Topology and Geometric Group Theory: Ohio State University, Columbus, USA, 2010–2011 PDF
Preview Topology and Geometric Group Theory: Ohio State University, Columbus, USA, 2010–2011
Springer Proceedings in Mathematics & Statistics Michael W. Davis James Fowler Jean-François Lafont Ian J. Leary Editors Topology and Geometric Group Theory Ohio State University, Columbus, USA, 2010–2011 Springer Proceedings in Mathematics & Statistics Volume 184 Springer Proceedings in Mathematics & Statistics This book series features volumes composed of selected contributions from workshops and conferences in all areas of current research in mathematics and statistics, including operation research and optimization. In addition to an overall evaluation of the interest, scientific quality, and timeliness of each proposal at the hands of the publisher, individual contributions are all refereed to the high quality standards of leading journals in the field. Thus, this series provides the research community with well-edited, authoritative reports on developments in the most exciting areas of mathematical and statistical research today. More information about this series at http://www.springer.com/series/10533 Michael W. Davis James Fowler (cid:129) ç Jean-Fran ois Lafont Ian J. Leary (cid:129) Editors Topology and Geometric Group Theory Ohio State University, Columbus, USA, – 2010 2011 123 Editors Michael W.Davis Jean-François Lafont Department ofMathematics Department ofMathematics OhioState University OhioState University Columbus, OH Columbus, OH USA USA James Fowler IanJ.Leary Department ofMathematics Mathematical Sciences OhioState University University of Southampton Columbus, OH Southampton USA UK ISSN 2194-1009 ISSN 2194-1017 (electronic) SpringerProceedings in Mathematics& Statistics ISBN978-3-319-43673-9 ISBN978-3-319-43674-6 (eBook) DOI 10.1007/978-3-319-43674-6 LibraryofCongressControlNumber:2016947207 MathematicsSubjectClassification(2010): 20-06,57-06,55-06,20F65,20J06,18F25,19J99,20F67, 57R67,55P55,55Q07,20E42 ©SpringerInternationalPublishingSwitzerland2016 Thisworkissubjecttocopyright.AllrightsarereservedbythePublisher,whetherthewholeorpart of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission orinformationstorageandretrieval,electronicadaptation,computersoftware,orbysimilarordissimilar methodologynowknownorhereafterdeveloped. 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Printedonacid-freepaper ThisSpringerimprintispublishedbySpringerNature TheregisteredcompanyisSpringerInternationalPublishingAGSwitzerland Preface During the academic year 2010–2011, the Ohio State University Mathematics Departmenthostedaspecialyearongeometricgrouptheory.Overthecourseofthe year, four-week-long workshops, two weekend conferences, and a week-long conference were held, each emphasizing a different aspect of topology and/or geometric group theory. Overall, approximately 80 international experts passed throughColumbusoverthecourse oftheyear, and thetalkscovered alarge swath of the current research in geometric group theory. This volume contains contri- butions from the workshop on “Topology and geometric group theory,” held in May 2011. One of thebasic questions in manifold topology isthe Borel Conjecture, which askswhetherthefundamentalgroupofaclosedasphericalmanifolddeterminesthe manifold up to homeomorphism. The foundational work on this problem was carriedoutinthelate1980sbyFarrellandJones,whoreformulatedtheproblemin terms of the K-theoretic and L-theoretic Farrell–Jones Isomorphism Conjectures (FJIC). In the mid-2000s, Bartels, Lück, and Reich were able to vastly extend the techniquesofFarrellandJones.Notably,theywereabletoestablishtheFJICs(and hence the Borel Conjecture) for manifolds whose fundamental groups were Gromovhyperbolic.Lückreportedonthisprogressatthe2006ICMinMadrid.At the Ohio State University workshop, Arthur Bartels gave a series of lectures explaining their joint work on the FJICs. The write-up of these lectures provides a gentle introduction to this important topic, with an emphasis on the techniques of proof. Staying on the theme of the Farrell–Jones Isomorphism Conjectures, Daniel Juan-PinedaandJorgeSánchezSaldañacontributedanarticleinwhichboththe K-andL-theoreticFJICareverifiedforthebraidgroupsonsurfaces.Thesearethe fundamentalgroupsofconfigurationspacesoffinitetuplesofpoints,movingonthe surface.Braidgroupshavebeenlongstudied,bothbyalgebraictopologists,andby geometric group theorists. A major theme in geometric group theory is the study of the behavior “at infinity”ofaspace(orgroup).Thisisasubjectthathasbeenstudiedbygeometric v vi Preface topologists since the 1960s. Indeed, an important aspect of the study of open manifolds is the topology of their ends. The lectures by Craig Guilbault present thestateoftheartonthesetopics.Theselecturesweresubsequentlyexpandedinto agraduatecourse,offeredinFall2011attheUniversityofWisconsin(Milwaukee). An important class of examples in geometric group theory is given by CAT(0) cubical complexes and groups acting geometrically on them. Interest in these has grown in recent years, due in large part to their importance in 3-manifold theory (e.g., their use in Agol and Wise’s resolution of Thurston’s virtual Haken conjec- ture).AnumberoffoundationalresultsonCAT(0)cubicalspaceswereobtainedin MichahSageev’sthesis.InhiscontributedarticleDanielFarleygivesanewproof of one of Sageev’s key results: any hyperplane in a CAT(0) cubical complex embeds and separates the complex into two convex sets. One of the powers of geometric group theory lies in its ability to produce, through geometric or topological means, groups with surprising algebraic proper- ties. One such example was Burger and Mozes’ construction offinitely presented, torsion-free simple groups, which were obtained as uniform lattices inside the automorphism group of a product of two trees (a CAT(0) cubical complex!). The articlebyPierre-EmmanuelCapraceandBertrandRémyintroducesageometric argumenttoshowthatsomenonuniformlatticesinsidetheautomorphismgroupof a product of trees are also simple. Animportantlinkbetweenalgebraandtopologyisprovidedbythecohomology functors. Our final contribution, by Peter Kropholler, contributes to our under- standingofthefunctorialpropertiesofgroupcohomology.Heconsiders,forafixed group G, the set of integers n for which the group cohomology functor HnðG;(cid:2)Þ commuteswithcertaincolimitsofcoefficientmodules.Foralarge classofgroups, he shows this set of integers is always either finite or cofinite. We hope these proceedings provide a glimpse of the breadth of mathematics coveredduringtheworkshop.Theeditorswouldalsoliketotakethisopportunityto thank all the participants at the workshop for a truly enjoyable event. Columbus, OH, USA Michael W. Davis December 2015 James Fowler Jean-François Lafont Ian J. Leary Acknowledgments The editors of this volume thank the National Science Foundation (NSF) and the Mathematics Research Institute (MRI). The events focusing on geometric group theoryattheOhioStateUniversityduringthe2010–2011academicyearwouldnot have been possible without the generous support of the NSF and the MRI. vii Contents 1 On Proofs of the Farrell–Jones Conjecture.... .... .... ..... .... 1 Arthur Bartels 2 The K and L Theoretic Farrell-Jones Isomorphism Conjecture for Braid Groups... .... .... .... .... .... ..... .... 33 Daniel Juan-Pineda and Luis Jorge Sánchez Saldaña 3 Ends, Shapes, and Boundaries in Manifold Topology and Geometric Group Theory .. .... .... .... .... .... ..... .... 45 Craig R. Guilbault 4 A Proof of Sageev’s Theorem on Hyperplanes in CAT(0) Cubical Complexes .. .... .... .... .... .... ..... .... 127 Daniel Farley 5 Simplicity of Twin Tree Lattices with Non-trivial Commutation Relations... ..... .... .... .... .... .... ..... .... 143 Pierre-Emmanuel Caprace and Bertrand Rémy 6 Groups with Many Finitary Cohomology Functors . .... ..... .... 153 Peter H. Kropholler Index .... .... .... .... .... ..... .... .... .... .... .... ..... .... 173 ix Contributors Arthur Bartels Mathematisches Institut, Westfälische Wilhelms-Universität Münster, Münster, Germany Pierre-Emmanuel Caprace IRMP, Université catholique de Louvain, Louvain-la-Neuve, Belgium DanielFarley DepartmentofMathematics,MiamiUniversity,Oxford, OH,USA Craig R. Guilbault Department of Mathematical Sciences, University of Wisconsin-Milwaukee, Milwaukee, WI, USA Daniel Juan-Pineda Centro de Ciencias Matemáticas, Universidad Nacional Autónoma de México, Campus Morelia, Morelia, Michoacan, Mexico PeterH.Kropholler Mathematics,UniversityofSouthampton,Southampton,UK Bertrand Rémy École Polytechnique, CMLS, UMR 7640, Palaiseau Cedex, France Luis Jorge Sánchez Saldaña Centro de Ciencias Matemáticas, Universidad Nacional Autónoma de México, Morelia, Michoacan, Mexico xi
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