BivariantK-theory of groupoids and the

noncommutative geometry of limit sets

Dissertation

zur

Erlangung des Doktorgrades

der

Mathematisch-Naturwissenschaftlichen Fakult at

der

Rheinischen Friedrich-Wilhelms-Universit at Bonn

Vorgelegt von

Bram Mesland

aus

Amstelveen, Niederlande

Bonn 2009Diese Dissertation ist auf dem Hochschulschriftenserver der ULB Bonn unter

http :==hss:ulb:uni-bonn:de=diss online

elektronisch publiziert.Angefertigt mit Genehmigung der

Mathematisch-Naturwissenschaftlichen Fakult at der

Rheinischen Friedrich-Wilhelms-Universit at Bonn.

1.Gutachter: Prof. Dr. Matilde Marcolli

2.Gutachter: Prof.Dr. Matthias Lesch

Tag der Promotion: Freitag, 17.Juli 2009

iiiSummary

We present a categorical setting for noncommutative geometry in the sense of Connes.

This is done by introducing a notion of morphism for spectral triples. Spectral triples are

the unbounded cycles for K-homology ([11]), and their bivariant generalization are the

cycles for Kasparov’sKK-theory ([32]). The central feature ofKK-theory is the Kasparov

product

KK (A;B)

KK (B;C)!KK (A;C):i j i+j

Here A;B and C are C -algebras, and the product allows one to view KK as a category.

The unbounded picture of this theory was introduced by Baaj and Julg ([4]). In this

picture the external product

0 0 0 0

KK (A;B)

KK (A;B )!KK (A

B;A

B );i j i+j

is given by an algebraic formula, as opposed to Kasparov’s original approach, which is

more analytic in nature, and highly technical.

In order to describe the internal Kasparov product of unbounded KK-cycles, we in-

troduce a notion of connection for unbounded cycles (E;D). This is a universal connection

1~r : E! E

(B);B

in the sense of Cuntz and Quillen ([20]), such that [r;D] extends to a completely bounded

operator. The topological tensor product used here is the Haagerup tensor product for

operator spaces. Blecher ([7]) showed this tensor product coincides with the C -module

tensor product, in case both operator spaces areC -modules. His work plays a crucial role

in our construction. The product of two cycles with connection is given by an algebraic

formula and the product of connections can also be de ned. Thus, cycles with connection

form a category, and the bounded transform

12

2(E;D;r)7! (E;D(1 +D ) );

de nes a functor from this category to the category KK.

We also describe a general construction for obtaining KK-cycles from real-valued

groupoid cocycles. IfG is a locally compact Hausdor groupoid with Haar system and

c :G!R a continuous closed cocycle, we show that pointwise multiplication by c in the

convolution algebra C (G), extends to an unbounded regular operator on the completionc

ofC (G) as a C -module over C (H), whereH is the kernel of c. It gives a KK-cycle forc

(C (G);C (H)). In case the groupoidH is unimodular with respect to a quasi-invariant

measure, or more general, if C (H) carries a trace, this KK-cycle gives rise to an index

map K (C (G))!C.1

This result is general enough to be applied in a wide variety of examples. We use it

to obtain the noncommutative torus as a smooth quotient (in the above categorical sense)

of the irrational rotation action on the circle. In the last chapter we sketch the promising

range of applications the above categorical setting and cocycle construction may have in

the noncommutative geometry of limit sets.Acknowledgements

My appreciation goes out to Matilde Marcolli, for guiding me through this

project, suggesting the subject to me, and stimulating me to develop and work

through the technicalities that were involved. On a human level, this thesis would

not have come into existence without her kindness and understanding of the fact

that time is needed to overcome di culties in personal life.

I thank Alain Connes and Nigel Higson for valuable email correspondence dur-

ing crucial phases of the project. Georges Skandalis, Jean Renault and Klaas Lands-

man I thank for guiding me through some of the literature on groupoids.

My Ph.D. studies were supported by an IMPRS-grant from the Max Planck

Gesellschaft, for which I am grateful. The Max Planck Institut fur Mathematik is

a very inspiring environment for doing mathematics, and I have been lucky to have

been able to spend such a long time here. I thank the sta of the MPI for their

kindness and for providing these excellent conditions.

Important parts of the work in this thesis were done during my visits to Florida

State University and the California Institute of Technology. I thank them for their

hospitality and support.

I thank Branimir Cacic, Ioanna Dimitriou, Ivan Dynov, Eugenia Ellis, Tobias

Fritz, Nikolay Ivankov, Javier Lopez, Elke Markert, Nicola Mazzari, Rafael Torres

and Dapeng Zhang for the many inspiring and useful conversations we had, and for

providing some relaxation and fun now and then.

I thank all my friends for the good times we had, for being there, and for keep-

ing their faith in me along the way.

Bob, Annemieke, Lot en Joost, zonder jullie was ik hier nooit gekomen.

viiContents

iii

Summary v

Acknowledgements vii

Introduction 1

Chapter 1. Unbounded bivariant K-theory 5

1. C -modules 5

2. KK-theory 12

3. Operator modules 17

4. Smoothness 22

5. Universal connections 28

6. Correspondences 37

Chapter 2. Groupoids 45

1. A category of groupoids 45

2. C -algebras and -modules 51

3. Cocycles and K-theory 55

4. Crossed products 62

5. Semidirect products 65

Chapter 3. Limit sets 71

1. Limit sets and Patterson-Sullivan measures 71

2. Hyperbolic manifolds 75

3. Fuchsian groups 81

Bibliography 85

ix