NON R-COVERED ANOSOV FLOWS IN HYPERBOLIC 3-MANIFOLDS ARE QUASIGEODESIC SERGIO R. FENLEY

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NON R-COVERED ANOSOV FLOWS IN HYPERBOLIC

3-MANIFOLDS ARE QUASIGEODESIC

SERGIO R. FENLEY

Abstract. The main result is that if an Anosov ﬂow in a closed hyperbolic

three manifold is not R-covered, then the ﬂow is a quasigeodesic ﬂow. We also

prove that if a hyperbolic three manifold supports an Anosov ﬂow, then up

to a double cover it supports a quasigeodesic ﬂow. We prove the continuous

extension property for the stable and unstable foliations of any Anosov ﬂow in

a closed hyperbolic three manifold, and the existence of group invariant Peano

curves associated with any such ﬂow.

Keywords: Anosov ﬂows, quasigeodesic ﬂows, geometric properties of ﬂow

lines, freely homotopic orbits, large scale geometry of ﬂows.

Mathematics Subject Classiﬁcation 2020: Primary: 57R30, 37E10,

37D20, 37C85; Secondary: 53C12, 37C27, 37D05, 37C86.

1. Introduction

This article studies geometric properties of Anosov ﬂows in 3-manifolds, with

a particular interest when the manifold is hyperbolic. The goal is to study the

quasigeodesic property for such ﬂows. A quasigeodesic is a rectiﬁable curve so

that length along the curve is a uniformly eﬃcient measure of distance, up to

a bounded multiplicative distortion and additive constant, when lifted to the

universal cover. Quasigeodesics are extremely important when the manifold is

hyperbolic [Th1,Th2,Gr]. A ﬂow with no point orbits is a quasigeodesic ﬂow if

all its orbits are quasigeodesics.

A suspension Anosov ﬂow, or any suspension ﬂow in any manifold for that

matter, is quasigeodesic [Ze]. Any closed 3-manifold ﬁbering over the circle with

pseudo-Anosov monodromy is hyperbolic [Th2], hence the class of quasigeodesic

ﬂows in hyperbolic 3-manifolds is very large. What should a quasigeodesic ﬂow

in a hyperbolic 3-manifold look like? Calegari [Cal3] started the study of such

ﬂows, and this was greatly extended by work of Frankel [Fra1,Fra2,Fra3]. Frankel

analyzed the universal circle associated with such ﬂows and also the stable and

unstable decomposition sets in the orbit space of such ﬂows. He used the stable

and unstable decompositions to prove that such ﬂows always have closed orbits

[Fra3].

Hence it is very natural to consider pseudo-Anosov ﬂows in this setting and

ask: given a pseudo-Anosov ﬂow in a closed hyperbolic 3-manifold when is it

quasigeodesic? In this article we will answer this question in the case of Anosov

ﬂows, that is, when there are no p-prong singularities with p≥3. More speciﬁ-

cally we will consider topological Anosov ﬂows: it is a weakening of the Anosov

property allowing for less regularity. See precise deﬁnition in Section 2. Still the

orbits of topological Anosov ﬂows are C1curves, hence rectiﬁable.

Research partially supported by Simons foundation grant 637554, by National Science Foun-

dation grant DMS-2054909 and by the Institute for Advanced Study.

arXiv:2210.09238v1 [math.DS] 17 Oct 2022

2 S.R. FENLEY

In general it is very hard to decide whether an Anosov ﬂow is quasigeodesic,

or not, and this question has remained open until now. In fact the ﬁrst result

previously obtained is a negative one. A codimension one foliation is called R-

covered if the leaf space of the foliation lifted to the universal cover of the manifold

is homeomorphic to the real line R[Fe1]. An Anosov ﬂow in a 3-manifold is

called R-covered if the stable and the unstable 2-dimensional foliations of the

ﬂow are R-covered [Fe1]. It is enough to assume that one of these foliations is

R-covered [Ba1]. For Anosov ﬂows in 3-manifolds, there are many examples:

suspensions and geodesic ﬂows for example. Using Dehn surgery on periodic

orbits of suspensions and geodesic ﬂows [Go,Fr], it was proved in [Fe1] that

there are inﬁnitely many examples of R-covered Anosov ﬂows in hyperbolic 3-

manifolds.

The ﬁrst result in hyperbolic manifolds was that any R-covered Anosov ﬂow

in a closed hyperbolic 3-manifold is not quasigeodesic [Fe1]. The reason is as

follows: An R-covered Anosov ﬂow is necessarily transitive [Ba1]. Let Φ be an

R-covered Anosov ﬂow in a closed hyperbolic 3-manifold. Suppose that Φ is

quasigeodesic. Since Φ is transitive then the ﬂow is uniformly quasigeodesic,

meaning that the quasigeodesic constants are uniform over all orbits. For any

R-covered Anosov ﬂow in a hyperbolic 3-manifold, then up to a double cover

where the stable foliation is transversely orientable, the following happens: every

periodic orbit is freely homotopic to inﬁnitely many other periodic orbits [Fe1].

Coherent lifts of the freely homotopic orbits all have the same ideal points in

the sphere at inﬁnity S2

∞of the canonical compactiﬁcation of the universal cover.

The uniform quasigeodesic behavior implies that the coherent lifts of the periodic

orbits are a globally bounded Hausdorﬀ distance from each other [Th1,Th2,Gr].

Hence they accumulate in the universal cover. This is impossible for coherent

lifts of freely homotopic periodic orbits.

In this article we obtain a deﬁnitive answer to the quasigeodesic question for

Anosov ﬂows in hyperbolic 3-manifolds, by proving that R-covered is the only

obstruction:

Main Theorem. Let Φbe a topological Anosov ﬂow in a closed hyperbolic 3-

manifold. Suppose that Φis not R-covered. Then Φis a quasigeodesic ﬂow.

Convention. Any topological Anosov ﬂow in a hyperbolic 3-manifold is orbitally

equivalent to a (smooth) Anosov ﬂow by recent results of Shannon [Sha]. Our

results and techniques are not dependent on the regularity of the ﬂow and its

foliations. Hence we will abuse terminology, and many times refer to our ﬂows

as Anosov ﬂows, even when they may only be topological Anosov ﬂows.

In addition the Main Theorem concerns behavior of the foliations in the univer-

sal cover, so it is unaﬀected by taking ﬁnite lifts. Therefore whenever necessary

we assume that the manifold is orientable or the foliations are transversely ori-

entable. But orientability is not necessary for many intermmediate steps in the

proof.

The Main theorem has many consequences:

Theorem 1.1. Let Mbe a hyperbolic 3-manifold admitting an Anosov ﬂow.

Then up to perhaps a double cover, Madmits a quasigeodesic pseudo-Anosov

ﬂow. In any case Madmits a one dimensional foliation by quasigeodesics, with

a dense leaf.

Roughly this goes as follows: if the ﬂow is not R-covered we use the Main

Theorem. Otherwise up to a double cover if necessary, the stable foliation is

QUASIGEODESIC ANOSOV FLOWS IN HYPERBOLIC 3-MANIFOLDS 3

R-covered and transversely orientable. Then one can produce a quasigeodesic

pseudo-Anosov ﬂow transverse to the stable foliation [Th3,Cal1,Fe6]. We explain

in detail in Section 12 that the double cover may be necessary: this happens for

example for many Fried surgeries on geodesic ﬂows of closed, non orientable

hyperbolic surfaces.

Another consequence concerns the continuous extension property. Suppose a

topological Anosov ﬂow Φ in a 3-manifold Mhas stable leaves which are C1.

This can always be achieved up to orbit equivalence [BFP]. The leaves of the

stable foliation are Gromov hyperbolic [Pla,Sul,Gr]. Hence if Fis a leaf of the

stable foliation in the universal cover f

M, then it has a canonical compactiﬁcation

into a closed disk F∪S1(F). The continous extension property asks whether the

embedding F→f

Mextends continuously to a map F∪S1(F)→f

M∪S2

∞, see

[Ga] and Question 10.2 of [Cal2]:

The continuous extension property was ﬁrst proved for ﬁbrations over the

circle in the celebrated article by Cannon and Thurston [Ca-Th]. This was an

unexpected and very surprising result. Since then it has been extended to some

classes of foliations: ﬁnite depth foliations [Fe5], R-covered foliations [Th3,Fe6],

and foliations with one sided branching [Fe6]. In general it is a very hard property

to prove. For Anosov foliations this question is very complex, in part because

these foliations do not have compact leaves. In this article we prove:

Theorem 1.2. Let Φbe an Anosov ﬂow in a hyperbolic 3-manifold. Then the

stable and unstable foliations of Φhave the continuous extension property.

The non R-covered case is entirely new in terms of techniques: it is the ﬁrst

result proving the continuous extension property, where the key tool is not a

transverse or almost transverse pseudo-Anosov ﬂow which is quasigeodesic.

The quasigeodesic property also has consequences for the computation of the

Thurston norm. We refer the reader to [Mos2] for that.

Finally the main theorem implies the existence of group invariant Peano curves

as follows. Given an Anosov or pseudo-Anosov ﬂow Φ in a closed 3-manifold M,

let Obe the orbit space of the lifted ﬂow e

Φ in the universal cover f

M. This orbit

space is always homeomorphic to the plane R2[Fe1,Fe-Mo]. The orbit space has

one dimensional, possibly singular foliations Os,Ouwhich are the projections

to Oof the two dimensional stable and unstable foliations in f

M. In [Fe6] we

produce an ideal boundary ∂Oand ideal compactiﬁcation O ∪ ∂Ousing only the

one dimensional foliations Os,Ou. The ideal boundary is always homeomorphic

to a circle and the compactiﬁcation is homeomorphic to a closed disk. In addition

the fundamental group π1(M) naturally acts on all these objects. The results of

this article lead to Peano curves associated with Anosov ﬂows:

Theorem 1.3. Let Mbe a hyperbolic 3-manifold admitting an Anosov ﬂow Φ.

Then there is a group invariant Peano curve η:∂O → S2

∞where ∂Ois boundary

of the orbit space Oof a quasigeodesic ﬂow in either Mor a double cover of M.

The quasigeodesic ﬂow in Theorem 1.3 is the original ﬂow Φ in the case that

Φ is quasigeodesic −that is, when Φ is not R-covered; or it is a pseudo-Anosov

ﬂow which is transverse to (say) the stable foliation of either Φ in Mor the lift

to a double cover of M−in the case that Φ is R-covered.

The ﬁrst result concerning group invariant Peano curves was again proved by

Cannon and Thurston in the same seminal paper [Ca-Th]. They considered ﬁbers

of ﬁbrations, which lift to properly embedded planes in f

M. They proved that

the embeddings in f

Mextend to the ideal compactiﬁcations, and proved that the

4 S.R. FENLEY

ideal maps are group invariant Peano curves. There are results concerning Peano

curves for quasigeodesic pseudo-Anosov ﬂows in hyperbolic 3-manifolds [Fe6,Fe7].

There are similar results for general quasigeodesic ﬂows in hyperbolic 3-manifolfds

[Fra1,Fra2]. Finally there are also many results on Cannon-Thurston maps from

the Kleinian group setting, see for example [Mj].

Remark 1.4. To prove the Main theorem we only use that Mis atoroidal.

By the geometrization theorem of Perelman this implies that Mis hyperbolic,

but that is an extremely hard result. In this article we will invariably state

results and do arguments using atoroidal as an hypothesis. By atoroidal we

mean “homotopically atoroidal”, that is, there is no π1-injective immersion of

a torus into M. There is a slight diﬀerence between homotopically atoroidal

and “geometrically atoroidal” −the second condition means that there is an

embedded incompressible torus in M. If Mhas an Anosov ﬂow, then Mis

irreducible, so the diﬀerence is only for small Seifert ﬁbered spaces. But there

are small Seifert ﬁbered spaces that admit Anosov ﬂows. For example let Sbe

a compact hyperbolic 2-dimensional orbifold which has a ﬁnite cover which is a

surface, and Sis a sphere with 3 cone points. Then M=T1Sis a small Seifert

ﬁbered space and the geodesic ﬂow of Sin Mis an Anosov ﬂow. These small

Seifert ﬁbered spaces are geometrically atoroidal but not homotopically atoroidal.

1.1. Examples of R-covered and non R-covered Anosov ﬂows in hyper-

bolic 3-manifolds. The results of this article obviously lead naturally to the

question as to the existence of both of these classes and how widespread they are.

It turns out that both R-covered and non R-covered Anosov ﬂows are very com-

mon in hyperbolic 3-manifolds. We already mentioned the existence of inﬁnitely

many R-covered examples obtained by some Dehn surgeries on closed orbits of

suspensions and geodesic ﬂows [Fe1]. On the other hand any Anosov ﬂow in a

non orientable hyperbolic 3-manifold is non R-covered [Fe3] and one can obtain

these by doing Dehn surgery on suspensions of orientation reversing hyperbolic

diﬀeomorphisms of the torus.

The wealth of examples was enormously extended by a recent article of Bonatti

and Iakovouglou [Bo-Ia]. We just mention some results of [Bo-Ia] that show that

both classes are extremely large.

To obtain examples of non R-covered Anosov ﬂows: Theorems 2 and 3 of [Bo-Ia]

state that starting with a non R-covered Anosov ﬂow and doing appropriate Fried

Dehn surgeries one obtains a non R-covered Anosov ﬂow. To get examples in

hyperbolic 3-manifolds, start with a non R-covered Anosov ﬂow in a hyperbolic

3-manifold. Most Dehn surgeries will produce hyperbolic 3-manifolds. Theorems

5 and 6 of [Bo-Ia] provide many more examples which are non R-covered.

To obtain examples of R-covered Anosov ﬂows: Theorems 4 and 7 of [Bo-Ia]

produce a wide class of R-covered Anosov ﬂows by doing Dehn surgeries on sus-

pensions. If the surgery coeﬃcients are big in absolute value, then the resulting

manifold is hyperbolic. Corollary 4.1 and Proposition 4.1 of [Bo-Ia] produce

R-covered examples starting with geodesic ﬂows.

There are many more results in [Bo-Ia] showing either the R-covered behavior

or the non R-covered behavior for Anosov foliations in 3-manifolds.

Many other possible examples can possibly be obtained as follows: start with

an example as constructed in [BBY] using hyperbolic plugs. This is an extremely

general construction producing an enormous amount of Anosov ﬂows. Do Dehn

surgeries on some periodic orbits to eliminate all tori. The resulting manifold

is hyperbolic, and one can possibly determine whether the resulting ﬂow is R-

covered or not using the methods of [Bo-Ia].

QUASIGEODESIC ANOSOV FLOWS IN HYPERBOLIC 3-MANIFOLDS 5

1.2. Some ideas on the proof of the Main Theorem. We will prove the

contrapositive of the statement of the Main theorem. As a starting point of the

analysis we use the following result: It was proved in [Fe7] that a topological

Anosov ﬂow Φ (even a pseudo-Anosov ﬂow) in a hyperbolic 3-manifold is quasi-

geodesic if and only if there is a global bound K, so that any set of pairwise freely

homotopic periodic orbits of Φ has cardinality bounded above by K. The strat-

egy of the proof of our Main theorem is to assume that Φ is not quasigeodesic,

hence this boundedness condition fails. We then prove that this implies that Φ

is R-covered.

In very rough terms what we will do is the following: assume that the ﬂow is

not quasigeodesic. Then we will produce an appropriate essential lamination in

Mwhich will imply that the ﬂow is R-covered.

To do that we will employ a standard form for the ﬂow as follows. Using resuls

of Candel [Cand] it was proved previously in [BFP] that Φ is orbitally equivalent

to a topological Anosov ﬂow so that the stable leaves are C1with a leafwise

Riemannian metric making every leaf a hyperbolic surface (meaning gaussian

curvature constant and equal to −1). In addition the ﬂow lines are geodesics in

the stable leaves.

By assumption we have arbitrarily big sets of pairwise freely homotopic peri-

odic orbits. We realize the free homotopies between the periodic orbits so that

intersections with stable leaves are geodesics in the respective leaves. These

are immersed annuli in M. These immersed annuli are chosen so that they are

uniquely determined by the periodic orbits. If there are more orbits freely homo-

topic to a given orbit then the annuli can be put together forming bigger annuli.

In other words one can concatenate the free homotopies in a nice way. We take

limits of these big annuli when the number of freely homotopic orbits goes to

inﬁnity. We can take limits and they are well behaved in part because of the

canonical form of the free homotopy annuli: the stable leaves are hyperbolic sur-

faces and the intersections of the annuli with these stable leaves are geodesics in

the leaves. This is one main reason for the setup with the metric on the stable

leaves and the ﬂow lines geodesics in these stable leaves. Even with this setup it

is very diﬃcult to understand the limits. We will explain more in Subsection 3.3

once we have deﬁned lozenges and other objects associated with free homotopies.

The limits of the bigger and bigger free homotopy annuli, when lifted to the uni-

versal cover, are what we call walls. The walls are properly embedded planes in

the universal cover, and intersect an Rworth set of stable leaves. The goal is to

prove that this is the set of all stable leaves in f

M, and hence Φ is R-covered. To

do that we study properties of these walls and certain subsets of the orbit space

in f

M, called bi-inﬁnite blocks, associated with the walls. A lot of the analysis is

done without assuming that Mis atoroidal. But the atoroidal hypothesis is used

in many places to obtain stronger results.

One goal is to understand the projection of the walls in M. There are two main

cases: either the projection of a wall does not self intersect transversely or it does.

In the ﬁrst case one shows, in the case that Mis atoroidal, that the projection

of a wall to Mhas closure which is a lamination −in other words any tangent

intersection of two walls leads to equality of these walls. This last fact in general

holds only when Mis atoroidal. We study the complementary regions of this

lamination and eventually prove that Φ is R-covered. In the second case there

are transverse self intersections when projecting to M. We then study a type of

“convex hull” in f

Massociated with such a wall. This convex hull is obtained

using the hyperbolic metric in the stable leaves and using that walls intersect

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NONR-COVEREDANOSOVFLOWSINHYPERBOLIC3-MANIFOLDSAREQUASIGEODESICSERGIOR.FENLEYAbstract.ThemainresultisthatifanAnosovowinaclosedhyperbolicthreemanifoldisnotR-covered,thentheowisaquasigeodesicow.WealsoprovethatifahyperbolicthreemanifoldsupportsanAnosovow,thenuptoadoublecoveritsupportsaquasigeodesicow.We...

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