590 In the Tradition of Ahlfors-Bers, VI 5th Ahlfors-Bers Colloquium March 24–27, 2011 Rice University, Houston, TX Ursula Hamenstädt Alan W. Reid Rubí Rodríguez Steffen Rohde Michael Wolf Editors AmericanMathematicalSociety In the Tradition of Ahlfors-Bers, VI 5th Ahlfors-Bers Colloquium March 24–27, 2011 Rice University, Houston, TX Ursula Hamenstädt Alan W. Reid Rubí Rodríguez Steffen Rohde Michael Wolf Editors 590 In the Tradition of Ahlfors-Bers, VI 5th Ahlfors-Bers Colloquium March 24–27, 2011 Rice University, Houston, TX Ursula Hamenstädt Alan W. Reid Rubí Rodríguez Steffen Rohde Michael Wolf Editors AmericanMathematicalSociety Providence,RhodeIsland EDITORIAL COMMITTEE Dennis DeTurck, managing editor Michael Loss Kailash Misra Martin J. Strauss 2010 Mathematics Subject Classification. Primary20-XX,30-XX,31-XX,32-XX,33-XX, 42-XX, 51-XX, 53-XX, 57-XX, 58-XX. Library of Congress Cataloging-in-Publication Data Ahlfors-BersColloquium(5th: 2011: RiceUniversity)InthetraditionofAhlfors-Bers,VI:5th Ahlfors-BersColloquium,March24–27,2011,RiceUniversity,Houston,TX/UrsulaHamenstadt etal.,editors. p.cm–(ContemporaryMathematics;volume590) Includesbibliographicalreferences. ISBN978-0-8218-7427-1(alk. paper) 1. Functions–Congresses. 2. Riemann surfaces–Congresses. 3. Mappings (Mathematics)– Congresses. 4. Eigenvalues–Congresses. I.Hamensta¨dt,Ursula,editor. II.Title. QA331.A4622011 510–dc23 2013005153 ContemporaryMathematicsISSN:0271-4132(print);ISSN:1098-3627(online) Copying and reprinting. Materialinthisbookmaybereproducedbyanymeansfor edu- cationaland scientific purposes without fee or permissionwith the exception ofreproduction by servicesthatcollectfeesfordeliveryofdocumentsandprovidedthatthecustomaryacknowledg- ment of the source is given. 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(cid:2)∞ Thepaperusedinthisbookisacid-freeandfallswithintheguidelines establishedtoensurepermanenceanddurability. VisittheAMShomepageathttp://www.ams.org/ 10987654321 181716151413 Contents Preface vii Plenary talks ix Spheres in the curve complex Spencer Dowdall, Moon Duchin, and Howard Masur 1 Character varieties and moduli of quiver representations Carlos Florentino and Sean Lawton 9 Periodicity of asymptotic Teichmu¨ller modular transformation Ege Fujikawa 39 Hyperbolization of locally compact non-complete metric spaces Zair Ibragimov 51 Quasiregular maps and the conductivity equation in the Heisenberg group Anton Isopoussu, Kirsi Peltonen, and Jeremy T. Tyson 61 Pushing points on pillowcase covers Chris Judge 77 Entropy of automorphisms, homology and the intrinsic polynomial structure of nilpotent groups Thomas Koberda 87 Continuity of radial and two-sided radial SLE at the terminal point Gregory F. Lawler 101 No proper conjugation for quasiconvex cocompact groups of Gromov hyperbolic spaces Katsuhiko Matsuzaki and Yasuhiro Yabuki 125 A differential formula for extremal length Hideki Miyachi 137 On convergence to the Denjoy-Wolff point in the parabolic case Olena Ostapyuk 153 Construction of a closed hyperbolic surface of arbitrarily small eigenvalue of prescribed serial number Susovan Pal 161 A simple method to compute volumes of even-dimensional Coxeter polyhedra Julien Paupert 167 v vi CONTENTS Some remarks about analytic functions defined on an annulus Pietro Poggi-Corradini 177 On injectivity radius in configuration space and in moduli space Hiroshige Shiga 183 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, Teichmu¨ller theory, hyperbolic geometry, and partial differential equations. However,theworkofAhlforsandBershasimpactedandcreatedinterac- tionswithmanyotherfieldsofmathematics,suchasalgebraicgeometry,dynamical systems, topology, geometricgrouptheory,mathematicalphysics, andnumber the- ory. Recent years have seen a flowering of this legacy with an increased interest in their work. This current volume contains articles on a wide variety of subjects that are central to this legacy. These include papers in Kleinian groups, classical Riemann surface theory, translation surfaces, algebraic geometry and dynamics. The major- ity of the papers present new research, but there are survey articles as well. There have been five volumes of these colloquia previously published by AMS in the Contemporary Mathematics Series: volumes 211, 256, 335, 432, and 510. Ursula Hamenst¨adt Alan W. Reid Rub´ı E. Rodr´ıguez Steffen Rohde Michael Wolf vii Plenary Talks Tadeusz Iwaniec (Syracuse) The Hopf-Laplace Equation Steve Kerckhoff (Stanford) Life without Ahlfors-Bers: Hyperbolic structures in higher dimensions BruceKleiner(NYU)Asymptoticgeometry,finitegenerationoffundamentalgroups, and harmonic functions GregLawler(Chicago)GeometricandFractalPropertiesoftheSchramm-Loewner Evolution Vlad Markovic (Warwick) The Good Pants and the Ehrenpreis Conjecture MartinMoeller(Frankfurt)Non-varyingsumofLyapunovexponentsfortheTeich- muller geodesic flow Eero Saksman (Helsinki) On random conformal weldings Anna Wienhard (Princeton) Domains of Discontinuity for Anosov representations Amie Wilkinson (Northwestern) The Weil Petersson geodesic flow is ergodic ix