Table Of ContentLONDONMATHEMATICALSOCIETYLECTURENOTESERIES
ManagingEditor:ProfessorM.Reid,MathematicsInstitute,UniversityofWarwick,CoventryCV47AL,UnitedKingdom
Thetitlesbelowareavailablefrombooksellers,orfromCambridgeUniversityPressatwww.cambridge.org/mathematics
216 Stochasticpartialdifferentialequations, A.ETHERIDGE(ed)
217 Quadraticformswithapplicationstoalgebraicgeometryandtopology, A.PFISTER
218 Surveysincombinatorics,1995, P.ROWLINSON(ed)
220 Algebraicsettheory, A.JOYAL&I.MOERDIJK
221 Harmonicapproximation, S.J.GARDINER
222 Advancesinlinearlogic, J.-Y.GIRARD,Y.LAFONT&L.REGNIER(eds)
223 Analyticsemigroupsandsemilinearinitialboundaryvalueproblems, K.TAIRA
224 Computability,enumerability,unsolvability, S.B.COOPER,T.A.SLAMAN&S.S.WAINER(eds)
225 Amathematicalintroductiontostringtheory, S.ALBEVERIOetal
226 Novikovconjectures,indextheoremsandrigidityI, S.C.FERRY,A.RANICKI&J.ROSENBERG(eds)
227 Novikovconjectures,indextheoremsandrigidityII, S.C.FERRY,A.RANICKI&J.ROSENBERG(eds)
228 ErgodictheoryofZdactions, M.POLLICOTT&K.SCHMIDT(eds)
229 Ergodicityforinfinitedimensionalsystems, G.DAPRATO&J.ZABCZYK
230 Prolegomenatoamiddlebrowarithmeticofcurvesofgenus2, J.W.S.CASSELS&E.V.FLYNN
231 Semigrouptheoryanditsapplications, K.H.HOFMANN&M.W.MISLOVE(eds)
232 ThedescriptivesettheoryofPolishgroupactions, H.BECKER&A.S.KECHRIS
233 Finitefieldsandapplications, S.COHEN&H.NIEDERREITER(eds)
234 Introductiontosubfactors, V.JONES&V.S.SUNDER
235 Numbertheory:Se´minairedethe´oriedesnombresdeParis1993–94, S.DAVID(ed)
236 TheJamesforest, H.FETTER&B.G.DEBUEN
237 Sievemethods,exponentialsums,andtheirapplicationsinnumbertheory, G.R.H.GREAVESetal(eds)
238 Representationtheoryandalgebraicgeometry, A.MARTSINKOVSKY&G.TODOROV(eds)
240 Stablegroups, F.O.WAGNER
241 Surveysincombinatorics,1997, R.A.BAILEY(ed)
242 GeometricGaloisactionsI, L.SCHNEPS&P.LOCHAK(eds)
243 GeometricGaloisactionsII, L.SCHNEPS&P.LOCHAK(eds)
244 Modeltheoryofgroupsandautomorphismgroups, D.M.EVANS(ed)
245 Geometry,combinatorialdesignsandrelatedstructures, J.W.P.HIRSCHFELDetal(eds)
246 p-Automorphismsoffinitep-groups, E.I.KHUKHRO
247 Analyticnumbertheory, Y.MOTOHASHI(ed)
248 TametopologyandO-minimalstructures, L.VANDENDRIES
249 Theatlasoffinitegroups-Tenyearson, R.T.CURTIS&R.A.WILSON(eds)
250 Charactersandblocksoffinitegroups, G.NAVARRO
251 Gro¨bnerbasesandapplications, B.BUCHBERGER&F.WINKLER(eds)
252 Geometryandcohomologyingrouptheory, P.H.KROPHOLLER,G.A.NIBLO&R.STO¨HR(eds)
253 Theq-Schuralgebra, S.DONKIN
254 Galoisrepresentationsinarithmeticalgebraicgeometry, A.J.SCHOLL&R.L.TAYLOR(eds)
255 Symmetriesandintegrabilityofdifferenceequations, P.A.CLARKSON&F.W.NIJHOFF(eds)
256 AspectsofGaloistheory, H.VO¨LKLEIN,J.G.THOMPSON,D.HARBATER&P.MU¨LLER(eds)
257 Anintroductiontononcommutativedifferentialgeometryanditsphysicalapplications(2ndEdition), J.MADORE
258 Setsandproofs, S.B.COOPER&J.K.TRUSS(eds)
259 Modelsandcomputability, S.B.COOPER&J.TRUSS(eds)
260 GroupsStAndrews1997inBathI, C.M.CAMPBELLetal(eds)
261 GroupsStAndrews1997inBathII, C.M.CAMPBELLetal(eds)
262 Analysisandlogic, C.W.HENSON,J.IOVINO,A.S.KECHRIS&E.ODELL
263 Singularitytheory, W.BRUCE&D.MOND(eds)
264 Newtrendsinalgebraicgeometry, K.HULEK,F.CATANESE,C.PETERS&M.REID(eds)
265 Ellipticcurvesincryptography, I.BLAKE,G.SEROUSSI&N.SMART
267 Surveysincombinatorics,1999, J.D.LAMB&D.A.PREECE(eds)
268 Spectralasymptoticsinthesemi-classicallimit, M.DIMASSI&J.SJO¨STRAND
269 Ergodictheoryandtopologicaldynamicsofgroupactionsonhomogeneousspaces, M.B.BEKKA&M.MAYER
271 Singularperturbationsofdifferentialoperators, S.ALBEVERIO&P.KURASOV
272 Charactertheoryfortheoddordertheorem, T.PETERFALVI.TranslatedbyR.SANDLING
273 Spectraltheoryandgeometry, E.B.DAVIES&Y.SAFAROV(eds)
274 TheMandelbrotset,themeandvariations, T.LEI(ed)
275 Descriptivesettheoryanddynamicalsystems, M.FOREMAN,A.S.KECHRIS,A.LOUVEAU&B.WEISS(eds)
276 Singularitiesofplanecurves, E.CASAS-ALVERO
277 Computationalandgeometricaspectsofmodernalgebra, M.ATKINSONetal(eds)
278 Globalattractorsinabstractparabolicproblems, J.W.CHOLEWA&T.DLOTKO
279 Topicsinsymbolicdynamicsandapplications, F.BLANCHARD,A.MAASS&A.NOGUEIRA(eds)
280 CharactersandautomorphismgroupsofcompactRiemannsurfaces, T.BREUER
281 Explicitbirationalgeometryof3-folds, A.CORTI&M.REID(eds)
282 Auslander–Buchweitzapproximationsofequivariantmodules, M.HASHIMOTO
283 Nonlinearelasticity, Y.B.FU&R.W.OGDEN(eds)
284 Foundationsofcomputationalmathematics, R.DEVORE,A.ISERLES&E.SU¨LI(eds)
285 Rationalpointsoncurvesoverfinitefields, H.NIEDERREITER&C.XING
286 Cliffordalgebrasandspinors(2ndEdition), P.LOUNESTO
287 TopicsonRiemannsurfacesandFuchsiangroups, E.BUJALANCE,A.F.COSTA&E.MART´INEZ(eds)
288 Surveysincombinatorics,2001, J.W.P.HIRSCHFELD(ed)
289 AspectsofSobolev-typeinequalities, L.SALOFF-COSTE
290 QuantumgroupsandLietheory, A.PRESSLEY(ed)
291 Titsbuildingsandthemodeltheoryofgroups, K.TENT(ed)
292 Aquantumgroupsprimer, S.MAJID
293 SecondorderpartialdifferentialequationsinHilbertspaces, G.DAPRATO&J.ZABCZYK
294 Introductiontooperatorspacetheory, G.PISIER
295 Geometryandintegrability, L.MASON&Y.NUTKU(eds)
296 Lecturesoninvarianttheory, I.DOLGACHEV
297 Thehomotopycategoryofsimplyconnected4-manifolds, H.-J.BAUES
298 Higheroperads,highercategories, T.LEINSTER(ed)
299 Kleiniangroupsandhyperbolic3-manifolds, Y.KOMORI,V.MARKOVIC&C.SERIES(eds)
300 IntroductiontoMo¨biusdifferentialgeometry, U.HERTRICH-JEROMIN
301 StablemodulesandtheD(2)-problem, F.E.A.JOHNSON
302 DiscreteandcontinuousnonlinearSchro¨dingersystems, M.J.ABLOWITZ,B.PRINARI&A.D.TRUBATCH
303 Numbertheoryandalgebraicgeometry, M.REID&A.SKOROBOGATOV(eds)
304 GroupsStAndrews2001inOxfordI, C.M.CAMPBELL,E.F.ROBERTSON&G.C.SMITH(eds)
305 GroupsStAndrews2001inOxfordII, C.M.CAMPBELL,E.F.ROBERTSON&G.C.SMITH(eds)
306 Geometricmechanicsandsymmetry, J.MONTALDI&T.RATIU(eds)
307 Surveysincombinatorics2003, C.D.WENSLEY(ed.)
308 Topology,geometryandquantumfieldtheory, U.L.TILLMANN(ed)
309 Coringsandcomodules, T.BRZEZINSKI&R.WISBAUER
310 Topicsindynamicsandergodictheory, S.BEZUGLYI&S.KOLYADA(eds)
311 Groups:topological,combinatorialandarithmeticaspects, T.W.MU¨LLER(ed)
312 Foundationsofcomputationalmathematics,Minneapolis2002, F.CUCKERetal(eds)
313 Transcendentalaspectsofalgebraiccycles, S.MU¨LLER-STACH&C.PETERS(eds)
314 Spectralgeneralizationsoflinegraphs, D.CVETKOVIC´,P.ROWLINSON&S.SIMIC´
315 Structuredringspectra, A.BAKER&B.RICHTER(eds)
316 Linearlogicincomputerscience, T.EHRHARD,P.RUET,J.-Y.GIRARD&P.SCOTT(eds)
317 Advancesinellipticcurvecryptography, I.F.BLAKE,G.SEROUSSI&N.P.SMART(eds)
318 Perturbationoftheboundaryinboundary-valueproblemsofpartialdifferentialequations, D.HENRY
319 DoubleaffineHeckealgebras, I.CHEREDNIK
320 L-functionsandGaloisrepresentations, D.BURNS,K.BUZZARD&J.NEKOVA´Rˇ(eds)
321 Surveysinmodernmathematics, V.PRASOLOV&Y.ILYASHENKO(eds)
322 Recentperspectivesinrandommatrixtheoryandnumbertheory, F.MEZZADRI&N.C.SNAITH(eds)
323 Poissongeometry,deformationquantisationandgrouprepresentations, S.GUTTetal(eds)
324 Singularitiesandcomputeralgebra, C.LOSSEN&G.PFISTER(eds)
325 LecturesontheRicciflow, P.TOPPING
326 ModularrepresentationsoffinitegroupsofLietype, J.E.HUMPHREYS
327 Surveysincombinatorics2005, B.S.WEBB(ed)
328 Fundamentalsofhyperbolicmanifolds, R.CANARY,D.EPSTEIN&A.MARDEN(eds)
329 SpacesofKleiniangroups, Y.MINSKY,M.SAKUMA&C.SERIES(eds)
330 Noncommutativelocalizationinalgebraandtopology, A.RANICKI(ed)
331 Foundationsofcomputationalmathematics,Santander2005, L.M.PARDO,A.PINKUS,E.SU¨LI&
M.J.TODD(eds)
332 Handbookoftiltingtheory, L.ANGELERIHU¨GEL,D.HAPPEL&H.KRAUSE(eds)
333 Syntheticdifferentialgeometry(2ndEdition), A.KOCK
334 TheNavier–Stokesequations, N.RILEY&P.DRAZIN
335 Lecturesonthecombinatoricsoffreeprobability, A.NICA&R.SPEICHER
336 Integralclosureofideals,rings,andmodules, I.SWANSON&C.HUNEKE
337 MethodsinBanachspacetheory, J.M.F.CASTILLO&W.B.JOHNSON(eds)
338 Surveysingeometryandnumbertheory, N.YOUNG(ed)
339 GroupsStAndrews2005I, C.M.CAMPBELL,M.R.QUICK,E.F.ROBERTSON&G.C.SMITH(eds)
340 GroupsStAndrews2005II, C.M.CAMPBELL,M.R.QUICK,E.F.ROBERTSON&G.C.SMITH(eds)
341 Ranksofellipticcurvesandrandommatrixtheory, J.B.CONREY,D.W.FARMER,F.MEZZADRI&
N.C.SNAITH(eds)
342 Ellipticcohomology, H.R.MILLER&D.C.RAVENEL(eds)
343 AlgebraiccyclesandmotivesI, J.NAGEL&C.PETERS(eds)
344 AlgebraiccyclesandmotivesII, J.NAGEL&C.PETERS(eds)
345 Algebraicandanalyticgeometry, A.NEEMAN
346 Surveysincombinatorics2007, A.HILTON&J.TALBOT(eds)
347 Surveysincontemporarymathematics, N.YOUNG&Y.CHOI(eds)
348 Transcendentaldynamicsandcomplexanalysis, P.J.RIPPON&G.M.STALLARD(eds)
349 ModeltheorywithapplicationstoalgebraandanalysisI, Z.CHATZIDAKIS,D.MACPHERSON,A.PILLAY&
A.WILKIE(eds)
350 ModeltheorywithapplicationstoalgebraandanalysisII, Z.CHATZIDAKIS,D.MACPHERSON,A.PILLAY&
A.WILKIE(eds)
351 FinitevonNeumannalgebrasandmasas, A.M.SINCLAIR&R.R.SMITH
352 Numbertheoryandpolynomials, J.MCKEE&C.SMYTH(eds)
353 Trendsinstochasticanalysis, J.BLATH,P.MO¨RTERS&M.SCHEUTZOW(eds)
354 Groupsandanalysis, K.TENT(ed)
355 Non-equilibriumstatisticalmechanicsandturbulence, J.CARDY,G.FALKOVICH&K.GAWEDZKI
356 EllipticcurvesandbigGaloisrepresentations, D.DELBOURGO
357 Algebraictheoryofdifferentialequations, M.A.H.MACCALLUM&A.V.MIKHAILOV(eds)
358 Geometricandcohomologicalmethodsingrouptheory, M.BRIDSON,P.KROPHOLLER&I.LEARY(eds)
359 Modulispacesandvectorbundles, L.BRAMBILA-PAZ,S.B.BRADLOW,O.GARC´IA-PRADA&
S.RAMANAN(eds)
360 Zariskigeometries, B.ZILBER
361 Words:Notesonverbalwidthingroups, D.SEGAL
362 Differentialtensoralgebrasandtheirmodulecategories, R.BAUTISTA,L.SALMERO´N&R.ZUAZUA
363 Foundationsofcomputationalmathematics,HongKong2008, M.J.TODD,F.CUCKER&A.PINKUS(eds)
364 Partialdifferentialequationsandfluidmechanics, J.C.ROBINSON&J.L.RODRIGO(eds)
365 Surveysincombinatorics2009, S.HUCZYNSKA,J.D.MITCHELL&C.M.RONEY-DOUGAL(eds)
366 Highlyoscillatoryproblems, B.ENGQUIST,A.FOKAS,E.HAIRER&A.ISERLES(eds)
367 Randommatrices:Highdimensionalphenomena, G.BLOWER
368 GeometryofRiemannsurfaces, F.P.GARDINER,G.GONZA´LEZ-DIEZ&
C.KOUROUNIOTIS(eds)
369 Epidemicsandrumoursincomplexnetworks, M.DRAIEF&L.MASSOULIE´
LondonMathematicalSocietyLectureNoteSeries:368
Geometry of Riemann Surfaces
Proceedings of the Anogia Conference to celebrate
the 65th birthday of William J. Harvey
Editedby
FREDERICK P. GARDINER
BrooklynCollege,CUNY
GABINO GONZA´ LEZ-DIEZ
UniversidadAuto´nomadeMadrid
CHRISTOS KOUROUNIOTIS
UniversityofCrete
cambridge university press
Cambridge,NewYork,Melbourne,Madrid,CapeTown,Singapore,
Sa˜oPaulo,Delhi,Dubai,Tokyo
CambridgeUniversityPress
TheEdinburghBuilding,CambridgeCB28RU,UK
PublishedintheUnitedStatesofAmericabyCambridgeUniversityPress,NewYork
www.cambridge.org
Informationonthistitle:www.cambridge.org/9780521733076
(cid:1)C CambridgeUniversityPress2010
Thispublicationisincopyright.Subjecttostatutoryexception
andtotheprovisionsofrelevantcollectivelicensingagreements,
noreproductionofanypartmaytakeplacewithoutthewritten
permissionofCambridgeUniversityPress.
Firstpublished2010
PrintedintheUnitedKingdomattheUniversityPress,Cambridge
AcataloguerecordforthispublicationisavailablefromtheBritishLibrary
ISBN978-0-521-73307-6Paperback
Additionalresourcesforthispublicationatwww.cambridge.org/9780521733076
CambridgeUniversityPresshasnoresponsibilityforthepersistenceor
accuracyofURLsforexternalorthird-partyinternetwebsitesreferredto
inthispublication,anddoesnotguaranteethatanycontentonsuch
websitesis,orwillremain,accurateorappropriate.
Contents
Preface vii
Foreword W. J. Harvey ix
Semisimple actions of mapping class groups on CAT(0)
spaces M. R. Bridson page 1
A survey of research inspired by Harvey’s theorem on
cyclic groups of automorphisms
E. Bujalance, F. J. Cirre and G. Gromadzki 15
Algorithms for simple closed geodesics P. Buser 38
Matings in holomorphic dynamics S. Bullett 88
Equisymmetric strata of the singular locus of the moduli
space of Riemann surfaces of genus 4
A. F. Costa and M. Izquierdo 120
Diffeomorphisms and automorphisms of compact hyper-
bolic 2-orbifolds C. J. Earle 139
Holomorphic motions and related topics
F. P. Gardiner, Y. Jiang and Z. Wang 156
Cutting sequences and palindromes
J. Gilman and L. Keen 194
On a Schottky problem for the singular locus of A
5
V. González-Aguilera 217
Non-special divisors supported on the branch set of a
p-gonal Riemann surface G. González-Diez 238
A note on the lifting of automorphisms
R. Hidalgo and B. Maskit 260
v
vi Contents
Simple closed geodesics of equal length on a torus
G. McShane and H. Parlier 268
On extensions of holomorphic motions—a survey
S. Mitra 283
Complex hyperbolic quasi-Fuchsian groups
J. R. Parker and I. D. Platis 309
Geometry of optimal trajectories
M. Pontani and P. Teofilatto 356
Actions of fractional Dehn twists on moduli spaces
R. Silhol 376
Preface
ThisconferenceontheGeometryofRiemannSurfacesandrelatedtopics
washeldinthebeautifulhilltownofAnogiaattheConferenceCentreof
the University of Crete, spanning four days in June and July 2007. The
pretextwasthecelebrationofBillHarvey’s65thbirthdayandretirement
from teaching. About 50 mathematicians and friends came, many from
far-flung points of the globe, to enjoy this opportunity to refresh mind,
body and spirit.
We invited all participants to contribute articles based on their talks
orrelatedmaterial; theresponsewaswholeheartedandexpert,withthe
result that you see before you. The broad range of topics addressed
by the articles reflects the pervasive influence of the theory of Riemann
surfaces and the remarkable variety of geometric ideas and methods
which flow from it; this expansive aspect of the field will be discussed
by Bill Harvey in the introduction which follows.
We take this opportunity also to thank Professor Harvey himself,
whose supportive and knowledgeable comments provided foil and in-
spiration for all the participants.
OnbehalfofthoseluckyenoughtobeatAnogiaforthisconference,we
thank all the sponsors, the Universidad Autónoma de Madrid and Co-
munidadAutónomadeMadrid(GrantC-101),theSpanishGovernment
Ministerio Español de Educación y Ciencia (Grants MTM2006-01859 &
MTM2006-28257-E), the Department of Mathematics of the University
of Crete and the Anogia Academic Village, Crete, for generous financial
supportwhichmadethemeetingpossible;inparticular,thefundingpro-
vided subsistence and travel expenses for graduate students and others
lacking support.
We are also very grateful to the local organisers and staff at the Uni-
versity of Crete, in Iraklio and at the Conference Centre, and especially
vii
viii Preface
theconferencesecretaryMarinaVasilaki,fortheirworkinplanningand
preparation and for friendly assistance during the meeting. Significant
editorial assistance with the Proceedings was given by David Torres
Teigell (UAM) and is gratefully acknowledged. Finally, the appearance
of this volume is thanks to the unselfish hard work of many people in-
cluding all the anonymous referees, testament in itself to the continuing
good health of this section of the mathematical community.
Fred Gardiner
Gabino González-Diez
Christos Kourouniotis
Foreword: Riemann surfaces and a little
history
William Harvey
King’s College London
A Riemann surface is a thing of beauty, possessing geometric shape as
well as analytic or algebraic structure. From its introduction in 1851 in
Riemann’s inaugural dissertation, his first great work establishing the
foundations of geometric complex analysis, the concept has exerted an
unusual influence as a powerful clarifying mental tool.
Today,thepervasiveroleofcomplexanalysisinthemathematicaland
physical sciences has brought these ideas into a significance wider than
eventheirfoundercouldhavepredicted. Inthepresentbook,thereader
willfindaselectionofresultswhichcanonlyindicatethepartcurrently
played by surfaces and their spaces of deformations: just as a single
convergent power series is enough to generate by continuation an entire
Riemann surface structure, so the foundational ideas of our discipline
extend and evolve beyond our present view of them.
Central to the contemporary study of Riemann surfaces is the inter-
play between different aspects, geometric ideas and algebraic or ana-
lytical calculations, leading to insights into the deeper properties these
objectspossess. Thebasicnotionprovidesatopologicalbasefordeploy-
ingthemostpowerfulideasofalgebra, geometryandanalysis: indeedit
establishes a central role for topology in bringing about a unique math-
ematical synthesis. A single accessible theory serves to interconnect
complex analysis and the various algebraic invariants, the fundamental
group, field of functions, homology and period lattices. In the reas-
suring familiarity of a two dimensional framework, we have a global
base for complex analytic and covering space methods, interacting with
Galois-theoretic properties of the function field. And awaiting new de-
velopments, there are so many areas where Riemann surfaces are di-
rectcontributors: hyperbolicmanifoldsandkleiniangroups,iterationof
polynomialorholomorphicfunctions,crystallographicgroups,geometric
ix
