I am (update: was) currently (update: but am no longer) in Brisbane for the “New directions in geometric group theory” conference, which has been an entirely enjoyable and educational experience. I got to eat fish and chips, to watch Australia make 520 for 7 (declared) against the West Indies at the WACA, and to hear Masato Mimura give a very nice talk about his recent results on rigidity of the “universal lattice”.
His talk included a quick and beautiful survey of some geometric aspects of the theory of rigidity for infinite groups, which I will attempt to partially reproduce (despite the limitations of the wordpress format). In this context, rigidity is expressed in terms of isometric affine actions of groups on Banach spaces. This means the following. Suppose is a Banach space (i.e. a complete, normed vector space) and
is a group. A linear isometric action is a representation
from
to the group of linear isometries of
— i.e. linear norm-preserving automorphisms. An affine action is a representation from
to the group of affine isometries of
— i.e. isometries as a metric space that do not necessarily fix the zero element. The group of isometries of a Banach space
is a semi-direct product
where
is the group of linear isometries, and
is the Banach space, thought of as an Abelian group, acting on itself by (isometric) translations. Such an action is usually encoded by a pair
which records the “linear” part of the action, and a 1-cocycle with coefficients in
, i.e. a function
satisfying
for every
. This formula might look strange if you don’t know where it comes from: it is just the way that factors transform in semi-direct products. The affine action is given by sending
to the transformation that sends each
to
. Consequently,
is sent to the transformation that sends
to
and the fact that this is a group action becomes the formula
Equating the left and right hand sides gives the cocycle condition. Given one affine isometric action, one can obtain another in a silly way by conjugating by an isometry for some
. Under conjugation by such an isometry, a cocycle
transforms by
. A function of the form
is called a 1-coboundary, and the quotient of the space of 1-cocycles by the space of 1-coboundaries is the 1 dimensional cohomology of
with coefficients in
. This is usually denoted
, where
is suppressed in the notation. In particular, an affine isometric action of
on
with linear part
has a global fixed point iff it represents
in
. Contrapositively,
admits an affine isometric action on
without a global fixed point iff
for some
.
A group is said to satisfy Serre’s Property (FH) if every affine isometric action of
on a Hilbert space has a global fixed point. In 2007, Bader-Furman-Gelander-Monod introduced a property (FB) for a group
to mean that every affine isometric action of
on some (out of a class of) Banach space(s)
has a global fixed point. Mimura used the notation property (FL_p) for the case that
is allowed to range over the class of
spaces (for some fixed
).
Intimately related is Kazhdan’s Property (T), introduced by Kazhdan in this paper. Let be a locally compact topological group (for example, a discrete group). The set of irreducible unitary representations of
is called its dual, and denoted
. This dual is topologized in the following way. Associated to a representation
, a unit vector
, a positive number
and a compact subset
there is an open neighborhood of
consisting of representations
for which there is a unit vector
such that
whenever
. With this topology (called the Fell topology), one says that a group
has property (T) if the trivial representation is isolated in
. Note that this topology is very far from being Hausdorff: the trivial representation fails to be isolated exactly when there are a sequence of representations
, unit vectors
, numbers
and compact sets
exhausting
so that
for any
. The vectors
are said to be (a sequence of) almost invariant vectors. Hence (informally) a group has property (T) if some compact subset must move some unit vector a definite amount in every irreducible nontrivial unitary representation. If a group fails to have property (T), one can rescale a sequence of irreducible actions near a sequence of almost invariant vectors in such a way that one obtains in the geometric limit a nontrivial isometric action on
without a global fixed point. A famous theorem of Delorme–Guichardet says that property (T) and property (FH) are equivalent for (locally compact second countable) groups. Property (T) passes to quotients, and to lattices (i.e. finite covolume discrete subgroups of a topological group). Kazhdan already showed in his paper that
has property (T) for
at least
, and therefore the same is true for lattices in this groups, such as
, a fact which is not easy to see directly from the definition. One beautiful application, already pointed out by Kazhdan, is that this means that all lattices in
, for instance the groups
(and in fact, all discrete groups with property (T)) are finitely generated. Kazhdan’s proof of this is incredibly short: let
be a discrete group and
and sequence of elements. For each
, let
be the subgroup of
generated by
. Notice that
is finitely generated iff
for all sufficiently large
. On the other hand, consider the unitary representations of
induced by the trivial representations on the
. Every compact subset of
is finite, and therefore eventually fixes a vector in every one of these representations; thus there is a sequence of almost fixed vectors. If
has property (T), this sequence eventually contains a fixed vector, which can only happen if
is finite, in which case
is finitely generated, as claimed.
Property (FL_p) generalizes (FH) (equivalently (T)) in many significant ways, with interesting applications to dynamics. For example, Navas showed that if is a group with property (T) then every action of
on a circle which is at least
factors through a finite group. Navas’s argument can be generalized straightforwardly to show that if
has (FL_p) for some
then every action of
on a circle which is at least
factors through a finite group. The proof rests on a beautiful construction due to Reznikov (although a similar construction can be found in Pressley-Segal) of certain functions on a configuration space of the circle which are not in
but have coboundaries which are; this gives rise to nontrivial cohomology with
coefficients for groups acting on the circle in a sufficiently interesting way.
(Update: Nicolas Monod points out in an email that the “function on a configuration space” is morally just the derivative. In fact, he made the nice remark that if is any elliptic operator on an
-manifold, then the commutator
is of Schatten class
whenever
is a sufficiently smooth function; morally this should give rise to nontrivial cohomology with suitable coefficients for groups acting with enough regularity on any given
-manifold, and one would like to use this e.g. to approach Zimmer’s conjecture, but nobody seems to know how to make this work as yet; in fact the work of Monod et. al. on (FL_p) is at least partly motivated by this general picture.)
Mimura discussed a spectrum of rigid behaviour for infinite groups, ranging from most rigid (property (FL_p) for every ) to least rigid (amenable) (note: every finite group is both amenable and has property (T), so this only really makes sense for infinite groups; moreover, every reasonable measure of rigidity for infinite groups is usually invariant under passing to subgroups of finite index). Free groups,
and so on are very non-rigid. However, it is well-known that certain infinite families of (word) hyperbolic groups, including lattices in groups of isometries of quaternion-hyperbolic symmetric spaces, and “random” groups with relations having density parameter
(see Zuk or Ollivier) are both hyperbolic and have property (T). Nevertheless, these groups are not as rigid as higher rank lattices like
for
. The latter have property (FL_p) for every
, whereas Yu showed that every hyperbolic group admits a proper affine isometric action on
for some
(the existence of a proper affine isometric action on a Hilbert space is called “a-T-menability” by Gromov, and the “Haagerup property” by some. Groups satisfying this property, or even Yu’s weaker property, are known to satisfy some version of the Baum-Connes conjecture, the subject of a very nice minicourse by Graham Niblo at the same conference).
It is in this context that one can appreciate Mimura’s results. His first main result is that the group (i.e. the “universal lattice”) has property (FL_p) for every
provided
is at least 4. Since property (FL_p) (like (T)) passes to quotients, this implies that
has (FL_p) for every unital, commutative, finitely generated ring
.
His second main result concerns a “quasification” of FL_p, to a property called (FFL_p). Without getting too technical, this property concerns “quasi-actions” of a group on a Banach space by affine isometries; algebraically these are encoded by 1-cochains for which there is a universal constant
so that
as measured in the Banach norm on
. Any bounded map
defines a 1-cochain; such (bounded) 1-cochains corresponds to quasi-action with a bounded orbit. Associated to
one defines in a similar way a complex of bounded cochains; quasi-actions modulo bounded quasi-actions are parameterized by the kernel of the comparison map
from bounded to ordinary cohomology. Mimura’s second main result is that when
is the universal lattice as above, and
has no invariant vectors, the comparison map from bounded to ordinary cohomology in dimension 2 is injective.
The fact that as above is required to have no invariant vectors is a technical necessity of Mimura’s proof. When
is trivial, one is studying “ordinary” bounded cohomology, and there is an exact sequence
with real coefficients for any (here
denotes the vector space of homogeneous quasimorphisms on
). In this context, one knows by Bavard duality that
is injective if and only if the stable commutator length is identically zero on
. By quite a different method, Mimura shows that for
at least
, and for any Euclidean ring
(i.e. a ring for which one has a Euclidean algorithm; for example,
) the group
has vanishing stable commutator length, and therefore one has injectivity of bounded to ordinary cohomology in dimension
.
(Update 1/9/2010): Nicholas Monod sent me a nice email commenting on a couple of points in this blog entry, and I have consequently modified the language a bit in a few places. Ta much!