Preface
In the spring of 2002 I gave a series of graduate lectures at Pen n State 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 , with 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 the 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 'coarse
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 sets 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 from 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 asymptotic 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 for their support unde r grant s
DMS-9800765 an d DMS-0100464.
John Ro e
Previous Page Next Page