639

Geometry, Groups andDynamics

ICTS ProgramGroups, Geometry and Dynamics

December 3–16, 2012CEMS, Kumaun University, Almora, India

C. S. AravindaWilliam M. Goldman

Krishnendu GongopadhyayAlexander Lubotzky

Mahan MjAnthony Weaver

Editors

American Mathematical Society

Geometry, Groups andDynamics

ICTS ProgramGroups, Geometry and Dynamics

December 3–16, 2012CEMS, Kumaun University, Almora, India

C. S. AravindaWilliam M. Goldman

Krishnendu GongopadhyayAlexander Lubotzky

Mahan MjAnthony Weaver

Editors

KashyapRajeevasarathy

Participants from the ICTS program:Groups, Geometry and Dynamics

639

Geometry, Groups andDynamics

ICTS ProgramGroups, Geometry and Dynamics

December 3–16, 2012CEMS, Kumaun University, Almora, India

C. S. AravindaWilliam M. Goldman

Krishnendu GongopadhyayAlexander Lubotzky

Mahan MjAnthony Weaver

Editors

American Mathematical SocietyProvidence, Rhode Island

EDITORIAL COMMITTEE

Dennis DeTurck, Managing Editor

Michael Loss Kailash Misra Martin J. Strauss

2010 Mathematics Subject Classification. Primary 51MXX, 32-XX, 20-XX.

Library of Congress Cataloging-in-Publication Data

Geometry, groups and dynamics : ICTS program : groups, geometry and dynamics, December3–16, 2012, Almora, India / C. S. Aravinda [and five others], editors.

pages cm. – (Contemporary mathematics ; volume 639)Includes bibliographical references.ISBN 978-0-8218-9882-6 (alk. paper)1. Group theory. 2. Discrete groups. 3. Hyperbolic spaces. 4. Geometry, Non-Euclidean.

I. Aravinda, C. S., 1963– editor.

QA174.2.G47 2015512′.2–dc23 2014040086

Contemporary Mathematics ISSN: 0271-4132 (print); ISSN: 1098-3627 (online)

DOI: http://dx.doi.org/10.1090/conm/639

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10 9 8 7 6 5 4 3 2 1 20 19 18 17 16 15

This volume is dedicated to Ravi S. Kulkarni on his seventieth birthday

Contents

Foreword ix

Preface xi

Lecture Notes

Complex Kleinian groupsW. Barrera, A. Cano, J. P. Navarrete, and J. Seade 1

Complete Lorentzian 3-manifoldsVirginie Charette and Todd A. Drumm 43

The Goldman bracket and the intersection of curves on surfacesMoira Chas 73

An introduction to flows on homogeneous spacesS. G. Dani 85

Quantitative geometry of hyperbolic manifoldsPeter B. Shalen 103

Discrete groups and Riemann surfacesAnthony Weaver 129

Research Expositions

A note on Chern’s theorem on invariant measuresVikram T. Aithal 161

Random affine latticesJayadev S. Athreya 169

Upper central series for the group of unitriangular automorphisms of a freeassociative algebra

Valeriy G. Bardakov and Mikhail V. Neshchadim 175

Hermitian structure and bundles on G/ΓIndranil Biswas 189

Log-Riemann surfaces, Caratheodory convergence and Euler’s formulaKingshook Biswas and Ricardo Perez-Marco 197

Uniformization of simply connected finite type Log-Riemann surfacesKingshook Biswas and Ricardo Perez-Marco 205

vii

viii CONTENTS

The Euler characteristic of a Haken 4-manifoldAllan L. Edmonds 217

A discreteness criterion for groups containing parabolic isometriesViveka Erlandsson and Saeed Zakeri 235

On the z-classes in a centrally finite division ringRony Gouraige 243

On Lorentz spacetimes of constant curvatureFrancois Gueritaud 253

On the asymptotic behavior of complex earthquakes and Teichmuller disksSubhojoy Gupta 271

Characteristically simple Beauville groups, I: Cartesian powers of alternatinggroups

Gareth A. Jones 289

Relatively hyperbolic spacesAbhijit Pal and Akshay Kumar Singh 307

Complex hyperbolic free groups with many parabolic elementsJohn R. Parker and Pierre Will 327

On Fatou components and omitted valuesTarakanta Nayak 349

Some dynamical properties of certain meromorphic functionsShenglan Yuan 359

Foreword

This volume is an outcome of the ICTS program: Groups, Geometry and Dy-namics (GGD-2012) that was held at Almora, India, during December, 2012. Theactivity was an academic tribute to Ravi S. Kulkarni on his turning seventy. There-fore, the program focussed on those areas of mathematics that have been influencedsignificantly by Ravi’s ideas and are closely related to his work and contribution,which fall in the broad area of differential geometry that encompasses a large por-tion of group theory and dynamics in its proximity.

The program was divided into two parts. The first part was an advancedschool, December 3–12, consisting of mini-courses. The second part was a discussionmeeting, December 13–16, whose purpose was to connect the lecture topics of theschool to modern research areas. The present volume reflects this plan. It consistsof lecture notes on the topics of the advanced school, as well as surveys and researchpapers on current topics that connect the lecture notes with cutting edge research.In both cases the purpose is to invite mathematicians into the exciting areas ofcontemporary research that Ravi helped create. Many of the results in this volumeare new. We hope that the volume will be a useful guide to students, as well as agood reference for professional researchers and teachers.

Ravi has been a permanent source of inspiration to us. He has always en-couraged us to see the larger picture. We hope that this volume will successfullyprovide the reader a flavor of his broad view. It is an extreme pleasure to dedicatethis volume to Ravi as a token of appreciation from all of us.

The Volume Editors

ix

Preface

It gives me immense pleasure to write the preface of the volume. This volumeis a documented account of the ICTS program: Groups, Geometry and Dynamicsthat was held at Almora, December 3–16, 2012.

The International Centre for Theoretical Sciences (ICTS) of the Tata Instituteof Fundamental Research is a multi- and inter-disciplinary effort with a mandateto take new initiatives on the frontiers of science, and catalyze and promote collab-oration in research. It is also involved in high level education and training and inreaching out to the larger society by serving as a node for scientific information andvalues. Its aim is to foster excellence in the basic sciences through its programs,interactions and cross-fertilization between disciplines.

The local organization was supervised by the Centre of Excellence in Mathe-matical Sciences (CEMS) Almora headed by H. S. Dhami and assisted by SanjayPant. The CEMS is an effort by the Kumaun University, Almora and is supportedby the Uttarakhand Council of Science and Technology. The aim of the CEMS is topromote mathematics vigorously at all levels in Uttarakhand and its neighboringstates and to serve as an academic support for students and teachers of mathemat-ics in the region. The CEMS aims to create a resource center to serve national andinternational communities as a focal point for advanced research and training inthe mathematical sciences and its applications.

The entire program took place at the Kasaar Jungle Resort that is about 10 kmaway from the main town of Almora and is away from the hustle-bustle of the mainlocality. The peaceful atmosphere during the activity was enhanced by the breath-taking view of the Himalayan valley and the Himalayan range from just outside theresort. We managed to accommodate all the participants in the resort and the ideawas to keep all the participants together so that the peace and spirituality of thelocation foster interaction and mutual discussions among the participants. Further,to catalyze the intellectual urge of the participants, many discussion sessions wereconducted in the open air in the picturesque backdrop of the Himalaya.

The program and this volume would not have been a reality without the help ofmany colleagues who I must acknowledge. I would like to thank my co-organisers ofthe Almora program: H. S. Dhami (Vice Chancellor, Kumaun University, Almora),Sanjay Pant (DDU College, Delhi) and Siddhartha Sarkar (IISER Bhopal). With-out their hard work and support, it would not have been possible to conduct theprogram smoothly. I also thank the ICTS for approving our proposal for the pro-gram, which assured us the necessary resources for organizing it successfully in aremote place like Almora.

I am grateful to the scientific committee of the program that consisted of theeditorial board members of this volume. My deep gratitude to Bill Goldman and

xi

xii PREFACE

C. S. Aravinda for their constant encouragements and advice in all sorts of mattersrelated to the organization. Bill Goldman also took the forefront to arrange an NSFgrant for the US participants. This NSF grant helped a lot in bringing a numberof participants from USA and Europe to Almora. The left-over NSF fund willsupport follow-ups of the Almora activity at Delhi University and IISER Mohali inDecember 2014.

I am indebted to IISER Mohali, especially our director N. Sathyamurthy anddepartmental chair Kapil Paranjape, for being generous in granting me the nec-essary leave during the program. I am thankful to my colleagues Lingaraj Sahuand Mahender Singh for taking the pains of managing my teaching responsibilitiesduring my participation at Almora.

I thank all speakers of the Almora program and all contributors to this volume.I thank all the anonymous referees for their attentive and vigilant work. Thanks alsoto Christine M. Thivierge for her help and suggestions during the entire editorialand production process of this volume.

Finally, my indepth gratitude to Ravi S. Kulkarni for introducing me to moderngeometry and guiding me to appreciate the beauty of the subject. It is Ravi’sinfluence that has cultivated my interest in geometry and its relationship withgroup theory and dynamics. I am thankful to him for his support, affection andencouragements over the years. It is indeed a great contentment to dedicate thisvolume to him.

Krishnendu GongopadhyayIISER Mohali, S.A.S. Nagar, India

5th September 2014

PREFACE xiii

Personal Remarks by Bill Goldman, read atRavi’s felicitation in Almora 1

We first met in 1976 in Princeton. (Has it really been 36 years?) I reallyenjoyed spending the spring semester of 1981 together in Colorado, where we triedto decipher what Dennis Sullivan and Bill Thurston were talking about. Aftermoving East, I enjoyed many pleasant visits with Ravi, first in Indiana and thenin New York City. After Ravi returned to India, I got to see him in both my tripsto India. I was particularly happy that he was chosen to chair my ICM talk inHyderabad two years ago.

Ravi’s papers on the uniformization of flat conformal structures where he in-troduced the connected sum construction, proper actions of discrete groups onhomogeneous space, and 3-dimensional anti-de Sitter geometry were extremely in-fluential on my research. Of course this is just a small part of his mathematicalcontributions, which are uniformly marked by depth and exquisite good taste! Hewas one of the first people I met who appreciated the richness of locally homoge-neous geometric structures on manifolds. Ravi must feel deeply gratified to see howhis seminal work in this subject has taken off so successfully in recent years.

I remember in Boulder, how Dennis commented after Ravi spoke on his jointwork with Allan Edmonds and John Ewing on hyperbolic surfaces, why nobodydid this decades earlier. This work, which I still revisit, opened up a rich field ofmathematics with contacts to geometry, topology, algebra and dynamics.

The wide range of Ravi’s work provided a big challenge for us to organize theactivity, as well as to edit this volume! The diversity of of his research horizon madeit difficult to restrain ourselves with so many interesting mathematicians workingin areas to which he contributed. We didn’t know where to stop. Finally, we aredone and the volume is coming up. By looking at the volume one would get afeel of the integrated picture that Ravi likes to see in Mathematics and that hasinfluenced a whole new generation of Mathematicians including two of the editors,Krishnendu Gongopadhyay and Anthony Weaver of this volume, who are his formerPhD students.

Even though I was one of the scientific organisers of the Almora activity, I couldnot attend. (I was, however, fortunate to be able to attend a similar conference inhonor of Ravi’s birthday at Queens College in New York in October 2013.) So, withextreme pleasure we dedicate this volume to Ravi S. Kulkarni and to acknowledgehis influence in the broad area of Geometry.

William M. GoldmanCollege Park, Maryland, USA

5th September 2014

1The felicitation program was supported by the CEMS Almora.

SELECTED PUBLISHED TITLES IN THIS SERIES

639 C. S. Aravinda, William M. Goldman, Krishnendu Gongopadhyay, AlexanderLubotzky, Mahan Mj, and Anthony Weaver, Editors, Geometry, Groups andDynamics, 2015

632 Gohar Kyureghyan, Gary L. Mullen, and Alexander Pott, Editors, Topics inFinite Fields, 2015

631 Siddhartha Bhattacharya, Tarun Das, Anish Ghosh, and Riddhi Shah, Editors,Recent Trends in Ergodic Theory and Dynamical Systems, 2015

630 Pierre Albin, Dmitry Jakobson, and Frederic Rochon, Editors, Geometric andSpectral Analysis, 2014

629 Milagros Izquierdo, S. Allen Broughton, Antonio F. Costa, and Rubı E.Rodrıguez, Editors, Riemann and Klein Surfaces, Automorphisms, Symmetries andModuli Spaces, 2014

628 Anita T. Layton and Sarah D. Olson, Editors, Biological Fluid Dynamics:Modeling, Computations, and Applications, 2014

627 Krishnaswami Alladi, Frank Garvan, and Ae Ja Yee, Editors, Ramanujan 125,2014

626 Veronika Furst, Keri A. Kornelson, and Eric S. Weber, Editors, OperatorMethods in Wavelets, Tilings, and Frames, 2014

625 Alexander Barg and Oleg R. Musin, Editors, Discrete Geometry and AlgebraicCombinatorics, 2014

624 Karl-Dieter Crisman and Michael A. Jones, Editors, The Mathematics ofDecisions, Elections, and Games, 2014

623 Pramod N. Achar, Dijana Jakelic, Kailash C. Misra, and Milen Yakimov,Editors, Recent Advances in Representation Theory, Quantum Groups, AlgebraicGeometry, and Related Topics, 2014

622 S. Ejaz Ahmed, Editor, Perspectives on Big Data Analysis, 2014

621 Ludmil Katzarkov, Ernesto Lupercio, and Francisco J. Turrubiates, Editors,The Influence of Solomon Lefschetz in Geometry and Topology, 2014

620 Ulrike Tillmann, Søren Galatius, and Dev Sinha, Editors, Algebraic Topology:Applications and New Directions, 2014

619 Gershon Wolansky and Alexander J. Zaslavski, Editors, Variational and OptimalControl Problems on Unbounded Domains, 2014

618 Abba B. Gumel, Editor, Mathematics of Continuous and Discrete Dynamical Systems,2014

617 Christian Ausoni, Kathryn Hess, Brenda Johnson, Wolfgang Luck, and Jerome

Scherer, Editors, An Alpine Expedition through Algebraic Topology, 2014

616 G. L. Litvinov and S. N. Sergeev, Editors, Tropical and Idempotent Mathematicsand Applications, 2014

615 Plamen Stefanov, Andras Vasy, and Maciej Zworski, Editors, Inverse Problemsand Applications, 2014

614 James W. Cogdell, Freydoon Shahidi, and David Soudry, Editors, AutomorphicForms and Related Geometry, 2014

613 Stephan Stolz, Editor, Topology and Field Theories, 2014

612 Patricio Cifuentes, Jose Garcıa-Cuerva, Gustavo Garrigos, Eugenio Hernandez,Jose Marıa Martell, Javier Parcet, Keith M. Rogers, Alberto Ruiz, FernandoSoria, and Ana Vargas, Editors, Harmonic Analysis and Partial DifferentialEquations, 2014

611 Robert Fitzgerald Morse, Daniela Nikolova-Popova, and Sarah Witherspoon,Editors, Group Theory, Combinatorics, and Computing, 2014

For a complete list of titles in this series, visit theAMS Bookstore at www.ams.org/bookstore/conmseries/.

This volume contains the proceedings of the ICTS Program: Groups, Geometry and Dy-namics, held December 3–16, 2012, CEMS, Kumaun University, Almora, India. The ac-tivity was an academic tribute to Ravi S. Kulkarni on his turning 70.

Articles included in this volume, both introductory and advanced surveys, representthe broad area of geometry that encompasses a large portion of group theory (finite orotherwise) and dynamics in its proximity. These areas have been influenced by Kulkarni’sideas and are closely related to his work and contribution.

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