Table Of ContentProgress in Mathematics
Volume 197
Series Editors
H. Bass
J. Oesterle
A. Weinstein
Pierre-Alain Cherix
Michael Cowling
Paul Jolissaint
Pierre Julg
Alain Valette
Groups with the
Haagerup Property
Gromov's a-T-menability
Springer Basel AG
Authors:
Pierre-Alain Cherix Michael Cowling
Section de Mathematiques School of Mathematics
Universite de Geneve University of New South Wales
Rue du Lievre 2--4 Sydney NSW 2052
C.P. 240 Australia
1211 Geneve 24 [email protected]
Switzerland
[email protected] Pierre Julg
Departement de Mathematiques
Paul Jolissaint Universite d'Orleans
Institut de Mathematiques B.P.6759
Universite de Neuchâtel 45067 Orleans Cedex 2
Emile-Argand II France
2000 Neuchâtel [email protected]
Switzerland
[email protected]
Alain Valette
Institut de Mathematiques
Universite de Neuchâtel
Emile-Argand II
2000 Neuchâtel
Switzerland
[email protected]
2000 Mathematics Subject Classification 20-XX, 22Dxx, 22Exx, 43-XX, 46Lxx, 51-XX
A CIP catalogue record for this book is available from the Library of Congress, Washington D.C.,
USA
Deutsche Bibliothek Cataloging-in-Publication Data
Groups with the Haagerup property : Gromov's a-T-menability / Pierre-Alain Cherix ...
2001 Springer Basel AG
(Progress in mathematics ; VoI. 197)
ISBN 978-3-0348-9486-9 ISBN 978-3-0348-8237-8 (eBook)
DOI 10.1007/978-3-0348-8237-8
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© 2001 Springer Basel AG
Originally published by Birkh!iuser Verlag, Basel-Boston -Berlin in 2001
Softcover reprint of the hardcover 1s t edition 2001
Member of the BertelsmannSpringer Publishing Group
Printed on acid-free paper produced of chlorine-free pulp. TCF
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ISBN 978-3-0348-9486-9
987654321 www.birkhauser-science.com
Contents
1 Introduction
by Alain Valette
1.1 Basic definitions ................... 1
1.1.1 The Haagerup property, or a-T-menability . 1
1.1.2 Kazhdan's property (T) 2
1.2 Examples ........... 2
1.2.1 Compact groups .... 3
1.2.2 SO(n, l) and SU(n, 1) 3
1.2.3 Groups acting properly on trees. 3
1.2.4 Groups acting properly on lR-trees 4
1.2.5 Coxeter groups ........... 4
1.2.6 Amenable groups . . . . . . . . . . 4
1.2.7 Groups acting on spaces with walls . 4
1.3 What is the Haagerup property good for? 5
1.3.1 Harmonic analysis: weak amenability . 6
1.3.2 K-amenability ......... 7
1.3.3 The Baum-Connes conjecture . 8
1.4 What this book is about ........ 10
2 Dynamical Characterizations
by Paul Jolissaint
2.1 Definitions and statements of results 15
2.2 Actions on measure spaces . 21
2.3 Actions on factors ..... 25
3 Simple Lie Groups of Rank One
by Pierre Julg
3.1 The Busemann co cycle and the
Gromov scalar product. . ... 34
3.2 Construction of a quadratic form 35
3.3 Positivity ............. 37
3.4 The link with complementary series 38
vi Contents
4 Classification of Lie Groups with
the Haagerup Property
by Pierre-Alain Cherix, Michael Cowling
and Alain Valette
4.0 Introduction.............. 41
4.1 Step one . . . . . . . . . . . . . . . . 42
4.1.1 The fine structure of Lie groups. 42
4.1.2 A criterion for relative property (T) 44
4.1.3 Conclusion of step one ....... . 46
4.2 Step two . . . . . . . . . . . . . . . . . . . . 46
4.2.1 The generalized Haagerup property. 46
4.2.2 Amenable groups . 54
4.2.3 Simple Lie groups 56
4.2.4 A covering group . 56
4.2.5 Spherical fu~tions 57
4.2.6 The group SU(n, 1) . 59
4.2.7 The groups SO(n, 1) and SU(n, 1). 60
4.2.8 Conclusion of step two . . . . . . . 60
5 The Radial Haagerup Property
by Michael Cowling
5.0 Introduction............... 63
5.1 The geometry of harmonic N A groups 64
5.2 Harmonic analysis on H-type groups 66
5.3 Analysis on harmonic N A groups . . 72
5.4 Positive definite spherical functions . 75
5.5 Appendix on special functions. 82
6 Discrete Groups
by Paul Jolissaint, Pierre Julg
and Alain Valette
6.1 Some hereditary results 85
6.2 Groups acting on trees . 91
6.3 Group presentations . . 97
6.4 Appendix: Completely positive maps
on amalgamated products,
by Paul Jolissaint. . . . . . . . . . . . . . . . . . . . . . . .. 99
7 Open Questions and Partial Results
by Alain Valette
7.1 Obstructions to the Haagerup property. 105
7.2 Classes of groups . . . . . 105
7.2.1 One-relator groups. . . . . . . . 105
Contents vii
7.2.2 Three-manifold groups . 106
7.2.3 Braid groups ... . 107
7.3 Group constructions ... . 107
7.3.1 Semi-direct products 107
7.3.2 Actions on trees .. 107
7.3.3 Central extensions . 108
7.4 Geometric characterizations 109
7.4.1 Chasles' relation .. 109
7.4.2 Some cute and sexy spaces III
7.5 Other dynamical characterizations 112
7.5.1 Actions on infinite measure spaces 112
7.5.2 Invariant probability measures 113
Bibliography 115
Index .... 125
Chapter 1
Introduction
by Alain Valette
1.1 Basic definitions
1.1.1 The Haagerup property, or a-T-menability
For a second countable, locally compact group G, consider the following four
properties:
(1) there exists a continuous function 'ljJ: G -> lR.+ which is conditionally
negative definite and proper, that is, limg--->oo 'ljJ(g) = 00;
(2) G has the Haagerup approximation property, in the sense of C.A. Ake
mann and M. Walter [AW81] and M. Choda [Cho83], or property Co
in the sense of V. Bergelson and J. Rosenblatt [BR88]: there exists a
sequence (¢n)nEN of continuous, normalized (i.e., ¢n(1) = 1) positive
definite functions on G, vanishing at infinity on G, and converging to 1
uniformly on compact subsets of G;
(3) Gis a-T-menable, as M. Gromov meant it in 1986 ([Gro88, 4.5.C]): there
exists a (strongly continuous, unitary) representation of G, weakly con
taining the trivial representation, whose matrix coefficients vanish at in
finity on G (a representation with matrix coefficients vanishing at infinity
will be called a Co-representation);
(4) G is a-T-menable, as Gromov meant it in 1992 ([Gro93, 7.A and 7.E]):
there exists a continuous, isometric action a of G on some affine Hilbert
space 71, which is metrically proper (that is, for all bounded subsets B
of 71, the set {g E G : a(g)B n B i- 0} is relatively compact in G).
It was gradually realized that these conditions are actually equivalent,
hence define a unique class of groups (see [AW81] for the equivalence of (1) and
P.-A. Cherix et al., Groups with the Haagerup Property
© Birkhäuser Verlag 2001
2 Chapter 1. Introduction
(2), [JolOO] for the equivalence of (1) and (3), and [BCV95] for the equivalence
of (1) and (4), proved there for discrete groups, but the proof goes over without
change to the general case). We believe that Gromov was already aware in 1992
of the equivalence of (3) and (4), but that he had no formal proofl.
Definition 1.1.1. A second countable, locally compact group has the Haagerup
property if it satisfies one and hence all of the equivalent conditions (1) to (4)
above.
A short proof of the equivalence of conditions (1) to (4) will be given in
Theorem 2.1.1 below.
This set of cognate papers is devoted to the study of the class of groups
with the Haagerup property.
1.1.2 Kazhdan's property (T)
It is patently obvious that each of the conditions (1) to (4) above is designed
as a strong negation to Kazhdan's property (T); indeed, here are the four
corresponding equivalent formulations of property (T) for G (see [HV89] for
the proofs of the equivalences, and for many examples of groups with prop
erty (T)):
(1) every continuous, conditionally negative definite function on G is boun
ded;
(2) whenever a sequence of continuous, normalized, positive definite functions
on G converges to 1 uniformly on compact subsets of G, then it converges
to 1 uniformly on G;
(3) if a representation of G contains the trivial representation weakly, then
it contains it strongly (that is, the representation has G-fixed vectors);
(4) every continuous, isometric action of G on an affine Hilbert space has a
fixed point.
1.2 Examples
We give here a list of examples (roughly in chronological order) of groups with
the Haagerup property. They illustrate how large this class of groups is. It will
be noticed that most of these groups are of geometric origin. As advocated by
Julg [Jul98] and YA. Neretin [Ner98], we will describe, whenever possible, an
explicit proper isometric action on some affine Hilbert space.
lOtherwise, Gromov would not have asked, as he did in [Gro93, 7.E], whether amenable
groups are a-T-menable; indeed, given characterization (3), this fact is obvious, as observed
in [JoIOO].
1.2. Examples 3
1.2.1 Compact groups
Compact groups have the Haagerup property, trivially. They also have prop
erty (T). Conversely, it is clear from the conditions above that a group with
both the Haagerup property and property (T) has to be compact. As a con
sequence, any continuous homomorphism from a group with property (T) to
a group with the Haagerup property has relatively compact image (this is the
guiding principle in [HV89, Chap. 6]).
1.2.2 SO(n, 1) and SU(n, 1)
The Lie groups SO(n,l) and SU(n,l) (the isometry groups of the n-dimen
sional real and complex hyperbolic spaces respectively) have the Haagerup
property. Proper affine isometric actions were constructed by A.M. Vershik,
I.M. Gel'fand and M.1. Graev [VGG73], [VGG74] (see also [Ner98, 1.1-1.3], as
well as Chapter 3 below). On the other hand, it was proved by J. Faraut and K.
Harzallah [FH74] (see also [HV89, 6p. 79]) that, denoting by d the hyperbolic
distance and by Xo any point in real or complex hyperbolic space, the function
9 ~ d(gxo,xo) is conditionally negative definite on SO(n, 1) and on SU(n, 1);
for SO(n, I), A.G. Robertson ([Rob98, Cor. 2.5]) constructed the associated
affine action on the L2 space of the space of half-spaces of real hyperbolic
space. Note that Y. Shalom [Sha99] has proved that, if ex is an affine isometric
action of SO(n, 1) or SU(n, I), then either ex is proper or ex has a fixed point.
1.2.3 Groups acting properly on trees
Free groups on a finite set of generators have the Haagerup property; indeed,
U. Haagerup [Haa79] established the seminal result that the word length with
respect to a free generating subset is a conditionally negative definite func
tion on the group. (Since the Haagerup property is clearly inherited by closed
subgroups, this means that the free group on countably many generators also
has the Haagerup property.) Haagerup's result was reinterpreted in terms of
group actions on trees by several people, see [Wat81]' [Alp82], [JV84, Lem. 2.3]
(see also [HV89, p. 69]), [Mar91, Prop. 3.6], [Pav91]. Denote the distance func
tion on a tree X by d, and an arbitrary vertex in X by Xo; then the function
9 ~ d(gxo, xo) is conditionally negative definite on the automorphism group of
X. The corresponding affine isometric action on the £2 space of the set of ori
ented edges of the tree is implicit in most of the references above2. In particular,
any group acting properly on a locally finite tree, has the Haagerup property.
2This affine action is used in [Sha99J to prove a superrigidity result for actions of lattices
on trees. For groups acting on homogeneous, locally finite trees, a different affine isometric
action is constructed in [Ner98, l.4J.
