n}^{+} + q_{n}^{−}

where a_{n}, b_{n} ∈ ℝ , p_{n}^{−}, q_{n}^{−} ∈ 𝕍^{−}, and p_{n}^{+}, q_{n}^{−} ∈ 𝕍^{+}. Since

DΦ_{t}(ν) = ν

and Φ_{t_{n}}(P_{n}) = Q_{n}, taking 𝕍^{0}-components in (5.8) yields:

(5.9)  ∫_{0}^{t_{n}}(F_{u,s} ∘ ϕ_{t})(p_{n}) dt = b_{n} − a_{n}.

Since s is continuous, (5.7) implies s(p_{n}) → s(p_{∞}) and s(q_{n}) → s(q_{∞}). By (5.6),

lim_{n⟶∞}(b_{n} − a_{n}) = b_{∞} − a_{∞}.

Thus (5.9) implies

lim_{n→∞} ∫_{0}^{t_{n}}(F_{u,s} ∘ ϕ_{t})(p_{n}) dt = b_{∞} − a_{∞}

so

(5.10)  lim_{n→∞} \frac{1}{t_{n}} ∫_{0}^{t_{n}}(F_{u,s} ∘ ϕ_{t})(p_{n}) dt = 0

(because t_{n} ↗ +∞).

Define approximating probability measures

μ_{n} ∈ P(UK_{R})

by pushing forward Lebesgue measure on the orbit through p_{n} and dividing by t_{n}:

μ_{n} := \frac{1}{t_{n}} (ϕ_{[0,t_{n}]}^{p_{n}})_{∗}μ_{[0,t_{n}]}

where ϕ_{[0,t_{n}]}^{p_{n}} is defined in (5.1). Compactness of UK_{R} guarantees a weakly convergent subsequence μ_{n} in P(UK_{R}). Thus

Ψ_{[u]}(μ_{∞}) = lim_{n→∞} \frac{1}{t_{n}} ∫_{0}^{t_{n}}(F_{u,s} ∘ ϕ_{t})(p_{n})dt = 0

by (5.10).

μ_{∞} is ϕ-invariant: For any f ∈ L^{∞}(UK_{R}) and λ ∈ ℝ,

| ∫ fdμ_{n} − ∫ (f ∘ ϕ_{λ})dμ_{n}| < \frac{2λ}{t_{n}}∥f∥_{∞}

for n sufficiently large. Passing to the limit,

| ∫ f dμ_{∞} − ∫ (f ∘ ϕ_{λ}) dμ_{∞}| = 0.
□

6. Properness

Now we prove that Ψ_{[u]}(μ) ≠ 0 for a proper deformation Γ_{u}. The proof uses a lemma ensuring that the section s can be chosen only to vary in the neutral direction ν. The proof uses the hyperbolicity of the geodesic flow. The uniform positivity or negativity of Ψ_{[u]} implies Margulis’s “Opposite Sign Lemma’ (Margulis’ [28, 29], Abels [1], Drumm [14, 16]) as a corollary.

Lemma 6.1. There exists a section s ∈ S(𝔼) such that ∇_{ξ}(s) ∈ 𝕍^{0}.

Proof. Let s ∈ S(𝔼). Decompose ∇_{ξ}(s) by the splitting (4.3) into components:

∇_{ξ}s = ∇_{ξ}^{−}s + ∇_{ξ}^{0}s + ∇_{ξ}^{+}s.

where ∇_{ξ}^{±}(s) ∈ 𝕍^{±}. Since DΦ exponentially contracts V^{−} as t → +∞

∥DΦ_{t}(v)∥ ≤ Ce^{−kt}∥v∥

for v ∈ V^{−}. Then

∥ ∫_{0}^{T} DΦ_{t}(∇_{ξ}^{−}(s(ϕ_{−t}(p))))dt∥ ≤ ∫_{0}^{T} ∥DΦ_{t}(∇_{ξ}^{−}(s(ϕ_{−t}(p))))∥dt

≤ \frac{1 − e^{−kT}}{k} ∥∇_{ξ}^{−}(s)∥_{∞}

so

∫_{0}^{∞} DΦ_{t}^{∗}(∇_{ξ}^{−}(s))dt = lim_{T →∞} ∫_{0}^{T} DΦ_{t}^{∗}(∇_{ξ}^{−}(s))dt

and the convergence is uniform on compact subsets. Similarly

∫_{0}^{∞} ϕ_{−t}^{∗}∇_{ξ}^{+}(s)dt

converges uniformly on compact subsets. Thus

s − ∫_{0}^{∞} DΦ_{t}^{∗}∇_{ξ}^{−}(s)dt − ∫_{0}^{∞} DΦ_{−t}^{∗}∇_{ξ}^{+}(s)dt

is a section in S(𝔼) whose covariant derivative with respect to ξ lies in 𝕍^{0}, as claimed. □

Proposition 6.2. Suppose that Φ defines a proper action. Then

Ψ_{[u]}(μ) ≠ 0

for all μ ∈ C(Σ).

Proof. Let μ ∈ C(Σ) and p ∈ supp(μ) ⊂ U_{rec}Σ. Applying Lemma 6.1, choose a section s with ∇_{ξ}(s) ∈ 𝕍^{0}. Then

(6.1)  Φ_{T} (s(p)) = s(ϕ_{T} (p)) + ( ∫_{0}^{T} F_{u,s}(ϕ_{t}(p)) dt)ν

Since Φ is proper, for every pair of compact subsets K_{1}, K_{2} ∈ 𝔼, the set of t ∈ ℝ for which

Φ_{t}(K_{1}) ∩ K_{2} ≠ ∅

is compact. Take K_{1} = s(supp(μ)). Thus Φ_{t}s(p) → ∞ uniformly for p ∈ supp(μ), as t → ∞. Compactness of supp(μ) implies that s(ϕ_{t}p) − s(p) is bounded. Thus (6.1) implies

∫_{0}^{T} F_{u,s}(ϕ_{t}p) dt ⟶ ∞

uniformly in p ∈ supp(μ), as T ⟶ ∞. Furthermore, connectedness of α-limit sets and boundedness of F_{u,s} implies that

∫_{0}^{∞} F_{u,s}(ϕ_{t}p) dt = +∞.

Thus

(6.2)  lim_{T ⟶∞} ∫ dμ ∫_{0}^{T} F_{u,s}(ϕ_{t}p) dt = +∞.

Fubini’s theorem implies

(6.3)  ∫ dμ(p) ∫_{0}^{T} F_{u,s}(ϕ_{t}p) dt = ∫_{0}^{T} ( ∫ F_{u,s}(ϕ_{t}p)dμ(p)) dt.

Invariance of μ implies

(6.4)  ∫ F_{u,s}(ϕ_{t}p)dμ(p) = ∫ F_{u,s}dμ.

Thus (6.3) and (6.4) imply

∫ dμ ∫_{0}^{T} F_{u,s}(ϕ_{t}p)dt = ∫_{0}^{T} ( ∫ F_{u,s}(ϕ_{t}p)dμ)dt

= ∫_{0}^{T} ( ∫ F_{u,s}dμ)dt = T ∫ F_{u,s}dμ. (6.5)

Suppose that ∫ F_{u,s}dμ = 0. Taking the limit in (6.5) as T ⟶ ∞ contradicts (6.2). Thus ∫ F_{u,s}dμ ≠ 0, as desired. □

Corollary 6.3 (Margulis [28, 29]). Suppose that γ_{1}, γ_{2} ∈ Γ_{0} satisfy

α(γ_{1}) < 0 < α(γ_{2}).

Then Γ does not act properly.

Proof. Proposition 5.2 implies that

Ψ_{[u]}(μ_{γ_{1}}) < 0 < Ψ_{[u]}(μ_{γ_{2}})

Convexity of C(Σ) implies a continuous path μ_{t} ∈ C(Σ) exists, with t ∈ [1, 2], for which

μ_{1} = μ_{γ_{1}},
μ_{2} = μ_{γ_{2}}.

The function

C(Σ) ⟶^{Ψ_{[u]}} ℝ

is continuous. The intermediate value theorem implies Ψ_{[u]}(μ_{t}) = 0 for some 1 < t < 2. Proposition 6.2 implies that Γ does not act properly. □

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Department of Mathematics, University of Maryland, College Park, MD 20742 USA

E-mail address: wmg@math.umd.edu

Topologie et Dynamique, Université Paris-Sud F-91405 Orsay (Cedex) France

E-mail address: francois.labourie@math.u-psud.fr

Department of Mathematics, 10 Hillhouse Ave., P.O. Box 208283, Yale University, New Haven, CT 06520 USA

E-mail address: margulis@math.yale.edu