Table Of ContentPROMOTING ESSENTIAL LAMINATIONS
7
0 DANNYCALEGARI
0
2
ABSTRACT. We show thata co–orientable taut foliation of aclosed, orientable,
n
algebraicallyatoroidal3–manifoldiseithertheweakstablefoliationofanAnosov
a
flow, or else there are apair of veryfull genuinelaminations transversetothe
J
foliation.
0
1
]
T 1. INTRODUCTION
G
A topological manifold is a very flabby object. It has no local internal struc-
h. ture, and except in very special cases, the group of automorphisms is transitive
t on the set of subsets of a fixed finite cardinality. The same manifold can appear
a
inamyriadof differentforms, andthequestion of recognizingordistinguishing
m
manifolds, or of certifyingausefulproperty,isinthe bestcaseveryhard,andin
[
theworst(typical)casealgorithmicallyunsolvable.
3 It is therefore desirable to stiffen or rigidify the structure of a manifold, by in-
v troducing geometry in some form or other, in order to reduce this ambiguity of
8
formtoamanageableamount. Butexactlywhatsortofgeometricconstraintsare
4
neithertoomuch(sothattherearenoexamples)ortoolittle(sothatthegeomet-
1
0 ricstructuredoesnothelpwiththeproblemofunderstandingorrecognizingthe
1 underlyingmanifold)isverydimensiondependent. Asageneralrule,smallerdi-
2 mensionalobjectsareeasiertounderstand. Importantprinciplesbecomeeasierto
0
applyandyieldmoreandricherstructure.Inthispaper,thenotionofmonotonicity
/
h is very important, especially as it relates to natural order or partial-order struc-
t turesoncertainsets. Moregenerally,suchorderstructuresprovideabridgefrom
a
m geometricproblemstoalgebraiclanguage,andpermitonetoperformexperiments
andconstructcertificateswiththeuseofcomputers.
:
v As an organizational tool, monotonicity loses effectivenessas dimension goes
i
X up;consequentlyitismostpowerfulwhenusedinthecontextofcertaindynami-
calsystems, whichcaneffectivelyreducethestudyofamanifoldtotwocomple-
r
a mentaryproblemsofstrictlysmallerdimension: thestudyoftheorbitsofthesystem,
andthestudyoftheparameterspaceorleafspaceoftheorbits. Geometricoranalytic
qualitiesofthedynamicalsystemarereflectedinthepropertiesofthedimension-
allyreducedsystems.
1.1. Foliationsandarborealgrouptheory. Inthecaseofthestudyof3–manifolds,
averyeffectivetoolfordimensionalreductionisthestructureofa2–dimensional
foliation, especially a taut foliation F of M which, at least when M is atoroidal,
maybedefinedasa2–dimensionalfoliationwithoutsphericalortorusleaves. A
basicstructuretheoremofNovikov impliesthattheleafspaceLof the universal
Date:4/11/2006.Version0.30.
Keywordsandphrases. tautfoliation,essentiallamination,genuinelamination,Anosovflow,word
hyperbolic.
1
2 DANNYCALEGARI
coverF ofsuchafoliationisatypicallynon–Hausdorffsimply–connected1–manifold,
f
onwhichπ (M)actsnaturally.
1
Such1–manifoldsarenot unlike R–treesinsome ways, and manyofthe tools
ofarborealgrouptheory(e.g.[49],[1])canbeusedtostudytheactionofπ (M). If
1
F isco–orientable,theleafspaceLisanoriented1–manifold,andthisorientation
defines a partial order on the elements of L. This global partial order structure
adds extra nuances to the arboreal theory, and is the source of many important
constructions. Forexample,inaremarkabletourdeforce,Roberts,Shareshianand
Stein([47])recentlymanagedtogiveexamplesofaninfinitefamilyofhyperbolic
3–manifoldswhichdonotadmittautfoliations,simplybystudyingtheactionof
theirfundamentalgroupson(non–Hausdorff)simply–connected1–manifolds.
1.2. Theclassificationofsurfacehomeomorphisms. IfLandtheactionofπ (M)
1
are understood, it remains to understand the leaves of F themselves, and the
way they fill out the manifold M; the relevant subject is the theory of surface
homeomorphisms.
Inthissubsectionwediscussthesimplestcaseofthetheoryofsurfacehomeo-
morphisms. WederiveThurston’sfamoustheoremontheclassificationofsurface
homeomorphismsbyaroutewhichisnearlytheoppositeofthehistoricaland,for
thatmatter,thelogicaldirection. Thereasonismainlypedagogical: thisorderof
expositionmoreclearlyrevealstheorderofdevelopmentofsomeanalogousideas
for more general taut foliations. We will necessarily cover a lot of material very
briefly. Mostofthedetailscanbefoundinthepapers[55],[60]and[16].
Firstwerecallthestatementofthetheorem,initsmostbasicform.
Theorem1.2.1(Thurston, [55] Classificationof surfacehomeomorphisms). LetΣ
beaclosed,orientablesurfaceofgenusatleast2,andletφ:Σ→Σbeahomeomorphism.
Thenoneofthefollowingthreealternativesholds:
(1) φisperiodic;thatis,somefinitepowerofφisisotopictotheidentity.
(2) φ is reducible; that is, thereis some finite collection ofdisjoint essential simple
closedcurvesinΣwhicharepermuteduptoisotopybyφ.
(3) φispseudo–Anosov.
For the moment, we postpone the definition of a pseudo–Anosov diffeomor-
phismofasurface,sincethiswillbethepunchlineofourrevisioniststory.
Giventhepair(Σ,φ)oneformsthemappingtorusM whichisthequotientof
φ
theproductΣ×I bythe equivalencerelation(s,1) ∼ (φ(s),0). M isafibration
φ
overS1,withfiberΣandmonodromyφ. Byanalogywiththenotationforashort
exactsequence,wedenotethis
Σ→M →S1
φ
andthereisacorrespondingshortexactsequenceofgroups
π (Σ)→π (M )→Z
1 1 φ
which represents π (M ) as an HNN extension. The automorphism φ of Σ in-
1 φ
ducesanautomorphismφ∗ ofπ1(Σ),well–defineduptoinnerautomorphisms. A
presentationforπ (M )isthengivenby
1 φ
π1(Mφ)=hπ1(Σ),t|t−1αt=φ∗(α)foreachαinπ1(Σ)i
Thehomeomorphismtypeofthis3–manifoldonlydependsontheisotopyclassof
φ. Thentheclassificationofφneatlyreflectsthegeometryofthemappingtorus.
PROMOTINGESSENTIALLAMINATIONS 3
RecallthatforM aclosed,topological3–manifold,andX asimply–connected
locally symmetric Riemannian 3–manifold, an X geometry on M is a homeomor-
phism
ϕ:M →X/Γ
whereΓisafree,discrete,cocompact,properlydiscontinuoussubgroupofIsom(X).
See[56]formoredetails.
Theorem1.2.2(Thurston,[60]Geometrizationofsurfacebundles). LetΣbeasur-
faceofgenusatleast2,andletφ:Σ→Σbeahomeomorphism. Thenthemappingtorus
M satisfiesthefollowing:
φ
(1) Ifφisperiodic,M admitsanH2×Rgeometry.
φ
(2) Ifφisreducible,M hasanon–trivialJSJdecomposition.
φ
(3) Ifφispseudo–Anosov,M admitsanH3geometry.
φ
From now on we consider the case where φ is pseudo–Anosov, and therefore
M ishyperbolic,andwecanidentifyitsuniversalcoverwithhyperbolic3–space
φ
M =H3
φ
g
Theactionofπ (M )onM extendscontinuouslytoanactionontheidealboundary
1 φ φ
of H3, which is a topologgical sphere which we denote by S2 , and the action of
∞
π (M )onthissphereisbyMo¨biustransformations. Wedenotetherepresentation
1 φ
inducingthisactionby
ρ :π (M )→Homeo(S2 )
geo 1 φ ∞
ThereisanotherviewofM whichcomesfromthefoliatedstructureofM . To
φ φ
g
describe this point of view, we make use of some ideas of coarse geometry from
Gromovasdevelopedin[30].
The foliation of Σ ×I descends to a (taut) foliation of M by surfaces which
φ
arethefiberofthefibrationoverS1. ThisgivesM thestructureofanopensolid
φ
g
cylinder
M =Σ×R
φ
g e
The universal cover of each fiber Σ is quasi–isometric with its pulled back in-
θ
trinsicmetrictothehyperbolicplaneH2,andcanthereforebecompactifiedbyits
idealboundary,whichisatopologicalcircleS1 .
∞
This circle S1 can just as well be thought of as the Gromov boundary of the
∞
groupπ (Σ). The group π (M ) actson π (Σ) in the obvious way: the subgroup
1 1 φ 1
π (Σ) acts on the left by multiplication, and the element t acts by the automor-
1
phism φ∗. This action on π1(Σ) induces an action of π1(Mφ) on S∞1 (π1(Σ)), and
togetherwiththeactiononRgivenbythehomomorphismtoZ,thisgivesa(prod-
uctrespecting)actionofπ (M )onS1×Rwhichpartiallycompactifiestheaction
1 φ
ontheopencylinderΣ×R.
Theactionofπ (Me)onRisboring;alltheinformationisalreadycontainedin
1 φ
theactiononS1 . Wedenotetherepresentationinducingthisactionby
∞
ρ :π (M )→Homeo(S1 )
fol 1 φ ∞
Theorem 1.2.3(Cannon–Thurston [16] Continuity of Peano map). Suppose M is
φ
a hyperbolic surface bundle over S1 with fiber Σ and monodromy φ. Then there is a
continuous,surjectivemap
P :S1 →S2
∞ ∞
4 DANNYCALEGARI
which is a semiconjugacy between the two actions of π (M ). That is, for each α ∈
1 φ
π (M ),
1 φ
P ◦ρ (α)=ρ (α)◦P
fol geo
Since the image of S1 under P is closed and invariant under the action of
∞
π (M ), it is equal to the entire sphere S2 ; that is, it is a Peano curve, or sphere–
1 φ ∞
fillingmap.
ThefactthatP issphere–fillingisdisconcertingandbeautiful,butitisnotthe
wholestory. MoreinterestingisthefactthatP canbeapproximatedbyembeddings
inanaturalway.
If Σ denotes the universal cover of a fiber, then Σ is a properly embedded
θ θ
planefinM = H3. Bytheorem1.2.3, theembeddingfofΣ extendscontinuously
φ θ
g f
tothePeanomapontheboundary,bythecanonicalidentificationofS1 (Σ )with
∞ θ
S1 (π (Σ)). Inthe unit ballmodelof H3, letp ∈ Σ bea basepointatthfe origin.
∞ 1 θ
f
LetT ⊂Σ be(acomponentof)theintersectionofΣ withafamilyofconcentric
i θ θ
f f
spheresaboutp. RadialprojectionfrompinΣ identifieseachT withS1 (Σ),and
θ i ∞
radialprojectionfrompinH3identifieseachT withanembeddedcircleinSe2 . We
i ∞
denotethecompositionoftheseidentificationsby
P :S1 →S2
i ∞ ∞
which gives a family of maps which converge in the compact–open topology to
P. EachP decomposesthecomplementofitsimageintotwosides,whichwecan
i
consistently labelasthe positive andnegative sides, compatiblywithanorienta-
tiononS1 andS2 .
∞ ∞
Defineapositivepairtobeapairofelementsp,q ∈S1,andachoice,foreachP ,
i
of an arc γ ⊂ S2 from P (p) to P (q) whose interior is disjoint from P (S1) and
i ∞ i i i
containedonthepositiveside,andwhichsatisfies
lim diameter(γ )→0
i
i→∞
Wedenoteapositivepairby(p,q,{γ }).
i
Now,if(p ,p ,{γ })isonepositivepairand(q ,q ,{δ })isanother,theneither
1 2 i 1 2 i
{p ,p } and{q ,q } areunlinked ascopiesof S0 inS1 , or else allfour points are
1 2 1 2 ∞
mappedtothesamepointbyP.Thereasonisthatif{p ,p }and{q ,q }arelinked
1 2 1 2
inS1 ,thenγ andδ lyingonthesamesideoftheimageofP mustintersect. Since
∞ i i i
theirlengthsconvergeto0asi→∞,theclaimfollows.
The positive pairs define a subset of S1 ×S1 which generates a closed equiv-
alencerelation, whichwedenoteby∼+. Similarly,we candefine∼− intermsof
negativepairs. Notethatdistinctequivalenceclassesof∼+ sayhavetheproperty
thattheyareunlinkedassubsetsofS1,inthesensethatifS0,S0aretwoembedded
1 2
copiesofS0 inS1 whichareeachcontainedindistinctequivalenceclassesof∼+,
thenthehomologicallinkingnumberoftheS0’sis0.
i
ApplyingthisfacttothemapP :S1 →S2 letsusconstructapairofgeodesic
∞ ∞
laminationsΛ± ofΣasfollows(seesection§2foradefinitionandadiscussionof
geodesic lameinatioens). The lamination Λ+ is the union, over equivalence classes
e
[p]of∼+,oftheboundaryoftheconvexhullof[p],thoughtofasasubsetofS1 (Σ).
∞
e
Itiscrucialherethatdistinctequivalenceclassesareunlinked,sothattheresultis
a lamination, and not merelya collection of geodesics. The action of π (Σ) on Σ
1
preservesthese laminations, and they descend to geodesic laminations Λ± on Σe
PROMOTINGESSENTIALLAMINATIONS 5
which are preservedby the action of (a homeomorphism isotopic to) φ. It is not
hardto show that these laminations are transverse, and bind Σ, in the sense that
complementary regions are (compact) finite sided polygons. The usual Perron–
Frobeniustheory shows that Λ± admittransversemeasuresµ± which aremulti-
plied by λ,λ−1 respectively by φ, for some λ > 1. This is one of the definitions
ofapseudo–Anosovmap,andThurston’stheoremontheclassificationofsurface
homeomorphismsisrecoveredviaaverynon–standardroute.
Note that a posteriori, it can be seen that the laminations Λ± are determined
uniquelybytheactionofπ (M )onthecircleS1 ,andcanbeerecoveredfromthe
1 φ ∞
fixedpointdataofφ∗ anditsconjugates.
Of course this is not a logical deduction, since the usual proofs of both theo-
rem1.2.2andtheorem1.2.3dependessentiallyontheorem1.2.1.
More useful information can be derived from this picture. Each of the invari-
ant geodesic laminations Λ± on Σ suspend in the mapping torus to two dimen-
sional laminations. Notice that such laminations have some useful properties.
The leaves are covered in M by planes. The finitely many complementary re-
φ
f
gions are topologically open solid tori, which have the extra structure of finite
sided ideal polygon bundles over S1. They are the prototypical example of very
full genuine laminations, a particularly well behaved subclass of the class of gen-
uinelaminations,introducedbyGabaiandOertelin[28]. Suchlaminationscertify
important properties of their ambient manifold. In [25],[26] and [27] Gabai and
Kazezshowthatanatoroidal3–manifoldM withagenuinelaminationhasword–
hyperbolic fundamentalgroup, hasa finite mappingclass group, and that every
self–homeomorphismhomotopictotheidentityisisotopictotheidentity.
Inanotherdirection,thelaminationsΛ± canbeusedtoproduceaparticularly
niceflowX transversetothefibration. TheprojectivelymeasuredlaminationsΛ±
aredualtoapairoftopologicalR–treesT±. Thenthetautologicalquotientmaeps
ofΣtoT+andT−defineamaptotheproductT+×T−whoseimageistopologi-
calelyaplane. Moreover,theimageinheritsapairofsingularfoliationsF± bythe
intersectionwithfactorsT+×pointandpoint×T−oftheproductstructure. This
structureisequivariant,anddefinesapairoftransverselymeasuredsingularfoli-
ationsonΣwhicharetransversetoeachotherandinvariantbyasuitableelement
intheisotopyclassofφ.
ThesuspensionflowX ofthishomeomorphismispseudo–Anosov.Thatis,away
fromfinitelymanyorbits, thereisadecomposition ofthetangentspaceTM into
a sum TX ⊕TEs ⊕TEu which is preserved by the flow, and where the time t
flow multiplies the vectors in the sub–bundles TEs and TEu by factors O(etλ)
andO(e−tλ)respectively,forsome λ > 0. Moreover,the singularorbitslooklike
branched covers of the ordinary orbits, with branch index n/2 for some integer
n ≥ 3. A pseudo–Anosov without such singular orbits is Anosov. This pseudo–
Anosov flow has the propertyof being the minimalentropy flow transverse to the
foliation, and it also has the propertyof being quasigeodesic. That is, flowlines of
theliftX intheuniversalcoverareaboundeddistancefromhyperbolicgeodesics
inM =eH3.
φ
g
1.3. Circleofideas. Thispencilsketchof the theoryof surfacediffeomorphisms
outlines the application of this dimensional reductionideato 3–manifoldtheory.
A certain kind of foliation — namely a fibration — reduces a 3–manifold M to
φ
6 DANNYCALEGARI
a 2–manifold Σ together with some dynamics φ. Ideal geometry reduces the 2–
manifoldΣtoa1–manifoldS1 togetherwithsomefurtherdynamics,namelythe
∞
actionofπ (Σ). TherelationshipbetweenS1 andS2 canbeencodedinanother
1 ∞ ∞
pairof1–dimensionalobjects,namelythelaminationsΛ±,whichareactuallyen-
codedindatalivingonlyonS1 ,andwhichcanbereceoveredinprinciplepurely
∞
fromthedynamicsofπ (M )onthis1–dimensionalobject.
1 φ
The goal of this paper is to reproduce asmuch of this structure as possible in
thecontextof amoregeneralkind of foliation, namelyatautfoliation. We follow
the principlethatsmaller isbetterwhen itcomestodimension. Accordingly, we
aimtoreduceour3–manifold,viatheuseofsomeauxiliarydynamicaldata,toa
canonicalcircleS1 calledauniversalcircle,togetherwithanaturalrepresentation
univ
ρ :π (M)→S1
univ 1 univ
Thiscircleandrepresentationencodestheoriginaldynamicaldata,orasmuchof
itasisimportant. Inparticular, fromS1 and ρ we canreconstructthe orig-
univ univ
inal 3–manifold M and certify important topological, geometric and dynamical
propertiesofit.
Looselyspeaking,thesourcesofuniversalcirclesarethreefold: theyarisefrom
thefollowingthreeobjects,whichareallpresentintheexampleofsurfacebundle
overacircle.
(1) Tautfoliations
(2) Veryfullgenuinelaminations
(3) Quasigeodesicpseudo–Anosovflows
Precisedefinitionsofthesestructureswillbedeferreduntil§3.
Inthebestsituation,allthreestructuresgiverisetoandcanberecoveredfrom
theuniversalcircle,andtheirinteractionsareencodedinauniformway. Forde-
tails,consult[8]. Inthispaperweaimtoshowhow,undersuitablecircumstances,
oneofthestructures—atautfoliation—givesrisetoanother: a(pairof)veryfull
genuinelaminations.
1.4. Atoroidalversusalgebraicallyatoroidal. Throughoutthispaper,weusethe
termatoroidalinaslightlynonstandardwayasshorthandforalgebraicallyatoroidal.
A 3–manifold M is algebraically atoroidal if there is no Z ⊕ Z in π (M). A 3–
1
manifold M is geometricallyatoroidalif everyessential embeddedtorus is bound-
ary parallel. For closed 3–manifolds, the two terms are interchangeable except
whenM isasmallSeifertfiberedspace;i.e. aSeifertfiberedspaceoveratriangle
orbifold.
Instatementsofimportanttheorems,inordertominimizeconfusion,wetryto
usethelongertermalgebraicallyatoroidal.
1.5. Statementofresults. Inthissectionwestateourresultsprecisely.
§2gathersbasicresultsandconstructionsinthepointsettopologyofS1which
are used again and again throughout the rest of the paper. We define lamina-
tions of S1, laminar relations on S1, and geodesic laminations of the hyperbolic
planeH2,andweshowhowtomovebackandforwardbetweenthesethreekinds
of objects. We also define monotone maps between circles, which are degree one
mapswhose pointpreimagesareconnected. Themostimportanttheoremisthe-
orem2.2.8,whichconcernscontinuousfamiliesofmonotonemaps. Thisisatech-
nicaltheoremwhichisusedinlatersections,especially§5.
PROMOTINGESSENTIALLAMINATIONS 7
§3ismostlyexpository,beingabriefintroductiontothetheoryoftautfoliations
andtheir cousins, essentialand genuine laminations in3–manifolds. It ishardly
acomprehensivesurvey,butitgivesthedefinitionsofthemostimportantobjects
andconstructions, andgivesstatementsofandreferencestoallthebasicfounda-
tional results that we make use of in this paper. The subsection § 3.3 describes
Candel’suniformization theorem 3.3.1for hyperbolic laminations, and describes
howtousethistheoremtoconstructthecirclebundleE∞ overtheleafspaceLof
F foratautfoliationF. ThefiberofE∞overaleafλofF isjusttheidealbound-
f f
ary of λ in the sense of Gromov (see [30]). The more precise details of Candel’s
theoremarenecessarytodefinethecorrecttopologyonE∞. ThecirclebundleE∞
isusedrepeatedlythroughouttherestofthepaper.
§4isalsoexpository. WepresenttheoutlinesofproofsoftheLeafPockettheo-
remandtheUniversalCircletheoremfrom[8](theorems5.2and6.2respectively
in[8]). Formostof§5,wedonotneedthedetailsoftheproofsofthesetheorems,
and we proceed as far as possible from the axiomatic statements of these theo-
rems. However,laterinthepaperweneedtomakeuseofsomeoftheproperties
of the universal circlesconstructed in [8], andthereforeit isnecessaryto explain
theconstructionsinsomedetail.
§5containsthereallynewresultsinthispaper. Weconstructapairoflamina-
tionsΛ± oftheuniversalcircleS1 constructedin§4,andusetheselaminations
univ univ
±
toconstructapairof2–dimensionallaminationsΛ ofM whicharetransverse
split
toF. We thengoon toestablishbasicpropertiesof these laminations. We saya
foliationF has2–sidedbranchingiftheleafspaceLofthepullbackfoliationF on
f
theuniversalcoverbranchesinboththepositiveandthenegativedirections. Our
mainresultisthefollowing:
TheoremA. LetF bea co–orientabletautfoliationofaclosed, orientable algebraically
atoroidal 3–manifold M. Then either F has 2–sided branching and is the weak stable
±
foliationofanAnosovflow, orelsethereareapairofveryfullgenuinelaminationsΛ
split
transversetoF.
It follows by work of Gabai and Kazez, that a closed 3–manifold with a taut
foliationeithercontainsaZ⊕Zinitsfundamentalgroup,orcontainsanAnosov
flow whose stable and unstable foliation have 2–sided branching, or else it has
word–hyperbolicfundamentalgroup,themappingclassgroupisfinite,andevery
self–homeomorphismhomotopictotheidentityisisotopictotheidentity.
±
Finally, in § 6 we discuss the dynamics of the laminations Λ . This section
split
ismainly descriptive, andservesto illustrate some of the structure developedin
earliersections.
1.6. Notation. We make use of certain conventions for notation throughout this
paper, and try to be consistent throughout. For an object or structure X in a 3–
manifold M, X will denote the pull back of X to the universal cover M, where
e f
this makessense. Surfacesand manifolds will be denoted by upper caseRoman
letters S,M,N etc. and points by lower case Roman letters p,q,r etc. Foliations
will be denoted by script letters F,G etc. and laminations by upper case Greek
lettersΛ etc. Leaveswillbe denoted bylower caseGreeklettersλ,µ,ν etc. Guts
ofgenuinelaminationswillbedenotedbyGothicGandcorecirclesofinterstitial
annulibylowercaseGothiclettersc.
8 DANNYCALEGARI
1.7. Acknowledgements. IamindebtedtoAndrewCassonandSe´rgioFenleyfor
conversations and discussions concerning this material over a number of years.
More recently, I am greatly indebted to Nathan Dunfield for careful comments
about the content and readabilityof this manuscript. I would also like to thank
the anonymous refereefor his verycarefulreadingof this manuscript. Finally, it
shouldbestressedthatIlearnedmuchoftheoverarchingphilosophyandorganiz-
ingprinciplesofthetheoryoffoliationsandgeometrizationwhichpermeatethis
paperanditssiblings[5],[6]and[8]fromverylong,extensiveconversationswith
BillThurston,mainlyduring1996–1999.Manyofthetheoremsandconstructions
inthispaperareatleastinspiredbyideasandcommentsfromBill.
Whilewritingthispaper,IwaspartiallysupportedbyaSloanResearchFellow-
ship,andNSFgrantDMS0405491.
2. THE TOPOLOGYOFS1
InthissectionweestablishbasicpropertiesofthepointsettopologyofS1which
willbeusedintherestofthepaper.Goodgeneralreferencesforpointsettopology
inlowdimensionsare[35],[4]and[40].
2.1. LaminationsofS1.
Definition2.1.1. WeletS0 denotethe0sphere;i.e. thediscrete,twoelementset.
Twodisjoint copies of S0 in S1 arehomologicallylinked, or just linked if the points
inoneoftheS0’sarecontainedindifferentcomponentsofthecomplementofthe
other. Otherwisewesaytheyareunlinked.
Notethatthedefinitionoflinkingissymmetric.
Definition2.1.2. AlaminationΛofS1isaclosedsubsetofthespaceofunordered
pairs of distinct points in S1 with the property that no two elements of the lam-
ination are linked as S0’s in S1. The elements of Λ are called the leaves of the
lamination.
The space of unordered pairs of distinct points in S1 may be thought of as a
quotientofS1×S1\diagonalbytheZ/2Zactionwhichinterchangesthetwocom-
ponents. Topologically,thisspaceishomeomorphictoaMo¨biusband.
Mostreaderswillbefamiliarwiththe conceptofageodesiclaminationonahy-
perbolicsurface.
Definition2.1.3. A geodesic laminationΛ on a complete hyperbolic surface Σ is a
closedunionofdisjointembeddedcompletegeodesics.
Forathoroughdevelopmentoftheelementarytheoryofgeodesiclaminations,
see[17]. AgeodesiclaminationofΣpullsbacktodefineageodesiclaminationof
H2. Geodesic laminations of H2 and laminations of S1 areessentially equivalent
objects,asthefollowingconstructionshows:
Construction2.1.4. LetΛbealaminationofS1. WethinkofS1astheboundaryof
H2 intheunitdiskmodel. ThenweconstructageodesiclaminationofH2 whose
leavesarejustthegeodesicswhoseendpointsareleavesofΛ. Wewillsometimes
denotethisgeodesiclaminationbyΛ . Conversely,givenageodesiclamination
geo
Λof H2, we geta laminationof the idealboundaryS1 whose leavesarejustthe
∞
pairsofendpointsoftheleavesofΛ.
PROMOTINGESSENTIALLAMINATIONS 9
There is another perspective on circle laminations, coming from equivalence
relations. The correct class of equivalence relations for our purposes are upper
semi–continuousdecompositions.
Definition2.1.5. AdecompositionofatopologicalspaceX isapartitionintocom-
pact subsets. A decomposition G is upper semi–continuous if for every decompo-
sition element ζ ∈ G and every open set U with ζ ⊂ U, there exists an open set
V ⊂ U with ζ ⊂ V such that every ζ′ ∈ G with ζ′ ∩ V 6= ∅ has ζ′ ⊂ U. The
decompositionismonotoneifitselementsareconnected.
ApropermapfromaHausdorffspaceX toaHausdorffspaceY inducesade-
composition of X by its point preimageswhich is upper semi–continuous. Con-
versely,thequotientofaHausdorffspacebyanuppersemi–continuousdecompo-
sition is Hausdorff,and the tautological mapto the quotient spaceis continuous
andproper.Seee.g. [35].
Definition 2.1.6. An equivalence relation ∼ on S1 is laminar if the equivalence
classes are closed, if the resulting decomposition is upper semicontinuous, and if
distinctequivalence classesareunlinked assubsetsof S1. Thatis, if S0,S0 ⊂ S1
1 2
aretwoS0’swhicharecontainedindistinctequivalenceclasses,thentheyarenot
homologicallylinkedinS1.
Wenowshowhowtomovebackandforthbetweencirclelaminationsandlam-
inarrelations.
Construction2.1.7. Givenalaminarequivalencerelation∼ofS1,wethinkofS1
astheidealboundaryofH2. Thenforeveryequivalenceclass[p]of∼weformthe
convexhull
H([p])⊂H2
andtheboundaryoftheconvexhull
Λ([p])=∂H([p])⊂H2
WeletΛdenotetheunionoverallequivalenceclasses[p]:
Λ=[Λ([p])
[p]
Thenthefactthattheequivalenceclassesareunlinkedimpliesthatthegeodesics
makingupΛaredisjoint. Moreover,thefactthat∼isuppersemicontinuous im-
plies that Λ is closed as a subset of H2. That is, it is a geodesic lamination, and
determinesalaminationofS1byconstruction2.1.4.
Conversely, givena laminationΛ of S1, we mayformthe quotient Q of S1 by
thesmallestequivalencerelationwhichcollapseseveryleaftoapoint. Thisisnot
necessarilyHausdorff;weletQ′denotetheHausdorffification.Thenthemapfrom
S1 toQ′ inducesanuppersemi–continuousdecompositionofS1. Moreover,this
equivalencerelationisobviouslyunlinked;inparticular,itislaminar.
We abstractpartof construction 2.1.7toshow thateverysubsetK ⊂ S1 gives
risetoalamination,asfollows
Construction2.1.8. LetK ⊂ S1 bearbitrary. ThinkofS1 as∂H2,andletH(K) ⊂
H2betheconvexhulloftheclosureofKinS1. Thentheboundary∂H(K)isageo-
desiclaminationofH2,whichdeterminesalaminationofS1byconstruction2.1.4.
WedenotethislaminationofS1byΛ(K).
10 DANNYCALEGARI
2.2. Monotonemaps.
Definition2.2.1. LetS1,S1 behomeomorphictoS1. Acontinuousmapφ:S1 →
X Y X
S1 ismonotoneif itisdegreeone, andifitinducesamonotone decomposition of
Y
S1,inthesenseofdefinition2.1.5.
X
Notethatthetargetandimagecircleshouldnotbethoughtofasthesamecircle.
Equivalently,amapbetweencirclesismonotoneifthepointpreimagesarecon-
nected and contractible. Said yet another way, a map is monotone if it does not
reversethecyclicorderontriplesofpointsforsomechoiceoforientationsonthe
targetandimagecircle.
Definition2.2.2. Letφ : S1 → S1 be monotone. The gapsof φ are the maximal
X Y
openconnectedintervalsinS1 inthepreimageofsinglepointsofS1. Thecoreof
X Y
φisthecomplementoftheunionofthegaps.
Note that the gaps of φ are the connected components of the set where φ is
locallyconstant.
Recallthatasetisperfectifnoelementisisolated.
Lemma2.2.3. Letφ:S1 →S1 bemonotone. Thenthecoreofφisperfect.
X Y
Proof. The coreof φ is closed. If it is not perfect, there is some point p ∈ core(φ)
whichisisolatedincore(φ). Letp± bethenearestpointsincore(φ)toponeither
side, so that the open oriented intervals p−p and pp+ are gaps of φ. But then by
definition,
φ(p−)=φ(p)=φ(p+)=φ(r)
foranyrintheorientedintervalp−p+. Sobydefinition,theinteriorofthisinterval
iscontainedinasinglegapofφ. Inparticular,piscontainedinagapofφ,contrary
tohypothesis. (cid:3)
Itfollowsthatthesetofpointsincore(φ)whicharenontriviallimitsfromboth
directionsisdenseincore(φ).
Example2.2.4(TheDevil’sstaircase). Letf : [0,1] → [0,1]bethefunctiondefined
asfollows. Ift∈[0,1],let
0·t t t ···
1 2 3
denotethebase3expansionoft. Letibethesmallestindexforwhicht =1.Then
i
f(t)=sisthenumberwhosebase2expansionis
0·s s s ···s 00···
1 2 3 i
whereeachs =1ifft =1or2andj ≤i,ands =0otherwise. Thegraphofthis
j j j
functionisillustratedinfigure1.
ThecoreofthismapistheusualmiddlethirdCantorset.
Definition 2.2.5. Let B be a topological space, and E a circle bundle over B. A
monotonefamilyofmapsisacontinuousmap
φ:S1×B →E
whichcoverstheidentitymaponB,andwhichrestrictsforeachb∈Btoamono-
tonemapofcircles
φb =φ|S1×b :S1×b→Eb
Wedenoteamonotonefamilybythetriple(E,B,φ).
