Mirzakhani’s Curve Counting and Geodesic Currents

233 Pages·2022·3.231 MB·English

by  Viveka Erlandsson, Juan Souto

Progress in Mathematics
345
Viveka Erlandsson
Juan Souto
Mirzakhani’s
Curve Counting
and Geodesic
Currents

Progress in Mathematics
Volume 345
SeriesEditors
AntoineChambert-Loir ,UniversitéParis-Diderot,Paris,France
Jiang-HuaLu,TheUniversityofHongKong,HongKongSAR,China
MichaelRuzhansky,GhentUniversity,Belgium
QueenMaryUniversityofLondon,London,UK
YuriTschinkel,CourantInstituteofMathematicalSciences,NewYork,USA
ProgressinMathematicsisaseriesofbooksintendedforprofessionalmathemati-
ciansandscientists,encompassingallareasofpuremathematics.Thisdistinguished
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Viveka Erlandsson (cid:129) Juan Souto
Mirzakhani’s Curve
Counting and Geodesic
Currents
VivekaErlandsson JuanSouto
SchoolofMathematics IRMAR
UniversityofBristol UniversitéRennes1
Bristol,UK RennesCedex,France
ISSN0743-1643 ISSN2296-505X (electronic)
ProgressinMathematics
ISBN978-3-031-08704-2 ISBN978-3-031-08705-9 (eBook)
https://doi.org/10.1007/978-3-031-08705-9
MathematicsSubjectClassification:57M50,37C85,37E99,57K20
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Preface
This book is devoted to proving a theorem of Maryam Mirzakhani and explaining
someofitsapplications.Inherdissertation,Mirzakhaniestablishedtheasymptotic
behavior,whenLtendsto∞,ofthenumberofsimplegeodesicsofgiventypeand
of length at most L in a hyperbolic surface. Here, two geodesics are of the same
type if they differ, up to homotopy, by a homeomorphism. Some time later, using
differentmethods,sheextendedthisresulttonon-simplegeodesics.Ourmainaim
istogiveanewunifiedproofofthesetheorems,theoneforsimpleandtheonefor
non-simplecurves.
Although our proof has some philosophical similarities with Mirzakhani’s
original argument for simple curves, it is actually quite different. The two main
differencesbeingthekeyrolethatthespaceofgeodesiccurrentsplayforus—this
iswhatallowsustotreatthesimpleandthenon-simplecaseatonce—andthefact
that our argument bypasses any discussion of moduli space or the Weil–Petersson
metric.
We also present a number of applications of Mirzakhani’s curve counting
theorem, relating to, for example, counting square-tiled surfaces, the genericity of
pseudo-Anosovelementsinthemappingclassgroup,andthegeometryofrandom
pantsdecompositionsinafixedsurface.
Ourpresentationisprettymuchself-contained.Forexample,agraduatestudent
familiarwithtwo-dimensionaltopologyandhyperbolicgeometry,andhavingsome
understanding of what train tracks or measured laminations are, should be able to
read the proof of Mirzakhani’s theorem we present here. Only when it comes to
applicationsdowemakeuseofmoreadvancedconcepts,justremindingthereader
brieflyoftheneededresultsandreferringtotherelevantliterature.
Bristol,UK/ZoomuponAvon VivekaErlandsson
RennesCedex,France/ZoomsurVilaine JuanSouto
May2022
v
Acknowledgments
Friends, colleagues, and institutions, we thank you all for your love, money, and
support. The first author is grateful for the support by the University of Bristol,
UiT The Artic University of Norway, and the Engineering and Physical Sciences
ResearchCouncil(grantEP/T015926/1).Thesecondauthorissimilarlygratefulto
UniversitéRennes1andCentreNationaldelaRechercheScientifique.
This book would not have been possible without the help from many. First off,
we would never have written a book without being nudged to do so by Loretta
BartoliniandErikoHironaka,andtheresultwouldhavebeenmuchworsewithout
themanycommentswereceived.LetusspecificallythankVincentDelecroix,Alex
Eskin, Elise Goujard, Didac Martinez Granado, Howard Masur, Dylan Thurston,
Scott Wolpert, Alex Wright, and anonymous referees. The referees were great.
Besides that we are particularly grateful to Anton Zorich for spending hours and
hours discussing factorsof 2 withus,and toMark Bellfor providing us withdata
indicatingwhenthesefactorswerecorrect.Thisbookalsobenefitedalotfromour
discussionsandcollaborationswithChrisLeininger,GabrieleMondello,JingTao,
and Caglar Uyanik, and we cannot emphasize enough our gratitude to Kasra Rafi,
withoutwhosecontributionthemathematicsherewouldneverhavecrystallized.
Over the years, we have benefited greatly, in terms of math, cash, and social
life, from our friends Pekka Pankka and Hugo Parlier, and they have also made
invaluabledirectcontributionstothisbook:theillustrationswouldbeincomparably
worse if it were not for Hugo, and Pekka decisively helped us in the search for
appropriatescientificterminology.Notonlythat,bothofthemhaveattimeskeptthe
first author busy and—most importantly—employed. Who has not directly played
anypartinthewritingofthisbookisJean-PierreOtal,butthesecondauthorwishes
tothankhimformakinghimappreciatecurrentsandotherthings,andforotherwise
servingasarolemodel.
Finally,wewouldliketoaddtheobvious:Thisworkwouldhavebeenimpossible
without the earlier work of Maryam Mirzakhani. In fact, when rereading her
papersafterworkingonthistopic,onehasthefeelingofhavingonlyrediscovered
something Mirzakhani already knew. On a more personal note, Maryam was very
nicetous,anditmakesushappytohavewrittenabookwithhernameinthetitle.
vii
viii Acknowledgments
Comotepuedesimaginar,loquemesaldríadelalmaesponerunodelosmuchos
nombresqueusonormalmenteparallamarte,nombresquealgunaveztendríansu
razón de ser, pero de la que ya no se acuerda ni Dios. Pero siendo esto un libro,
creoquetendréqueusarelqueapareceenelDNI.Adèle,yasabesquetupadrete
quieremucho,mucho,mucho,queereslomejorquemehapasadonunca.Yaséque
telodigomuchasveces,quesoyunpesado,peroesqueesverdad.Estoloescribo
eldíaquehasaprendidolosnúmerosromanos.
Contents
1 Introduction................................................................. 1
2 ReadMe ..................................................................... 11
3 GeodesicCurrents.......................................................... 35
4 TrainTracks................................................................. 57
5 Radallas...................................................................... 79
6 SubconvergenceofMeasures.............................................. 101
7 ApproximatingtheThurstonMeasure................................... 117
8 TheMainTheorem......................................................... 131
9 CountingCurves............................................................ 151
10 CountingSquare-TiledSurfaces.......................................... 159
11 StatisticsofSimpleCurves ................................................ 169
12 Smörgåsbord................................................................ 189
A RadonMeasures............................................................ 207
B ComputingThurstonVolumes............................................ 213
References......................................................................... 221
Index............................................................................... 225
ix
Notation
We collect here some of the most commonly used notation throughout the book.
To keep it manageable, we did not include notation that is fixed and used only in
individualchapters.
(cid:2) is a compact connected orientable surface of negative Euler characteristic, of
genusg andwithr boundarycomponents.Onecanalwaysthinkof(cid:2) asbeing
equipped with a hyperbolic metric with totally geodesic boundary. The length
withrespecttothismetricisdenotedby(cid:3) (·).Theuniversalcover(cid:2)(cid:2)isequipped
(cid:2)
(cid:2)
withtheliftedmetricanditsGromovboundaryisdenotedby∂∞(cid:2).Wedenote
byG((cid:2))thespaceofunorientedbi-infinitegeodesicsin(cid:2)(cid:2),byG ((cid:2))⊂G((cid:2))
np
thesetofnon-peripheralgeodesicsasdefinedin(2.15),andfinallybyG ((cid:2))⊂
K
G ((cid:2)) the set of geodesics which project to geodesics contained in a compact
np
setK ⊂(cid:2)\∂(cid:2).
S is a connected orientable surface of negative Euler characteristic with empty
boundary,ofgenusg andwithr ends.WeoftenletS = (cid:2)\∂(cid:2) betheinterior
of some (cid:2) as above. We endow S however with a complete hyperbolic metric
of finite volume. Length in S is denoted by (cid:3) (·). The universal cover of S is
S
denoted by (cid:2)S, G(S) is the set of all unoriented bi-infinite geodesics in (cid:2)S, and
G (S)⊂G(S)isthesetofnon-peripheralgeodesics(2.14).
np
ι(·,·)standsforboththeintersectionformonthespaceofcurrents(Definition3.10)
andgeometricintersectionnumberofcurves(2.4).Avariant,ι (·,·),appearsin
X
(5.1).
Curv, Arc, and S are the sets of all topological multicurves, topological arc sys-
tems,andweightedmulticurves.Curv ,Arc ,andS arethesubsetsconsisting
k k k
ofthosewithatmostk self-intersections(2.6).MLZ isthesetofallintegrally
weightedsimplemulticurves.Aisthearccomplex.
L and ML are the spaces of geodesic laminations and measured laminations.
Again, MLZ is the set of all integral weighted simple multicurves. L(τ) and
ML(τ) are the laminations and measured laminations carried by a train-track
τ andLs(X)andMLs(X)arethosestronglycarriedbyathickeningX ofτ.
Finally,ML isthesetofthosemeasuredlaminationswhichfillwithacurveα.
α
xi