closed orbits then we show that π1(M) acts on ∂P as a hyperbolic Möbius- like group that .... into three perpendicular sub-bundles: the bundle TF tangent to the flow, a stable ... lifts to a plane CΣ that is transverse to the flow, and (CΛs, CΛu) restrict ...... (a, x) is nonempty then it contains some point in S0, so let a be this point.
Quasigeodesic ows from in nity Steven Frankel Magdalene College University of Cambridge
Supervised by Danny Calegari
This dissertation is submitted for the degree of Doctor of Philosophy. May 2013
This dissertation is the result of my own work and includes nothing which is the outcome of work done in collaboration except where speci cally indicated in the text.
Abstract A ow is called quasigeodesic if each owline is uniformly e cient at measuring distances on the large scale.
We study quasigeodesic ows on
closed hyperbolic 3-manifolds from the perspective at in nity. If space
M P
is a closed hyperbolic 3-manifold with a quasigeodesic ow, the orbit of the lifted ow on
H3
is planar, and it inherits an action by
We use Calegari's universal circle
Su1
to compactify
P
to a closed disc
way that is compatible with this action. We show that the maps
∂H3 to
P
in a
e± : P →
that send each orbit to its forward/backward limit extend continuously
P,
is a
π1 (M ).
and
e+ = e−
π1 -equivariant
on
∂P .
Consequently, the restriction
e : ∂P → ∂H3
sphere- lling curve, generalizing the Cannon-Thurston
theorem which produces such curves for suspension ows. If the ow has no closed orbits then we show that
π1 (M )
acts on
∂P
as a hyperbolic Möbius-
like group that is not Möbius, which conjecturally cannot occur. Finally, we show that
π1 (M )
acts on
∂P
with pseudo-Anosov dynamics.
This completes a major part of Calegari's program to show that quasigeodesic ows are homotopic (through quasigeodesic ows) to pseudo-Anosov ows.
i
Acknowledgements Foremost, I would like to thank Danny Calegari. While your mathematical in uence on this thesis is obvious, I'm most grateful for your encouragement and advice over the last ve years. And Therese, thanks for the hot tea and long conversations that kept me going in the winter.
The two of
you have made me feel like part of the family. A nephew or cousin, I mean; you're far too young to be parents. Thanks to Sylvain Cappell. You're the reason I wanted to study mathematics. Thanks to Matt Day, Sergio Fenley, Dave Gabai, Nik Makarov, Curt McMullen, and Dongping Zhuang for helpful conversations and correspondence. Thanks Benson Farb for your advice while in Chicago. Thanks to Jake Rasmussen and Henry Wilton for their careful examination of this dissertation, especially on such short notice. Thanks to my climbing buddies for making me work on beaches and mountains instead of my o ce.
In particular, to J and Matt, for holding
the rope on Pinnacle Gully while I worked out the proof of the continuous extension theorem. Thanks to all of my friends at Caltech, Cambridge, and Chicago. Rafa, Maurice, and Bapt, can we go for a pitcher at Amigos? Himanshu, sure, I'll come to that party I wasn't invited to. Dave and Jeannie, you always made LA interesting. Helge, we should watch more British spy movies together. Giulio, Goldie, Jack, Job, John, Steven, Zeb, and the rest of the CUMC, I'd be more speci c if I could remember more of the time we spent together. Brad, that's how I always am, outside of Chicago winter. Finally, thanks to my family for your love and support, even though I'm going to be the wrong kind of doctor. Please send money.
iii
Contents Abstract
i
Acknowledgements
Chapter 1.
iii
Introduction
1
1.
Overview
1
2.
Statement of results
7
Chapter 2.
Flows
9
1.
The orbit space
2.
Quasigeodesics
10
3.
Quasigeodesic ows
11
Chapter 3.
9
Circular Orders
17
1.
Order completion
19
2.
Universal circles
20
3.
Circularly ordered groups
23
Chapter 4.
Decompositions
25
1.
Ends
26
2.
Circular orders
30
3.
Universal circles
38
4.
The end compacti cation
42
Chapter 5.
Quasigeodesic Flows
49
1.
Extending the endpoint maps
50
2.
A more convenient circle
58
3.
Properties of the decompositions
59
4.
Dynamics on the universal circle
60
5.
Closed orbits
62 v
vi
Bibliography
CONTENTS
67
CHAPTER 1
Introduction The goal of this thesis is to outline the theory of quasigeodesic ows from the point of view at in nity.
The guiding conjecture for this, due
to Danny Calegari, is that every quasigeodesic ow on a closed hyperbolic 3-manifold is homotopic (through quasigeodesic ows) to a pseudo-Anosov ow.
Both quasigeodesic and pseudo-Anosov ows give rise to universal
circles at in nity, and our program attempts to relate quasigeodesic and pseudo-Anosov ows using their dynamics on the universal circle.
1. Overview Much of this thesis is about how certain 3-dimensional smooth dynamical objects give rise to an array of intricately related 1- and 2-dimensional discrete dynamical objects. The 3-dimensional objects we're concerned with are quasigeodesic and pseudo-Anosov ows, and these give rise to discrete group actions on 1-dimensional universal circles and 2-dimensional orbit spaces that preserve certain laminations, singular foliations, and unbounded decompositions. Some of these objects are built out of others by topological trickery, some come from quotient maps, and at the end they all t together nicely in a group-equivariant way.
We'll start informally in order to get to this
picture before plodding through all of the technical ingredients. To preserve continuity we'll refrain from citing sources here, but precise citations will accompany the precise statements in the main chapters. Let ric to
M
H3 ,
be a closed hyperbolic 3-manifold. Its universal cover is isometand
π1 (M )
acts on
H3
by loxodromic isometries with
quotient. Hyperbolic space has a natural compacti cation adding a sphere at in nity, and the action of the boundary. 1
3
H =
M
as its
2 by H3 ∪ S∞
π1 (M ) extends continuously to
2
1. INTRODUCTION
A ow on
M
lifts to a ow on
H3 ,
and the action of
transformations preserves the foliation by owlines. won't behave well with respect to the boundary of close to in nitely many points in
2 . S∞
π1 (M )
by deck
In general, owlines
3
H
, and might spiral
An obvious way to avoid this would
be to insist that each owline is a geodesic, but a theorem of Zeghib implies that no such ows exist. It turns out that quasigeodesic ows, where each owline tracks a geodesic in the large scale, are exactly the ows whose owlines have well-de ned and continuously varying endpoints in A ow
F
2 . S∞
on a 3-manifold is called Anosov if the tangent bundle splits
into three perpendicular sub-bundles: the bundle a stable bundle
Es,
and an unstable bundle
Eu.
TF
tangent to the ow,
Nearby orbits along the
stable direction converge exponentially in forwards time, and nearby orbits along the unstable direction converge exponentially in backwards time. The (weak) stable and unstable bundles integrate to a pair of transverse foliations
(Λs , Λu ),
where
T Λs
T Λu
spans the tangent and stable directions, and
spans the tangent and unstable directions.
Leaves of
Λs
and
Λu
intersect
transversely, and the intersection of two leaves is either empty or an orbit of
F.
The orbits in a single leaf of
orbits in a single leaf of
Λu
Λs
are all forwards asymptotic, and the
are all backwards asymptotic.
A ow is called pseudo-Anosov if it is Anosov except near some isolated singular orbits, where the ow looks like the suspension of a
p-prong
in the
plane (see Figure 1). A pseudo-Anosov ow has a pair of singular foliations
(Λs , Λu ).
These are true transverse foliations away from the singular orbits,
and they look like Figure 2 near singular orbits.
1.1. Suspension ows.
The prototype example for both quasigeodesic
and pseudo-Anosov ows is the suspension of a pseudo-Anosov homeomorphism. Let
Σ
be a surface of genus at least two, and consider
with the unit ow by gluing
F0
in the positive direction along
I.
The manifold
Σ × {0} to Σ × {1} with a homeomorphism f
when the mapping class of
f
M0 = Σ × I M
built
is hyperbolic exactly
is aperiodic and irreducible. In this case,
isotopic to a pseudo-Anosov homeomorphism, so it preserves a pair of 1-dimensional measured singular foliations on
Σ.
f
is
(λs , λu )
The monodromy
f
is
1. OVERVIEW
3
Figure 1. A 4-prong singularity in the plane.
Figure 2. Stable and unstable foliations near a singular orbit.
contracting along leaves of on
F
M0
λs
and expanding along leaves of
glues up to form a ow
F
on
M
λu .
The ow
called the suspension ow of
f,
F0
and
is both pseudo-Anosov and quasigeodesic. The 2-dimensional singular fo-
liations of by
f
on
F
Σ.
on
M
restrict to the 1-dimensional singular foliations preserved
That is,
Λs ∩ (Σ × {0}) = λs and
Λu ∩ (Σ × {0}) = λu . The ow
F
lifts to a pseudo-Anosov ow
unstable singular foliations are the lifts
e F
e s, Λ e u ). (Λ
on
H3
whose stable and
A copy of the surface, say
4
1. INTRODUCTION
e that is transverse to the ow, and (Λ e s, Λ e u ) restrict Σ × { 12 }, lifts to a plane Σ to the lifts
es , λ eu ) (λ
e. Σ
on
See Figure 3.
2 S∞
H3
e Σ i
M
Σ id × { 12 }
Figure 3. A surface bundle and lifted surface. The surface
e Σ
is transverse to the ow, and it hits every owline, so
there's a natural quotient map
e π : H3 → Σ that sends each owline to its point of intersection with naturally homeomorphic to the orbit space of the ow
F
e. F
e. Σ
That is,
e Σ
is
Quasigeodesity of
implies that the maps
2 e → S∞ e± : Σ that send each orbit to its forward/backward limit are well-de ned and continuous. The fact that orbits in a stable leaf are forwards asymptotic means that
e+ (p) = e+ (q)
e− (p) = e− (q)
if
p
if
p
and
and
q
q
are both contained in a leaf of
are both contained in a leaf of
eu . λ
es . λ
Similarly,
1. OVERVIEW
The fundamental group
π1 (M )
acts on
5
H3
preserving the 1-dimensional
foliation by owlines, as well as the 2-dimensional foliations Pushing forward under the map preserves
es and λ eu . λ
The
π,
we obtain an action of
es Λ
π1 (M )
e u. Λ
and
on
e Σ
that
e± maps are equivariant with respect to this action.
The surprising behavior of suspension ows becomes apparent when we compactify
e. Σ
e Σ
The surface
is homeomorphic to the hyperbolic plane
since it's the universal cover of the hyperbolic surface etry as a subspace of
H3
Σ.
H2 ,
However, its geom-
is very dierent from the geometry it gets as a lift of
a hyperbolic surface. The hyperbolic plane has a natural compacti cation to a closed disc
2
1 H = H2 ∪ S∞
homeomorphism
e ' H2 Σ
by adding a circle at in nity, so we can use the
e. Σ
to compactify our transversal
this can be done in a way that respects the action of
π1 (M ).
The famous
e → H3 i:Σ
Cannon-Thurston theorem states that the inclusion map to the boundary circle, where it restricts to a
It turns out that
π1 -equivariant
extends
curve
1 2 i : S∞ → S∞
that lls the sphere. In fact, the
e±
maps also extend continuously to the
− e+ |S∞ 1 = e |S 1 = i|S 1 . ∞ ∞
boundary, and
1.2. Pseudo-Anosov and quasigeodesic ows in general.
Much
of this structure remains in place for a general pseudo-Anosov ow. Fix a closed hyperbolic 3-manifold
P
The owspace endow
P
M
and let
F
be a pseudo-Anosov ow on
is the set of owlines of the lifted ow on
to the plane. can think of
e. Σ
We can
3 with a topology by thinking of it as the quotient of H obtained
by collapsing each owline to a point. It turns out that
of
H3 .
M.
If we take a section
P
i
of the quotient map
as a transversal to the ow
e. F
That is,
As before, the 2-dimensional singular foliations
1-dimensional singular foliations induced action of
π1 (M ).
P
es , λ eu ) (λ
on
P,
is homeomorphic
π : H3 → P ,
P
takes the place
e s, Λ e u) (Λ
restrict to
which are preserved by the
However, note that these are not generally lifts
of singular foliations on a compact surface, since the quotient of generally a surface. We lose the in general.
we
e±
maps since
F
P
is not
is no longer quasigeodesic
6
1. INTRODUCTION
Although
P
that the pair of foliations
P = P ∪ Su1
H2 ,
no longer has a natural homeomorphism to
es , λ eu ) (λ
it turns out
can be used to build a compacti cation
by adding a universal circle
Su1
at in nity. This universal circle
is built by introducing a point at in nity for each end of each leaf of and
eu λ
the nonsingular leaves contribute two points and
contribute
p
π1 (M )
on
P
eu λ
π1 -equivariant
implies that
i : P → H3
and
are
does not generally extend to the boundary of The action of
π1 (M )
acts on
π1 (M )
Su1
leaves
1 extends to Su . However, the section
points. The fact that
the action of
es λ
p-pronged
es λ
on
Su1
P.
takes a particular dynamical form. Each
g∈
with an even number of xed points that are alternately
attracting and repelling. Now let
F
be a quasigeodesic ow on
M.
The owspace
P
is de ned
similarly. Once again it turns out to be a plane, and once again it can be interpreted as a transversal to
π : H3 → P .
ow, we will use
lie at
e+
2 . e± : P → S∞
and
and consider
p.
by choosing a section
We lose the pair of singular foliations
get to keep the maps
2 p ∈ S∞
e F
e−
And, inspired by the case of a suspension
to build replacements for
es λ
and
eu . λ
Fix a point
(e+ )−1 (p), the set of owlines whose positive endpoints
Each component of
(e+ )−1 (p)
is closed and unbounded, and we
− decomposition D is built similarly using the are invariant under the action of
a generalization of
of the quotient map
e s, Λ e u ), but this time we (Λ
can collate these to form the positive decomposition
D±
i
es , λ e u ). (λ
π1 (M ),
D+
of
P.
The negative
e− map. The decompositions and we will treat
(D+ , D− )
as
Where in the pseudo-Anosov case a stable leaf
and an unstable leaf intersect transversely in a single orbit, here a positive decomposition element and a negative decomposition element intersect in a compact collection of orbits, all of which share a pair of endpoints in
2 . S∞
Remarkably, there's still enough structure to build a compacti cation for
P.
The decomposition elements are arbitrarily complicated closed un-
bounded sets instead of just lines and
p-prongs,
so we need to be more
sophisticated when talking about ends. Still, there is a workable notion of an end, and the ends of each decomposition element appear as points in a universal circle
Sv1 .
Here's where the new part of the story begins.
We'll
2. STATEMENT OF RESULTS
7
show that this universal circle compacti es the orbit space, and the maps
i, e+ ,
and
e−
all extend continuously to (and agree on) the boundary. This
vastly generalizes the Cannon-Thurston theorem.
π1 (M )
We'll also show that the dynamics of pseudo-Anosov case. Each
g ∈ π1 (M )
on
Su1
acts on
Sv1
looks just like the
with an even number of
xed points that are alternately attracting and repelling. Finally, we'll show that the action of amount of information about
π1 (M )
on
Sv1
contains a surprising
F, even though it's just a discrete 1-dimensional
image of the smooth 3-dimensional dynamics of a quasigeodesic ow. particular, we can use it to nd closed orbits in ment
g ∈ π1 (M )
homotopy class of
π1 (M )
into
We'll show that if an ele-
acts, up to conjugacy, as anything other than a hyperbolic
Möbius transformation, then the ow
of
F.
In
g.
Consequently, if
F
F
contains a closed orbit in the free
has no closed orbits then the action
1 on Sv is an example of a Möbius-like group that is not conjugate
P SL(2, R).
2. Statement of results For convenience, we'll collect our main original theorems. If geodesic ow on a 3-manifold
M
3
then
P
is the orbit space of
F is a quasie. F
Calegari
[ ] constructs a universal circle
Su1 ,
which turns out to be larger than we
need. We'll work with a quotient
Sv1
on which it's more natural to study the
dynamics of
π1 (M ).
Still, our rst four theorems hold for both
Su1
and
Sv1 .
3
The following idea is due to Calegari [ ]. Calegari's original construction of the universal circle involves approximating the ends of sets in the plane by proper rays. Our more direct handling of ends allows us to complete the proof.
Compactification Theorem. Let closed hyperbolic 3-manifold
P = P ∪ Sv1 P
M,
and let
F Sv1
be a quasigeodesic ow on a be the universal circle.
Then
has a natural topology making it into a closed disc with interior
and boundary
Sv1 .
The action of
to the universal circle action on
The space
P
π1 (M )
on
P
extends to
∂P .
is called the end compacti cation of
P.
P
and restricts
8
1. INTRODUCTION
Continuous Extension Theorem. Let a closed hyperbolic 3-manifold
M.
unique continuous extensions to
F
be a quasigeodesic ow on
P,
and
e+
agrees with
2 e : Sv1 → S∞
Consequently, the restriction
2 e ± : P → S∞
The endpoint maps
of
e−
e±
admit
on the boundary.
is a
π1 -equivariant
sphere- lling curve, generalizing the Cannon-Thurston theorem. A group
Γ
acting on the circle is called hyperbolic Möbius-like if each
element is conjugate to a hyperbolic Möbius transformation.
Möbius-like Theorem. Let perbolic 3-manifold
M.
F
be a quasigeodesic ow on a closed hy-
Suppose that
acts on the universal circle
Sv1
F
has no closed orbits. Then
π1 (M )
as a hyperbolic Möbius-like group.
In contrast:
Conjugacy Theorem. Let perbolic 3-manifold of
π1 (M )
M.
F
be a quasigeodesic ow on a closed hy-
Suppose that
F has no closed orbits.
on the universal circle is not conjugate into
Then the action
P SL(2, R).
Finally:
Pseudo-Anosov Dynamics Theorem. Let on a closed hyperbolic 3-manifold
M.
Then each
F
be a quasigeodesic ow
g ∈ π1 (M ) acts on Sv1
with
an even number of xed points (possibly zero) that are alternately attracting and repelling.
CHAPTER 2
Flows A ow point
F
on a manifold
x ∈ M
and
t ∈ R
M
is a continuous
there is a point
R-action.
t · x ∈ M,
That is, for each
and this satis es the
following conditions: (1) If
x∈M
(2) the map
and
t, s ∈ R,
R×M →M
then
(t + s) · x = t · (s · x),
de ned by
(t, x) 7→ t · x x∈M
We will call a ow nonsingular if for each that
and
is continuous. there is some
t
such
t · x 6= x.
We will restrict attention to nonsingular ows. The orbit of each point
x∈M
is the owline
R · x,
and we'll use the same symbol to refer to both
the ow and the associated oriented foliation
F = {R · x|x ∈ M } by owlines.
1. The orbit space Given a nonsingular ow set of owlines in
M,
F
on a manifold
M,
the orbit space
OF
with the topology obtained as a quotient of
is the
M
by
collapsing each owline to a point. We denote the universal cover of ow
e F
on
f. M
The owspace of
F
e F
by
f. M
π1 (M )
on
f by M
OFe
π1 (M )
on the owspace 9
OFe .
on
M
lifts to a
of the lifted ow.
deck transformations
by owlines, so the quotient map
f → Oe π:M F induces an action of
F
The ow
is the orbit space
The action of the fundamental group preserves the foliation
M
10
2. FLOWS
While the orbit space of a ow on a closed 3-manifold is often not very nice, in general not even Hausdor, the owspace is often much nicer.
In
particular, the ows we're concerned with will be product covered:
Definition 2.1. A ow if the foliation
R3
foliation of to
F
M
on a 3-manifold
is called product covered
e by lifted owlines is topologically equivalent to the product F as
R2 × R.
Equivalently, if the owspace
OFe
is homeomorphic
R2 .
2. Quasigeodesics Definition 2.2. Let in a metric space
(X, d)
k, be non-negative constants. (k, )-quasigeodesic
is a
A curve
γ:R→X
if it satis es
1/k · d(γ(x), γ(y)) − ≤ |x − y| ≤ k · d(γ(x), γ(y)) + for all
x, y ∈ R.
A curve is called a quasigeodesic if it is a
for some constants
If
X
(k, )-quasigeodesic
k, .
is a geodesic space and
γ
is a curve parametrized by arc length then
the left-hand inequality always holds, so we can think of the quasigeodesic condition as ensuring that a curve makes de nite progress on a large scale.
Hn
Recall that hyperbolic space
has a natural compacti cation as a
closed disc by adding a sphere at in nity de ned endpoints in
n−1 . S∞
n−1 . S∞
Geodesics in
Hn
have well-
In hyperbolic space, quasigeodesics are qualita-
tively similar to geodesics:
Proposition 2.3 (see [1], pp. 399404 or [18]). quasigeodesic in
Hn ,
then
γ
(1) If
γ
is a
(k, )-
has well-de ned, distinct endpoints in
n−1 . S∞ (2) There is a constant
every
C,
depending only on
(k, )-quasigeodesic
in
Hn
k , ,
and
is contained in the
n,
such that
C -neighborhood
of the geodesic with the same endpoints.
Quasigeodesity can also be formulated as a local condition.
γ : R → H3 all
x, y ∈ R
is called a with
c-local k -quasigeodesic
|x − y| < c.
if
d(γ(x), γ(y)) ≥
A curve
|x−y| k
−k
for
3. QUASIGEODESIC FLOWS
Lemma 2.4. (Gromov, [16]) For every stant
11
k ≥ 1,
there is a universal con-
c(k) such that every c(k)-local k -quasigeodesic is a (2k, 2k)-quasigeodesic.
3. Quasigeodesic ows Definition 2.5. A nonsingular ow
F
on a manifold
quasigeodesic if each owline lifts to a quasigeodesic in quasigeodesic if there are uniform constants to a
k,
M
f. M
is said to be
It is uniformly
such that each owline lifts
(k, )-quasigeodesic.
It turns out that we don't need to distinguish between quasigeodesic and uniformly quasigeodesic ows:
Lemma 2.6 (Calegari. [3], Lemma 3.10). Let 3-manifold. Then every quasigeodesic ow on
M
M
be a closed hyperbolic
is uniformly quasigeodesic.
Theorem 2.7 (Calegari. [3], Theorem 3.12). Let bolic 3-manifold. Then every quasigeodesic ow on
In particular, if manifold
M,
then
use the notation
F
OFe
M
be a closed hyper-
is product covered.
is a quasigeodesic ow on a closed hyperbolic 3is homeomorphic to the plane.
P := OFe
From now on we'll
for the owspace of a quasigeodesic ow.
Example 2.8 (Surface bundles). Let the circle.
M
Zeghib shows in [
37]
M
be a closed surface bundle over
that any ow that is transverse to the
foliation by surfaces is quasigeodesic.
Example 2.9 (Mosher's examples). In [25], Mosher constructs a class of quasigeodesic ows that do not come from the surface bundle construction. Start with the ow on
R3
de ned by
(x, y, z) · t = (x · λt , y · λ−t , z + t) for some
λ > 1.
and
y = ±1
ow
FT
Cut along the eight planes de ned by
and mod out by a translation in the
on an octagonal torus
T.
z
xy = ± 12 , x = ±1,
direction. The result is a
Actually, this is just a semi- ow, i.e. the
ow is not de ned for all time, as some orbits leave
T.
The boundary of
T
consists of eight annuli, and the ow points inward on two of these, outward
12
2. FLOWS
on two, and is tangent to the remaining four. There is a single closed orbit
σ,
T.
the core of
Every other owline either spirals around
σ
or runs from
one of the inward-pointing annuli to one of the outward-pointing annuli. Now let
Σ
be a compact surface with four boundary components, and
Σ×I
consider the unit ow on The boundary of
Σ×I
points inward along
that moves along the positive
outward along
four annuli. The ow
Σ × {1},
and is tangent along
the four annuli. We can glue the four boundary annuli of annuli of ow
F0
T
along which
FT
closed 3-manifold is the image of
∂+
M
∂+ .
to the four
∂−
M0
with a
and outward along
Glue these two surfaces together to build a
F,
with a ow
and
Σ×I
is tangent. The result is a 3-manifold
that points inward along one boundary surface
another boundary surface
direction.
Σ and
consists of two copies of
Σ × {0},
I
and let
S
be the embedded surface that
∂− .
If we make the right choices during this construction, we can ensure that
M
is a closed hyperbolic 3-manifold and
ow
F
has one closed orbit,
σ,
either eventually tracks If it tracks
σ
the lifts of
S
σ.
S
is an incompressible surface. The
If we follow any owline
or it crashes through
S
γ,
we nd that it
with de nite frequency.
then quasigeodesity follows easily. Otherwise, we can see that in the universal cover
H3
obey a certain uniform separation
property. This is enough to see that the orbits that crash through
S
makes
2 de nite progress towards a single point in S∞ , and quasigeodesity follows.
Example 2.10 (Finite depth foliations). Mosher's ows are particular examples of a more general phenomenon. The ow above is transverse to a depth-1 foliation. If
M
is a nontrivial class in
is a closed, orientable, irreducible 3-manifold and
ξ
H2 (M ) then Gabai [15] constructs a taut, nite-depth
foliation whose compact leaves represent bolic manifold, Mosher [
ξ.
Given such a foliation on a hyper-
26] constructs an almost
transverse pseudo-Anosov
ow, where almost transverse means that it is transverse after blowing up
9
nitely many closed orbits. In [ ], Fenley and Mosher show that these ows are also quasigeodesic.
3.1. Endpoint maps. a quasigeodesic ow
F.
Let
M
be a closed hyperbolic 3-manifold with
Since each owline is a quasigeodesic, there are
3. QUASIGEODESIC FLOWS
13
well-de ned endpoint maps
2 e± : P → S∞ that send each owline to its positive/negative endpoint. These endpoint maps are continuous. In fact, this characterizes quasigeodesic ows:
Theorem 2.11. Let
F.
Then
F
M
be a closed hyperbolic 3-manifold with a ow
is quasigeodesic if and only if the maps
+ continuous and satisfy e (p)
6=
e− (p) for all
e±
are well de ned and
f. p∈M
9
3
The if direction is [ ], Theorem B and the only if direction is [ ], Lemma 4.3.
Note that this theorem does not presuppose that the orbit
space is planar. Much of the peculiar behavior of quasigeodesic ows is contained in the following two observation.
Lemma 2.12 (Calegari, [3], Lemma 4.4). Let 3-manifold with a quasigeodesic ow dense in
F.
M
be a closed hyperbolic
e+ (P )
Then
and
e− (P )
are both
2 . S∞
This is true simply because
π1 (M )
acts almost transitively on
2 , S∞
so
any nontrivial invariant subset is dense.
Lemma 2.13 (Calegari, [3], Lemma 4.5). Let 3-manifold with a quasigeodesic ow
F.
If
D⊂P
M
be a closed hyperbolic
is an embedded disc in the
owspace, then
e± (D) = e± (∂D). Proof. We'll sketch this. Picture to
F.
Let
S
That is, choose a section be the surface in
H3
D
embedded in
σ : D → H3
consisting of
S,
if
x ∈
as a transversal
of the quotient map
σ(D),
half- owlines emanating from the boundary of orientation on
H3
together with the positive
σ(D).
Choosing the correct
H3 is a point on the positive side of
positve half- owline emanating from the negative half- owline intersects
x
π : H3 → P .
S
then the
stays on the positive side of
σ(D).
S
and
Now suppose, contradicting our
14
2. FLOWS
hypothesis, that there is some point
e+ (∂D)
p∈D
encloses some non-empty region
there is some owline
negative half of
γ
γ
e+ (p) ∈ / e+ (∂D).
such that
U 3 e+ (p).
Since
whose negative endpoint is in
is on the positive side of
S,
e− (P )
U.
2 } ∆ = {(p, p)|p ∈ S∞
where
is dense,
But then the
a contradiction.
We can think of the endpoints of lifted owlines as a subspace of
∆,
Then
2 , S 2 )\ (S∞ ∞
is the diagonal. That is, there is a map
2 2 e+ × e− : P → (S∞ , S∞ )\∆ de ned by
x ∈ P 7→ (e+ (x), e− (x)). Recall that a map is called proper if preimages of compact sets are compact.
Lemma 2.14. Let
M.
manifold
F
be a quasigeodesic ow on a closed hyperbolic 3-
Then
2 2 e+ × e− : P → (S∞ , S∞ )\∆ is a proper map.
Proof. Let Then
B
Let
2 , S 2 )\∆ be compact, and set B = (e+ ×e− )−1 (A). A ⊂ (S∞ ∞
is closed since it's the preimage of a closed set.
X
be the set of all geodesics with endpoints in
compact set constant
C
K⊂
H3 that intersects every element of
A.
X.
Then we can nd a
Recall that there is a
such that each owline has Hausdor distance at most
C
from the
geodesic connecting its endpoints (Proposition 2.3). Therefore, each owline with endpoints in The image
π(K 0 )
A
intersects the closed
is compact, and
C -neighborhood K 0 = NC (K).
B ⊂ π(K 0 ),
so
B
is bounded. Hence
The following is a consequence of Lemma 2.13.
Lemma 2.15 (Calegari, [3], Lemma 4.8). Let manifold with a quasigeodesic ow of
e+ ,
for
F.
every connected component of
e− .
is
compact.
3.2. Decompositions.
B
M
be a closed hyperbolic 3-
For any point in
(e+ )−1 (p)
2 p ∈ S∞
in the image
is unbounded, and similarly
3. QUASIGEODESIC FLOWS
15
K ⊂ (e+ )−1 (p)
were bounded, then we
Roughly, if some component
could surround it by a simple closed curve
γ
disjoint from
(e+ )−1 (p),
which
contradicts the previous lemma.
Definition 2.16. Let 3-manifold
M.
F
be a quasigeodesic ow on a closed hyperbolic
The positive and negative decompositions of the owspace
P
are
D± = {components
of
2 (e± )−1 (p)|p ∈ S∞ }.
By the previous lemma, the elements of bounded subsets of the plane. The action of
D+
and
π1 (M )
on
D− P
are closed, un-
preserves
D+
and
D− . We will draw an analogy between the pair of decompositions that come from a quasigeodesic ow and the pair of transverse singular foliations that come from a pseudo-Anosov ow.
There's no notion of transversality for
general closed unbounded subsets of the plane, but the following will su ce:
Lemma 2.17. Let manifold
M.
If
K∈
Proof. Let
F
be a quasigeodesic ow on a closed hyperbolic 3-
D+ and
p = e+ (K)
(e+ × e− )−1 ((p, q)),
L ∈ D− and
then
K ∩L
q = e− (L).
is compact.
Then
K∩L
is contained in
which is compact by Lemma 2.14.
Our study of quasigeodesic ows centers around the following theorem.
Theorem 2.18 (Calegari, [3]). Let with a quasigeodesic ow on a topological circle
F.
Su1 ,
Then
M
π1 (M )
be a closed hyperbolic 3-manifold
acts faithfully by homeomorphisms
called the universal circle of
F.
We will prove this theorem in Chapter 5, where it follows easily from the technical machinery in Chapters 3 and 4.
Each (positive or negative)
decomposition element is unbounded, and we will think of the ends of a decomposition element as directions towards in nity.
The universal circle
is built by collating the ends of decomposition elements, taking a sort of completion, and collapsing some ends together.
CHAPTER 3
Circular Orders An ordered triple of distinct points
x, y, z
in the circle is said to be
positively ordered if it spans a positively oriented simplex in the disc, and negatively ordered otherwise. Four points
x, y, z, w ∈ S
span four unoriented
simplices, and the orientation on one pair of adjacent simplices determines the orientation on the other pair.
This structure, in abstract, is called a
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
• hx, y, zi •
if
τ
is nonzero exactly when
is a permutation of
x, y ,
and
z
are distinct,
x, y, z ∈ S , then hτ x, τ y, τ zi = −1sgn(τ ) hx, y, zi,
and
•
(cocycle condition) if
x, y, z, w ∈ S
are distinct then
hy, z, wi −
hx, z, wi + hx, y, wi − hx, y, zi = 0. A circularly ordered set is a set together with a circular order. An ordered triple of points
ordered if
x, y, z
hx, y, zi = +1
in a circularly ordered set is said to be positively
and negatively ordered if
hx, y, zi = −1.
In line
with tradition, we also refer to these, respectively, as being in counterclockwise and clockwise order.
An ordered
n-tuple
of points
x0 , x2 , x3 , ..., xn−1
in a circularly ordered set is said to be positively/negatively ordered if
hxi−1 , xi , xi+1 i = ±1
for all
i
mod
n.
Example 3.2. The oriented circle
S1
has a natural circular order deter-
mined by the choice of orientation. One way to think of an orientation of is a choice of generator points
x, y, z ∈ S 1
τ
for
H 1 (S 1 ; Z) ' Z.
determines a 1-cycle 17
∆xyz
A ordered triple of distinct
consisting of three 1-simplices,
18
3. CIRCULAR ORDERS
x
the ordered segments between of
hx, y, zi
as the value of
Remark 3.3. If a linear order
s, t ∈ S , p γ
if
This is well-de ned by the preceding lemma.
Remark 4.22. While the middle section
C − (γ+ ) ∩ C + (γ− )
in the pre-
ceeding lemma has no ends, it may be unbounded. For example, see Figure 2 where
K
is upper rectangle with the no-ended open set of Example 4.11 cut
out.
Proposition 4.23. Let
C
be a collection of disjoint, mutually nonsepa-
rating unbounded continua in the plane. Then on
C.
h, , i
de nes a circular order
2. CIRCULAR ORDERS
Proof. Let
hK, L, M i
be distinct.
We will start by showing that
does not depend on the choice of
K
arcs from
K, L, M ∈ C
to
M
35
γ.
Note that if
then we can nd another arc from
K
to
γ1
M
and
γ2
are
disjoint from
both of them. So we may assume without loss of generality that
γ1
γ2
and
are disjoint.
Relabel the arcs according to Lemma 4.20 and suppose that
L ⊂ C + (γ+ ).
By part (4) of the lemma,
C − (γ
− ), so
hK, L, M i
L
is contained in either
C + (γ+ )
or
is well-de ned by part (3) of the lemma.
Next we'll show that
hK, L, M i = (−1)sgn(τ ) hτ (K), τ (L), τ (M )i, for a permutation
τ
of
(K, L, M ).
It is immediate that
hK, L, M i = −hM, L, Ki,
so we just need to show
that
hK, L, M i = −hK, M, Li.
Indeed, assume that
that
L ⊂ C + (γ)
from
K
to
M
for some arc
that intersects
γ0
sub-arc of hence so is
L
Therefore,
K
M.
to
We can nd an arc
and lies on the positive side of
that runs from
M.
γ
hK, L, M i = +1,
K
to
L.
Then
hK, M, Li = −1
γ
γ.
Let
γ0
γ 00
i.e.
from
be the
is on the negative side of
γ 00 ,
as desired.
It remains to show that the circular order is compatible on quadruples, i.e.
to verify the cocycle condition.
hK, L, M i = +1 avoids from
L
L
and
to
N,
and
N.
hK, N, M i = −1.
Then
L ∈ C + (γ)
Choose an arc
while
Therefore we have
γ
γ
γ
K
Let
γ0
and
to
M
that
be an arc
only once and transversely.
is on the negative side of
hL, K, N i = −1
Suppose that
from
N ∈ C − (γ).
which we can choose to intersect
Then the initial segment of
similarly
K, L, M, N ∈ C .
Let
γ0
hK, L, N i = +1
hence so is
as desired, and
hL, M, N i = +1.
2.2. For ends.
K.
We can now de ne a circular order on the ends of a
generalized unbounded decomposition.
Definition 4.24. Let
C
be a collection of eventually disjoint unbounded
continua in the plane, and let bounded set
A ∈ P
κ1 , κ2 , κ3 , ... ∈ E(C)
is said to distinguish the
are disjoint and mutually nonseparating.
κi
if
be distinct ends.
A
κ1 (A), κ2 (A), κ3 (A), ...
36
4. DECOMPOSITIONS
We can always nd a distinguishing set for a nite collection of ends.
κ1 , κ2 , ..., κn
Indeed, if
are distinct, one can choose a bounded open disc
κ1 (D0 ), κ2 (D0 ), ..., κn (D0 )
so that the
κi (D0 )
that intersects all of the
Definition 4.25. Let
P,
continua in the plane
D
disc
C
are disjoint.
D0
D ⊃ D0
Then any disc
distinguishes.
be a collection of eventually disjoint unbounded
and let
κ, λ, µ ∈ E(C)
be distinct ends. Choose a
that distinguishes these ends and de ne
hκ, λ, µi = hκ(D), λ(D), µ(D)i. Proposition 4.26. Let
P.
continua in the plane
C
be a collection of eventually disjoint unbounded
Then
h·, ·, ·i
de nes a circular order on
Proof. We only need to check that on the choice of Let
E(D).
D
and
hκ(D), λ(D), µ(D)i does not depend
D.
D0
and
M = µ(D)
K0 ⊂ K,
Suppose that
L.
Then
arc
l
L
from
to
hK, L, M i = +1.
L0
M 0.
Let
γ
γ,
hence so is
L0 .
to
K0
to the initial point of
γ,
follows
M
exhibits that
L0
K 0 to
M 0 , and the fact that
is as well. Therefore,
Remark 4.27. Suppose that separating unbounded continua.
κ ∈ E(K), λ ∈ E(L),
and
l
γ,
γ.
w ∈ (x, y)
x, y ∈ A
and
and
L ∪ l.
Then
as desired.
and
Then
hK, L, M i = hκ, λ, µi
γ0
are disjoint, mutually nonfor any ends
µ ∈ E(M ).
points in a circularly ordered set or
be an
and then runs to
K , L,
2.3. Properties of the circular order. (y, x),
γ0
Let
is on the positive side of
hK 0 , L0 , M 0 i = +1 M
that avoids
So we can nd an
We can choose this so that the rst and last segments avoid
γ 0 is an arc from
are
K
be an arc from
whose interior lies on the positive side of
arc that runs from
D⊂
and
D0 , which
etc.
is on the positive side of
γ
K 0 , L0 ,
and de ne
M 0 similarly. Without loss of generality we may assume that implies that
κ, λ, µ ∈
be bounded open discs that distinguish the ends
K = κ(D), L = λ(D),
Let
E(C).
S
z ∈ (y, x).
z, w ∈ B
Two pairs
are linked if either
Two subsets
and
z ∈ (x, y)
A, B ⊂ S
that are linked. A subset
x, y
z, w
and
of
w ∈
are linked if there
A⊂S
separates the
2. CIRCULAR ORDERS
B, C ⊂ S
subsets
C ⊂ (a0 , a).
if there are points
37
a, a0 ∈ A
such that
B ⊂ (a, a0 )
and
Note that this is not the same as topological separation in
S
with the order topology. The separation properties of disjoint unbounded continua can be detected by their ends:
K, L ⊂ P
Proposition 4.28. Let plane. Then
E(K)
and
E(L)
be disjoint unbounded continua in the
do not link in the canonical circular order on
E({K, L}). Let
K, L, M ⊂ P L
separates
from
M
be disjoint unbounded continua in the plane. Then if and only if
canonical circular order on
E(K)
D
L
κ1 , κ2 ∈ E(K)
γ
κ1 (D)
from
λ2 (D) are on the same side of γ γ.
is connected and disjoint from
λ1 , λ2 ∈ E(L),
so
E(K)
from
E(L)
and
to
κ2 (D)
and
κi (D)
be a bounded open disc distinguishing the
choose an oriented arc and
E(L)
E(M )
that avoids
λ1 , λ2 ∈ E(L).
and
L.
λj (D)
Then
This applies for any
to
M
K
does not separate
disjoint from
K,
L
M.
and
κ1 , κ2 ∈ E(K)
and
Then we can choose an arc
X = L ∪ γ ∪ M.
induces an order-preserving bijection between
X
and
K
The inclusion
E(L) ∪ E(M )
Suppose
γ
from
L
(L ∪ M ) ,→ X
and
E(X).
Now
are disjoint unbounded continua, so by the rst statement their
ends do not link, hence
E(K)
does not separate
K
It remains to show that if
κ1 , κ2 ∈ E(K), λ ∈ E(L), (κ2 , κ1 ).
λ1 (D)
do not link.
from
and let
and
L,
since they are both contained in
One direction in the second statement follows from the rst. that
in the
E({K, L, M }).
Proof. For the rst statement, let Let
separates
K
Let
γ
and
be an arc from
sub-arc that contains
γ ∩ K.
separates
µ ∈ E(M )
L
to
M,
L
E(L)
from
M
such that
and let
γ0
from
E(M ).
then there are ends
λ ∈ (κ1 , κ2 )
and
µ ∈
be the minimal connected
Set
K 0 := K ∪ γ 0 and
0 K± := K 0 ∩ C ± (γ; L, M ).
38
4. DECOMPOSITIONS
Note that the
L
M.
from
separate
0 , K−
are both closed and connected and neither one separates
0 ∩ K0 K+ −
Also,
and
κ1
So we can choose
0 K−
κ1 , κ2 , κ3 ∈ E(K)
U ⊂P
distinguishes
κ4 .
κ1 (U )
κ2
to be any end of
0 ∪ K 0 ) → E(K). E(K+ −
and
be an unbounded continuum in the plane, and
That is,
from
from any end of
κ4 ∈ (κ3 , κ1 )
Proof. Let from
0 K+
be distinct ends, ordered counterclockwise.
κ2
K
in
κ1 , κ2 ,
Suppose and
κ3 .
(κ3 , κ1 ).
and suppose that
κ2 (U ) = κ4 (U ).
κ3 (U ),
to
36], Theorem
is a connected, bounded open set that distinguishes
U
Then
K⊂P
0 ∪ K0 K 0 = K+ −
must be unbounded by [
to be any end of
where we're using the bijection
Lemma 4.29. Let let
is connected, so in order for
0 L from M , both K+
4
II.5.23.
0 K±
U
does not distinguish
Then by Proposition 4.28,
contradicting the assumption that
U
κ2
κ2 (U ) separates
distinguishes.
Remark 4.30 (Singular foliations). A singular foliation of the plane is an unbounded decomposition, where each decomposition element consists of nitely many properly embedded rays that share an initial point. It's easier to work with the circular order on the ends of a singular foliation, since ends are represented by properly embedded rays. Let
K , L,
and
embedded circle
P.
with and
M
Let
with
be properly embedded rays in the plane
S ⊂P
k , l,
S.
M
and
Then
that intersects all three, and orient
m
P. S
Choose an
consistently
be, respectively, the last intersections of
k, l, m
has a circular order inherited from
determines the circular order of
S,
K , L,
and this
K, L, M .
3. Universal circles We now have a canonical circular order on the ends of a generalized unbounded decomposition, and we'd like to form a universal circle using Construction 3.12. First, we'll need to know a couple of things about the topology of the set of ends.
Proposition 4.31. Let
C
continua in the plane. Then 4
x
B,
and
If
and
and
y
are points of
A∩B
S2
be a collection of eventually disjoint unbounded
E(C)
is separable.
which are not separated by either of the closed sets
is connected, then
x
and
y
are not separated by
A ∪ B.
A
3. UNIVERSAL CIRCLES
39
Proof. Fix an exhaustion of the plane P by a nested sequence of bounded
Di .
open discs
Di−1 ⊂ Di
That is,
i
for each
and
S
i Di
= P.
i,
For each
consider
[
P i :=
KDi ,
K∈C where
KDi
is the union of the unbounded components of
a subspace of a separable space is separable. So
{pij }∞ j=1
and we can choose a countable set and
j
Kji ∈ C
let
κij (Di )
is the component of
Kji \ Di
Recall that
P i.
For each
and choose an end
containing
pij .
i,
is separable for each
that is dense in
pij ,
be the element containing
Pi
K \ Di .
κij
i
so that
We will show that
E 0 = {κij }i,j E(C).
is dense in
κ, µ ∈ E(C),
Let
We will show that distinguishes
κ, λ,
(κ, µ)
and assume that
κij ∈ (κ, µ) and
µ
for some
i, j .
contains at least one end,
Choose a bounded disc
D
λ.
that
and set
K = κ(D), M = µ(D). γ
Let
D
contains subset of
there is a
κij (Di )
K
be an arc from and
P i,
γ.
Set
to
M
and choose
C + := C + (γ; K, M ).
i
large enough so that
C+ ∩ P i
Now
and it is nonempty because it contains
pij ∈ C + .
The corresponding end
is contained in
κij
λ(Di ).
Di
is open as a
So for some
is contained in
(κ, µ),
C +.
j
since
If our unbounded decomposition were actually a singular foliation, then we could work with the circular orders more concretely using embedded circles as in Remark 4.30. rays.
Suppose
Choose an embedded circle
k, l, m ∈ S
crosses between
K
and
(k, l) ⊂ S
and
N.
and
M
are properly embedded
that intersects all three, and let
K, L, M
with
L
and
S.
If
hK, L, M i = +1,
is outermost, in the sense that
L on the positive side of (k, l).
outermost sub-arc between
M
S ⊂ P
be the last intersections of
then the oriented sub-arc
between
K , L,
M,
and
(m, n)
Similarly,
S
never
(l, m) is the
is the outermost sub-arc
40
4. DECOMPOSITIONS
In general, there's no natural way to de ne the last intersection between an unbounded continuum and a circle, but we can still nd outermost subarcs.
The rst of the following two lemmas can be taken as a de nition
of outermost sub-arcs, and the second shows that they behave as expected. We'll use them in the proof of Lemma 4.34.
K, M ⊂ P
Lemma 4.32. Let and let and
S⊂P
M.
be disjoint unbounded continua in the plane,
be a positively oriented embedded circle that intersects both
There is a unique sub-arc
orientation inherited from both ends in either
We'll call
γo
K
or
S)
γo ⊂ S
running from
such that every component of
to
M
(with the
S ∩ C + (γo )
has
M.
the outermost sub-arc from
Proof. The components of oriented open arcs. Let
K
K
Γ
K
S \ (K ∪ M )
to
M.
form a countable collection of
be the collection of closures of these arcs. We will
partition
Γ = ΓK ∪ ΓM ∪ ΓK,M ∪ ΓM,K where
ΓK = {sub-arcs
from
K
to itself},
ΓM = {sub-arcs
from
M
to itself},
ΓK,M = {sub-arcs
from
K
to
M },
ΓM,K = {sub-arcs
from
M
to
K}.
and
Let
Γs = ΓK,M ∪Γ− M,K , where the minus means to reverse the orientation
of each arc (the s stands for spanning). As in Remark 4.21, with a linear order. We will show that this maximal element lies in Suppose
γ1 < γ2 ...
γi
j
is endowed
has a maximal element, and that
rather than
Γ− M,K .
has no maximal element. Then we can nd a sequence
of arcs in
there exists of the
Γs
ΓK,M
Γs
Γs
Γs
that dominates
such that
and let
γ∞
γ < γj .
Γs ,
in the sense that for every
γ ∈ Γs
Take a Hausdor convergent subsequence
be the limit. Although
necessarily an element of
γ0
j (j
{Di }∞ i=1
integer such that gap
(κ, λ)
arc from
Skn
Dn
to
by nested bounded open discs
Dj ).
distinguishes
intersects both
with the open interval
κ(Dn )
P
is a dummy variable, i.e. for each
to keep a copy of every circle outside of the rst integer such that
of
If
κ
(κ, λ)
from
we want
is a gap then let
λ,
and let
κ(Dn ) and λ(Dn ).
Uκ,λ ⊂ Skn
j
k
n
be
be the rst
We will associate the
whose closure is the outermost
λ(Dn ).
The disjoint union
[
Sij
i>j is second countable, so any collection of disjoint open subsets is countable. Distinct gaps correspond to disjoint open intervals in
S
Sij
by Lemma 4.33,
hence there are only countably many gaps. Applying Lemma 3.8, we have:
Corollary 4.35. Let
C
continua in the plane. Then
be a collection of eventually disjoint unbounded
E(C)
is 2nd countable.
This lls a gap in the literature, as 2nd countability had not previously been shown.
Construction 4.36 (Universal circles). Let bounded decomposition of the plane
P,
able. Recall that this is immediate if decomposition by Lemma 4.3. universal circle for
E(D)
D
be a generalized un-
and suppose that
D
E(D)
is uncount-
is a (non-generalized) unbounded
The universal circle
Su1
of
D
is simply the
that we built in Construction 3.12.
Example 4.37. It is appealing to think that the ends of a decomposition element would map to a closed set in the universal circle, but this is not generally true. For example, the set in Figure 3 has no rightmost end.
4. The end compacti cation Let's x a generalized unbounded decomposition
D
of the plane
P,
and
1 let Su be its universal circle. We will show that there is a natural topology on the set
P := P ∪ Su1
boundary
Su1 .
that makes it into a closed disc with interior
P
and
4. THE END COMPACTIFICATION
43
Figure 3. An unbounded continuum with no rightmost end. Definition 4.38. Let
P.
the plane
D
A subordinate set is a tail of an element of
κ ∈ E(D)
subordinate set if there is a that
be a generalized unbounded decomposition of
D.
That is,
and a bounded open disc
K
is a
D⊂P
so
K = κ(D).
If
K = κ(D) K,
containing
K0
be the decomposition element
E(K)
is naturally identi ed with a
is a subordinate set, let
i.e.
K 0 = κ(∅).
Then
E(K 0 ).
subset of
The peripheral sets in the following de nition will serve as neighborhoods in
P
of points on the boundary circle
Su1 .
Here
φ : E(D) → Su1
is the
natural map that takes ends to their images in the universal circle.
Definition 4.39. Let the plane to
L.
from
Let
P. I
D
be a generalized unbounded decomposition of
Fix subordinate sets
K
and
L,
and let
1 be the maximal open interval in Su
φ(E(K))
to
φ(E(L)).
γ
be an arc from
\ φ(E(K) ∪ E(L))
The peripheral set determined by
K
that runs
K, L,
and
γ
is
de ned to be
O(K, L, γ) := I ∪ C + (γ; K, L). Remark 4.40. It is possible that
I
is empty. For example, let
K
O(K, L, γ)∩S = ∅, i.e.
that the interval
be the set in Example 4.37 and
L
its mirror
image.
Construction 4.41 (End Compacti cation). Let unbounded decomposition of the plane The end compacti cation of
P
P,
and let
with respect to
P := P ∪ Su1
D
Su1
D
be a generalized
be its universal circle.
is the set
44
4. DECOMPOSITIONS
with the topology generated by open sets in
P
and peripheral sets.
We'll start by showing how to construct a peripheral set to particular speci cations.
D
Lemma 4.42. Let universal circle
Su1 .
a peripheral set
O
a, b, c, d ∈ Su1
Let
such that
O0
as long as
(b, c) ⊂ (O ∩ Su1 ) ⊂ (a, d). A ⊂ P,
Su1 ,
positively ordered such that the
(c, d).
and
λ's.
Let
This way,
D
In addition,
O
can
and contained in any
O0 ∩ Su1 ⊃ [b, c].
Proof. Ends are dense in
image in
with
be positively ordered. Then there is
be chosen to be disjoint from any bounded set peripheral set
P
be an unbounded decomposition of the plane
so we can choose ends
κ's
have image in
(a, b)
κ1 , κ, κ2 , λ1 , λ, λ2 λ's
and the
have
be a bounded open disc that distinguishes the
K = κ(D)
respectively by Lemma 4.29.
5
and
L = λ(D)
Connect
K
have ends in
L
and
(a, b)
with an arc
γ
and
κ's
(c, d)
and set
O = O(K, L, γ). If we're provided with a bounded set shrinking where
in
O0
and a peripheral set
(a, d) so that it's contained in O0 ∩ Su1 .
K 0 = κ0 (DK 0 )
DL0 , A,
A
and
γ0,
L0 = λ0 (DL0 ).
and
and choose
and disjoint from
A
γ
Suppose
D
Choose
to lie inside of
O0 .
P
(2)
P
O0 = O(K 0 , L0 , γ 0 ),
so that it contains
Then
O
will be contained
P.
be the end compacti cation of the plane
P
DK 0 ,
respect to the generalized unbounded decomposition (1) the open sets in
start by
as desired.
Now we can get a handle on the topology on
Lemma 4.43. Let
O0 ,
D.
P
with
Then
and peripheral sets form a basis for
P,
is 1st countable,
(3) the inclusion maps
P ,→ P
and
Su1 ,→ P
are homeomorphisms onto
their images, and 5
If we hadn't chosen the helper ends
have some ends in
(a, b)
and
(c, d)
κ1 , κ 2
and
λ1 , λ 2 ,
K
and
L
would still
respectively, but they might also have ends outside of
these intervals. The helper ends ensure that we choose extra ends.
then
D
large enough to cut o these
4. THE END COMPACTIFICATION
P
(4)
45
is compact.
Proof.
(1) (2):
hood basis for
p
Let
p ∈ Su1 .
We'll start by constructing a neighbor-
that consists of peripheral sets.
Di , and a sequence
Fix an exhaustion of the plane by bounded open discs of open intervals
T
i (ai , bi )
= p.
(ai , bi ) ⊂ Su1
Oi ∩ Di = ∅ If
such that
for each
O0 = O(K 0 , L0 , γ 0 )
eventually contained in
(O0 ∩ Su1 )
if
O0 .
Indeed, simply choose
K0
and
P ,→ P
is open then for each
Su1 )
⊂ U.
Let
U
Ui = O(Ki , Li , γi ) n + 1).
and
for all
Lk
i.
are the discs
L0 . U
and
V
are peripheral
i, p ∈ Oi ⊂ (U ∩ V ).
p∈U
we can nd a peripheral set
P.
{U1 , U2 , ..., Un } ⊂ U
Since
Su1
such that
that covers
Ki , Li−1 , Ki+1 , Li
i, j, k ,
and that
γi
intersects
Su1 .
Ui
After reordering
is a peripheral set
γi+1
γi
i
do not intersect the
only once, transversely,
See Figure 4.
We can concatenate the sub-arcs between intersections of the a simple closed curve least one of the
Ui
nite sub-cover of by the
with
is positively ordered for all
Further, we can assume that the arcs
for any
O
is compact we can nd
and taking a re nement we can assume that each
Kj
DL0
are
(Oi ∩ Su1 ) ⊂
so that
and
Oi
is a homeomorphism onto its image. For
be an open cover of
a nite subcollection
(mod
then the
it is clear that the preimage of an open set is open. Conversely,
U ⊂ Su1
(4):
i
DK 0
where
then for large enough
It is clear that
p ∈ (O ∩
p,
is any peripheral set containing
Di ⊃ (DK 0 ∪ γ 0 ∪ DL0 ),
and
p ∈ U ∩ V ∩ Su1
Su1 ,→ P ,
and
i.
To prove (1) and (2) it su ces to note that if
(3):
i,
[ai+1 , bi+1 ] ⊂ (Oi ∩ Su1 ) ⊂ (ai , bi ), Oi+1 ⊂ Oi ,
determining the subordinate sets
and
for each
Using Lemma 4.42 we can nd a sequence of peripheral sets
Oi = O(Ki , Li , γi ) and
[ai+1 , bi+1 ] ⊂ (ai , bi )
such that
Ui .
γ.
Points on the negative side of
and the positive side of
U
γ
γ
γi
to form
are contained in at
is compact. So we can nd a
that covers this compact piece and the rest is covered
The end compacti cation behaves well when we take closures of decomposition elements.
46
4. DECOMPOSITIONS
P
...
Li Ki+1
γi+1 γi Li−1
γi−1 Ki ...
Figure 4. A re nement covering Lemma 4.44. Let
P
S.
be the end compacti cation of the plane
respect to the generalized unbounded decomposition
D.
If
K∈D
P
with
then
clP (K) = K ∪ clSu1 (φ(E(K)), where
φ
is the natural map from
Proof. It is clear that
clSu1 (φ(E(K))) intersect
K,
E(D)
to
Su1 .
clP (K) ∩ P = K ,
so let
p ∈ Su1 .
then we can nd a peripheral set containing
so
p
If
p
is not in
that does not
p∈ / clP (K).
We're ready for the punchline of the chapter.
Theorem 4.45. Let plane
P.
interior
D
be a generalized unbounded decomposition of the
The end compacti cation
P
and boundary
Su1 .
P
is homeomorphic to a closed disc with
Any homeomorphism of
extends to a homeomorphism of
P
that preserves
D
P.
Proof. The second statement follows from the fact that the image of a peripheral set under a
D-preserving
homeomorphism is again a peripheral
set. To prove that
P
of the disc. An arc
γ with endpoints a, b is said to span
and
is a closed disc we will use the following characterization a set
S⊂X
if
a, b ∈ S
˚ γ ⊂ X \ S. Theorem 4.46 (Zippin, [36], Theorem II.5.1). Let
compact, 1st countable Hausdor space. Suppose that
X
X
be a connected,
contains a 1-sphere
4. THE END COMPACTIFICATION
S
such that some arc in
X
S,
spans
S
homeomorphic to a closed 2-disc with boundary
S.
P
S
every arc that spans
and no closed proper subset of an arc spanning
It is clear that
47
separates
separates
X.
X,
X
Then
is
is connected and Hausdor and we have already shown
that it is 1st countable and compact. Let's check the remaining conditions.
Existence of a spanning arc: sequence of peripheral sets such that
i.
each
pi
to
Choose a point
pi+1
p i ∈ γi
with interior in
p ∈ Su1 .
Fix
∩i Oi = p, i
for each
Oi \ Oi+1 .
P
curve in
P
Oa = O(Ka , La , γa )
sets containing
Ob ∩ Su1 .
a
There is a
intersect either
b
and
Ka
t0
or
c0
and
t1
such that
such that
in
γ
with ends in
(y, z)
L
Suppose that
K
to
L
Oa ∪ Ob . (y, z):
separate
K
Note that ˚ γ is a properly embedded
K
and
L
that are separated by
γ.
be disjoint peripheral
(x, y) = Oa ∩ Su1
clP (Ka )
or
and
since otherwise
clP (La ).
γ 0 = γ([t0 , t1 ])
(z, w) = γ
would
γ
Similarly, there is
γ
γ
Ka , La , γa , Kb , Lb , γb (w, x)
did not separate
γ.
and
γ0.
that is disjoint from
K
But then
must intersect either
from
has the property that
So we can nd a subordinate set
from
L.
K
disjoint
Similarly, nd a
γ.
Then we could nd an arc
b ∈ O(K, L, µ)
while
µ
a∈ / O(K, L, µ),
K , L, or µ, a contradiction.
So
γ
must
L.
No subset of a spanning arc separates: and let
Su1
just choose a subordinate set whose ends lie
with ends in
disjoint from
implying that
be an arc spanning
in some proper in nite sequence, implying that
that is disjoint from
subordinate set
from
Construct two such rays
γ([t1 , 1]) ⊂ Ub .
γ 0 is contained in
from
for
P.
γ([0, t0 ]) ⊂ Oa ,
Therefore, the compact sub-arc
γ\
p.
Ob = O(Kb , Lb , γb )
accumulates on some point in either a
to
respectively, and let
La
Oi = O(Ki , Li , γi )
into two unbounded discs by the Jordan curve
theorem. We will nd subordinate sets Let
be nested
be an arc that connects
γ : [0, 1] → P
Let
a and terminal point b.
so it separates
ci
{Oi }∞ i=1
The concatenation of these arcs is a
and connect their endpoints with an arc in
with initial point
and
and let
proper ray whose closure is an arc from
Spanning arcs separate:
Let
γ 0 = γ \ {x}
for some point
x.
Let
γ
be a spanning arc
48
4. DECOMPOSITIONS
Suppose
x ∈ Su1 .
subordinate set that
Su1 \ γ 0
P \ γ0
K
Then
γ0
P
separates
into two discs, one containing a
and one containing a subordinate set
is connected, and
clP K
L
(as above). Note
clP L
both intersect
P \ γ0
is connected.
and
Su1 \ γ 0 .
Hence
is connected.
If on the other hand contained in
clP (P \ γ 0 ),
x ∈ P,
then
hence connected.
But
P \ γ0
is
CHAPTER 5
Quasigeodesic Flows Now that the technical machinery of Chapters 3 and 4 is out of the way, we can reprise our main topic. Throughout this chapter, closed hyperbolic 3-manifold with a quasigeodesic ow
F.
M
will be a
The owspace
P
is homeomorphic to the plane, and it comes with unbounded decompositions
D+
and
D− .
The union of these,
D := D+ ∪ D− , is a generalized unbounded decomposition. We'll write
E ± := E(D± ) and
E := E(D). The fundamental group induces an action on
P
π1 (M ) acts on H3
by deck transformations, and this
via the quotient map
π : H3 → P. The decompositions
D±
and
D
are preserved by this action
Calegari's universal circle is de ned to be the universal circle from
D
(Construction 4.36), which inherits an action of
π1 (M )
Su1
built
via the nat-
ural map
φ : E(D) → Su1 . This completes the proof of Theorem 2.18. The owspace
P
can be compacti ed to a disc
cle (Construction 4.41), and the action of
π1 (M )
P
on
using the universal cir-
P
extends to
P
(Theo-
rem 4.45. The restriction of this action to the boundary is the usual universal circle action. This completes one version of our rst main theorem. 49
50
5. QUASIGEODESIC FLOWS
Compactification Theorem. Let closed hyperbolic 3-manifold Then
P = P ∪ Su1
interior
P
M,
and let
F
be a quasigeodesic ow on a
Su1
be Calegari's universal circle.
has a natural topology making it into a closed disc with
and boundary
Su1 .
π1 (M )
The action of
on
P
P
extends to
and
∂P .
restricts to the universal circle action on
We will de ne another universal circle
Sv1
in Section 2 and show that this
theorem holds for it too.
Su1,+
Note that one could build universal circles
D+
and
D− .
and
Su1,−
using only
In fact, one could even use these universal circles to build
compacti cations
P
+
and
P
−
.
However, the compacti cation
+ and useful, as it behaves well with respect to D
P
is more
D− simultaneously.
1. Extending the endpoint maps In this section we'll show that the endpoint maps
2 e± : P → S∞
extend continuously to Given a point
P.
x ∈ Su1 , we might hope to nd a sequence of (say, positive)
decomposition elements that nest down to sequence of that
Ki
Ki ∈ D +
separates
that converge to
Ki−1
from
Ki+1
x
x.
That is, we'd like to nd a
in the Hausdor sense, arranged so
for each i. If such a sequence exists, then
+ it turns out that e (Ki ) converges to some point
e+
by setting
e+ (x) = p.
trapped between the
Ki ,
Any sequence of points and it follows that
2 , and we can extend p ∈ S∞
xi ∈ P
e+ (xi )
approaching
converges to
p
x
gets
(this is
Case 2 in the proof of Lemma 5.3) Therefore, we've found the unique way to extend
e+
continuously to
x.
In general, not all points in
Su1
will have such a nice structure.
One
way to prove continuous extension is to break the analysis into several cases, depending on how the decompositions
D±
look near each point in
Su1 .
The
proof that follows is less direct. However, it has the bene t of brevity, and we obtain a concise understanding of
D±
along the way.
1. EXTENDING THE ENDPOINT MAPS
1.1. Sequences in the owspace. H3
sequences of owlines look in
51
We'll start by understanding how
as we go to in nity in
There's no way to extend hyperbolic metric on
P.
H3
2 , S∞
to
since the
distance between points will be in nite. However, if we think about
2 S∞
unit ball model, then
is the unit sphere, and we can endow
H3 in the
3
2 H = H3 ∪S∞
with the Euclidean metric. This will be useful when we consider sets that are close to in nity.
Remark 5.1 (A note on notation). If we'll write
P
or
P,
A
then
for the closure of
B
is the closure of
to mean the closure of the set Recall that if
p,
A
p ∈ P,
then
C
in
B
3
H
is a subset of either
. Similarly, if
P.
in
A
B
H3
or
3
H
,
is a subset of either
In all other cases we'll use clX (C)
in the space
π −1 (p) ⊂ H3
X.
is the owline corresponding to
and the closure of a owline is just the owline plus its endpoints.
In
other words,
π −1 (p) = e− (p) ∪ π −1 (p) ∪ e+ (p). The following lemma says that sets of owlines that are close to in nity in
P
have uniformly small diameter in
Lemma 5.2. For each each
p ∈ P \ A,
>0
the diameter of
3
H
.
there is a compact set
π −1 (p)
is at most
Proof. Recall that there is a constant Hasudor distance at most
C
C
A⊂P
such that for
in the Euclidean metric.
such that every owline has
from the geodesic between its endpoints (in the
hyperbolic metric). Geodesics with endpoints close together in the Euclidean metric have small diameter in the Euclidean metric. So given constant
there is a
0 such that all owlines whose endpoints are 0 -close have Euclidean
diameter at most
d(p, q) ≥
.
0 , and let
Let
A=
2 × S2 ) \ ∆ B ⊂ (S∞ ∞
(e+
be the set of pairs
(p, q)
with
× e− )−1 (B), which is compact by Lemma 2.14.
The following lemma is one of the main technical ingredients in our proof of continuous extension. Recall that if a topological space
X
every neighborhood of
then
x
lim sup Ai
(Ai )∞ i=1
is a sequence of sets in
is the set of points
intersects in nitely many of the
x∈X
Ai
such that
17]).
(see [
52
5. QUASIGEODESIC FLOWS
(Ui )∞ i=1
Lemma 5.3. Let
Ai = Fr Ui .
frontiers
be a sequence of disjoint open sets in
P
with
Then
lim sup e+ (Ui ) ⊆ lim sup e+ (Ai ), and similarly for
Proof. Let
e− . (xi )∞ i=1
xi ∈ Ui
be a sequence of points with
where we might have already taken a subsequence of the show that if
e+ (xi )
converges, then
Fix such a sequence and set
It su ces to
lim e+ (xi ) ∈ lim sup e+ (Ai ).
p := lim e+ (xi )
Q := lim sup e+ (Ai ).
and
xi
Suppose that in nitely many of the
are contained in a
bounded set. Then after taking a subsequence we can assume that the converge to some point since
Ai
separates
also converges to
xi
x.
i,
p ∈ Q.
We need to show that
Case 1:
Ui .
for each
x,
and note that
from
xi−1
But then
for each
e+ (x) = p
i,
we can nd points
lim e+ (yi ) = p,
U1
U2
U3 · · ·
x1
x2
x3
A1
A2
(see Figure 1).
so
p∈Q
xi
Then
yi ∈ Ai
that
as desired.
···
···
···
x
A3
Figure 1. Case 1. So assume that the
Case 2:
xi
escape to in nity.
Suppose for the moment that the
Ai
escape to in nity, i.e. that
they are eventually disjoint from every bounded set. If nd disjoint open sets
U
and
−1 (x ) and By Lemma 5.2, π i
V
respectively (see Figure 2).
separates
π −1 (xi )
from
V
in
3
H
π −1 (A
that contain
p
p∈ /Q
then we can
Q
respectively.
and
i ) are eventually contained in
But this contradicts the fact that
π −1 (xi−1 )
for each
i.
So we must have
U
and
π −1 (Ai )
p ∈ Q.
1. EXTENDING THE ENDPOINT MAPS
···
53
π
Q
··· p
V
U x3 x2 x1
Figure 2. Case 2.
Case 3:
a compact set out of each this might change that
Q
Ai
Now suppose that the
Ui
do not escape to in nity. We will cut
to make the
Q = lim sup Ai ,
Ai
escape to in nity. In general
but by choosing carefully we can ensure
shrinks.
Choose an exhaustion intersects all of the
Ui ,
(Dj )∞ j=1
of
P
by compact discs, each of which
where we might have to pass to a subsequence. The
argument in Case 1 shows that that for a bounded sequence of points
e+ (yi ) is eventually close to Q. contained in the bounded set
Note that for xed
Dj .
So for xed
j,
D j ∩ Ui
the sets
j , e+ (Dj ∩ Ui )
is eventually
It follows that we can nd a diagonal sequence: For each
(b)
i(j)
su ciently large so that (a)
lim supj→∞ e+ (Dj ∩ Ui(j) ) ⊂ Q.
component of
Ai(j)
The
p∈Q
Bj
or
xi(j)
is outside of
For each
j,
let
Dj
j
choose an
and such that
yj := xi(j) ,
let
Vj
P \ (Dj ∪ Ui(j) ) that contains yj , and let Bj := Fr Vj .
Q0 := lim supj→∞ Bj . either
are all
Q.
close to
integer
yi ∈ Ui ,
Observe that since each point in
(Dj ∩ Ui(j) ),
property (b) ensures that
escape to in nity, so by case 2
Bj
be the
Also, set
is contained in
Q0 ⊂ Q.
lim e+ (yj ) ⊂ Q0
See Figure 3. and therefore
as desired.
Corollary 5.4. Let
2 be disjoint compact sets. B, C ⊂ S∞
intersects at most nitely many components of for
e− .
Then
P \ (e+ )−1 (C),
(e+ )−1 (B)
and similarly
54
5. QUASIGEODESIC FLOWS
···
··· xi(j)
. . .
Bj Dj ∩ Ui(j) Dj
. . .
Fr
Figure 3. Case 3: Fixing the
Ai .
Proof. Otherwise we would have a sequence of points each of which is contained in a dierent component of the preceding lemma,
lim sup e+ (bi ) ∈ C .
contradicts the assumption that
B
and
C
K∈D
Lemma 5.5. For each point decomposition elements
P \ (e+ )−1 (C).
closure intersects
as
K∈D
x ∈ Su1 ,
so this
Recall (Lemma 4.44) that
there are at most countably many
whose closures intersect
x
either has an end at
or has ends approaching
i → ∞).
By
are disjoint.
x.
Proof. By Lemma 4.44, each decomposition element
x
(e+ )−1 (B),
K = K ∪ φ(E(K)).
then
φ(κ) = x)
in
lim sup e+ (bi ) ∈ B ,
But
1.2. Unions of decomposition elements. if
bi
x
x
(i.e.
there is a
(i.e. there are
K ∈ D
whose
κ ∈ E(K)
κi ∈ E(K)
with
with
φ(κi ) →
In the construction of the universal circle, at most countably
many ends are collapsed into one point, so there are at most countably many decomposition elements with an end at
x.
Therefore, it su ces to
show that there are at most countably many decomposition elements with ends approaching
x.
In fact there are at most four.
We will say that a positive decomposition element approaching
φ(κi ) → x
x
as
in the positive direction if there are
i → ∞,
and the
κ1 , κ2 , κ3 , ...
K ∈ D+
κi ∈ E(K)
has ends such that
are ordered counterclockwise.
There can be at most one decomposition element with ends approaching
x
in the positive direction, since if there were two then their ends would link,
1. EXTENDING THE ENDPOINT MAPS
55
implying that they intersect (Proposition 4.28). Similarly, there is at most one negative decomposition element with ends approaching
x in the positive
direction, and at most one positive and one negative decomposition element
x
with ends approaching
Lemma 5.6. Let
x
and
y.
A be a closed subset of the plane that separates the points A
Then some component of
Proof. Let
U
in the negative direction.
separates
be the component of
be the component of
P \U
x
P \A
that contains
y.
and
y.
that contains
Then Fr
V
x,
and let
separates
x
and
It is connected because the plane satis es the Brouwer property (see [
M
De nition I.4.1): if a component of
A
P \M
containing Fr
V
Recall that if
lim inf Ai
is a closed, connected subset of the plane then Fr
x
separates
(Ai )∞ i=1
x
write
Ai
36],
V
is
is connected. Therefore, the component of
and
y.
such that every neighborhood of
Ai .
all but nitely many of the then the
and
y.
is a sequence of sets in a topological space, then
is the set of points
lim sup Ai ,
V
P
V
If
lim inf Ai
x
intersects
is nontrivial and agrees with
are said to converge in the Hausdor sense, and we
lim Ai = lim sup Ai = lim inf Ai
17]).
(see [
If the
Ai
live in the plane,
and in nitely many intersect some compact set, then we can always nd a Hausdor convergent subsequence.
Lemma 5.7. Let
A ⊂ P
be a closed, connected set that is a union of
positive decomposition elements and let Fr
U
U
be a component of
P \ A.
Then
is contained in a single decomposition element.
Proof. Note that Fr su ces to show that Suppose that
is connected by the Brouwer property, so it
e+ (Fr U )
e+ (Fr U )
choose three points
K, L, M ⊂ A
U
is a point.
is not a point.
k, l, m ∈
Fr
U
Then it is in nite, so we can
whose images under
e+
are distinct. Let
be the positive decomposition elements that contain
k, l, m
respectively. These are mutually nonseparating, so by Proposition 4.28 their ends are mutually nonseparating. Assume that
K, L, M
element contained in
U
are positively ordered, and if
then
M, N, M
N
is a decomposition
is positively ordered. Let
ni ∈ P
be a
56
5. QUASIGEODESIC FLOWS
sequence of points in
U
that limit to some point in
be the decomposition element that contains
Ni
of the
ni .
Observe that
M
the
that avoids
Ni
N∞ ∪L separates K L
then the
ni
E(M )
have ends between
from
M.
i
and
B
E(K), of
N∞ ⊂ P .
Indeed, if
so the
N∞ ∪ L
let
Ni
Note that in nitely many
γ
K
to
M)
Ni
after
See Figure 4. is any arc from
are eventually on the positive side of
By Lemma 5.6, some component
γ,
separates
K
from
Therefore,
M.
K
and
eventually intersect
E(B) separates E(K) from E(M ), a contradiction.
then
and for each
intersect a compact set (for example, any arc from
taking a subsequence they converge to some set
to
L,
γ.
But
e+ (Fr U )
is a single point as desired.
L A K
M
Fr
U
N∞
Figure 4
Combining the last two lemmas, and using the fact that if rates
x, y ∈ P
then so does Fr
Corollary 5.8. Let elements. If
K⊂A
X
A⊂P
x
and
sepa-
1
:
be a closed union of positive decomposition
A separates the points x, y ∈ P
separates
X⊂P
then some decomposition element
y.
1.3. Extending the endpoint maps.
We can now prove the main
theorem of this section. 1
Let
both
x
U
and
be the component of
P \X
containing
y
U
P \ U,
must intersect both
and
x.
Any connected set that intersects
hence it must intersect Fr
U ⊂ Fr X .
1. EXTENDING THE ENDPOINT MAPS
Continuous Extension Theorem. Let
M.
a closed hyperbolic 3-manifold
x ∈ Su1 ,
Proof. Fix converges to as
i→∞
x.
and let
be a quasigeodesic ow on
The endpoint maps
P,
unique continuous extensions to
F
e+
and
(xi )∞ i=1
57
agrees with
2 e ± : P → S∞
e−
on the boundary.
2 . p ∈ S∞
P
be a sequence of points in
After taking a subsequence we can assume that
for some
admit
that
e+ (xi ) → p
e+ (x) = p.
Set
To see that this is well-de ned, suppose we have two such sequences
(xi )∞ i=1
and
(yi )∞ i=1
p 6= q .
Let
2 A0 ⊂ S ∞
A = (e+ )−1 (A0 ) Lemma 5.3, the
yi .
as are the separates in
P.
converging to
with
e+ (xi ) → p
and
be a simple closed curve separating
{xi }∞ i=I
separates
xi
x,
from
{yi }∞ i=I
e+ (xi ) → q p
are eventually contained in a single component of
from
{yi }∞ i=I .
Therefore,
K
separates
{xi }∞ i=I
Then
I.
for large enough
So by Corollary 5.8, some decomposition element
{xi }∞ i=I
q.
and
for
By
P \ A,
K ∈ D+
from
{yi }∞ i=I
x ∈ K.
This implies that
There are uncountably many disjoint simple closed curves separating and
q,
p
so the argument above would produce uncountably many distinct
decomposition elements
K
+ to so the extension of e
with
P
x ∈ K.
This would contradict Lemma 5.5,
is well-de ned. The same applies for
e− ,
and
continuity and uniqueness are immediate. To see that points and
xi ∈ P
e− (xi )
e+
and
e−
agree on
converging to
approaches
0
as
x.
Su1 ,
let
x ∈ Su1
and choose a sequence of
By Lemma 5.2, the distance between
i → ∞,
so
lim e+ (xi ) = lim e− (yi ).
e+ (xi )
From now on,
e± : P → Su1 will denote the extended endpoint maps, and
2 e : Su1 → S∞ their restriction to the universal circle. We'll restate the continuous extension theorem in a way that matches the Cannon-Thurston theorem. Let hyperbolic 3-manifold embedded
M.
F
be a quasigeodesic ow on a closed
A complete transversal to the lifted ow is an
2-manifold N ⊂ H3
that is transverse to the foliation by owlines.
58
5. QUASIGEODESIC FLOWS
P
Any complete transversal can be identi ed with the orbit space quotient map
π.
of
π : H3 → P ,
so we can think of
Therefore, we can compactify
universal circle by setting
Su1 ,
i=e
N
and the embedding
on
Su1 .
N
as a section
N
to a closed disc
i : N ,→ H3
via the
ψ : P → H3
by adding the
extends to the boundary
This is continuous since the diameter of owlines
goes to zero as we go to the boundary (Lemma 5.2). We have:
Theorem 5.9 (Generalized Cannon-Thurston Theorem). Let
M,
quasigeodesic ow on a closed hyperbolic 3-manifold
F
be a
N ⊂ H3
and let
be a complete transversal to the lifted ow on the universal cover. There is a natural compacti cation of a
π1
sion
N
action, and the embedding
i:N∪
Su1
→
H3
∪
as a closed disc
i : N ,→ H3
N = N ∪ Su1
that inherits
has a unique continuous exten-
2 . The restriction of S∞
i
Su1
to
is a
π1 -equivariant
2 . S∞
space- lling curve in
2. A more convenient circle Calegari's universal circle may contain more information than we need. For clarity, we'll momentarily add subscripts to some of our spaces. Let be the compacti cation of
P
built from
Su1 .
Ends of decomposition elements
φu : E → Su1 .
correspond to points in the universal circle via the map endpoint maps
2 eu : Su1 → S∞ If
2 e± : P → S∞
extend to
and these restrict to
it is possible that some components of
1 intervals. Let Sv be the quotient of to a point, for every
to an action on
φv : E → Sv1 .
2 e± u : P u → S∞ ,
The
on the boundary.
2 , p ∈ S∞
e−1 u (p)
Pu
Sv1 .
2 . p ∈ S∞
are
Su1 after collapsing each component of The action of
π1 (M )
Composing the quotient with
The map
1 e−1 u (p) ⊂ Su
φu ,
on
Su1
descends
we obtain a map
2 . eu factors through the quotient to form ev : Sv1 → S∞
We can use Construction 4.41 and Theorem 4.45 to produce a compacti cation
P v = P ∪ Sv1
by simply replacing
Su1
with
Sv1
proofs. We can extend the endpoint maps by setting with
ev
φu
with
φv
2 e± v : P → S∞
in the
to agree
on the boundary.
Note that
Pv
and
e−1 v (p)
is now totally disconnected for any
is the smallest compacti cation of
P
2 . p ∈ S∞
Therefore,
for which the endpoint maps extend
3. PROPERTIES OF THE DECOMPOSITIONS
59
to the boundary. This is exactly the property that we want from the universal circle.
In addition, the fact that point preimages are totally disconnected
will be useful when we study the dynamics of
π1 (M )
Sv1 .
on
In the sequel, we will drop the subscripts and let context determine which
P , e± , e,
version of
In many cases,
φ
and
we're referring to.
Su1 = Sv1 .
For example, if
F
is both quasigeodesic and
e± determine a dense lampseudo-Anosov then the endpoints of leaves of λ ination of
Su1 ,
and it follows that point preimages are already totally dis-
connected. In fact, as far as we know, it is possible that
Su1 = Sv1
for every
quasigeodesic ow.
2.1. Upper-semicontinuous decompositions. Sv1 compacti es
P
without referring back to the construction of the end-
compacti cation. Let of
X
X
semicontinuous if for any open set elements contained in
24]
of the sphere
U
The decomposition is called upper
U ⊂ X,
the union of decomposition
is open.
proved that if
S2
D be a decomposition
be a topological space, and let
into closed, connected subsets.
Moore [
We can also see that
D
is an upper semicontinuous decomposition
into non-separating sets, then the quotient obtained by
collapsing each decomposition element to a point is still homeomorphic to
S2. Double the disc
Pu
to obtain a sphere
whose elements are the components of in each copy of still
S2,
P.
S2.
Let
D
be the decomposition
2 e−1 u (p) for each p ∈ S∞ , and the points
This is clearly upper semicontinuous, so the quotient is
consisting of two copies of
Pv
glued along
Sv1 .
3. Properties of the decompositions We'll collect some useful observations concerning the regularity of and
D.
The statements in this section hold for both
Lemma 5.10. Let
Proof. Let at
p.
Set
2 . p ∈ S∞ 3
Ai ⊂ H
Then
Su1
and
D±
Sv1 .
(e+ )−1 (p)∪(e− )−1 (p) ⊂ P
is connected.
be a nested sequence of closed horoballs centered
Bi = π(Ai ∩ H3 )
for each
i.
The
Bi 's
are connected subsets of
P,
60
5. QUASIGEODESIC FLOWS
Bi
so their closures
are connected, compact subset of the compact space
P.
Therefore,
B=
\
Bi
i is connected. Suppose
x ∈ B.
Then there is a sequence of points
x (after taking subsequences). the
Ai ,
If
Bi ,
The corresponding owlines
so after taking a subsequence either
Therefore, either
x∈ /B
e+ (x) = p
or
xi ∈ Bi
e− (x) = p
e+ (xi )
or
approaching
π −1 (xi ) intersect
e− (xi )
approaches
p.
as desired.
−1 (x) is eventually disjoint from the horoballs then the owline π
so neither
e+ (x)
nor
e− (x)
Corollary 5.11. Let endpoint. Then
e−1 (p)
p.
is
2 , p ∈ S∞
and suppose that no owline has
p
as an
is a point.
Proof. Saying that no owline has
p as an endpoint is the same as say-
ing that
(e+ )−1 (p)∪(e− )−1 (p) is contained in Su1 , i.e. (e+ )−1 (p)∪(e− )−1 (p) =
e−1 (p).
By the preceding lemma,
e−1 (p) is connected, so it is either a point or
an interval. Images of ends are dense in closure of some
K∈D
Su1 , so if e−1 (p) were an interval, the
would intersect it. But then
which was supposed to be empty, would contain
(e+ )−1 (p) ∪ (e− )−1 (p),
K.
4. Dynamics on the universal circle We can now analyze the dynamics of the fundamental group action on the universal circle. Suppose
f : S1 → S1
is a homeomorphism. Let
xed point, and choose an interval called attracting if Each
f (I) ⊂ I ,
g ∈ π1 (M )
repelling pair.
Let
ag
repelling if
acts on and
x ∈ I ⊂ S1
rg
2 S∞
x ∈ S1
that isolates
f (I) ⊃ I ,
be an isolated
x.
Then
x
is
and indierent otherwise.
with two xed points in an attracting-
be the attracting and repelling xed points,
respectively. Set
1 Ag = e−1 v (ag ) ⊂ Sv , and
1 Rg = e−1 v (rg ) ⊂ Sv .
4. DYNAMICS ON THE UNIVERSAL CIRCLE
Note that
Ag
and
Lemma 5.12. If
Rg
61
are disjoint, closed, and invariant under
g ∈ π1 (M )
Sv1 ,
has a xed point on
g.
then it has at least
two.
p ∈ Sv1
Proof. Suppose
p.
take any point to
g.
is the only xed point of
Therefore,
Ag
and
Rg
must both contain
p
since they
are closed and invariant. This is impossible since they are disjoint.
Let
Fg
2 e(x) ∈ S∞
be the set of xed points of is xed by
g,
g
on
Sv1 .
If
x ∈ Sv1
g
The iterates of
is xed by
g,
then
so
Fg ⊂ Ag ∪ Rg . We'll partition
Fg
into
F Ag
and
F Rg ,
where
F Ag = Fg ∩ Ag , and
F Rg = Fg ∩ Rg . x, y ∈ F Rg .
Suppose we have distinct nected, so there is some point takes
e(z)
point
z 0 ∈ (x, y) with e(z 0 ) = ag .
A
and
g
point in
towards
by
g −1 .
F Ag ,
each
x.
i
let
F Ag
yi ∈ (xi , xi+1 )
Fg
g n (z)
is nite.
be in
F Rg ,
x ∈ F Rg ,
F Ag
z ∈ Sv1
limi→−∞ g n (z) ∈ F Rg .
gn
approaches some
F Rg
R
by
there is a
there is a point in
F Rg .
Otherwise we could nd a sequence of
x,
with
and observe that the
a contradiction. Similarly,
that is not a xed point,
limi→−∞ g n (e(z)) = rg .
F Rg
Iterating by
yi
Therefore,
Hence the points in
are repelling. In summary:
x ∈ F Ag .
F Rg
is nite.
F Ag
and
F Rg .
limi→∞ g n (e(z)) =
limi→∞ g n (z) ∈ F Ag ,
F Ag
For
also converge to
is nite, and xed points alternate between
Note that for any
points in
Rg .
A similar statement applies replacing
and between any two point in
This implies that
and
so after taking a subsequence,
that converges monotonically to some point
Therefore,
ag ,
that is not in
is totally discon-
In other words, between any two points in
It follows that
xi ∈ F Ag
rg ,
z ∈ (x, y)
Rg
Note that
and
are attracting, and the
62
5. QUASIGEODESIC FLOWS
Pseudo-Anosov Dynamics Theorem. Let on a closed hyperbolic 3-manifold
M.
Then each
F
be a quasigeodesic ow
g ∈ π1 (M ) acts on Sv1
with
an even number of xed points (possibly zero) that are alternately attracting and repelling.
5. Closed orbits We now turn to the question of whether quasigeodesic ows have closed orbits. The general question of nding closed orbits in
3-manifold dates back
to 1950, when Seifert asked whether every nonsingular ow on the has a closed orbit [
29].
3-sphere
Schweizer provided a counterexample in 1974, and
showed that every homotopy class of nonsingular ows on a
3-manifold
has
1 a C representative without closed orbits. Schweitzer's examples have since been generalized to smooth [
22] and volume-preserving ows [21].
On the other hand, Taubes' 2007 proof of the 3-dimensional Weinstein conjecture shows that Reeb ows on closed 3-manifolds do have closed orbits [
31].
Reeb ows are geodesible, i.e. there is a Riemannian metric in which the
owlines are geodesics. Complementary to this result, though by dierent methods, Rechtman showed in 2010 that the only geodesible real analytic ows on closed 3-manifolds that contain no closed orbits are on torus bundles over the circle with reducible monodromy [
27].
Geodesibility is a local condition, and furthermore one that is not stable under perturbations. By contrast, quasigeodesity is a macroscopic condition, and it is a stable condition under
C0
perturbations when the ambient 3-
manifold is hyperbolic. Calegari conjectured in 2006 that quasigeodesic ows on closed hyperbolic 3-manifolds should all have closed orbits. In this section we will provide evidence for this conjecture.
5.1. Möbius-like groups.
A Möbius transformation is a conformal
automorphism of the plane that preserves the unit circle.
The group of
orientation-preserving Möbius transformations is isomorphic to
P SL(2, R).
Each Möbius transformation is determined by its action on the unit circle,
5. CLOSED ORBITS
so we can think of
63
P SL(2, R) as a subgroup of Homeo+ (S 1 ), the orientation-
preserving homeomorphisms of the circle. An orientation-preserving Möbius transformations is classi ed as either elliptic, parabolic, or hyperbolic by whether it has no xed points, one indierent xed point, or two xed points in an attracting-repelling pair. A group
Γ
