Lectures on · Growth and amenability 3.1. Bounded geometry 3.2. Growth 3.3. Amenable spaces 3.4. Examples of amenable and nonamenable spaces 3.5. Amenable groups 3.6. F0lner's Theorem

Lectures o n Coarse Geometr y

http://dx.doi.org/10.1090/ulect/031

University

LECTURE Series

Volume 3 1

Lectures o n Coarse Geometr y

John Ro e

American Mathematica l Societ y Providence, Rhod e Islan d

EDITORIAL COMMITTE E Jer ry L . Bon a (Chair ) Eri c M . Friedlande r Nigel J . Hitchi n Pe te r Landwebe r

2000 Mathematics Subject Classification. P r i m a r y 20F65 , 51K05 , 53C24 , 46L85 , 54E15 .

For addi t iona l informatio n an d upda te s o n thi s book , visi t w w w . a m s . o r g / b o o k p a g e s / u l e c t - 3 1

Library o f Congres s Cataloging-in-Publicat io n Dat a

Roe, John , 1959 -Lectures o n coars e geometr y / Joh n Roe .

p. cm . — (Universit y lectur e series , ISS N 1047-399 8 ; v. 31 ) Includes bibliographica l references . ISBN 0-8218-3332- 4 (alk . paper ) 1. Metri c spaces . 2 . Algebrai c topology . I . Title : Coars e geometry . II . Title . III . Univer -

sity lectur e serie s (Providence , R . I. ) ; 31.

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Contents

Preface

Chapter 1 . Metri c Space s 1.1. Legendr e o n hyperboli c geometr y 1.2. Metri c space s an d lengt h space s 1.3. Th e coars e perspectiv e o n metri c space s 1.4. Group s an d leave s 1.5. Tree s an d complexe s 1.6. Hyperboli c spac e 1.7. Nilpoten t example s — Heisenber g grou p

Chapter 2 . Coars e Space s 2.1. Th e abstrac t notio n o f coarse structur e 2.2. Topologica l coars e structur e 2.3. Th e Higso n coron a 2.4. Metrizatio n o f coars e structure s 2.5. Hyperbolizatio n

Chapter 3 . Growt h an d amenabilit y 3.1. Bounde d geometr y 3.2. Growt h 3.3. Amenabl e space s 3.4. Example s o f amenabl e an d nonamenabl e space s 3.5. Amenabl e group s 3.6. F0lner' s Theore m 3.7. Amenabilit y an d analysi s

Chapter 4 . Translatio n Algebra s 4.1. Translatio n Algebra s 4.2. Amenabilit y an d th e Translatio n Algebr a 4.3. Finitenes s o f Grou p Algebra s 4.4. Translatio n C*-Algebra s 4.5. Translatio n Algebra s a s Crosse d Product s

Chapter 5 . Coars e Algebrai c Topolog y 5.1. Coars e cohomolog y theor y 5.2. Produc t structur e o n coars e theor y 5.3. Computatio n o f coars e cohomolog y

vii

1 1 3 5 7

11 13 18

21 21 26 29 33 35

39 39 42 44 47 51 55 58

59 59 60 63 66 68

71 71 76 78

V

vi CONTENT S

5.4. Covers , nerve s an d metrizatio n 8 0 5.5. Coars e homolog y theorie s 8 2 5.6. Th e Coars e Baum–Conne s conjectur e 8 4

Chapter 6 . Coars e Negativ e Curvatur e 8 7 6.1. Curvatur e condition s 8 7 6.2. Th e Rip s propert y an d Gromo v hyperbolicit y 8 8 6.3. Controllin g quasigeodesic s 9 2 6.4. Th e Gromo v boundar y o f a hyperboli c spac e 9 3 6.5. Bolicit y 9 7

Chapter 7 . Limit s o f Metri c Space s 9 9 7.1. Convergenc e o f metri c space s 9 9 7.2. Th e rescale d limi t o f metric space s 10 1 7.3. Group s o f polynomial growt h 10 4 7.4. Ultralimit s 10 5 7.5. Asymptoti c Cone s 10 7

Chapter 8 . Rigidit y 11 1 8.1. Wha t i s rigidity? 111 8.2. Th e quasi-isometr y grou p o f hyperboli c spac e 11 2 8.3. Proo f o f Mostow Rigidit y 11 7 8.4. Quasi-Isometri c Rigidit y Fo r Product s o f Hyperboli c Space s 12 1

Chapter 9 . Asymptoti c Dimensio n 12 9 9.1. Th e asymptoti c dimensio n o f a coarse spac e 12 9 9.2. Compositio n propertie s o f asymptoti c dimensio n 13 2 9.3. Mor e Example s 13 6 9.4. Analyti c implication s o f finit e asymptoti c dimensio n 13 9

Chapter 10 . Groupoid s an d coars e geometr y 14 1 10.1. Reminder s abou t topologica l groupoid s 14 1 10.2. Th e pai r produc t an d th e Stone-Cec h boundar y 14 3 10.3. Th e translatio n groupoi d o f a coarse spac e 14 5 10.4. Translatio n groupoi d an d translatio n algebr a 14 8

Chapter 11 . Coars e Embeddabilit y 15 1 11.1. Coars e embeddin g 15 1 11.2. Kernel s an d embedding s i n Hilber t spac e 15 3 11.3. Embeddabilit y an d Propert y T 15 7 11.4. Propert y T an d coars e equivalenc e 16 2 11.5. Propert y A and exactnes s fo r grou p C*-algebra s 16 5

Bibliography 173

Preface

In the spring of 2002 I gave a series of graduate lectures a t Pen n Stat e on 'coars e geometry'. Thes e ar e th e edite d lectur e note s fro m tha t course . Th e intentio n wa s to discus s variou s aspect s o f the theor y o f 'larg e scal e structures ' o n spaces , wit h a particular focu s on the notions of asymptotic dimension an d uniform embeddabilit y into Hilber t space , whic h hav e recentl y prove d o f significanc e fo r th e Noviko v conjecture. O n the othe r hand , s o far a s is consistent wit h th e precedin g objective , the study o f C*-algebras arisin g from coars e geometry has been de-emphasized ; thi s has alread y bee n writte n abou t a t lengt h elsewher e [59] .

The first fe w chapters of the book are devoted to a general perspective on 'coars e structures' whic h was first se t ou t i n the pape r [34] . Thi s notion ha s the advantag e of including unde r on e heading man y o f the differen t notion s o f 'control ' tha t hav e been use d b y topologist s (fo r exampl e [2]) ; an d eve n whe n onl y metri c coars e structures ar e i n view , th e abstrac t framewor k bring s th e sam e simplificatio n a s does the passage from epsilon s and delta s to open set s when speaking o f continuity . In thi s mor e genera l contex t on e ca n stil l discus s idea s lik e growth , amenability , and coars e cohomology , an d thes e ar e addresse d i n chapter s 3 through 5 .

The middl e sectio n o f th e note s review s notion s o f negativ e curvatur e an d rigidity. Moder n interest i n large scale geometry derives in large part fro m Mostow' s rigidity theorem , wit h it s crucia l insigh t tha t th e coars e structur e o f hyperboli c space determines the quasiconformal structur e o f the boundary, an d from Gromov' s subsequent 'larg e scale ' renditio n o f th e crucia l propertie s o f negativel y curve d spaces. Ther e ar e man y excellen t exposition s o f thi s materia l an d ou r accoun t i s brief i n places .

In the fina l section s we discuss recent result s o n asymptoti c dimensio n (mostl y due t o Bel l an d Dranishnikov ) an d unifor m embeddin g int o Hilber t space . W e also tak e th e opportunit y t o revie w th e beautifu l constructio n o f Skandalis , T u and Y u [62 ] whic h allow s on e t o encod e th e larg e scal e structur e o f a (bounde d geometry) spac e b y mean s o f a suitable groupoid .

The larg e scal e geometr y o f discrete groups i s a beautifu l an d activ e are a o f research, an d i n thes e note s w e barel y scratc h it s surface . Th e reade r wh o want s to lear n mor e abou t geometri c grou p theor y i s referred t o th e book s [13 , 25 , 17] .

I am grateful t o the Nationa l Scienc e Foundation fo r thei r suppor t unde r grant s DMS-9800765 an d DMS-0100464 .

John Ro e

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