Geometry and Dynamics of Groups and Spaces: In Memory of Alexander Reznikov PDF
Preview Geometry and Dynamics of Groups and Spaces: In Memory of Alexander Reznikov
Progress in Mathematics Volume 265 Series Editors H. Bass J. Oesterlé A. Weinstein Geometry and Dynamics of Groups and Spaces In Memory of Alexander Reznikov Mikhail Kapranov Sergiy Kolyada Yuri I. Manin Pieter Moree Leonid Potyagailo Editors Birkhäuser Basel · Boston · Berlin Editors: Mikhail Kapranov Yuri Ivanovich Manin Department of Mathematics Pieter Moree Yale University Max-Planck-Institut für Mathematik 10 Hillhouse Avenue Vivatsgasse 7 New Haven CT 06520, USA 53111 Bonn, Germany e-mail: [email protected] e-mail: [email protected] [email protected] Sergiy Kolyada Leonid Potyagailo Institute of Mathematics UFR de Mathématiques National Academy of Sciences of Ukraine Université de Lille 1 Tereshchenkivs’ka, 3 59655 Villeneuve d’Ascq Cedex, France 01601 Kyiv, Ukraine e-mail: [email protected] e-mail: [email protected] 2000 Mathematics Subject Classification: 08, 11, 18, 20, 22, 37, 51, 54, 55, 60 Library of Congress Control Number: 2007938509 Bibliographic information published by Die Deutsche Bibliothek. Die Deutsche Bibliothek lists this publication in the Deutsche Nationalbibliografie; detailed bibliographic data is available in the Internet at http://dnb.ddb.de ISBN 978-3-7643-8607-8 Birkhäuser Verlag AG, Basel · Boston · Berlin This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, re-use of illustrations, broadcasting, reproduction on microfilms or in other ways, and storage in data banks. For any kind of use whatsoever, permission from the copyright owner must be obtained. © 2008 Birkhäuser Verlag AG Basel · Boston · Berlin P.O. Box 133, CH-4010 Basel, Switzerland Part of Springer Science+Business Media Printed on acid-free paper produced from chlorine-free pulp. TCF ∞ Printed in Germany ISBN 978-3-7643-8607-8 e-ISBN 978-3-7643-8608-5 9 8 7 6 5 4 3 2 1 www.birkhauser.ch Contents Preface.......................................................................vii Short Recollections about Sasha..............................................xv Pieter Moree Sipping Tea with Sasha ............................................... xxv Part I. Analytic Topology of Groups, Actions, Strings and Varieties Alexander Reznikov Analytic Topology of Groups, Actions, Strings and Varieties ............. 3 Part II. Research Articles J. Vernon Armitage and John R. Parker Jørgensen’s Inequality for Non-Archimedean Metric Spaces ............. 97 Jean-Michel Bismut The Hypoelliptic Dirac Operator ..................................... 113 Dennis V. Borisov and Yuri I. Manin Generalized Operads and Their Inner Cohomomorphisms ............. 247 Paul Bressler, Alexander Gorokhovsky, Ryszard Nest and Boris Tsygan Chern Character for Twisted Complexes .............................. 309 Alexandre I. Danilenko (C,F)-Actions in Ergodic Theory ..................................... 325 Thomas Delzant and Rostislav Grigorchuk Homomorphic Images of Branch Groups, and Serre’s Property (FA) ... 353 H´el`ene Esnault, Phu`ng Hˆo Hai and Xiaotao Sun On Nori’s Fundamental Group Scheme ................................ 377 vi Contents Alexander Fel’shtyn and Daciberg L. Gon¸calves The Reidemeister Number of Any Automorphism of a Baumslag–Solitar Group is Infinite .................................... 399 Alexander B. Goncharov Pentagon Relation for the Quantum Dilogarithm and Quantized Mcyc 415 0,5 Brendan Guilfoyle and Wilhelm Klingenberg Geodesic Flow on the Normal Congruence of a Minimal Surface ....... 429 Jaya N. Iyer and Carlos T. Simpson The Chern Character of a Parabolic Bundle, and a Parabolic Corollary of Reznikov’s Theorem ..................................... 439 Michael Kapovich Kleinian Groups in Higher Dimensions ................................ 487 Volodymyr Lyubashenko and Oleksandr Manzyuk A∞-bimodules and Serre A∞-functors ................................ 565 Vitali D. Milman Geometrization of Probability ........................................ 647 Masanori Morishita Milnor Invariants and l-Class Groups ................................. 669 Jean-Pierre Otal Three Topological Properties of Small Eigenfunctions on Hyperbolic Surfaces .............................................................. 685 Yan Soibelman Quantum p-adic Spaces and Quantum p-adic Groups ................. 697 Mikhail Zaidenberg Convolution Equations on Lattices: Periodic Solutions with Values in a Prime Characteristic Field ........................... 721 Preface Thisbookisacollectionofselectedpapersonrecenttrendsinthestudyofvarious branches of mathematics. Most of the authors who contributed to this volume were invited speakers at the International Conference “Geometry and Dynamics of Groups and Spaces. In Memory of Alexander Reznikov” which was held at the Max Planck Institute for Mathematics (Bonn), Germany, in September 22–29, 2006. Alexander (Sasha)Reznikov (1960–2003)wasa brilliantmathematicianwho died unfortunately very early. This conference in his remembrance focused on topics Sasha made a contribution to. In particular: hyperbolic, differential and complexgeometry;geometricgrouptheory;threedimensionaltopology;dynamical systems. The list of participants: Belolipetsky Mikhail; Bismut Jean-Michel; Boileau Michel; Breuillard Emmanuel; Danilenko Alexandre; Delzant Thomas; Deninger Christopher;EsnaultH´el`ene;Franks John;GalS.R.; GuneschRoland;Hai Phung Ho;KaimanovichVadim;KapovichMichael;KlingenbergWilhelm;Makar-Limanov Leonid; Milman Vitali; Morishita M.; Moree Pieter; Navas Andres; Neretin Yuri; PapazogluPanagiotis;ParkerJohnR.;PortiJoan;RosellenMarkus;SimpsonCar- los; Swenson Eric L.; Szczepanski A.; Tomanov Georges;Tsygan Boris;Verjovsky Sola Alberto; Wang Shicheng; ZakrzweskiWojtek. Talks took place in an informal and constructive atmosphere, and it was a pleasure to see discussions taking place between groups of participants all over the Institute and at all times of day. The exceptional quality of the lectures and the great interest they generated among the conference participants gave us, the organizersoftheconference,theideaofthisbook.Theeditorswishtorecordtheir thanks to the staff at the Max-Planck Institute and to the many researchers who participated for all their efforts in making this such a stimulating experience. Thisvolumeisacollectionofpaperswhichaimsatreflectingthepresentstate of the art in a most active area of research at the intersection of several branches of mathematics. In the volume we include an unpublished manuscript “Analytic TopologyofGroups,Actions,StringsandVarietes”bySasha(thisarticleisposted, essentially, in the form it appeared as the ArXive preprint math.DG/0001135 in January of 2000; the referee corrected obvious typos, updated and added few references, and made several comments in the form of footnotes and italicized remarks) and some short speeches/recollections about Sasha. viii Preface Topics discussed in the book include analytic topology of groups, actions, strings and varieties, category theory, homological algebra, the quantum dilog- arithm, Chow groups, parabolic bundles, quantum spaces over non-archimedean fields, Nori’s fundamental group scheme, Baumslag-Solitar groups, finitely gen- erated branch groups, Serre’s property (FA), hypoelliptic equations, Hodge the- ory,indextheoryandrelatedfixedpointtheorems,determinantsanddeterminant bundles, analytic torsion, Kleinian groups, hyperbolic manifolds, geometric func- tionalanalysis,Cherncharacterandcyclichomology,Milnorinvariantsandl-class groups, geodesic flow, operads, algebras, inner cohomomorphisms, symmetry and deformationsinnoncommutativegeometry,Chebyshev-Dicksonpolynomials,con- volution operators, lattices, discrete harmonic functions, ergodic transformations and rank-one actions, hyperbolic surfaces, eigenfunctions, non-Archimedean met- ric spaces. The articles collected in this volume should be of interest to specialists in suchareasofmathematicsasalgebra,dynamicalsystems,geometry,grouptheory, functional analysis, number theory, probability theory and topology. The broad spectrumoftopicscoveredshouldalsopresentanexcitingopportunityforgraduate students and young researchers working in any of these areas who are willing to put their researchin a wider mathematical perspective. Alexander (Sasha) Reznikov (1960–2003)
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