Topology and Geometric Group Theory: Ohio State University, Columbus, USA, 2010–2011

179 Pages·2016·2.35 MB·English

by  Michael W. Davis, James Fowler, Jean-Francois Lafont, Ian J. Leary (eds.)

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
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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
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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
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