CONTEMPORARY MATHEMATICS

262

Crystallographic Groups and Their Generalizations

Workshop Katholieke Universiteit Leuven

Campus Kortrijk, Belgium May 26-28, 1999

Paullgodt Herbert Abels

Yves Felix Fritz Grunewald

Editors

Selected Titles in This Series

262 Paul Igodt, Herbert Abels, Yves Felix, and Fritz Grunewald, Editors, Crystallographic groups and their generalizations, 2000

261 Gregory Budzban, Philip Feinsilver, and Arun Mukherjea, Editors, Probability on algebraic structures, 2000

260 Salvador Perez-Esteva and Carlos Villegas-Blas, Editors, First summer school in analysis and mathematical physics: Quantization, the Segal-Bargmann transform and semiclassical analysis, 2000

259 D. V. Huynh, S. K. Jain, and S. R. Lopez-Permouth, Editors, Algebra and its applications, 2000

258 Karsten Grove, Ib Henning Madsen, and Erik Kjmr Pedersen, Editors, Geometry and topology: Aarhus, 2000

257 Peter A. Cholak, Steffen Lempp, Manuel Lerman, and Richard A. Shore, Editors, Computability theory and its applications: Current trends and open problems, 2000

256 Irwin Kra and Bernard Maskit, Editors, In the tradition of Ahlfors and Bers: Proceedings of the first Ahlfors-Bers colloquium, 2000

255 Jerry Bona, Katarzyna Saxton, and Ralph Saxton, Editors, Nonlinear PDE's, dynamics and continuum physics, 2000

254 Mourad E. H. Ismail and Dennis W. Stanton, Editors, q-series from a contemporary perspective, 2000

253 Charles N. Delzell and James J. Madden, Editors, Real algebraic geometry and ordered structures, 2000

252 Nathaniel Dean, Cassandra M. McZeal, and Pamela J. Williams, Editors, African Americans in Mathematics II, 1999

251 Eric L. Grinberg, Shiferaw Berhanu, Marvin I. Knopp, Gerardo A. Mendoza, and Eric Todd Quinto, Editors, Analysis, geometry, number theory: The Mathematics of Leon Ehrenpreis, 2000

250 Robert H. Gilman, Editor, Groups, languages and geometry, 1999 249 Myung-Hwan Kim, JohnS. Hsia, Yoshiyuki Kitaoka, and Rainer

Schulze-Pillot, Editors, Integral quadratic forms and lattices, 1999· 248 Naihuan Jing and Kailash C. Misra, Editors, Recent developments in quantum

affine algebras and related topics, 1999 247 Lawrence Wasson Baggett and David Royal Larson, Editors, The functional and

harmonic analysis of wavelets and frames, 1999 246 Marcy Barge and Krystyna Kuperberg, Editors, Geometry and topology in

dynamics, 1999 245 Michael D. Fried, Editor, Applications of curves over finite fields, 1999 244 Leovigildo Alonso Tarrfo, Ana Jeremias Lopez, and Joseph Lipman, Studies in

duality on noetherian formal schemes and non-noetherian ordinary schemes, 1999 243 Tsit Yuan Lam and Andy R. Magid, Editors, Algebra, K-theory, groups, and

education, 1999 242 Bernheim Booss-Bavnbek and Krzysztof Wojciechowski, Editors, Geometric

aspects of partial differential equations, 1999 241 Piotr Pragacz, Michal Szurek, and Jaroslaw Wisniewski, Editors, Algebraic

geometry: Hirzebruch 70, 1999 240 Angel Carocca, Victor Gonzalez-Aguilera, and Rubi E. Rodriguez, Editors,

Complex geometry of groups, 1999 239 Jean-Pierre Meyer, Jack Morava, and W. Stephen Wilson, Editors, Homotopy

invariant algebraic structures, 1999 (Continued in the back of this publication)

http://dx.doi.org/10.1090/conm/262

Crystallographic Groups and Their Generalizations

CoNTEMPORARY MATHEMATICS

262

Crystallographic Groups and Their Generalizations

Workshop Katholieke Universiteit Leuven

Campus Kortrijk, Belgium May 26-28, 1999

Paullgodt Herbert Abels

Yves Felix Fritz Grunewald

Editors

American Mathematical Society Providence. Rhode Island

Editorial Board

Dennis DeTurck, managing editor

Andreas Blass Andy R. Magid Michael Vogelius

The Proceedings of a conference on Crystallographic Groups and Their Generalizations II, held at Katholieke Universiteit Leuven, Campus Kortrijk, Belgium, May 26-28, 1999, supported by the Fund for Scientific Research Flanders, the F. W. 0. Scientific Research Network W0.011.96N, and the Katholieke Universiteit Leuven Campus Kortrijk (Faculty of Sciences).

2000 Mathematics Subject Classification. Primary 20H15, 20E42, 22E25, 20F16, 57S30, 55P60, 53C50, 17B30, 51E24.

Library of Congress Cataloging-in-Publication Data Crystallographic groups and their generalizations : workshop, Katholieke Universiteit Leuven Campus Kortrijk, Belgium, May 26-28, 1999 / Paul Igodt ... [et a!.], editors.

p. em. - (Contemporary mathematics, ISSN 0271-4132; 262) ISBN 0-8218-2001-X (alk. paper) 1. Crystallography, Mathematical-Congresses. 2. Group theory-Congresses. I. Igodt,

Paul, 1956- H. Contemporary mathematics (American Mathematical Society) ; v. 262. QD905.2 .C83 2000 548'.7-dc21 00-044744

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Contents

Preface ix

List of participants xi

Titles of talks and posters xiv

Tores affines YVES BENOIST 1

On structures preserved by idempotent transformations of groups and homotopy types CARLES CASACUBERTA 39

Affine Schottky groups and crooked tilings VIRGINIE CHARETTE AND WILLIAM M. GOLDMAN 69

Polynomial structures on polycyclic groups: Recent developments KAREL DEKIMPE 99

Problems on the geometry of finitely generated solvable groups BENSON FARB AND LEE MOSHER 121

Flat Lorentz 3-manifolds and cocompact Fuchsian groups WILLIAM M. GOLDMAN AND GREGORY A. MARGULIS 135

Varieties of discontinuous groups OLIVER BAUES 147

Affine cohomology classes for filiform Lie algebras DIETRICH BURDE 159

Hyperbolic automorphisms for groups in 7(4, 2) CHARLES CASSIDY, NEIL KENNEDY, AND DIRK SCEVENELS 171

VariE~tes projectives a holonomie dans le groupe Aff+ (R) SERGE DUPONT 177

Classification of homogeneous complex affinely flat surfaces with compact quotients and applications to complex projective structures YOSHINOBU KAMISHIMA 195

On the fundamental groups of compact complete quaternionic affinely flat 2-manifolds YOSHINOBU KAMISHIMA AND TETSURO UDONO

vii

225

viii CONTENTS

An obstruction to the existence of Anosov diffeomorphisms on infra-nilmanifolds WIM MALFAIT 233

Genus and localization of virtually nilpotent groups NIAMH O'SULLIVAN 253

Immeubles affines: construction par les normes et etude des isometries ANNE PARREAU 263

Isoclinic spheres and flat homogeneous pseudo-Riemannian manifolds JOSEPH A. WOLF 303

Preface

"Crystallographic Groups and their Generalizations II" was held at Kortrijk from May 26 to May 28, 1999. The main aim of the meeting was to report on and to discuss recent developments, achievements, and problems concerning crys-tallographic groups and all concepts which can be seen as generalizations of them. With 7 one-hour invited talks, 22 short communications, several posters, and all together over 65 participants, very internationally composed, the organisers have the fair impression that the conference has met these aims.

As expressed by several participants, the general feeling was that of a suc-cessful and stimulating meeting, presenting high-level research mathematics and a discussion forum for many young researchers.

It is clear today again, upon proposing these proceedings containing survey as well as original research papers, that also in this respect the workshop can be called a success, in fact quite beyond the organisers' initial estimations. It is particularly encouraging to see that even a few of the articles contain or at least announce results obtained (in cooperation) during the conference itself. All papers have been thoroughly refereed, which resulted more than once in most valuable suggestions to improve the quality of the manuscripts.

On behalf of the organizing scientific committee, I want to express gratitude to the invited speakers who, by their excellent talks combining insight, overview, and lists of challenging problems, created the mathematical outline of the conference, and moreover, of forthcoming research. This gratitude extends to all speakers and authors for their contributions and manuscripts and to all participants for the very positive and cooperative atmosphere during the conference. We express the hope that this meeting created real opportunities to them for shaping future research work. Our thanks go also to the K. U. Leuven Campus Kortrijk, the Fund for Scientific Research (Flanders), and its Research Network "Fundamental methods and techniques in mathematics" for their financial support, and to the American Mathematical Society for their interest in the publication of this volume.

It is with gratitude and recognition to all the authors of proceedings papers and to the referees that we present this volume and invite all readers to enter the subject and/or enjoy the new results.

On behalf of the organising scientific committee Paul Igodt

K. U. Leuven Campus Kortrijk March 2000

ix

List of participants

Herbert Abels: Fakultiit fur Mathematik, Universitat Bielefeld, Postfach 100131, D-33501 Bielefeld, Germany; abels@mathematik.uni-bielefeld.de

Thierry Barbot: Ecole Normale Superieure de Lyon, UMPA- UMR 5669 du CNRS, 46 allee d'Italie, 69367 Lyon, France; barbot@umpa.ens-lyon.fr

Gemma Bastardas: Dept. de Matematiques, Facultat de Ci{mcies, Universi-tat Autonoma de Barcelona, 08193 Bellaterra, Spain; gemmab@mat.uab.es

Oliver Baues: Dept. Mathematik, ETH-Zentrum, CH-8092 Zurich, Switzer-land; oliver@math.ethz.ch

Igor Belegradek: Dept. of Mathematics and Statistics, McMaster Univer-sity, 1280 Main Street West, Hamilton, ON, L8S 4K1, Canada; belegi@icarus.math.mcmaster .ca

Yves Benoist: Ecole Normale Superieure, 45 Rue d'Ulm, 75005 Paris, France; Yves.Benoist@ens.fr

Peter Boonants: Faculteit E.T.E.W., Monitoraat K.U.Leuven, Dekenstraat 2,3000 Leuven, Belgium; Peter.Boonants@econ.kuleuven.ac.be

Natasha Bozovic: Dept. Mathematics and Computer Science, San Jose State University, 208 MacQuarrie Hall, San Jose, California, USA; bozovic@mathcs.sjsu.edu (also n. bozovic@uumail.de)

Martin Bridson: Mathematical Institute, University of Oxford, 24-29 St. Giles', Oxford OX1 3LB, UK; bridson@maths.ox.ac.uk

Dietrich Burde: Mathematisches Institut, Universitat Dusseldorf, Univer-sitat-strasse 1, D-40225 Dusseldorf, Germany; dietrich@math. uni-duesseldorf.de

Leo Butler: Dept. of Mathematics, Jeffery Hall, Queen's University, Kingston, ON, K7L 3N6, Canada; lbutler@mast.queensu.ca

Carles Casacuberta: Dept. de Matematiques, Universitat Autonoma de Barcelona, E-08193 Bellaterra, Spain; casac@manwe.mat.uab.es

Heekyung Cho: Dept. of Mathematics, Seoul National University, Kwanak-gu, Shinlim-dong, 151-742 Seoul, Korea; hkcho@math.snu.ac.kr

Suhyoung Choi: Dept. of Mathematics, Seoul National University, Kwanak-gu, Shinlim-dong, 151-742 Seoul, Korea; shchoi@math.snu.ac.kr

Fran~,;oise Dal'bo: Dept. de Mathematiques, Universite Rennes 1, Campus Beaulieu, 35000 Rennes, France; Francoise.Dalbo@univ-rennesl.fr

Alexei Davydov: Dept. of Mathematics, National University of Singapore, 10 Kent Ridge Crescent, 119260 Singapore; matad@nus.edu.sg

Karel Dekimpe: K.U.Leuven Campus Kortrijk, Universitaire Campus, B-8500 Kortrijk, Belgium; Karel.Dekimpe@kulak.ac. be

An Descheemaeker: K.U.Leuven Campus Kortrijk, Universitaire Campus, B-8500 Kortrijk, Belgium; An.Descheemaeker@kulak.ac.be

Cornelia Drutu: UFR de Mathematiques, Universite de Lille 1, 59655 Vil-leneuve d'Ascq, France; drutu@gat.univ-lillel.fr

Serge Dupont: Universite Paris 7; Avenue de Flandres 119, 75019 Paris, France; sdupont@math.jussieu.fr

Benson Farb: Dept. of Mathematics, The University of Chicago, Eckhart Hall,5734 S.University Avenue,Chicago,Il60637, USA;farb@math.uchicago.edu

Yves Felix: Dept. de Mathematiques, Universite Catholique de Louvain, Chemin du Cyclotron 2, 1348 Louvain-la-Neuve, Belgium; felix@agel.ucl.ac.be

xi

xii LIST OF PARTICIPANTS

William M. Goldman: Dept. of Mathematics, University of Maryland, Col-lege Park, MD 20742, USA; wmg@math.umd.edu

Fritz Grunewald: Mathematisches Institut, Universitat Dusseldorf, Univer-sitat-strasse 1, D-40225 Dusseldorf, Germany; fritz@math. uni-duesseldorf.de

Manfred Hartl: ISTV, Universite de Valenciennes, B.P. 311, F-59304 Valen-ciennes, France; hartl@univ-valenciennes.fr

Toshiaki Hattori: Laboratoire de Topologie, Departement de Mathematiques, Batiment 425, Universite de Paris-Sud, F-91405 Orsay Cedex, France; hat tori @topo.math. u-psud.fr

Paul lgodt: K.U.Leuven Campus Kortrijk, Universitaire Campus, B-8500 Ko-rtrijk, Belgium; Paul.Igodt@kulak.ac.be

Yoshinobu Kamishima: Dept. of Mathematics, Tokyo Metropolitan Uni-versity, Minami-Osawa 1-1, Hachioji, Tokyo 192-0397, Japan; kamiCQlcomp.metro-u.ac.jp

Hyun Jung Kim: Dept. of Mathematics, University of Manchester, Oxford Road, Manchester M13 9PL, UK; hjkim@ma.man.ac.uk

Kyunghee Kim: 207-43 Cheongryangri-dong, Dongdaemun-gu, KIAS, Seoul 130-012, Korea; khkim@kias.re.kr

Daan Krammer: Mathematisches Institut, Universitat Basel, Rheinsprung 21, CH-4051 Basel, Switzerland; krammer@math.unibas.ch

Thilo Kuessner: Math. Fakultat, Universitat Tubingen, Auf der Morgen-stelle 10, D-72076 Tubingen, Germany; kuesnner@picard. ups-tlse.fr

Joonhyoung Lee: Dept. of Mathematics, Seoul National University, Gua-nakgu, Shillim-dong San 56-1, Seoul, Korea; jhJee@math.snu.ac.kr

Isabelle Liousse: UMRAGAT, UFRMathematiques, UniversiteLille 1, 59655 Villeneuve d'Ascq, France; liousse@gat.univ-lillel.fr

Nicolas Louvet: Institut de Mathematiques, Universite de Neuchatel, Case Postale 2, CH-2007 Neuchatel, Switzerland; nicolas.louvet@maths.unine.ch

Wim Malfait: K.U.Leuven Campus Kortrijk, Universitaire Campus, B-8500 Kortrijk, Belgium; Wim.Malfait@kulak.ac. be

Gregory Margulis: Dept. of Mathematics, Yale University, 10 Hillhouse Av-enue, New Haven, Connecticut 06520-8283, USA; margulis@math.yale.edu

A. W. Mason: Mathematics Department, Glasgow University, Glasgow G12, 8QW, Scotland, UK; awm@maths.gla.ac.uk

Franc;ois Maucourant: Universite de Lille 1, 134 Rue de Paris, 91120 Palaiseau, France; fmaucour@aglae. univ-lille1.fr

Igor Mineyev: Max-Planck-Institute fur Mathematik, Vivatsgasse 7, 53111 Bonn, Germany; mineyev@mpim-bonn.mpg.de

Lee Mosher: Dept. of Mathematics and Computer Sciences, Rutgers Uni-versity, Newark NJ 07102, USA; mosher@andromeda.rutgers.edu

Guennadi Noskov: Mathematisches Institut, Universitat Dusseldorf, Uni-versitiit-strasse 1, D-40225 Dusseldorf, Germany; noskov@ bart. cs. uni-d uesseldorf. de

Hee Oh: Dept. of Mathematics, Hebrew University, Jerusalem 91904, Israel; heeoh@math.huji.ac.il

Niamh O'Sullivan: Dept. of Mathematics, N.U.I. Galway, Galway, Ireland; niamh.eleanor.osullivan@ucg.ie

LIST OF PARTICIPANTS xiii

Joannis Panagopoulos: Dept. of Mathematics, Section of Algebra and Ge-ometry, University of Athens, Panepistemiopolis, Athens 15784, Greece; ipanagop@atlas. uoa.gr

Paul-Eugene Parent: Dept. de Mathematiques, Universite Catholique de Louvain, Chemin du Cyclotron 2, 1348 Louvain-la-Neuve, Belgium; parent@agel. ucl.ac. be

Anne Parreau: Dept. de Mathematiques, Universite Paris-Sud, 91405 Orsay Cedex, France; parreau@topo.math.u-psud.fr

Frederic Paulin: Dept. de Mathematiques, Universite Paris XI, 91405 Orsay Cedex, France; Frederic.Paulin@math. u-psud.fr

Leonid Potyagailo: UFR de Mathematiques, Universite de Lille 1, 59655 Villeneuve d' Ascq cedex, France; Leonid.Potyagailo@univ-lillel.fr

Evagelos Raptis: Dept. of Mathematics, University of Athens, Panepis-temiopolis 15784, Greece; eraptis@atlas.uoa.gr

Juan Pablo Rossetti: FaMAF, Ciudad Universitaria, 5000 Cordoba, Argentina; rossetti@mate.uncor.edu

Dirk Scevenels: Dept. Wiskunde, K.U.Leuven, Celestijnenlaan 200 B, B-3001 Heverlee, Belgium; Dirk.Scevenels@wis.kuleuven.ac.be

Eveline Soetaert: K.U.Leuven Campus Kortrijk, Universitaire Campus, B-8500 Kortrijk, Belgium; Eveline.Soetaert@kulak.ac.be

Gregory Soifer: Dept. of Mathematics and Computer Science, Bar-Ilan Uni-versity, Ramat Gan 52900, Israel; soifer@macs.biu.ac.il

Andrzej Strojnowski: Institute of Mathematics, Warsaw University, ul. Ba-nacha 2, 02-097 Warszawa, Poland; stroa@mimuw.edu.pl

Andrzej Szczepanski: Institute of Mathematics, University of Gdansk, ul. Wita Stwosza 57, 80-955 Gdansk, Poland; matas@paula.univ.gda.pl

Olympia Talelli: Dept. of Mathematics, University of Athens, Athens 15784, Greece; otalelli@atlas. uoa.gr

Paulo Tirao: Mathematisches Institut, Universitat Dusseldorf, D-40225 Dusseldorf, Germany; ptirao@bart.cs.uni-duesseldorf.de

Aristide Tsemo: Dept. de Mathematiques, Universite de Montpellier II, Place Eugene Bataillon, 34095 Montpellier, France; tsemo@math.univ-montp2.fr

Alain Valette: Institut de Mathematiques, Universite de Neuchatel, Rue Emile Argand 11, CH-2007 Neuchatel, Switzerland; alain.valette@maths.unine.ch

Dimitrios Varsos: Dept. of Mathematics, University of Athens, Panepis-temiopolis 15784, Greece; dvarsos@atlas.uoa.gr

Alfred Warrinnier: Dept. Wiskunde, K.U.Leuven, Celestijnenlaan 200 B, B-3001 Heverlee, Belgium: alfred. warrinnier@wis.kuleuven.ac. be

Dave Witte: Dept. of Mathematics, 401 Math Sciences, Oklahoma State University, Stillwater, OK 74078, USA; dwitte@math.okstate.edu

Joseph A. Wolf: Dept. of Mathematics, University of California, Berkeley, CA 94720-3840, USA; jawolf@math.berkeley.edu

xiv

Titles of talks and posters Invited talks

• Y. Benoist: Automorphisms of convex cones. • M. Bridson: Complexity and the geometry of groups. • C. Casacuberta: Idempotent constructions preserving virtually nilpotent groups. • K. Dekimpe: Polynomial structures: recent developments. • B. Farb: The asymptotic geometry of solvable groups. • W. Goldman: Crooked tilings and affine Schottky groups. • G. Margulis: Multiplicative ergodic theorem and nonpositively curved spaces.

Short communications • T. Barbot: Goldman flows and flag manifolds. • 0. Baues: Varieties of discontinuous groups. • I. Belegradek: Precompactness for finite volume negatively curved manifolds. • D. Burde: Second cohomology spaces for filiform Lie algebras. • L. Butler: Liouville-integrable geodesic flows on a fami ly of 2-step nilman-

ifolds. • S. Choi: Flat real projective and affine structures on 3-manifolds. • A. Descheemaeker: Localization of virtually nilpotent groups. • S. Dupont: Projective structures on three-dimensional solvmanifolds. • Y. Kamishima: Homogeneous complex affine fiat surfaces and its compact

quotients. • K. Kim: Left invariant affinely fiat structures on solvable Lie groups. • T. Kuessner: Ideal reflection groups and Gromov's simplicial volume. • J. Lee: Affine manifolds with partial dilations. • W. Malfait: An obstruction to the existence of Anosov diffeomorphisms on

infra-nilmanifolds. • I. Mineyev: Straightening and bounded cohomology of hyperbolic groups. • H. Oh: Compact Clifford-Klein forms of homogeneous spaces of S0(2, n). • A. Parreau: Compactification of representation spaces of discrete groups. • L. Potyagailo: Hierarchical accessibility of finitely presented groups and ap-

plications to geometry. • J.P. Rossetti: Isospectral Hantzsche- Wendt manifolds. • G. Soifer: Crystallographic semigroups and Margulis' conjecture. • 0. Talelli: On groups which act freely and properly on finite dimensional

homotopy spheres. • A. Valette: Markov operators on solvable Baumslag-Solitar groups. • J. Wolf: Flat homogeneous pseudo-Riemannian manifolds.

Posters • G. Bastardas, C. Casacuberta, T.G. Choo: Localization of free groups. • N. Bozovic, I. Bozovic, M. Damnjanovic: Line group symmetry and optical

spectra of carbon nanotubes. • H. Cho: A classification of convex affine 3-manifolds. • A. Davydov, J. Berrick: Multiplicative structure of the cohomology of the

discrete Heisenberg group. • G. Noskov: Walls in a Moussong complex of a Coxeter group. • N. O'Sullivan: Localization of virtually nilpotent groups. • D. Scevenels, C. Cassidy, N. Kennedy: Hyperbolic automorphisms for groups

in T(4, 2).

TITLES OF TALKS AND POSTERS XV

• A. Strojnowski: Crystallographic group rings. • A. Tsemo: Dynamique des varietes affines.

Selected Titles in This Series (Continued from the front of this publication)

238 Gui-Qiang Chen and Emmanuele DiBenedetto, Editors, Nonlinear partial differential equations, 1999

237 Thomas Branson, Editor, Spectral problems in geometry and arithmetic, 1999 236 Bruce C. Berndt and Fritz Gesztesy, Editors, Continued fractions: From analytic

number theory to constructive approximation, 1999 235 Walter A. Carnielli and Itala M. L. D'Ottaviano, Editors, Advances in

contemporary logic and computer science, 1999 234 Theodore P. Hill and Christian Houdre, Editors, Advances in stochastic

inequalities, 1999 233 Hanna Nencka, Editor, Low dimensional topology, 1999 232 Krzysztof Jarosz, Editor, Function spaces, 1999 231 Michael Farber, Wolfgang Liick, and Shmuel Weinberger, Editors, Tel Aviv

topology conference: Rothenberg Festschrift, 1999 230 Ezra Getzler and Mikhail Kapranov, Editors, Higher category theory, 1998 229 Edward L. Green and Birge Huisgen-Zimmermann, Editors, Trends in the

representation theory of finite dimensional algebras, 1998 228 Liming Ge, Huaxin Lin, Zhong-Jin Ruan, Dianzhou Zhang, and Shuang Zhang,

Editors, Operator algebras and operator theory, 1999 227 John McCleary, Editor, Higher homotopy structures in topology and mathematical

physics, 1999 226 Luis A. Caffarelli and Mario Milman, Editors, Monge Ampere equation:

Applications to geometry and optimization, 1999 225 Ronald C. Mullin and Gary L. Mullen, Editors, Finite fields: Theory, applications,

and algorithms, 1999 224 Sang Geun Hahn, Hyo Chul Myung, and Efim Zelmanov, Editors, Recent

progress in algebra, 1999 223 Bernard Chazelle, Jacob E. Goodman, and Richard Pollack, Editors, Advances

in discrete and computational geometry, 1999 222 Kang-Tae Kim and Steven G. Krantz, Editors, Complex geometric analysis in

Pohang, 1999 221 J. Robert Dorroh, Gisele Ruiz Goldstein, Jerome A. Goldstein, and Michael

Mudi Tom, Editors, Applied analysis, 1999 220 Mark Mahowald and Stewart Priddy, Editors, Homotopy theory via algebraic

geometry and group representations, 1998 219 Marc Henneaux, Joseph Krasil'shchik, and Alexandre Vinogradov, Editors,

Secondary calculus and cohomological physics, 1998 218 Jan Mandel, Charbel Farhat, and Xiao-Chuan Cai, Editors, Domain

decomposition methods 10, 1998 217 Eric Carlen, Evans M. Harrell, and Michael Loss, Editors, Advances in differential

equations and mathematical physics, 1998 216 Akram Aldroubi and EnBing Lin, Editors, Wavelets, multiwavelets, and their

applications, 1998 215 M. G. Nerurkar, D. P. Dokken, and D. B. Ellis, Editors, Topological dynamics

and applications, 1998 214 Lewis A. Coburn and Marc A. Rieffel, Editors, Perspectives on quantization, 1998 213 Farhad Jafari, Barbara D. MacCluer, Carl C. Cowen, and A. Duane Porter,

Editors, Studies on composition operators, 1998

For a complete list of titles in this series, visit the AMS Bookstore at www.ams.org/bookstorej.

Crystallographic Groups and Their Generalizations

Paul Igodt, Herbert Abels, Yves Felix, and Fritz Grunewald, Editors

This volume contains articles written by the invited speakers and workshop participants from the conference on "Crystallographic Groups and Their Generalizations", held at Katholieke Universiteit Leuven, Kortrijk (Belgium). Presented are recent developments and open problems. Topics include the theory of affine structures and polynomial struc-tures, affine Schottky groups and crooked tilings, theory and problems on the geometry of finitely generated solvable groups, flat Lorentz 3-manifolds and Fuchsian groups, filiform Lie algebras, hyperbolic automorphisms and Anosov diffeomorphisms on infra-nilrnani-folds , localization theory of virtually nilpotent groups and aspherical spaces, projective varieties, and results on affine appartment systems. Participants delivered high-level research mathematics and a discussion forum for new researchers was held. The survey results and original papers contained in this volume offer a comprehensive view of current developments in the field.

ISBN 0-8218-2001-X

9 780821 820018 CONM/262