Quasi–geodesic flows

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Quasi–geodesic flows

Maciej Czarnecki

Uniwersytet L´odzki, Katedra Geometrii ul. Banacha 22, 90-238 L´od´z, Poland

e-mail: maczar@math.uni.lodz.pl

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Introduction

These are cosy notes of Erasmus+ lectures at Universidad de Granada, Spain on April 16–18, 2018 during the author’s stay at IEMath–GR.

After Danny Calegari and Steven Frankel we describe a structure of quasi–

geodesic flows on 3–dimensional hyperbolic manifolds.

A flow in an action of the addirtive group R o on given manifold. We concentrate on closed 3–dimensional hyperbolic manifolds i.e. having locally isometric covering by the hyperbolic space H³. Such flow is quasi–geodesic if every flowline of the lifted flow is a quasi–geodesic in H³. Quasi–geodesic flows are probably the only reasonable metric objects which are foliations of hyperbolic 3–manifolds which do not carry neither geodesic foliation in any dimension (Zeghib) nor quasi–geodesic foliations of dimension 2 (Fenley).

We start with foundations of hyperbolic manifolds, hyperbolic groups and their asympotic properties. Then we describe shortly notions for foliations and flows mentioning their type like (quasi)–isometric, (quasi)–geodesic, (pseudo)–Anosov etc.

We take care of topology of the plane focusing on decompositions into continua. For such decompositions we construct a circular order in the set of their topological ends.

Since after Calegari any quasi–geodesic flow on a hyperbolic 3 manifold has the Hausdorff flowspace (i.e. the plane) we are able to apply decompositions for a compactification the flowspace by ends of flowlines making it a closed disc.

Finally, we study new results of Frankel on extension properties of quasi–

geodesic flow. In particular we take a look for its proof of Calegari conjecture stating that such flows need to have closed orbits.

At the end, we add some remarks about (quasi)–geodesic flows in non- compact case.

I would like to thank Prof. Antonio Mart´ınez López and Prof. Joaqu´ın Pérez Muñoz for organizing my visit to IEMath–GR and their overall hospi- tality.

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1 Hyperbolic manifolds and hyperbolic groups

1.1 Hyperbolic space

[BP]

Definition 1.1. In the Minkowski space Rⁿ⁺¹1 with the Lorentz form h.|.i given by

hx|yi = −x₀y₀+ x₁y₁+ . . . x_ny_n we define the n–dimensional hyperbolic space as

Hⁿ=x ∈ Rⁿ⁺¹1 | hx|xi = −1, x₀ > 0

On every tangent space T_xHⁿ = x^⊥ (⊥ means Lorentz orthogonality) the Lorentz form in an inner product making Hⁿ a Riemannian manifold with constant sectional curvature equal −1.

The most useful models of Hⁿ are

(B) the ball model in the unit ball Bⁿ ⊂ Rⁿ together with standard Riemannian metric multiplied at x by ⁴

(1−kxk²)²,

(Π) the half–space model in the open upper half–space Π^n,+ ⊂ Rⁿ together with standard Riemannian metric multiplied at x by _x¹2

n. In the ball and the half–space models complete geodesci are circle arcs or rays perpendicular (in the Euclidean sense) to their topological boundaries Sⁿ⁻¹ and Rⁿ⁻¹× {0} ∪ {∞}, respectively.

Definition 1.2. We say that unit speed geodesic rays in Hⁿ are asymptotic iff the distance between them is bounded.

We associate to Hⁿ its ideal boundary Hⁿ(∞) consisting of asymptoticity class of geodesics.

The cone topology in Hⁿ ∪ Hⁿ(∞) is such that ideal points are close to each other if representing geodesic make small angle at their common origin, and close to ordinary points lying on such geodesics far awy from the origin.

The Hadamard–Cartan theorem states that every n–dimensional Hada- mard manifold i.e. connected, simply connected Riemannian n–manifold which is complete and nonpositively curved is diffeomorphic to Rⁿ and its ideal boundary is defined analogously.

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1.2 M¨ obius transformations

[R]

Definition 1.3. We call an inversion in a sphere S(c, r) ⊂ Rⁿ a transformation given by

ι_c,r(x) = r² x − c kx − ck² + c.

After obvious extension we could treat inversions as transformations of Sⁿ including reflections in hyperplanes.

Inversions imitate properties of standard reflection but they are only conformal i.e. preserve angles instead of distances.

Definition 1.4. The M¨obius group M¨ob_n is a group generated by all the inversions in (n − 1)–dimensional subspheres of Sⁿ.

1.5. Orientation preserving M¨obius transformations of the upper half plane Π^2,+ are simply homographies

z 7→ az + b cz + d

of the complex plane C with real coefficients and determinant equal 1.

1.3 Isometries of hyperbolic spaces

[BP], [R]

1.6. Isometries of Hⁿ in the hyperboloid model are easily observed as Isom(Hⁿ) = O⁺(1, n) =A ∈ Mn+1,n+1 | AJA^T = J

where J is diagonal matrix with entries −1, 1, . . . , 1. Thus matrices from O⁺(1, n) preserve the Lorentz form and upper sheet of hyperboloid hx|xi =

−1.

In the ball model isometries are compositions of an orthogonal transformation of Rⁿ with inversion in sphere orthogonal to Sⁿ⁻¹ or orthogonal transformations. Similarly, in the half–space model we compose an inversion or identity with an orthogonal transformation pereserving the last coordi- nate.

Thus isometries of Hⁿ have canonical extension to the ideal boundary as M¨obius transformations.

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1.7. In the case n = 2, the most popular description of Isom⁺(H²) comes from the half–plane model where isometries are described as homographies (cf. 1.5). Thus we obtain the full group of orientation preserving isometries as projectivization P SL(2, R) which is doubly covered by SL(2, R) (opposite matrices identify).

Definition 1.8. An isometry f of Hⁿ is elliptic if it has a fixed point in Hⁿ,

parabolic if it has no fixed points in Hⁿ and fixes a unique fixed point of Sⁿ⁻¹ = Hⁿ(∞)

hyperbolic if it has no fixed points in Hⁿand fixes two points of Sⁿ⁻¹= Hⁿ(∞).

1.9. Isom(Hⁿ) is a Lie group. For n = 2 its Lie algebra serving as tangent space to P SL(2, R) is sl(2, R) consisting of matrices of zero trace.

1.10. The group P SL(2, R) acts transitively on the unit tangent bundle T¹H² of H² and stabilizer of any vector is trivial then P SL(2, R) could be identified (but not canonically) with T¹H².

1.11. Since isometries of Hⁿ act on its ideal boundary Sⁿ⁻¹, the group P SL(2, R) could treated as a subgroup of Homeo⁺(S¹) of orientation preserving homeomorpshisms of the circle.

A subgroup of Homeo(S¹) is a M¨obius–like group if every its element is (topologically) conjugate to a M¨obius transformation.

1.12. Finally, observe that all M¨obius transformations could be described in terms of isometries of Hⁿ.

M¨ob_n' O⁺(1, n + 1)

In fact, if ι is an inversion in an (n − 1)–sphere S ⊂ Sⁿ then there is a unique n–sphere ˜S ⊂ Rⁿ⁺¹ which contains S and is orthogonal to Sⁿ and then the inversion ˜ι in ˜S is extension of ι. Think of Sⁿ as ideal boundary of hyperbolic space Hⁿ⁺¹ in the ball model. Thus ˜ι is an hyperbolic reflection and could be treated as element of O⁺(1, n+1). The converse implication follows that every isometry of Hⁿ⁺¹is a composition of at most n + 2 hyperbolic reflections.

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1.4 Gromov hyperbolicity

[BH]

Definition 1.13. A geodesic in a metric space (X, d) is an isometric embedding of an interval I ⊂ R in to X i.e. such a map c : I → X that

d(c(t), c(t⁰)) = |t − t⁰| for any t, t⁰ ∈ I.

In a geodesic metric space every two points could joined by a geodesic.

Definition 1.14. We say that a geodesic metric space X is δ–hyperbolic (in the sense of Gromov) if in any triangle made of geodesics between points x, y, z ∈ X the geodesic segment [y, z] in the δ–neighbourhood of [x, y]∪[x, z].

The above definition does not need assumption on geodesicity. Gromov hyperbolic condition could formulated using only distances between quadru- ples of points.

Example 1.15. Hⁿ is (ln(1 +√

2))–hyperbolic.

1.16. The definition of the ideal boundary ∂X of a Gromov hyperbolic space X is based on asymptotic geodesics as in case of Hadamard manifolds but the topology in X ∪ ∂X comes from the uniform convergence of rays on compact sets.

A special role in this theory plays the notion ”quasi–geodesic” which more flexible and more invariant. Even in case of Riemannian manifolds we know that ”(totally) geodesic” is fragile and could be lost even with small perturbations.

Definition 1.17. A function f between metric spaces (X, d) and (Y, ρ) is a (λ, ε)–quasi–isometric embedding if

1

λd(x, x⁰) − ε ≤ ρ(f (x), f (x⁰)) ≤ λ d(x, x⁰) + ε for any x, x⁰ ∈ X where λ ≥ 1 and ε ≥ 0. We say that such f is a (λ, ε)–quasi–isometry if the image of X is of bounded distance from Y .

A quasi–geodesic (of appropriate constants) in X is a quasi–isometric embedding of R into X.

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1.18. Quasi–isometric image of a Gromov hyperbolic space is still Gromov hyperbolic (constant δ could change and depends on quasi–isometry constants).

Theorem 1.19 (stability of quasi–geodesics). For any δ ≥ 0, λ ≥ 1 and ε ≥ 0 there is such R > 0 that the image of a (λ, ε)–quasi–geodesic γ : [a, b]

in a geodesic δ–hyperbolic space X is in R–neighbourhood of the geodesic joining γ(a) to γ(b).

Corollary 1.20. Every quasi–geodesic line in a Gromov hyperbolic space X has two different ends on the ideal boundary ∂X.

In particular, every quasi–geodesic line in H³ has two different ends on S_∞² = H³(∞).

1.21. If G is a finitely generated group and A its symmetric generating set we establish the Cayley graph of G with respect to A elements of G as vertices and edges between elements which differ of a generator from A.

The Cayley graph could metrized by the word metric in G (every edge has length 1). Cayley graphs of different set of generators are quasi–isometric.

Definition 1.22. We say that a finitely generated group is hyperbolic if its Cayley graph is Gromov hyperbolic.

1.5 Hyperbolic surfaces and hyperbolic manifolds

[BP], [R]

1.23. Recall that a closed (i.e. compact and without boundary) orientable topological surface is homeomorphic to sphere S², torus T² or Σ_g i.e. connected sum of g tori with g ≥ 2. On Σg could endowed with a hyperbolic Riemannia structure by an appropriate gluing of sides of 4g–gon in H² with total of internal angles equal 2π. This gluing is realized formally as a quotient of H² by a group generated by hyperbolic reflections in sides of the polygon.

Similarly, we obtain hyperbolic 3–manifolds (generally n–manifolds) as quotients of H³(resp. Hⁿ) by discrete subgroups of Isom(H³) (resp. Isom(Hⁿ)).

1.24. If Γ is such a subgroup of Isom(Hⁿ) that M = Hⁿ/Γ is a hyperbolic n–manifold then Γ is isomorphic to the group π1(M ), the fundamental group of M ; in a class from π₁(M ) we find a geodesic α and then attach to any point of x ∈ Hⁿ the end of the lift α from x.

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Therefore, elements of π₁(M ) act on Hⁿ as deck transformations i.e.

π ◦ g = g where π : Hⁿ→ M is the quotient projection.

1.25. Fundamental groups of hyperbolic manifolds are hyperbolic groups.

1.26. Lattices in P SL(2, R) i.e. discrete subgroups acting cocompactly, espe- cially the modular group P SL(2, Z) are also examples of hyperbolic groups.

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2 Foliations and flows

2.1 Foliations

[Ca2], [CanCon]

Definition 2.1. Let M be an n–dimensional differentiable manifold. On M we have a p–dimensional (or q = n − p codimensional) foliation if M splits into connected immersed p–dimensional submanifolds (called leaves) and there is an atlas on M consisting of such product charts that cross section of a leaf and a chart domain contains at mosta countably many components which are mapped into horizontal p-planes.

Image of R^q of the inverse of a product charts is called a local transversal of the foliation.

Example 2.2. Reeb foliation of S³ cf. [CanCon].

Theorem 2.3 (Frobenius). A p–dimensional distribution D on M i.e. collection of p–dimensional subspaces of T_p(M ) for any p ∈ M is tangent to a foliation iff for any vector fields X, Y tangent to D its Lie bracket [X, Y ] is tangent to D.

Corollary 2.4. Every nonsingular vector field on M is tangent to a 1–

dimensional foliation on M .

Definition 2.5. A codimension 1 foliation on 3–dimensional manifold M is taut if there is a topological circle in M transversal to every leaf.

2.2 Flows

[F1]

Definition 2.6. A flow on a space X is such a map Φ : R × X → X that ϕt◦ ϕs = ϕt+s for any t, s ∈ R

where ϕ_r= Φ(r, .).

In other words, a flow is an action of the additive group (R, +) on X.

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Definition 2.7. For a flow Φ = (ϕ_t) we call set O(x) = R·x = {ϕt(x) | t ∈ R}

an orbit (or flowline) of the point x.

An orbit space of the flow is the quotient space obtained by identifying points of the same orbits.

Definition 2.8. A flow on a manifold M is a continuous action of the additive group R on M. Such flow in nonsingular if every point of M is moved by some ϕ_t.

2.3 Anosov and pseudo–Anosov flows

[Ca2]

Definition 2.9. A differentiable flow Φ = (ϕ_t) on a Riemannian 3–manifold M is called an Anosov flow with 1–dimensional foliation L if it preserves continuous splitting of tangent bundle

T M = E^s⊕ T L ⊕ E^u

which is invariant under time t of Φ and the flow uniformly contracts E^s and uniformly expands E^u i.e. there are constants a ≥ 1 and b > 0 such that

kdϕ_t(v)k ≤ ae^−btkvk for any v ∈ E^s kdϕ−t(v)k ≤ ae^−btkvk for any v ∈ E^u and for any nonnegative t.

Example 2.10. We shall construct an Anosov flow on 3–dimensional Lie group P SL(2, R). First observe that matrices

H = 1 0 0 −1

, X = 0 1 0 0

, Y = 0 0 1 0

∈ sl(2, R) are such that

[H, X] = 2X, [H, Y ] = −2Y, [X, Y ] = H.

Thus from Frobenius theorem we obtain two 2–dimensional foliations F^ws and F^wu spanned respectively by H, X and H, Y . Then the splitting into subspaces spanned by X, H, Y gives an Anosov flow and foliations are F^ws, F^wu are weak stable and weak unstable foliations of the flow.

Details in [Ca2] Example 4.50.

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Definition 2.11. A flow on a Riemannian 3–manifold is pseudo–Anosov if away from finitely many closed orbits there is a continuous splitting T M = E^s ⊕ T L ⊕ E^u for 1–dimensional bundles which are stable, tangent to 1–dimensional foliation and unstable (cf 2.9). Moreover, E^s ⊕ T L and E^u⊕ T L are tangent to 2–dimensional foliations.

2.4 Geodesic and quasi–geodesic flows

[Pa], [F1]

Definition 2.12. A geodesic flow on a Riemannian manifold M is a flow on its unit tangent bundle T¹M attaching such that ϕ_t(v) is the vector tangent at time t to the unit speed geodesic starting from the point being projection of v, in direction of v.

Flowlines of the geodesic flow are simply unit vector fields tangent to geodesics. Nevertheless, the definition 2.8 is more useful for us and we say that a geodesic flow is a flow on a manifold (not on tangent bundle) with orbits being geodesics.

Now we introduce a notion which belongs to coarse geometry, namely a quasi–geodesic flow.

Definition 2.13. Let M be a closed hyperbolic 3–manifold and Φ be a continuous nonosingular flow on M . We say that Φ is a quasi–geodesic flow if the flow ˜Φ lifted to ˜M = H³ is such that all its flowlines are quasi–geodesics.

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3 Compactification of decomposed plane

[Ca1], [F1]

3.1 Circular order

Definition 3.1. A circular order on a set S is a map h., ., .i : S × S × S → {−1, 0, 1} such that

(i) hx, y, zi 6= 0 iff x, y, z ∈ S are pairwise distinct

(ii) for any permutation τ of x, y, z ∈ S, hτ (x), τ (y), τ (z)i = sgnτ hx, y, zi (iii) for any x, y, z, w ∈ S,

hy, z, wi − hx, z, wi + hx, y, wi − hx, y, zi = 0.

3.2. Using circular order in S we define an open interval (x, y) consisting of those z ∈ S for which hx, z, yi = 1 and a closed interval [x, y] = (x, y)∪{x, y}.

We introduce order topology of having all open intervals as its base.

3.3. One can build a completion ¯S of a circularly ordered set S (cf. [F1] Con- struction 5.3) in which every family of nested closed intervals has nonoempty intersection.

Theorem 3.4. If S circularly ordered, order complete and 2nd countable then S could be identified with a compact subset of the circle S¹ by preserving order homeomorphism.

3.2 Construction of universal circle

Theorem 3.5 (Cantor–Bendixson). Every closed subset Y of a separable, complete metric space is of a form Y = T ∪ U where T is closed and perfect while U is countable.

3.6. Now we constuct a universal circle for a circularly ordered set S which is new topology on the circle S¹ induced by S.

Assume that S is an uncountable circularly orderer set which is 2nd countable. The its completion is a compact subset of the circle S¹ and could decomposed like in the Cantor–Bendixson theorem. Remove the closed and perfect set T from S¹ and in S¹\ T collpase closures of all intervals. Such a new topological space S_u¹ is a universal circle for S.

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3.3 Decompositions of the plane and ordering ends

Recall that continuum is a closed and connected subset of the plane.

Definition 3.7. An unbounded decomposition of the plane P is a collection D of unbounded continua such that

(i) distinct K, L ∈ D are disjoint (ii) S D = P .

If we replace the codiition (i) by compactness of intesection of any two distinct elements of D (i.e. they are eventually disjoint ) then we have a generalized unbounded decomposition of P .

Every unbounded decomposition of the plane is uncountable which could be not true for generalized one.

Lemma 3.8. If K1, . . . , Kn are disjoint unbouded continua in the plane P such that any of them does not separate in P any two of remaining ones then there is exactly one connected component C(K₁, . . . K_n) of P \ (K₁∪ . . . ∪ K_n) which limits on every Ki. Any other component limits on only one of Ki’s.

3.9. A collection of disjont mutually nonseparating continua we order circularly as follows.

In the component C(K, M ) we take a continuous curve γ joining K to M which avoids L. Then L lies on the positive side C⁺(γ; K, L) or on the negative side C⁻(γ; K, L) of γ (orientation induced from the plane) and respectively we put hK, L, M i = 1 or hK, L, M i = −1.

One can prove ([F1] Proposition 6.6. with preceeding lemmas) that this way we define a circular order.

Definition 3.10. A end of a metric space X is a map ξ which assigns to any bounded set A ⊂ X a connected component of X \ A in such a way that if A ⊂ A⁰ then ξ(A⁰) ⊂ ξ(A).

Every unbouded continuum in the plane has at least one end

3.11. Now we order ends E (C) of a collection C consisting of eventually disjoint unbounded continua.

For κ, λ, µ ∈ E (C) we find a disc D such that κ(D), λ(D), µ(D) are disjoint and mutally nonseparating. Now we can apply the method from 3.9 and define a circular order in the set of end by

hκ, λ, µi = hκ(D), λ(D), µ(D)i.

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3.4 End compactification

3.12. Let D be an uncountable generalized unbounded decomposition of the plane P. In the set of its ends E (D) we constructed circular order. According to 3.6 we obtain universal circle S_u¹.

Now we inherit ¯P = P ∪ S_u¹ with topology wich compactifies P with respect to D.

Let K, L be tails of some elements of D i.e. images of some disc under end maps. Their ends are subsets of S_u¹. Let I be the maximal open interval that runs from E (K) to E (L) in S_u¹\ (E(K) ∪ E(L)) and γ a curve from K to L.

In ¯P we take topology generated by open sets of P and peripheral sets of the form I ∪ C⁺(γ; K, L).

Theorem 3.13 ([F1] Theorem 7.9). For an uncountable generalized unbounded decomposition D of the plane P the topology generated by open sets in P and peripheral sets makes ¯P = P ∪ S_u¹ a compact space homeomorphic to the closed disc of interior P and boundary S_u¹.

Moreover, any homeomorphism of P preserving D extends to a homeomorphism of ¯P .

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4 Quasi–geodesic flow and its end extension

4.1 Product covering

[Ca1]

4.1. In the definition of quasi–geodesic flow we assumed that every flowline of lifted flow is a quasi–geodesic. In fact, flowlines of such a flow on closed hyperbolic 3–manifold are uniformly quasi–geodesic as is in [Ca1] Lemma 10.20.

4.2. The flow space of a quasi–geodesic flow is Hausdorff as in [Ca1] Lemma 4.48 and Lemma 10.21 which together with the fact that ˜M = H³ is homeomorphic to R³ gives that the flow space of quasi–geodesic flow is homeomorphic to R².

In fact, suppose that P is not Hausdorff i.e. there are two leaves L, L⁰ ∈ ˜F and sequence of leaves (Li) coverging to both L and L⁰. Thus there are sequences of points p_i ∈ L and p⁰_i ∈ L_i coverging respectively to p and p⁰.

Let d(p, p⁰) = t. Because the flow is uniformly (λ, ε)–quasi–geodesic (4.1) then dLi(pi, p⁰_i) ≤ λd(pi, p⁰_i) + ε → λt + ε. Leaves are complete hence the distances d_L_i(p_i, p⁰_i) are realized by curves γ_i on L_i. Some subsequence converge to a curve of length λt + ε on one leaf joining p to p⁰, a contradiction.

4.2 Compactification of the flow space

[F1]

4.3. Now let Φ be a quasi–geodesic flow on a closed hyperbolic 3–manifold M and the plane P is the flow space of the lifted flow ¯Φ. Every flowline of ˜Φ has two different ends on S_∞² = H³ so the maps

e^± : P → S_∞²

are well defined. We have also two unbounded decompositions of the plane P

D^± = { e^±−1

(p) | p ∈ S_∞² } which are π₁(M )–invariant.

For two elements of distinct decompositions their intersection is compact hence D = D⁻∪ D⁺ is a generalized unbounded decomposition which is also π₁(M )–invariant.

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Using the end compactification method we obtain compactification of the flowspace with universal circle S_u¹. The action of π₁(M ) on P extends to S_u¹. Quasi–geodesics which are close to the boundary of P are small (from Euclidean point of view) in the ball model. Thus

Theorem 4.4. The maps e⁻ and e⁺ have continuous extensions on P ∪ S_u¹ and the extensions coincide on S_u¹.

4.3 Properties of extension and the Calegari conjec- ture

[F1], [F2]

4.5. Cannon and Thurston proved that if M is closed hyperbolic 3–manifold which is a closed hyperbolic surface Σ bundle over the circle then embedding Σ = H˜ ² → ˜M = H³ extends to the π1(Σ)–equivariant surjective map S¹ → S² of ideal boundaries.

Frankel generalized it as follows

If N is a complete transversal to lifted quasi–geodesic flow then its embedding in H³ has a unique continuous extension which restricted to S_u¹ is a π₁(M )–equivariant space filling curve.

4.6. Quasi–geodesic flows imitate pseudo–Anosov property in such a sense that every g ∈ π₁(M ) acts on the universal circle with an even (possibly 0) numer of pairs which are attracting–repelling.

4.7. Type of action of π_M on the universal circle is related to existence of closed orbits of quasi–geodesic flow on M .

If there is no such orbits the action is hyperbolic M¨obius–like but not M¨obius.

Current knowledge of M¨obius–like groups suggests evidence of the Cale- gari conjecture that every quasi–geodesic flow on a closed hyperbolic 3–

manifold has a closed orbit. Frankel proved it using more sophisticated methods in [F2].

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5 Non–compact case

5.1 Spaces of spheres

[H–J], [LW]

5.1. De Sitter space

Λⁿ⁺¹ = {x ∈ Rⁿ⁺²1 | hx|xi = 1}

(1–sheeted hyperboloid in n + 2 dimensional space) is a space of oriented (n − 1)–subspheres of Sⁿ.

Here Sⁿ is visible as interstection of the light cone hx|xi = 0 and hyper- plane x₀ = 1 and to σ ∈ Λⁿ⁺¹ we attach (n − 1)–sphere Σ = σ^⊥∩ Sⁿ. 5.2. The Lorentz form is responsible for angle intersection of (n−1)–spheres.

For (n − 1)–spheres Σ, T ⊂ Sⁿ its angle of intersection α is given as cos α = hsigma|τ i.

In particular, spheres are orthogonal (resp. tangent, disjoint) if the their Lorentz form is 0 (resp. ±1, of absolute value > 1).

5.2 Constant curvature flows on H

²

[Cz], [CzL]

5.3. In [Cz] there are explicit restictions for totally umbilical codimension 1 foliations of Hⁿ orthogonal to constant curvature curve in hyperbolic plane (geodesic or hypercycle).

Adapting it to n = 2 we obtain an information how constant curvature flows (which are quasi–geodesic) of H² look like.

5.4. In [CzL] there is conformal characterization of totally codimension 1 foliations of Hⁿ. It generalizes classical result of Ferus about totally geodesic ones.

A curve in Λⁿ⁺¹ represents a totally umbilical foliation iff at any point its tangent vector is in some boosted time cone.

This gives a full characterization of constant curvature flows in H².

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5.3 Remarks on geodesic flows in H

³

[H–J]

For studying flows on H³ the starting points is to understand structure of geodesic flows which are represented as families of circles orthogonal to 3–dimensional sphere or equivalently 0–spheres (pairs of points) on S².

There are modifications of classical Grasmannians (which describe k–

planes in the n–space) for spheres which could be useful.

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Quasi–geodesic flows