Tracts in Mathematics 25 PY Tracts in Mathematics 25 iv e e r rs e C do er n lau Yves Cornulier Hlie ar Yves Cornulier Pierre de la Harpe r p e Pierre de la Harpe Metric Geometry of Locally Compact Groups The main aim of this book is the study of locally compact groups from a LM Metric Geometry geometric perspective, with an emphasis on appropriate metrics that can be o e defined on them. The approach has been successful for finitely generated cat groups, and can favourably be extended to locally compact groups. Parts of r the book address the coarse geometry of metric spaces, where ‘coarse’ llyic of Locally refers to that part of geometry concerning properties that can be formulated G in terms of large distances only. This point of view is instrumental in C studying locally compact groups. oe Compact Groups o m m Basic results in the subject are exposed with complete proofs, others are p stated with appropriate references. Most importantly, the development of ae the theory is illustrated by numerous examples, including matrix groups with ct r entries in the the field of real or complex numbers, or other locally compact ty fields such as p-adic fields, isometry groups of various metric spaces, and, G o last but not least, discrete group themselves. r of The book is aimed at graduate students and advanced undergraduate u students, as well as mathematicians who wish some introduction to coarse p geometry and locally compact groups. s O G N R O A P M H D A R W A ISBN 978-3-03719-166-8 www.ems-ph.org Cornulier| Tracts in Mathematics 25 | Fonts Nuri /Helvetica Neue | Farben Pantone 116 / Pantone 287 | RB 24 mm EMS Tracts in Mathematics 25 EMS Tracts in Mathematics Editorial Board: Michael Farber (Queen Mary University of London, Great Britain) Carlos E. Kenig (The University of Chicago, USA) Michael Röckner (Universität Bielefeld, Germany, and Purdue University, USA) Vladimir Turaev (Indiana University, Bloomington, USA) Alexander Varchenko (The University of North Carolina at Chapel Hill, USA) This series includes advanced texts and monographs covering all fields in pure and applied mathematics. Tracts will give a reliable introduction and reference to special fields of current research. The books in the series will in most cases be authored monographs, although edited volumes may be published if appropriate. They are addressed to graduate students seeking access to research topics as well as to the experts in the field working at the frontier of research. For a complete listing see our homepage at www.ems-ph.org. 8 Sergio Albeverio et al., The Statistical Mechanics of Quantum Lattice Systems 9 Gebhard Böckle and Richard Pink, Cohomological Theory of Crystals over Function Fields 10 Vladimir Turaev, Homotopy Quantum Field Theory 11 Hans Triebel, Bases in Function Spaces, Sampling, Discrepancy, Numerical Integration 12 Erich Novak and Henryk Woz´niakowski, Tractability of Multivariate Problems. Volume II: Standard Information for Functionals 13 Laurent Bessières et al., Geometrisation of 3-Manifolds 14 Steffen Börm, Efficient Numerical Methods for Non-local Operators. 2-Matrix Compression, Algorithms and Analysis 15 Ronald Brown, Philip J. Higgins and Rafael Sivera, Nonabelian Algebraic Topology. Filtered Spaces, Crossed Complexes, Cubical Homotopy Groupoids 16 Marek Janicki and Peter Pflug, Separately Analytical Functions 17 Anders Björn and Jana Björn, Nonlinear Potential Theory on Metric Spaces 18 Erich Novak and Henryk Woz´niakowski, Tractability of Multivariate Problems. Volume III: Standard Information for Operators 19 Bogdan Bojarski, Vladimir Gutlyanskii, Olli Martio and Vladimir Ryazanov, Infinitesimal Geometry of Quasiconformal and Bi-Lipschitz Mappings in the Plane 20 Hans Triebel, Local Function Spaces, Heat and Navier–Stokes Equations 21 Kaspar Nipp and Daniel Stoffer, Invariant Manifolds in Discrete and Continuous Dynamical Systems 22 Patrick Dehornoy with François Digne, Eddy Godelle, Daan Kramer and Jean Michel, Foundations of Garside Theory 23 Augusto C. Ponce, Elliptic PDEs, Measures and Capacities. From the Poisson Equation to Nonlinear Thomas–Fermi Problems 24 Hans Triebel, Hybrid Function Spaces, Heat and Navier-Stokes Equations Yves Cornulier Pierre de la Harpe Metric Geometry of Locally Compact Groups Winner of the 2016 EMS Monograph Award Authors: Prof. Yves Cornulier Prof. Pierre de la Harpe Laboratoire de Mathématiques d’Orsay Section de mathématiques Université de Paris-Sud, CNRS Université de Genève Université de Paris-Saclay 2–4 rue du Lièvre 91405 Orsay C.P. 64 France 1211 Genève 4 E-mail: yves.cornulier @math.u-psud.fr Switzerland E-mail: Pierre.delaHarpe @unige.ch 2010 Mathematical Subject Classification: Primary: 20F65; Secondary: 20F05, 22D05, 51F99, 54E35, 57M07, 57T20. Key words: Locally compact groups, left-invariant metrics, s-compactness, second countability, compact generation, compact presentation, metric coarse equivalence, quasi-isometry, coarse connectedness, coarse simple connectedness, growth, amenability. ISBN 978-3-03719-166-8 The Swiss National Library lists this publication in The Swiss Book, the Swiss national bibliography, and the detailed bibliographic data are available on the Internet at http://www.helveticat.ch. 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, recitation, broad- casting, reproduction on microfilms or in other ways, and storage in data banks. For any kind of use permission of the copyright owner must be obtained. © 2016 European Mathematical Society Contact address: European Mathematical Society Publishing House Seminar for Applied Mathematics ETH-Zentrum SEW A27 CH-8092 Zürich Switzerland Phone: +41 (0)44 632 34 36 Email: [email protected] Homepage: www.ems-ph.org Typeset using the author’s TEX files: le-tex publishing services GmbH, Leipzig, Germany Printing and binding: Beltz Bad Langensalza GmbH, Bad Langensalza, Germany ∞ Printed on acid free paper 9 8 7 6 5 4 3 2 1 Foreword Thisbookofferstostudylocallycompactgroupsfromthepointofviewofappropri- ate metrics that can be defined on them, in other words to study “Infinite groups as geometric objects”, as Gromov writes it in the title of a famous article. The theme hasoftenbeenrestrictedtofinitelygeneratedgroups,butitcanfavourablybeplayed forlocallycompactgroups. The development of the theory is illustrated by numerous examples, including matrixgroupswithentriesinthethefieldofrealorcomplexnumbers,orotherlocally compactfields suchas p-adic fields, isometry groupsofvariousmetric spaces, and, lastbutnotleast,discretegroupthemselves. Word metrics for compactly generated groups play a major role. In the particu- lar case of finitely generatedgroups, they were introducedby Dehn around1910 in connectionwiththeWordProblem. Someoftheresultsexposedconcerngenerallocallycompactgroups,suchascri- teriafortheexistenceofcompatiblemetrics(Birkhoff–Kakutani,Kakutani–Kodaira, Struble). Otherresultsconcernspecialclassesofgroups,forexamplethosemapping ontoZ(theBieri–Strebelsplittingtheorem,generalizedtolocallycompactgroups). Prior to their applications to groups, the basic notions of coarse and large-scale geometryaredevelopedinthegeneralframeworkofmetricspaces. Coarsegeometry isthatpartofgeometryconcerningpropertiesofmetricspacesthatcanbeformulated in terms of large distances only. In particular coarse connectedness, coarse simple connectedness,metriccoarseequivalences,andquasi-isometriesofmetricspacesare givenspecialattention. The final chapters are devoted to the more restricted class of compactly pre- sentedgroups,whichgeneralizefinitelypresentedgroupstothelocallycompactset- ting. Theycanindeedbecharacterizedasthosecompactlygeneratedlocallycompact groupsthatarecoarselysimplyconnected. Contents 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.A Discretegroupsasmetricspaces . . . . . . . . . . . . . . . . . . . . 1 1.B Discretegroupsandlocallycompactgroups . . . . . . . . . . . . . . 3 1.C ThreeconditionsonLC-groups . . . . . . . . . . . . . . . . . . . . . 4 1.D Metriccharacterization . . . . . . . . . . . . . . . . . . . . . . . . . 6 1.E Oncompactpresentations . . . . . . . . . . . . . . . . . . . . . . . . 8 1.F Outlineofthebook . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 1.G Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 2 Basicproperties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 2.A Topologicalspacesandpseudo-metricspaces . . . . . . . . . . . . . 12 2.B Metrizabilityand(cid:2)-compactness . . . . . . . . . . . . . . . . . . . . 19 2.C Compactgeneration . . . . . . . . . . . . . . . . . . . . . . . . . . 26 2.D Miscellanea . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 2.E StructureofLC-groups . . . . . . . . . . . . . . . . . . . . . . . . . 39 3 Metriccoarseandlarge-scalecategories . . . . . . . . . . . . . . . . . . 53 3.A CoarselyLipschitzandlarge-scaleLipschitzmaps . . . . . . . . . . . 53 3.B Coarsepropertiesandlarge-scaleproperties . . . . . . . . . . . . . . 65 3.B.a Coarselyconnected,coarselygeodesic, andlarge-scalegeodesicpseudo-metricspaces . . . . . . . . . 65 3.B.b Coarselyultrametricpseudo-metricspaces . . . . . . . . . . . 70 3.B.c Asymptoticdimension . . . . . . . . . . . . . . . . . . . . . . 72 3.C Metriclatticesinpseudo-metricspaces . . . . . . . . . . . . . . . . . 73 3.D Growthandamenability . . . . . . . . . . . . . . . . . . . . . . . . . 77 3.D.a Growthforuniformlylocallyfinitepseudo-metricspaces . . . 77 3.D.b Growth for uniformly coarsely proper pseudo-metric spaces and(cid:2)-compactLC-groups. . . . . . . . . . . . . . . . . . . . 80 3.D.c Amenability . . . . . . . . . . . . . . . . . . . . . . . . . . . 88 3.E Thecoarsecategory . . . . . . . . . . . . . . . . . . . . . . . . . . . 92 4 Groupsaspseudo-metricspaces . . . . . . . . . . . . . . . . . . . . . . 95 4.A Adapted(pseudo-)metricson(cid:2)-compactgroups . . . . . . . . . . . . 95 4.B Pseudo-metricsoncompactlygeneratedgroups . . . . . . . . . . . . 99 4.C Actionsofgroupsonpseudo-metricspaces. . . . . . . . . . . . . . . 107 4.D Localellipticity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114 4.E Cappedsubgroups . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121 4.F AmenableandgeometricallyamenableLC-groups . . . . . . . . . . . 127 viii Contents 5 ExamplesofcompactlygeneratedLC-groups . . . . . . . . . . . . . . . 133 5.A Connected,abelian,nilpotent,Lie,andalgebraic . . . . . . . . . . . . 133 5.B Isometrygroups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139 5.C LatticesinLC-groups . . . . . . . . . . . . . . . . . . . . . . . . . . 146 6 Coarsesimpleconnectedness . . . . . . . . . . . . . . . . . . . . . . . . 153 6.A Coarselysimplyconnectedpseudo-metricspaces . . . . . . . . . . . 153 6.B Onsimplicialcomplexes . . . . . . . . . . . . . . . . . . . . . . . . 157 6.C TheRips2-complexofapseudo-metricspace . . . . . . . . . . . . . 160 7 Boundedpresentations . . . . . . . . . . . . . . . . . . . . . . . . . . . 163 7.A Presentationswithrelatorsofboundedlength . . . . . . . . . . . . . 163 7.B TheRipscomplexofaboundedpresentation . . . . . . . . . . . . . . 171 8 Compactlypresentedgroups . . . . . . . . . . . . . . . . . . . . . . . . 173 8.A Definitionandfirstexamples . . . . . . . . . . . . . . . . . . . . . . 173 8.B AmalgamatedproductsandHNN-extensions . . . . . . . . . . . . . . 184 8.C HomomorphismstoZandsplittings . . . . . . . . . . . . . . . . . . 192 8.C.a AtopologicalversionoftheBieri–Strebelsplittingtheorem . . 192 8.C.b Proofs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194 8.C.c Engulfingautomorphisms . . . . . . . . . . . . . . . . . . . . 197 8.D Furtherexamples . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201 8.D.a SemidirectproductswithZ . . . . . . . . . . . . . . . . . . . 201 8.D.b Moregeneralsemidirectproducts,andSL .K/ . . . . . . . . . 204 n Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213 Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 229 Chapter 1 Introduction 1.A Discrete groups as metric spaces Whenever a group (cid:3) appears in geometry, which typically means that (cid:3) acts on a metricspaceofsomesort(examplesincludeuniversalcoveringspaces,Cayleygraphs andRipscomplexes),thegeometryofthespacereflectssomegeometryofthegroup. ThisphenomenongoesbackatleasttoFelixKleinandHenriPoincare´,withtes- sellationsofthehalf-planerelatedtosubgroupsofthemodulargroups,around1880. Ithasthenbeenawell-establishedtraditiontostudypropertiesofgroupswhichcan beviewed,atleastinretrospect,asgeometricproperties. Asasample,wecanmen- tion: – “DehnGruppenbild”(also knownas Cayleygraphs), usedto picturefinitely gen- eratedgroupsandtheirwordmetrics,inparticularknotgroups,around1910. Note thatDehnintroducedwordmetricsforgroupsinhisarticlesondecisionproblems (1910–1911). – Amenability of groups (von Neumann, Tarski, late 20’s), and its interpretation in termsofisoperimetricproperties(Følner,mid50’s). – Properties “at infinity”, or ends of groups (Freudenthal, early 30’s), and struc- ture theorems for groups with two or infinitely many ends (Stallings for finitely generatedgroups,late 60’s, Abels’generalizationfortotally disconnectedlocally compactgroups,1974). – LatticesinLiegroups,andlaterinalgebraicgroupsoverlocalfields;firstacollec- tionofexamples,andfromthe40’sasubjectofgrowingimportance,withfounda- tionalwork by Siegel, Mal’cev, Mostow, L. Auslander, Borel & Harish-Chandra, Weil,Garland,H.C.Wang,Tamagawa,Kazhdan,Raghunathan,Margulis(toquote onlythem);leadingto: – Rigidity of groups, and in particular of lattices in semisimple groups (Mostow, Margulis,60’sand70’s). – Growth of groups, introduced independently (!) by A.S. Schwarz (also written Sˇvarc)in1955andMilnorin1968,popularizedbytheworkofMilnorandWolf, andstudiedlaterbyGrigorchuk,Gromov,andothers,includingGuivarc’h,Jenkins, andLosertforlocallycompactgroups. – Structureofgroupsactingfaithfullyontrees(Tits, Bass–Serretheory,Dunwoody decompositionsandaccessibilityofgroups,70’s);treelattices. – Propertiesrelatedtorandomwalks(Kesten,late50’s,Guivarc’h,70’s,Varopoulos). – And the tightly interwoven developmentsof combinatorialgroup theory and low dimensionaltopology,fromDehntoThurston,andsomanyothers. From1980onwards,forallthesereasonsandundertheguidanceofGromov,inpar- ticular of his articles [Grom–81b, Grom–84, Grom–87, Grom–93], the group com-