CONTEMPORARY MATHEMATICS 211

Lipa's Legacy

Proceedings of the Bers Colloquium October 19-20, 1995 Graduate School and University Center of CUNY

J6zef Dodziuk Linda Keen Editors http://dx.doi.org/10.1090/conm/211 Selected Titles in This Series

211 J6zef Dodziuk and Linda Keen, Editors, Lipa's legacy: Proceedings from the Bers Colloquium, 1997 210 V. Kumar Murty and Michel Waldschmidt, Editors, Number theory, 1997 209 Steven Cox and Irena Lasiecka, Editors, Optimization methods in partial differential equations, 1997 208 Michel L. Lapidus, Lawrence H. Harper, and Adolfo J. Rumbos, Editors, Harmonic analysis and nonlinear differential equations: A volume in honor of Victor L. Shapiro, 1997 207 Yujiro Kawamata and Vyacheslav V. Shokurov, Editors, Birational algebraic geometry: A conference on algebraic geometry in memory of Wei-Liang Chow (1911~1995), 1997 206 Adam Koranyi, Editor, Harmonic functions on trees and buildings, 1997 205 Paulo D. Cordaro and Howard Jacobowitz, Editors, Multidimensional complex analysis and partial differential equations: A collection of papers in honor of Fran<;ois Treves, 1997 204 Yair Censor and Simeon Reich, Editors, Recent developments in optimization theory and nonlinear analysis, 1997 203 Hanna Nencka and Jean-Pierre Bourguignon, Editors, Geometry and nature: In memory of W. K. Clifford, 1997 202 Jean-Louis Loday, James D. Stasheff, and Alexander A. Voronov, Editors, Operads: Proceedings of Renaissance Conferences, 1997 201 J. R. Quine and Peter Sarnak, Editors, Extremal Riemann surfaces, 1997 200 F. Dias, J.-M. Ghidaglia, and J.-C. Saut, Editors, Mathematical problems in the theory of water waves, 1996 199 G. Banaszak, W. Gajda, and P. Krason, Editors, Algebraic K-theory, 1996 198 Donald G. Saari and Zhihong Xia, Editors, Hamiltonian dynamics and celestial mechanics, 1996 197 J. E. Bonin, J. G. Oxley, and B. Servatius, Editors, Matroid theory, 1996 196 David Bao, Shiing-shen Chern, and Zhongmin Shen, Editors, Finsler geometry, 1996 195 Warren Dicks and Enric Ventura, The group fixed by a family of injective endomorphisms of a free group, 1996 194 Seok-Jin Kang, Myung-Hwan Kiln, and Insok Lee, Editors, Lie algebras and their representations, 1996 193 Chongying Dong and Geoffrey Mason, Editors, Moonshine, the Monster, and related topics, 1996 192 Tomek Bartoszynski and Marion Scheepers, Editors, Set theory, 1995 191 Thong Ton-That, Kenneth I. Gross, Donald St. P. Richards, and Paul J. Sally, Jr., Editors, Representation theory and harmonic analysis, 1995 190 Mourad E. H. Ismail, M. Zuhair Nashed, Ahmed I. Zayed, and Ahmed F. Ghaleb, Editors, Mathematical analysis, wavelets, and signal processing, 1995 189 S. A. M. Marcantognini, G. A. Mendoza, M. D. Moran, A. Octavio, and W. O. Urbina, Editors, Harmonic analysis and operator theory, 1995 188 Alejandro Adem, R. James Milgram, and Douglas C. Ravenel, Editors, Homotopy theory and its applications, 1995 187 G. W. Brumfiel and H. M. Hilden, SL(2) representations of finitely presented groups, 1995 186 Shreeram S. Abhyankar, Walter Feit, Michael D. Fried, Yasutaka Ihara, and Helmut Voelklein, Editors, Recent developments in the inverse Galois problem, 1995 185 Raul E. Curto, Ronald G. Douglas, Joel D. Pincus, and Norberto Salinas, Editors, Multivariable operator theory, 1995 (See the AMS catalog for earlier titles) Editorial Board Dennis DeTurck, managing editor Andy Magid Michael Vogelius Clark Robinson Peter M. Winkler

This volume contains proceedings from the Bers Colloquium held at the Graduate School and University Center of the City University of (CUNY) on October 19-20, 1995.

1991 Mathematics Subject Classification. Primary 3Q-06, 32-06, 35-06.

Library of Congress Cataloging-in-Publication Data Bers Colloquium (1st : 1995 : Graduate School and University Center of CUNY) Lipa's legacy : proceedings of the Bers Colloquium, October 19-20, 1995, Graduate School and University Center of CUNY / Jozef Dodziuk, Linda Keen, editors. p. em. -(Contemporary mathematics, ISSN 0271-4132; 211) Includes bibliographical references (p. ). ISBN 0-8218-0671-8 (alk. paper) 1. Functions-Congresses. I. Dodziuk, Jozef, 1947- . II. Keen, Linda. Ill. Title. IV. Se- ries: Contemporary mathematics (American Mathematical Society) ; v. 211. QA33l.B424 1995 515.9---dc21 97-25848 CIP

Copying and reprinting. Material in this book may be reproduced by any means for educational and scientific purposes without fee or permission with the exception of reproduction by services that collect fees for delivery of documents and provided that the customary acknowledgment of the source is given. This consent does not extend to other kinds of copying for general distribution, for advertising or promotional purposes, or for resale. Requests for permission for commercial use of material should be addressed to the Assistant to the Publisher, American Mathematical Society, P. 0. Box 6248, Providence, Rhode Island 02940-6248. Requests can also be made by e-mail to reprint-permission~ams.org. Excluded from these provisions is material in articles for which the author holds copyright. In such cases, requests for permission to use or reprint should be addressed directly to the author(s). (Copyright ownership is indicated in the notice in the lower right-hand corner of the first page of each article.) © 1997 by the American Mathematical Society. All rights reserved. The American Mathematical Society retains all rights except those granted to the United States Government. Printed in the United States of America. § The paper used in this book is acid-free and falls within the guidelines established to ensure permanence and durability. Visit the AMS home page at URL: http: I /wr;n;r. ams. org/ 10 9 8 7 6 5 4 3 2 1 02 01 00 99 98 97 Editorial Board Dennis DeTurck, managing editor Andy Magid Michael Vogelius Clark Robinson Peter M. Winkler

This volume contains proceedings from the Bers Colloquium held at the Graduate School and University Center of the City University of New York (CUNY) on October 19-20, 1995.

1991 Mathematics Subject Classification. Primary 30--06, 32-06, 35-06.

Library of Congress Cataloging-in-Publication Data Bers Colloquium (1st: 1995 : Graduate School and University Center of CUNY) Lipa's legacy: proceedings of the Bers Colloquium, October 19-20, 1995, Graduate School and University Center of CUNY / J6zef Dodziuk, Linda Keen, editors. p. cm. - (Contemporary mathematics, ISSN 0271-4132 ; 211) Includes bibliographical references (p. ). ISBN 0-8218-0671-8 (alk. paper) 1. Functions-Congresses. I. Dodziuk, J6zef, 1947-. II. Keen, Linda. III. Title. IV. Se- ries: Contemporary mathematics (American Mathematical Society) ; v. 211. QA331.B424 1995 515.9-dc21 97-25848 CIP

Copying and reprinting. Material in this book may be reproduced by any means for educational and scientific purposes without fee or permission with the exception of reproduction by services that collect fees for delivery of documents and provided that the customary acknowledgment of the source is given. This consent does not extend to other kinds of copying for general distribution, for advertising or promotional purposes, or for resale. Requests for permission for commercial use of material should be addressed to the Assistant to the Publisher, American Mathematical Society, P. O. Box 6248, Providence, Rhode Island 02940-6248. Requests can also be made bye-mail to [email protected]. Excluded from these provisions is material in articles for which the author holds copyright. In such cases, requests for permission to use or reprint should be addressed directly to the author(s). (Copyright ownership is indicated in the notice in the lower right-hand corner of the first page of each article.) © 1997 by the American Mathematical Society. All rights reserved. The American Mathematical Society retains all rights except those granted to the United States Government. Printed in the United States of America. €?! The paper used in this book is acid-free and falls within the guidelines established to ensure permanence and durability. Visit the AMS home page at URL: http://www . ams. orgl 10987654321 02 01 00 99 98 97 Contents

Preface vii Computing the Douady-Earle extension WILLIAM ABIKOFF AND TAIPING YE 1 Selected problems in Kleinian groups and hyperbolic geometry ARA BASMAJIAN 9 The law of the iterated logarithm for Kleinian groups CHRISTOPHER J. BISHOP AND PETER W. JONES 17 Weil-Petersson geometry and determinant bundles on inductive limits of moduli spaces INDRANIL BISWAS AND SUBHASHIS NAG 51 Iteration of mapping classes on a Bers slice: examples of algebraic and geometric limits of hyperbolic 3-manifolds JEFFREY F. BROCK 81 Lattice constants and a lemma of Zagier C. CAO, F. W. GEHRING, AND G. J. MARTIN 107 The quasi-Fuchsian space of four-punctured spheres YUNGYEN CHIANG 121 Approximating L2 invariants of amenable covering spaces: A heat kernel approach JOZEF DODZIUK AND VARGHESE MATHAI 151 Random walks on circle packings TOMASZ DUBEJKO 169 Some maximal holomorphic motions CLIFFORD J. EARLE 183 Vector fields for holomorphic motions of closed sets CLIFFORD J. EARLE, FREDERICK P. GARDINER, AND NIKOLA LAKIC 193 Theta constant identities with applications to combinatorial number theory H. M. FARKAS AND 1. KRA 227 The topology of real projective structures DANIEL M. GALLO 253

v vi CONTENTS

A discreteness condition for subgroups of PSL(2, C) JANE GILMAN 261 On principally polarized abelian varieties and Riemann surfaces associated to prisms and pyramids VICTOR GONZALEZ-AGUILERA AND RUBI E. RODRIGUEZ 269 Schur products of certain polynomials A. HINKKANEN 285 Inverse problems for fluid flows with a moving boundary YURI E. HOHLOV AND DMITRY Y. KLEINBOCK 297 On rigidity of one-dimensional maps YUNPING JIANG 319 Twist maps and Aubry-Mather sets Y. KATZNELSON AND D. S. ORNSTEIN 343 How to bend pairs of punctured tori LINDA KEEN AND 359 , complex analyst 1. KRA AND B. MASK IT 389 Strebel points NIKOLA LAKIC 417 Remarks on m-symmetric Riemann surfaces BERNARD MASKIT 433 Quasi-conformal mappings of Carnot-Caratheodory spaces G. D. MOSTOW 447 Lipman Bers and partial differential equations LOUIS NIRENBERG 455 Irrational rotations DAVID J. PINCHBECK 463

/1,- homeomorphisms URI SREBRO AND EDUARD YAKUBOV 473 Preface

The mathematical works of Lars Ahlfors and Lipman Bers are fundamental and lasting. They have influenced and altered the development of twentieth cen- tury mathematics. The personalities of these two scientists helped create a mathe- matical family and have had a permanent positive effect on a whole generation of mathematicians. Their mathematical heritage continues to lead succeeding genera- tions. In the fall of 1994 some members of this family decided to inaugurate a series of conferences, The Bers Colloquium, to be held every three years, whose theme would be a topic in the Ahlfors-Bers mathematical tradition, broadly interpreted. There was a precedent for such a series, the Nevanlinna Colloquium. It is held approximately every three years (usually in Europe) and it has helped to maintain contact among researchers working in classical complex analysis and function the- ory. Whereas the Nevanlinna Colloquia cover most of classical complex analysis, the intention of the Bers Colloquia is to cover topics in modern function theory such as Riemann surfaces, Kleinian groups and Teichmuller theory as well as complex dynamics, hyperbolic geometry and 2 and 3-dimensional manifolds. The first colloquium was held on October 19-20, 1995, at the Graduate Center of CUNY in New York. It coincided roughly with the second anniversary of Bers' death. The organizing committee consisted of Bill Abikoff, J6zef Dodziuk, Jane Gilman, Linda Keen (chair), and Irwin Kra. The inaugural meeting, retrospective in nature, focused on the influence of Lipman Bers' work on current research in complex analysis applicable to hyperbolic geometry and dynamics. There were six lectures and much time for informal mathematical discussion. This volume contains papers by most of the speakers and many of the participants. The approximately 120 participants came from far and wide; there were people from Asia, Europe and South America as well as from all corners of the United States. Many of them were graduate students who constitute the next generation carrying on the Ahlfors-Bers tradition. Three of the lectures were retrospective and three were on recent research much influenced by Bers' work. The broad range of papers in this volume indicates how strong and far reaching Bers' influence has been. The topics represented include Teichmuller theory, Kleinian groups, higher dimensional hyperbolic geometry, ge- ometry of numbers, circle packings, theory of discrete groups, classical complex function theory, one dimensional dynamics, fluid dynamics, quasiconformal map- pings in higher dimensions, partial differential equations and classical algebraic geometry.

vii viii PREFACE

Lars Ahlfors passed away in October 1996. Future meetings will be called Ahlfors-Bers colloquia. The second colloquium will be devoted to recent work and new directions in Teichmi.iller theory and Kleinian groups. It will be held at SUNY, Stony Brook in the fall of 1998.

J6zef Dodziuk and Linda Keen Lipa's Legacy J6zef Dodziuk and Linda Keen, Editors

The mathematical works of Lars Ahlfors and Lipman Bers are fundamental and lasting. They have influenced and altered the development of twentieth century mathematics. The personal- ities of these two scientists helped create a mathematical family and have had a permanent pos- itive effect on a whole generation of mathematicians. Their mathematical heritage continues to lead succeeding generations. In the fall of 1994, one year after Bers' death, some members of this family decided to inaugurate a series of conferences, The Bers Colloquium, to be held every three years. The theme was to be a topic in the Ahlfors-Bers mathematical tradition, broadly interpreted. Ahlfors died a year after the first colloquium; future colloquia in this series will be called The Ahlfors-Bers Colloquium. The first colloquium was held in October 1995 at the Graduate Center, CUNY in New York. It coincided roughly with the second anniversary of Bers' death. There were six lectures and much informal mathematical discussion. This volume contains papers by the speakers and many of the participants. The broad range of papers indicate how strong and far reaching Bers' influence has been. The topics represented in the book include Teichmtiller theory, Kleinian groups, higher dimensional hyperbolic geometry, geometry of numbers, circle packings, theo- ry of discrete groups, classical complex function theory, one dimensional dynamics, fluid dynamics, quasiconformal mappings in higher dimensions, partial differential equations, and classical algebraic geometry. Features: • Twenty-seven very high-level papers on related topics • Open problems • Expository articles

ISBN Q-8218-0671-8

9 780821 806715