Geometry of Riemann Surfaces

416 Pages·2010·6.236 MB·English

by  Frederick P. Gardiner, Gabino González-Diez, Christos Kourouniotis (eds.)

LONDONMATHEMATICALSOCIETYLECTURENOTESERIES
ManagingEditor:ProfessorM.Reid,MathematicsInstitute,UniversityofWarwick,CoventryCV47AL,UnitedKingdom
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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