CONTEMPORARY MATHEMATICS 432

In the Tradition of Ahlfors-Bers/ IV Ahlfors-Bers Colloquium May 19-22, 2005 University of Michigan Ann Arbor, Michigan

Dick Canary Jane Gilman Juha Heinonen Howard Masur Editors http://dx.doi.org/10.1090/conm/432

In the Tradition of Ahlfors-Bers, IV CoNTEMPORARY MATHEMATICS

432

In the Tradition of Ahlfors-Bersl IV

Ahlfors-Bers Colloquium May 19-22, 2005 University of Michigan Ann Arbor, Michigan

Dick Canary Jane Gilman Juha Heinonen Howard Masur Editors

American Mathematical Society Providence, Rhode Island Editorial Board Dennis DeTurck, managing editor George Andrews Andreas Blass Abel Klein

2000 Mathematics Subject Classification. Primary 14H15, 20H10, 28A75, 30F40, 30C62, 32G15, 54E40, 57M50.

Library of Congress Cataloging-in-Publication Data Ahlfors-Bers Colloquium (4th : 2005 : Ann Arbor, Michigan) In the tradition of Ahlfors-Bers, IV : the Ahlfors-Bers Colloquium, May 19-22, 2005, Ann Arbor, Michigan / Dick Canary ... [et a!.], editors. p. em. -(Contemporary mathematics, ISSN 0271-4132 ; 432) Includes bibliographical references. ISBN-13: 978-0-8218-4227-0 (alk. paper) ISBN-10: 0-8218-4227-7 (alk. paper) 1. Functions-Congresses. 2. Riemann surfaces-Congresses. 3. Mappings (Mathematics)- Congresses. I. Canary, Richard Douglas. II. Title.

QA331.A462 2005 515--dc22 2007060747

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Preface vii

Plenary Talks IX Workshops and Workshop Speakers xi Colloquium Participants xiii Uniformly exponential growth and mapping class groups of surfaces J. W. ANDERSON, J. ARAMOYONA, and K. J. SCHACKLETON 1

An A 1 weight not comparable with any quasiconformal Jacobian CHRISTOPHER J. BISHOP 7 Curvature, stretchiness, and dynamics MOON DUCHIN 19 Some special loci in the Siegel space of genus two CLIFFORD J. EARLE 31 Another approach to the automorphism theorem for Teichmiiller spaces EGE FUJIKAWA 39 Energy of twisted harmonic maps of Riemann surfaces WILLIAM M. GOLDMAN and RICHARD A. WENTWORTH 45 Isometries of some hyperbolic-type path metrics, and the hyperbolic medial axis PETER HASTO, ZAIR IBRAGIMOV, DAVID MINDA, SAMINATHAN PONNUSAMY, and 8WADESH 8AHOO 63 From left earthquakes to right JuN Hu 75 Small eigenvalues and maximal laminations on complete surfaces of negative curvature CHRISTOPHER M. JUDGE 93 A generalized hyperbolic metric for plane domains LINDA KEEN and NIKOLA LAKIC 107 Subgroups of mapping class groups from the geometrical viewpoint RICHARD P. KENT IV and CHRISTOPHER J. LEININGER 119 Determinants of Laplacians, quasifuchsian spaces, and holomorphic extensions of Laplacians YOUNG-HEON KIM 143

v vi CONTENTS

Hyperbolic and quasisymmetric structure of hyperspace LEONID V. KOVALEV and JEREMY T. TYSON 151 A classification of the modular transformations of infinite dimensional Teichmiiller spaces KATSUHIKO MATSUZAKI 167 Random hyperbolic surfaces and measured laminations MARYAM MIRZAKHANI 179 Extensions of holomorphic motions to quasiconformal motions SUDEB MITRA 199 Analyst's Traveling Salesman Theorems. A Survey RAANAN SHUL 209 Energy of harmonic maps and Gardiner's formula RICHARD A. WENTWORTH 221 Preface

The Ahlfors-Bers Colloquia commemorate the mathematical legacy of Lars Ahlfors and Lipman Bers. The core of this legacy lies in the fields of geometric function theory, Teichmiiller theory, hyperbolic manifolds, and partial differential equations. However, their work has impacted and created interactions with many other fields, such as algebraic geometry, mathematical physics, dynamics, geomet- ric group theory, number theory and topology. These triannual colloquia serve as a venue to disseminate the work since the previous colloquium to the wider mathematical community and bring the key participants together to ponder future directions in the field. The third Ahlfors-Bers Colloquium was held in Ann Arbor, Michigan in May 2005 and was attended by nearly 150 mathematicians from eight countries. This was a particularly exciting time for the colloquium to take place as many old and well known problems had been resolved since the last colloquium. Here are some high- lights. In the theory of hyperbolic 3-manifolds, Ahlfors Measure Conjecture, the Bers-Sullivan-Thurston Density Conjecture, Marden's Tameness Conjecture and Thurston's Ending Lamination Conjecture had all been resolved, and new phenom- ena in the deformation theory of Kleinian groups had been revealed. The study of mapping class groups had been re-invigorated by techniques from geometric group theory and Teichmiiller theory. In the theory of meromorphic functions, the spher- ical Bloch constant had been determined. Versions of the uniformization theorem had been found that were purely metric in formulation, with applications to geo- metric group theory. The conformal invariance of percolation had been established. In addition to the plenary talks, there were three pre-colloquium workshops that focused on early career mathematicians. The workshops were Conformal Analysis and Geometric Function Theory organized by Lukas Geyer and Sergiy Merenkov, Deformation Theory of Hyperbolic 3-manifolds organized by Dick Canary and Jane Gilman, and Weil-Petersson Geometry of Teichmiiller space organized by Scott Wolpert and Mike Wolf. The list of plenary and workshop speakers and their titles is given below. The first Colloquium in this series was the Bers Colloquium (1995). This was followed by the first Ahlfors-Bers colloquium (1998) and the second Ahlfors- Bers colloquium (2001). Proceeding of these three colloquia have been published in the AMS Contemporary Mathematics Series, Lipa's Legacy, volume 211, In the Tradition of Ahlfors-Bers, volume 256, In the Tradition of Ahlfors-Bers, III, volume 335, respectively. One important part of the Ahlfors-Bers legacy is to cast a wide net that includes old and young, new and established researchers. We hope that with this volume the legacy of Ahlfors and Bers will continue to have an impact.

vii viii 1. PREFACE

AB05 was supported by the NSF and the University of Michigan. The orga- nizing committee consisted of Bill A bikoff, Dick Canary (Chair), Alex Eremenko, Jane Gilman, Juha Heinonen, Howie Masur, Yair Minsky, Michael Wolf, and Scott Wolpert. The editors of this volume and the organizing committee thank the mem- bers of the local organizing committee, Dick Canary, Lukas Geyer, Juha Heinonen, and Sergiy Merenkov, for making the colloquium a success.

Dick Canary Jane Gilman J uha Heinonen Howard Masur Plenary Speakers with Titles

Agol, Ian Tameness of negatively curved 3-manifolds Astala, Kari Complex analysis, impedance tomography and quasiconformal mappings Bishop, Chris Conformal mapping in linear time Bonk, Mario Conformal Analysis on Fractals Calegari, Danny Shrinkwrapping and the taming of hyperbolic 3-manifolds Kerckhoff, Steve Boundary value problems in deformation theory Minsky, Yair Ends, Laminations and Families of Surfaces Mirzakhani, Maryam Random hyperbolic surfaces and measured laminations Rees, Mary The Ending Laminations Theorem Direct from Teichmuller Geodesics. Souto, Juan Volume of hyperbolic manifolds and distances of Heegaard splittings Traizet, Martin Riemann surfaces with nodes and minimal surfaces without symmetries.

ix Workshops and Speakers

Conformal Analysis and Geometric Function Theory organizers: Lukas Geyer and Sergiy Merenkov Deformation Theory of Hyperbolic 3-manifolds organizers: Dick Canary and Jane Gilman Weil-Petersson Geometry of Teichmiiller space organizers: Scott Wolpert and Mike Wolfe

Jeremy Best Huang, Zheng Constants for weighted The Canonical metric on a Riemann Moser-Trudinger inequalities on surface and its induced metric on the sub-Riemannian spaces moduli

Brock, Jeffrey Ito, Kentaro Chains of flats, minimal irrational 3-dimensional extension of the Maskit laminations, and divergent geodesics in slice for once-punctured tori the Weil-Petersson metric Kim, Young-Heon Determinants of Laplacians and their Choi, Young holomorphic extensions Lines of minima in Teichmuller space Kovalev, Leonid DeMarco, Laura Conformal dimension of metric spaces Polynomial dynamics and trees Leininger, Chris Shadows of mapping class groups: Duchin, Moon capturing convex co-compactness. Variation of Teichmuller geodesics Margalit, Dan Dumas, David Weil-Petersson isometries via the pants Grafting, the Schwarzian derivative, complex and complex projective structures on surfaces Marshall, Tim The smallest covolume Kleinian group Evans, Richard Mcintyre, Andrew Bounded balls in convex cores An explicit Belavin-Knizhnik formula Hasti::i, Peter Milley, Peter Isometries of some Mobius and Cusp diagrams and small hyperbolic similarity invariant metrics 3-manifolds

xi xii WORKSHOPS, ORGANIZERS, SPEAKERS AND TITLES

Miyachi, Hideki Moduli of continuity of Cannon-Thurston maps Namazi, Hossein Heegaard splittings and hyperbolic geometry Pilgrim, Kevin Coarse conformal dynamics Purcell, Jessica Geometry of complicated knots and links Raft, Kasra A combinatorial model for the Teichmuller metric Roeder, Roland Combinitorial aspects of Andreev's classification of Hyperbolic Polyhedra. Saric, Dragomir Decorated Teichmiiller space of the punctured solenoid Schul, Raanan A characterization of subsets of finite length curves in Hilbert space and the analyst's TSP Storm, Peter A group theoretic application of the Marden conjecture Sun, Xiaofeng Good metrics on the moduli space of Riemann surfaces Tyson, Jeremy Best constants for weighted Moser-Tridinger inequalities on sub-Riemannian spaces Uriarte-Thero, Ignacio On Marcinkiewicz integrals and harmonic measure Colloquium Participants

Bill Abikoff Dick Canary University of Connecticut University of Michigan Ilesanmi A de boye Christopher Cashen University of Michigan University of Illinois, Chicago Ian Agol Young-Eun Choi University of Illinois at Chicago University of California, Davis Jim Anderson Suhyoung Choi University of Southampton KAIST Kari Astala Adam Coffman University of Helsinki IU Purdue-Fort Wayne Christopher Atkinson Serban Costea University of Illinois at Chicago University of Michigan David Barrett David Crombecque University of Michigan University of Southern California Ara Basmajian Jason DeBlois University of Oklahoma University of Texas-Austin Thomas Bieske Laura DeMarco University of South Florida University of Chicago Chris Bishop Casey Douglas SUNY Stony Brook Rice Mario Bonk Moon Duchin University of Michigan University of Chicago Joshua Bowman David Dumas Cornell University Math Department Rice University Jeffrey Brock Nathan Dunfield Brown University Caltech Ken Bromberg Anirban Dutta U ofUT SUNY at Stony Brook Danny Calegari Clifford Earle California Institute of Technology Cornell University

xiii xiv PARTICIPANTS

Richard Evans Elmas Irmak U. of Auckland University of Michigan Erlend Fornaess Kentaro Ito Wold Graduate School of Mathematics, Nagoya University Ege Fujikawa Tokyo Institute of Technology Prabhu Janakiraman University of Illinois, Vaibhav Gadre Urbana-Champaign Caltech Lizhen Ji Frederick Gardiner Univ of Michigan Brooklyn College, CUNY Chris Judge Frederick Gehring Indiana University U of Michigan Jason Karcher Lukas Geyer University of Illinois, Chicago University of Michiga Robert Kaufman Jane Gilman University of Illinois Rutgers-Newark Linda Keen Jasun Gong CUNY, Lehman College and Graduate University of Michigan Center Hrant Hakobyan Richard Kent SUNY, Stony Brook University University of Texas at Austin Peter Hasto Steve Kerckhoff Norwegian University of Science and Stanford University Technolgy Young-Heon Kim Juha Heinonen Northwestern University U. Michigan Soomin Kim Chris Hiatt Rice University University of Southern California Youngju Kim Pat Hooper University of Oklahoma Yale and SUNY, Stony Brook Leonid Kovalev Jun Hu Washington University in St. Louis CUNY, Brooklyn College and Graduate Enrico Le Donne Center Yale University Zheng Huang Christopher Leininger University of Michigan Columbia U. Zair Ibragimov Anna Lenzhen University of Cincinnati University of Illinois at Chicago Nikolaev Igor Dean Leonardi University of Calgary University of Illinois at Chcago PARTICIPANTS XV

John Mackay Roman Muchnik U of Michigan University of Chicago John Maki Gregory Muller University of Illinois at Cornell University Urbana-Champaign Kei Nakamura Johanna Mangahas UC Davis University of Michigan Hossein N amazi Al Marden Yale & SUNY Stony Brook University of Minnesota Yusuke Okuyama Dan Margalit Kanazawa University & Helsinki University of Utah University Tim Marshall J ani Onninen North Dakota State University University of Michigan Howard Masur Barbara Paier UIC Rice University Katsuhiko Matsuzaki Seungkyu Park Ochanomizu University usc Andrew Mcintyre Kevin Pilgrim Ohio State University Indiana University Jean McKemie Jessica Purcell St. Edward's University University of Texas at Austin Ben McReynolds Kasra Rafi University of Texas University of Connecticut Ivana Medos Kai Rajala Columbia University University of Michigan Sergiy Merenkov Mary Rees University of Michigan University of Liverpool Peter Milley Roland Roeder University of California at Riverside Cornell University Yair Minsky Dragomir Saric Yale University SUNY Stony Brook Maryam Mirzakhani Saul Schleimer Princeton University /Clay Rutgers Sudeb Mitra Ben Schmidt CUNY, Queens College University of Michigan Hideki Miyachi Raanan Schul Tokyo Denki University Yale University Bryan Mosher Peter Scott University of Michigan U Michigan xvi PARTICIPANTS

Ali Shaker Matthias Weber University of Illinois at Chicago Indiana University, Marie Snipes Richard Wentworth University of Michigan Johns Hopkins University Juan Souto Adam Weyhaupt Laboratoire Emile Picard, CNRS, Indiana University- Bloomington Toulouse Kevin Wildrick University of Michigan Ralf Spatzier University of Michigan Michael Wolf Rice University Peter Storm Stanford Scott Wolpert University of Maryland Xiaofeng Sun Harvard University Jang-Mei Wu University of Illinois Perry Susskind Connecticut College/Williams College Shanshuang Yang Emory University Jing Tao Shenglan Yuan UIC GC-CUNY Edward Taylor Wei Zhu Wesleyan University Rice University Josh Thompson Eric Zupunski University of Utah University of Michigan Martin Traizet Universite de Tours, France Jeremy Tyson University of Illinois at Urbana-Champaign Ignacio Uriarte-Thero Helsinki University Adrian Vajiac Chapman University Sergey Vasilyev University of Chicago Liz Vivas University of Michigan Fang Wang Northwestern Unveristy Biao Wang Cornell University Titles in This Series

432 Dick Canary, Jane Gilman, Juha Heinonen, and Howard Masur, Editors, In the tradition of Ahlfors-Bers, IV, 2007 431 Michael Batanin, Alexei Davydov, Michael Johnson, Stephen Lack, and Amnon Neeman, Editors, Categories in algebra, geometry and mathematical physics, 2007 430 Idris Assani, Editor, Ergodic theory and related fields, 2007 429 Gui-Qiang Chen, Elton Hsu, and Mark Pinsky, Editors, Stochastic analysis and partial differential equations, 2007 428 Estela A. Gavosto, Marianne K. Korten, Charles N. Moore, and Rodolfo H. Torres, Editors, Harmonic analysis, partial differential equations, and related topics, 2007 427 Anastasios Mallios and Marina Haralampidou, Editors, Topological algebras and applications, 2007 426 Fabio Ancona, Irena Lasiecka, Walter Littman, and Roberto Triggiani, Editors, Control methods in POE-dynamical systems, 2007 425 Su Gao, Steve Jackson, and Yi Zhang, Editors, Advances in Logic, 2007 424 V. I. Burenko, T. Iwaniec, and S. K. Vodopyanov, Editors, Analysis and geometry in their interaction, 2007 423 Christos A. Athanasiadis, Victor V. Batyrev, Dimitrios I. Dais, Martin Henk, and Francisco Santos, Editors, Algebraic and geometric combinatorics, 2007 422 JongHae Keum and Shigeyuki Kondo, Editors, Algebraic Geometry, 2007 421 Benjamin Fine, Anthony M. Gaglione, and Dennis Spellman, Editors, Combinatorial group theory, discrete groups, and number theory, 2007 420 William Chin, James Osterburg, and Declan Quinn, Editors, Groups, rings and algebras, 2006 419 Dinh V. Huynh, S. K. Jain, and S. R. L6pez-Permouth, Editors, Algebra and Its applications, 2006 418 Lothar Gerritzen, Dorian Goldfeld, Martin Kreuzer, Gerhard Rosenberger, and Vladimir Shpilrain, Editors, Algebraic methods in cryptography, 2006 417 Vadim B. Kuznetsov and Siddhartha Sahi, Editors, Jack, Hall-Littlewood and Macdonald polynomials, 2006 416 Toshitake Kohno and Masanori Morishita, Editors, Primes and knots, 2006 415 Gregory Berkolaiko, Robert Carlson, Stephen A. Fulling, and Peter Kuchment, Editors, Quantum graphs and their applications, 2006 414 Deguang Han, Palle E. T. Jorgensen, and David Royal Larson, Editors, Operator theory, operator algebras, and applications, 2006 413 Georgia M. Benkart, Jens C. Jantzen, Zongzhu Lin, Daniel K. Nakano, and Brian J. Parshall, Editors, Representations of algebraic groups, quantum groups and Lie algebras, 2006 412 Nikolai Chernov, Yulia Karpeshina, Ian W. Knowles, Roger T. Lewis, and Rudi Weikard, Editors, Recent advances in differential equations and mathematical physics, 2006 411 J. Marshall Ash and Roger L. Jones, Editors, Harmonic analysis: Calder6n-Zygmund and beyond, 2006 410 Abba Gumel, Carlos Castillo-Chavez, Ronald E. Mickens, and Dominic P. Clemence, Editors, Mathematical studies on human disease dynamics: Emerging paradigms and challenges, 2006 409 Juan Luis Vazquez, Xavier Cabre, and Jose Antonio Carrillo, Editors, Recent trends in partial differential equations, 2006 408 Habib Ammari and Hyeonbae Kang, Editors, Inverse problems, multi-scale analysis and effective medium theory, 2006 TITLES IN THIS SERIES

407 Alejandro Adem, Jesus Gonzalez, and Guillermo Pastor, Editors, Recent developments in algebraic topology, 2006 406 Jose A. de Ia Pefia and Raymundo Bautista, Editors, Trends in representation theory of algebras and related topics, 2006 405 Andrew Markoe and Eric Todd Quinto, Editors, Integral geometry and tomography, 2006 404 Alexander Borichev, Hakan Hedenmalm, and Kehe Zhu, Editors, Bergman spaces and related topics in complex analysis, 2006 403 Tyler J. Jarvis, Takashi Kimura, and Arkady Vaintrob, Editors, Gromov-Witten theory of spin curves and orbifolds, 2006 402 Zvi Arad, Mariagrazia Bianchi, Wolfgang Herfort, Patrizia Longobardi, Mercede Maj, and Carlo Scoppola, Editors, Ischia group theory 2004, 2006 401 Katrin Becker, Melanie Becker, Aaron Bertram, PaulS. Green, and Benjamin McKay, Editors, Snowbird lectures on string geometry, 2006 400 Shiferaw Berhanu, Hua Chen, Jorge Hounie, Xiaojun Huang, Sheng-Li Tan, and Stephen S.-T. Yau, Editors, Recent progress on some problems in several complex variables and partial differential equations, 2006 399 Dominique Arlettaz and Kathryn Hess, Editors, An Alpine anthology of homotopy theory, 2006 398 Jay Jorgenson and Lynne Walling, Editors, The ubiquitous heat kernel, 2006 397 Jose M. Muii.oz Porras, Sorin Popescu, and Rub{ E. Rodriguez, Editors, The geometry of Riemann surfaces and Abelian varieties, 2006 396 Robert L. Devaney and Linda Keen, Editors, Complex dynamics: Twenty-five years after the appearance of the Mandelbrot set, 2006 395 Gary R. Jensen and Steven G. Krantz, Editors, 150 Years of Mathematics at Washington University in St. Louis, 2006 394 Rostislav Grigorchuk, Michael Mihalik, Mark Sapir, and Zoran Sunik, Editors, Topological and asymptotic aspects of group theory, 2006 393 Alec L. Matheson, Michael I. Stessin, and Richard M. Timoney, Editors, Recent advances in operator-related function theory, 2006 392 Stephen Berman, Brian Parshall, Leonard Scott, and Weiqiang Wang, Editors, Infinite-dimensional aspects of representation theory and applications, 2005 391 Jiirgen Fuchs, Jouko Mickelsson, Grigori Rozenblioum, Alexander Stolin, and Anders Westerberg, Editors, Noncommutative geometry and representation theory in mathematical physics, 2005 390 Sudhir Ghorpade, Hema Srinivasan, and Jugal Verma, Editors, Commutative algebra and algebraic geometry, 2005 389 James Eells, Etienne Ghys, Mikhail Lyubich, Jacob Palis, and Jose Seade, Editors, Geometry and dynamics, 2005 388 Ravi Vakil, Editor, Snowbird lectures in algebraic geometry, 2005 387 Michael Entov, Yehuda Pinchover, and Michah Sageev, Editors, Geometry, spectral theory, groups, and dynamics, 2005 386 Yasuyuki Kachi, S. B. Mulay, and Pavlos Tzermias, Editors, Recent progress in arithmetic and algebraic geometry, 2005 385 Sergiy Kolyada, Yuri Manin, and Thomas Ward, Editors, Algebraic and topological dynamics, 2005

For a complete list of titles in this series, visit the AMS Bookstore at www.arns.org/bookstore/. The Ahlfors-Bers Colloquia commemorate the mathematical legacy of Lars Ahlfors and Lipman Bers. The core of this legacy lies in the fields of geometric function theory, Teichmiiller theory, hyperbolic manifolds, and partial differential equations. However, the work of Ahlfors and Bers has impacted and created interactions with many other fields, such as algebraic geometry, mathematical physics, dynamics, geometric group theory, number theory, and topology. The triannual Ahlford-Bers colloquia serve as a venue to disseminate the relevant work to the wider mathematical community and bring the key participants together to ponder future directions in the field. The present volume includes a wide range of articles in the fields central to this legacy. The majority of articles present new results, but there are expository articles as well.

ISBN 978-0-8218-4227-0

9 780821 842270