PROPER AFFINE ACTIONS AND GEODESIC FLOWS OF HYPERBOLIC SURFACES

W. M. 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.

Date: September 26, 2008. 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 grants DMS-0103889, DMS0405605, DMS070781, the Mathematical Sciences Research Institute and the Oswald Veblen Fund at the Insitute for Advanced Study, and a Semester Research Award from the General Research Board of the University of Maryland. Margulis gratefully acknowledge partial support from National Science Foundation grant DMS-0244406. Labourie thanks l’Institut Universitaire de France.

2 GOLDMAN, LABOURIE, AND MARGULIS

Contents

Introduction  3
Acknowledgements  6
Notation and Terminology  7
1. Hyperbolic geometry  7
1.1. The hyperboloid model  8
1.2. The geodesic flow  9
1.3. The convex core  9
1.4. The recurrent subset  10
2. Affine geometry  12
2.1. Affine spaces and their automorphisms  12
2.2. Affine deformations of linear actions  13
2.3. Margulis’s invariant of affine deformations  14
3. Flat bundles associated to affine deformations  16
3.1. Semidirect products and homogeneous affine bundles  17
3.2. Homogeneous connections  18
3.3. Flatness  18
4. Sections and subbundles  19
4.1. The flow on the affine bundle  19
4.2. The neutral section  19
4.3. Stable and unstable subbundles  20
4.4. Bundles over UΣ  21
5. Proper Γ-actions and proper ℝ-actions  21
6. Labourie’s diffusion of Margulis’s invariant  23
6.1. The invariant is continuous  24
6.2. The invariant is independent of the section  24
6.3. Periodic orbits  25
7. Nonproper deformations  28
8. Proper deformations  30
8.1. Neutralizing sections  30
8.2. Conclusion of the proof  32
References  32

Introduction

It is well known that every discrete group of Euclidean isometries of ℝ^{n} contains a free abelian subgroup of finite index. In [36], 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 [42], this is equivalent to the existence of a proper affine action of a nonabelian free group. Margulis subsequently showed [33, 34] that proper affine actions of nonabelian free groups do indeed exist. The present paper describes the deformation space of proper affine actions on ℝ^{3} for a large class of discrete groups, as a bundle of convex cones over the Fricke-Teichmüller deformation space of hyperbolic structures on a compact surface with geodesic boundary.

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 [20] 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 [20]). Mess [35] proved the hyperbolic surface

Σ = Γ_{0}\O(2, 1)^{0}/SO(2)

is noncompact. Goldman-Margulis [23] and Labourie [31] gave alternate proofs of Mess’s theorem, using ideas which evolved to the present work.

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 [13, 14, 16, 12] 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 [18].

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 a 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 [31]. Proper actions occur only when V has dimension 4k + 3 (see Abels [1], Abels-Margulis-Soifer [2, 3] and Labourie [31]). In those dimensions, Abels-Margulis-Soifer [2, 3] constructed proper affine deformations of Fuchsian actions of free groups.

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 tool in this paper is a generalization of the invariant constructed by Margulis [33, 34] and extended to higher dimensions by Labourie [31]. Margulis’s invariant α_{u} is a class function Γ_{0} ⟶^{α_{u}} ℝ associated to an affine deformation Γ_{u}. It satisfies the following properties:

• α_{[u]}(γ^{n}) = |n|α_{[u]}(γ) ;

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

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

• The map

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

is injective ([19]).

• 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.

The significance of the sign of α_{[u]} is the following result, due to Margulis [33, 34]:

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 a normalized version of Margulis’s invariant α 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 Γ_{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 [31]. 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 φ be the vector field on UΣ generating the geodesic flow. For a sufficiently smooth section s of 𝔼, the covariant derivative ∇_{φ}(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 unitspacelike 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(∇_{φ}(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 [33, 34] 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 [21, 23] conjectured this necessary condition is sufficient: that is, properness is equivalent to the positivity (or negativity) of Margulis’s invariant. This has been proved when Σ is a three-holed sphere; see Jones [26] and Charette-Drumm-Goldman [11]. 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] showed that the positivity of α_{[u]} requires infinitely many inequalities, suggesting the original conjecture is generally false. Recently Goldman-Margulis-Minsky [24] have disproved the original conjecture, using ideas inspired by Thurston’s unpublished manuscript [41].

This paper is organized as follows. §1 collects facts about convex cocompact hyperbolic surfaces and the recurrent part of the geodesic flow. §2 reviews constructions in affine geometry such as affine deformations, and defines the Margulis invariant. §3 defines the flat affine bundles associated to an affine deformation. §4 introduces the horizontal lift of the geodesic flow to the flat affine bundle whose dynamics reflects the dynamics of the affine deformation of the Fuchsian group Γ_{0}. §5 extends the normalized Margulis invariant to the function Ψ_{[u]} on geodesic currents (first treated by Labourie [31]). §6 shows that for every non-proper affine deformation [u] there is a geodesic current μ ∈ C(Σ) for which Ψ_{[u]}(μ) = 0. §7 shows that, conversely, for every proper affine deformation [u], Ψ_{[u]}(μ) ≠ 0 for every μ ∈ C(Σ). The Opposite Sign Lemma follows as a corollary.

Acknowledgements

We thank Herbert Abels, Francis Bonahon, Virginie Charette, Todd Drumm, David Fried, Yair Minsky, 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, as well as the anonymous referee for helpful comments.

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} and its boundary by ∂H^{2} = S_{∞}^{1}. For distinct a, b ∈ S_{∞}^{1}, denote the geodesic having a, b as ideal endpoints by \overleftrightarrow{ab}.

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 ξ is a vector field on M, then

∇_{ξ}(s) ∈ R^{0}(𝕍).

denotes covariant derivative of s with respect to ξ. If f is a function, then ξf denotes the directional derivative of f with respect to ξ.

An affine bundle 𝔼 over M is a fiber bundle over M whose fiber is an affine space E with structure group the group Aff of affine automorphisms of E. The vector bundle 𝕍 associated to 𝔼 is the vector bundle whose fiber 𝕍_{x} over x ∈ M is the vector space associated to the fiber 𝔼_{x}. That is, 𝕍_{x} is the vector space consisting of all translations 𝔼_{x} ⟶ 𝔼_{x}.

Denote the space of continuous sections of a bundle 𝔼 by Γ(𝔼). If 𝕍 is a vector bundle, then Γ(𝕍) is a vector space. If 𝔼 is an affine bundle with underlying vector bundle 𝕍, then Γ(𝔼) is an affine space with underlying vector space Γ(𝕍). 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 the flow defined by a vector field ξ are denoted ξ_{t}, for t ∈ ℝ.

1. Hyperbolic geometry

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

B(v, w) := v_{1}w_{1} + v_{2}w_{2} − v_{3}w_{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.

1.1. The hyperboloid model. We define H^{2} as follows. We work in the irreducible representation ℝ^{2,1} (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 p_{0} lies in this hyperboloid, then its connected component equals

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

For clarity, we fix a timelike vector p_{0}, for example

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

and define H^{2} as the connected component (1.1).

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 p_{0}

K_{0} := PO(2).

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

G_{0}/K_{0} ⟶ ℝ^{2,1}
gK_{0} ⟼ gp_{0}

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

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

defines a unit tangent vector to H^{2} at p_{0}, which we denote:

(p_{0}, ṗ_{0}) ∈ UH^{2}.

Evaluation at (p_{0}, ṗ_{0})

G_{0} ⟶^{E} UH^{2}
(1.3)  g_{0} ⟼ g_{0}(p_{0}, ṗ_{0})

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 p_{0}.

1.2. 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)p_{0} ∈ H^{2}

describes the geodesic through p_{0} tangent to ṗ_{0}.

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 rightmultiplication by a one-parameter subgroup. At the basepoint (p_{0}, ṗ_{0}), the geodesic flow corresponds to right-multiplication by A_{0}. Hence this one-parameter subgroup must be A_{0}. Therefore right-multiplication 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 φ̃:

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

1.3. 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 generated 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

(1.4)  Σ ⟶^{Π_{core(Σ)}} core(Σ).

We use the following standard compactness results (see Kapovich [27], §4.17 (pp. 98–102), Canary-Epstein-Green [8], or Marden [32] for discussion and proof):

Lemma 1.1. Let Σ = H^{2}/Γ_{0} be the hyperbolic surface where Γ_{0} is convex cocompact and let R ≥ 0. Then

(1.5)  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.

1.4. The recurrent subset. Let U_{rec}Σ ⊂ UΣ denote the union of all recurrent orbits of φ.

Lemma 1.2. Let Σ be as above, UΣ its unit tangent bundle and φ_{t} the geodesic flow on UΣ. Then:

• U_{rec}Σ ⊂ UΣ is a compact φ-invariant subset.

• Every φ-invariant Borel probability measure on UΣ is suppoted in U_{rec}Σ ⊂ UΣ.

• Tthe space of φ-invariant Borel probability measures on UΣ is a convex compact space C(Σ) with respect to the weak ⋆-topology.

Proof. Clearly U_{rec}Σ is invariant.

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).

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 consisting of vectors x ∈ Ucore(Σ) with

η(x̃), η(−x̃) ∈ Λ.

Thus U_{rec}Σ is compact.

By the Poincaré recurrence theorem [25], every φ-invariant probability measure on UΣ is supported in U_{rec}Σ. Since U_{rec}Σ is compact, the space of Borel probability measures P(U_{rec}Σ) on U_{rec}Σ is a compact convex set with respect to the weak topology. 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. □

The following well known fact will be needed later.

Lemma 1.3. U_{rec}Σ is connected.

Proof. Let Λ ⊂ S_{∞}^{1} be the limit set as above. For every λ ∈ Λ, the orbit Γ_{0}λ is dense in Λ. (See for example Marden [32], Lemma 2.4.1.) Denote

Λ^{∗} := {(λ_{1}, λ_{2}) ∈ Λ × Λ | λ_{1} ≠ λ_{2}}.

The preimage { \atop U_{rec}Σ} of U_{rec}Σ ⊂ UΣ under the covering space UH^{2} ⟶^{Π} UΣ is the union

{ \atop U_{rec}Σ} = ⋃_{(λ_{1},λ_{2})∈Λ^{∗}} \overleftrightarrow{λ_{1}λ_{2}}

Choose a hyperbolic element γ ∈ Γ_{0} and let λ_{±} ∈ Λ denote its fixed points. The image

c_{γ} := Π(\overleftrightarrow{λ_{−}λ+})

is the unique closed geodesic in U_{rec}Σ corresponding to γ.

We claim that the image U_{+} under Π of the subset

⋃_{λ∈Λ} \overleftrightarrow{λ_{−}λ} ⊂ U_{rec}Σ

is connected. Suppose that W_{1}, W_{2} ⊂ UΣ are disjoint open subsets such that U_{+} ⊂ W_{1} ∪ W_{2}. Since c_{γ} is connected, either W_{1} or W_{2} contains c_{γ}; suppose that W_{1} ⊃ c_{γ}. We claim that U_{+} ⊂ W_{1}. Otherwise \overleftrightarrow{λ_{−}λ} meets W_{2} for some λ ∈ Λ. Connectedness of \overleftrightarrow{λ_{−}λ} implies \overleftrightarrow{λ_{−}λ} ⊂ W_{2}. Now

lim_{n→∞} γ^{n}λ = λ_{+},

which implies

lim_{n→∞} γ^{n}(\overleftrightarrow{λ_{−}λ}) = lim_{n→∞} (\overleftrightarrow{λ_{−}(γ^{n}λ)}) = \overleftrightarrow{λ_{−}λ_{+}}.

Thus for n >> 1, the geodesic \overleftrightarrow{λ_{λ}} lies in Π^{−1}(W_{1}), a contradiction.

Thus U_{+} ⊂ W_{1}, implying that U_{+} is connected. Density of Γ_{0}λ in Λ implies U_{+} is dense in U_{rec}Σ. Therefore U_{rec}Σ is connected. □

2. 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.

2.1. 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 GL(V) ⋉ 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:

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

for γ_{1}, γ_{2} ∈ Aff(E).

2.2. 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}), to which we also refer 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 (2.1). 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}

where v_{0} ∈ V. 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.

2.3. 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}.

The representation ℝ^{2,1} introduced in §1 is V_{1}, the case when r = 1. 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)
(2.2)  ∥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

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

with neutral vector

(2.4)  x^{0}(a(t)) = [\begin{matrix}0 \\ 1 \\ 0 \end{matrix}]

when t ≠ 0.

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_{g} ⊂ E which is g-invariant. The line l_{g} is parallel to the neutral vector x^{0}(γ). The restriction of g to l_{g} is a translation by the vector

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 [33, 34], 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)

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 [19]). 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, 18, 21, 22, 23, 31].)

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)).

3. 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 [29].) We define a vector field Φ on 𝔼 which is uniquely determined by the following properties:

• Φ is ∇-horizontal ;

• Φ covers the vector field φ defining the geodesic flow on UΣ. We derive an direct alternate description of the corresponding flow Φ_{t} in terms of the Lie group G_{0} and its semidirect products V ⋊ G_{0}.

3.1. 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

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

Projection

G ⟶^{Π} G_{0}
(v, g) ⟼ g

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 left-multiplication on G_{0} and the homomorphism 𝕃. Since 𝕃 is a homomorphism, (3.1) implies equivariance of Π.

When r = 1, that is, V = ℝ^{2,1}, then G is the tangent bundle T G_{0}, with its natural Lie group structure. Compare Goldman-Margulis [23] and [21].

Since the fiber of Π equals the vector spaceV, the fibration G ⟶^{Π} G_{0} has the structure of (trivial) affine bundle over G_{0}. Furthermore this structure is G-invariant: For (3.1) 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 the total space of this G-homogeneous affine bundle over G_{0} by 𝔼̃.

Consider Π also as a (trivial) vector bundle. By (3.1), this structure is G_{0}-invariant. Via 𝕃, this G_{0}-homogeneous vector bundle becomes a G-homogeneous vector bundle 𝕍̃ ⟶ G_{0}.

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 (3.1) 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}).

Of course both 𝔼̃ and 𝕍̃ identify with G, but each has different actions of the discrete group Γ ≅ Γ_{0}, imparting the different structures of a flat affine bundle and a flat vector bundle to the respective quotients.

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.

3.2. 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.

3.3. 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 ˜∇.

4. 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 + 1)-di-mensional representation V_{r}.

4.1. 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}, where a(−t) is defined by (2.3). That is, the diagram

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

commutes. The vector field Φ̃ on 𝔼̃ generating Φ̃_{t} covers the vector field φ̃ generating φ̃_{t}.

Furthermore

s̃_{v}(ga(−t)) = (v, ga(−t))
= (v, g)(0, a(−t))
= s̃_{v}(g)a(−t)

implies

s̃_{v} ∘ φ̃_{t} = Φ̃_{t} ∘ s̃_{v},

whence Φ̃ is the ˜∇-horizontal lift of φ̃.

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}

4.2. 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 defined in (2.4). Let ν̃ denote the section of 𝕍̃ defined by:

UH^{2} ≈ G_{0} ⟶^{ν̃} V ⋊ G_{0} ≈ 𝕍̃
g ⟼ (gv_{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 4.1. ˜∇φ̃(ν̃) = 0.

Proof. Let g ∈ G_{0}. Then

˜∇φ̃(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 §2.3. Another approach to the flat connection ∇ and the neutral section ν is given in Labourie [31].

4.3. 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^{rt}, e^{(r−1)t}, …, 1, …, e^{(1−r)t}, e^{−rt}

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

(4.1)  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}.

4.4. 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_{t} on 𝔼 which is the horizontal lift of the flow φ_{t} 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 (4.1) all descend to a flow DΦ_{t}, a section ν and a splitting

(4.2)  𝕍 = 𝕍^{−} ⊕ 𝕍^{0} ⊕ 𝕍^{+}

of the flat vector bundle 𝕍 over UΣ.

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

• The bilinear form 𝔹 is bounded with respect to g;

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

• Hyperbolicity: The flow DΦ_{t} exponentially expands the subbundle V^{+} and exponentially contracts V^{−}. Explicitly, there exist constants C, k > 0 exist so that

(4.3)  ∥D(Φ_{t})v∥ ≤ Ce^{kt}∥v∥ as t ⟶ −∞,

for v ∈ 𝕍^{+}, and

(4.4)  ∥D(Φ_{t})v∥ ≤ Ce^{−kt}∥v∥ as t ⟶ +∞

for v ∈ 𝕍^{−}. To construct such a metric, first choose a Euclidean inner product on the vector space V. Then extend it to a left G_{0}-invariant Euclidean metric on the vector bundle

𝕍̃ ≈ G = V ⋊ G_{0}.

Hyperbolicity follows from the description of the adjoint action of A_{0} as in (2.2). With this Euclidean metric, the space Γ(𝕍) of continuous sections of 𝕍 is a Banach space.

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 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 [30] and Palais [37].

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

Proposition 5.1. 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 [38, 39].)

Lemma 5.2. 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. However 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. Cartesian projection A × B ⟶ A maps

(5.1)  (A × B)_{K} ⟶ A_{(B·K)/B}.

A acts properly on X/B implies A_{(B·K)/B} is a compact subset of A. Because (5.1) is a proper map, (A × B)_{K} is compact, as desired.

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

□

Proof of Proposition 5.1. Apply Lemma 5.2 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. □

6. Labourie’s diffusion of Margulis’s invariant

In [31], Labourie defined a function

UΣ ⟶^{F_{u}} ℝ,

corresponding to the invariant α_{u} defined by Margulis [33, 34]. 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 O is the origin and x^{0}(γ) ∈ V is the neutral vector of γ (see §2.3 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} on V will be replaced by the geodesic flow φ_{t} on UΣ.

Let s be a C^{∞} section of 𝔼. In particular s is C^{1} along φ. That is, the function

ℝ ⟶ 𝔼
t ⟼ s(φ_{t}(p))

is C^{1} for all p ∈ UΣ. Its covariant derivative with respect to φ is a smooth section ∇_{φ}(s) ∈ R^{0}(𝕍). Pairing with ν ∈ R^{0}(𝕍) via

𝕍 × 𝕍 ⟶^{𝔹} ℝ

produces a continuous function UΣ ⟶^{F_{u,s}} ℝ defined by:

F_{u,s} := 𝔹(∇_{φ}(s), ν).

6.1. The invariant is continuous. Let S(𝔼) denote the space of continuous sections s of 𝔼 over U_{rec}Σ which are differentiable along φ and the covariant derivative ∇_{φ}(s) is continuous. If s ∈ S(𝔼), then F_{u,s} is continuous.

For each probability measure μ ∈ P(UΣ),

∫_{UΣ} F_{u,s} dμ

is a well-defined real number and the function

P(UΣ) ⟶ ℝ
μ ⟼ ∫_{UΣ} F_{u,s} dμ

is continuous in the weak ⋆-topology on P(UΣ). Furthermore its restriction to the subspace C(Σ) ⊂ P(UΣ) of Φ-invariant measures is also continuous in the weak ⋆-topology.

6.2. The invariant is independent of the section.

Lemma 6.1. Let s_{1}, s_{2} ∈ S(𝔼) and μ be a φ-invariant Borel probability measure on UΣ. Then

∫_{UΣ} F_{u,s_{1}} dμ = ∫_{UΣ} F_{u,s_{2}} dμ.

Proof. The difference s = s_{1} − s_{2} of the sections s_{1}, s_{2} of the affine bundle 𝔼 is a section of the vector bundle 𝕍 and

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

Therefore

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

The first term vanishes since μ is φ-invariant:

∫_{UΣ} φ𝔹(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 ∇_{φ}(ν) = 0 (Lemma 4.1). □

Thus

Ψ_{[u]}(μ) := ∫_{UΣ} F_{u,s} dμ

is a well-defined continuous function C(Σ) ⟶^{Ψ_{[u]}} ℝ which is independent of the section s used to define it.

6.3. Periodic orbits. Suppose

t ⟼ φ_{t}(x_{0})

defines a periodic orbit of the geodesic flow φ with period T > 0. That is, T is the least positive number such that φ_{T} (x_{0}) = x_{0}. Suppose that γ is the corresponding element of Γ_{0} ≅ π_{1}(Σ). Then T equals the length ℓ(γ) of the corresponding closed geodesic on Σ.

If x_{0} ∈ UΣ and T > 0, let φ_{[0,T]}^{x_{0}} denote the map

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

The period T of the periodic orbit corresponding to γ equals the length ℓ(γ) of the closed geodesic in Σ corresponding to γ. Then

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

defines the geodesic current associated to the periodic orbit γ, where μ_{[0,T]} denotes Lebesgue measure on [0, T].

Proposition 6.2 (Labourie [31], Proposition 4.2). Let γ ∈ Γ_{0} be hyperbolic and let μ_{γ} ∈ C_{per}(Σ) be the corresponding geodesic current. Then

(6.3)  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 (6.1) and the geodesic current (6.2).

Choose a section s ∈ S(𝔼). Pull s back by the covering space

UH^{2} ≈ G_{0} ⟶^{Π} UΣ

to a section of 𝔼̃ = Π^{∗}𝔼. Since 𝔼̃ ⟶ UH^{2} is a trivial E-bundle, this section is the graph of a map

G_{0} ⟶^{ṽ} E

satisfying

ṽ ∘ γ = ρ(γ)ṽ.

Then

(6.4)  ∫_{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 x̃_{0} corresponds via E to some g_{0} ∈ G_{0}, where E is defined in (1.3). The path φ_{[0,T]}^{x_{0}} lifts to the map

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

The periodic orbit lifts to the trajectory

ℝ ⟶ G_{0}
t ⟼ g_{0}a(−t).

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

γg_{0} = g_{0}a(−T)

which implies

(6.5)  γ = g_{0}a(−T)g_{0}^{−1}.

PROPER AFFINE ACTIONS 27

Evaluate ∇_{φ}s and ν along the trajectory φ̃x̃_{0} ; [0,T] by lifting to the covering space and computing in G:

(6.6)  (∇_{φ}s̃)(φ_{t}x̃_{0}) = \frac{D}{dt}ṽ(g_{0}a(−t)).

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

G_{0} ⟶^{ν̃} V
(6.7)  g ⟼ 𝕃(g)v_{0}.

Lemma 6.3. For any g ∈ G_{0} and t ∈ ℝ,

ν̃(ga(−t)) = x^{0}(γ).

Proof.

ν̃(ga(−t)) = 𝕃(ga(−t)) v_{0}  (by (6.7))
= 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 (6.5)).
□

Conclusion of proof of Proposition 6.2. Evaluate the integrand in (6.4):

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

= 𝔹(∇_{φ}s̃(g_{0}a(−t)), ν̃(g_{0}a(−t))

= B(\frac{D}{dt}ṽ(g_{0}a(−t)), x^{0}(γ)) (by Lemma 6.3 and (6.6))

= \frac{d}{dt}B(ṽ(g_{0}a(−t)), x^{0}(γ))

and

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

as claimed. □

7. Nonproper deformations

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

Proposition 7.1. Suppose that Γ_{u} is a non-proper affine deformation. Then there exists a geodesic current μ ∈ C(Σ)(that is, a φ-invariant Borel probability measure) such that Ψ_{[u]}(μ) = 0.

Proof. By Proposition 5.1, we may assume that the flow Φ_{t} defines a non-proper action on 𝔼. Choose compact subsets K_{1}, K_{2} ⊂ 𝔼 for which the set of t ∈ ℝ such that

Φ_{t}(K_{1}) ∩ K_{2} ≠ ∅

is noncompact. Choose 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, assume that t_{n} ↗ +∞ and

(7.1)  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

(7.2)  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 assume that all p_{n}, q_{n} lie in the compact set UK_{R} where K_{R} is the R-neighborhood of the core, defined in (1.5). Since K_{R} is geodesically convex, the curves

{φ_{t}(p_{n}) | 0 ≤ t ≤ t_{n}}

also lie in UK_{R}(Lemma 1.1).

Choose a section s ∈ S(𝔼). We use the splitting (4.2) of 𝕍 and the section s of 𝔼 to decompose the points P_{1}, …, P_{∞} and Q_{1}, …, Q_{∞}. For n = 1, …, ∞, write

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

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

DΦ_{t}(ν) = ν

and Φ_{t_{n}}(P_{n}) = Q_{n}, taking 𝕍^{0}-components in (7.3) yields:

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

Since s is continuous, (7.2) implies s(p_{n}) → s(p_{∞}) and s(q_{n}) → s(q_{∞}). By (7.1),

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

Thus (7.4) implies

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

so

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

(because t_{n} ↗ +∞).

Now we construct the measure μ ∈ C(Σ) such that

Ψ_{[u]}(μ) = ∫_{UΣ} F_{u,s} dμ = 0.

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 (6.1). Compactness of UK_{R} guarantees a weakly convergent subsequence μ_{n} in P(UK_{R}). Let

μ_{∞} := lim_{n⟶∞} μ_{n}.

Thus by (7.5).

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

Finally we show that 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

as desired. □

8. Proper deformations

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’ [33, 34], Abels [1], Drumm [15, 17]) as a corollary.

Proposition 8.1. Suppose that Φ defines a proper action. Then

Ψ_{[u]}(μ) ≠ 0

for all μ ∈ C(Σ).

Corollary 8.2 (Margulis [33, 34]). Suppose that γ_{1}, γ_{2} ∈ Γ_{0} satisfy

α(γ_{1}) < 0 < α(γ_{2}).

Then Γ does not act properly.

Proof. Proposition 6.2 implies

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 8.1 implies that Γ does not act properly. □

8.1. Neutralizing sections. A section s ∈ S(𝔼) is neutralized if and only if

∇_{φ}(s) ∈ 𝕍^{0}.

This will enable us to relate the properness of the flow {Φ_{t}}_{t∈ℝ} to the invariant Ψ_{[u]}(μ). To construct neutralized sections, we need the following technical facts.

The flow {DΦ_{t}}_{t∈ℝ} on 𝕍 and the flow {Φ_{t}}_{t∈ℝ} on 𝔼 induce (by push-forward) one-parameter groups of continuous bounded operators (DΦ_{t})_{∗} and (Φ_{t})_{∗} on Γ(𝕍) and Γ(𝔼) respectively. We begin with an elementary observation, whose proof is immediate. (Compare §4.4.)

Lemma 8.3. Let ξ ∈ Γ(𝕍). Suppose that

t ⟼ (DΦ_{t})_{∗}(ξ)

is a path in the Banach space Γ(𝕍) which is differentiable at t = 0. Then ξ ∈ S(𝔼) and

\frac{d}{dt}|_{t=0}(DΦ_{t})_{∗}(ξ) = ∇_{φ}(ξ).

Here is our main lemma:

Lemma 8.4. A neutralized section exists.

Proof. By the hyperbolicity condition (4.3),

∥(DΦ_{t})_{∗}(ξ^{+})∥_{+} ≤ Ce^{−kt}∥ξ^{+}∥

for any section ξ^{+} ∈ Γ(𝕍^{+}). Consequently the improper integral

ζ^{+} := ∫_{0}^{∞}(DΦ_{t})_{∗}(ξ^{+})dt

converges. Moreover, Lemma 8.3 implies that ζ^{+} ∈ S(𝔼) with

∇_{φ}(ζ^{+}) = −ξ^{+}.

Similarly (4.4) implies that, for any section ξ^{−} ∈ Γ(𝕍^{−}), the improper integral

ζ^{−} := ∫_{−∞}^{0}(DΦ_{t})_{∗}(ξ^{−})dt

converges, and ζ^{−} ∈ S(𝔼) with

∇_{φ}(ζ^{−}) = ξ^{−}.

Now let s ∈ S(𝔼). Decompose ∇_{φ}(s) by the splitting (4.2):

∇_{φ}(s) = ∇_{φ}^{−}(s) + ∇_{φ}^{0}(s) + ∇_{φ}^{+}(s),

where ∇_{φ}^{±}(s) ∈ Γ(𝕍^{±}). Apply the above discussion to ξ^{−} = ∇_{φ}^{−}(s) and ξ^{+} = ∇_{φ}^{+}(s). Then

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

lies in S(𝔼) with ∇_{φ}(s_{0}) = ∇_{φ}^{0}(s). Hence s_{0} is neutralized, as claimed. □ 8.2. Conclusion of the proof. Let μ ∈ C(Σ) so that

Ψ_{[u]}(μ) = ∫ F_{u,s}dμ = 0.

Define, for T > 0 and p ∈ U_{rec}Σ,

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

Since μ is φ-invariant, by Fubini’s Theorem,

∫ g_{T} dμ = 0.

Therefore, for every T > 0, since U_{rec}Σ is connected by Lemma 1.3, there exists p_{T} ∈ U_{rec}Σ such that

g_{T} (p_{T}) = 0.

We may assume that s ∈ S(𝔼) is neutralized. Then

\frac{d}{dt}(DΦ_{t})_{∗}(s) = ∇_{φ}(s) = F_{u,s}ν,

and

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

for any T > 0. Thus

(Φ_{T} s)(p_{T}) = s(φ_{T} p_{T}).

Let K be the compact set s(U_{rec}Σ). Then, for all T > 0,

Φ_{T} (K) ∩ K ≠ ∅,

and {Φ_{t}}_{t∈ℝ} is not proper, as claimed. This completes the proof.

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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