FH, T, FLp and all that

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 B is a Banach space (i.e. a complete, normed vector space) and G is a group. A linear isometric action is a representation \rho from G to the group of linear isometries of B — i.e. linear norm-preserving automorphisms. An affine action is a representation from G to the group of affine isometries of B — i.e. isometries as a metric space that do not necessarily fix the zero element. The group of isometries of a Banach space B is a semi-direct product B \rtimes U(B) where U(B) is the group of linear isometries, and B 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 \rho:G \to U(B) which records the “linear” part of the action, and a 1-cocycle with coefficients in \rho, i.e. a function c:G \to B satisfying c(gh) = c(g) + \rho(g)c(h) for every g,h \in G. 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 g \in G to the transformation that sends each b \in B to \rho(g)b + c(g). Consequently, gh is sent to the transformation that sends b to \rho(gh)b + c(gh) and the fact that this is a group action becomes the formula

\rho(gh)b + c(gh) = \rho(g)(\rho(h)b + c(h)) + c(g) = \rho(gh)b + \rho(g)c(h) + c(g)

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 b \to b + b' for some b' \in B. Under conjugation by such an isometry, a cocycle c transforms by c(g) \to c(g) + \rho(g)b' - b'. A function of the form c(g) = \rho(g)b' - b' 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 G with coefficients in \rho:G \to U(B). This is usually denoted H^1(G,\rho), where B is suppressed in the notation. In particular, an affine isometric action of G on B with linear part \rho has a global fixed point iff it represents 0 in H^1(G,\rho). Contrapositively, G admits an affine isometric action on B without a global fixed point iff H^1(G,\rho) \ne 0 for some \rho.

A group G is said to satisfy Serre’s Property (FH) if every affine isometric action of G on a Hilbert space has a global fixed point. In 2007, Bader-Furman-Gelander-Monod introduced a property (FB) for a group G to mean that every affine isometric action of G on some (out of a class of) Banach space(s) B has a global fixed point. Mimura used the notation property (FL_p) for the case that B is allowed to range over the class of L_p spaces (for some fixed 1 < p < \infty).

Intimately related is Kazhdan’s Property (T), introduced by Kazhdan in this paper. Let G be a locally compact topological group (for example, a discrete group). The set of irreducible unitary representations of G is called its dual, and denoted \hat{G}. This dual is topologized in the following way. Associated to a representation \rho:G \to U(L), a unit vector X \in L, a positive number \epsilon > 0 and a compact subset K \subset G there is an open neighborhood of \rho consisting of representations \rho':G \to U(L') for which there is a unit vector Y \in L such that |\langle \rho(g)X,X\rangle - \langle \rho(g')Y, Y\rangle| < \epsilon whenever g \in K. With this topology (called the Fell topology), one says that a group G has property (T) if the trivial representation is isolated in \hat{G}. 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 \rho_i:G \to U(L_i), unit vectors X_i \in L_i, numbers \epsilon_i \to 0 and compact sets K_i exhausting G so that |\langle\rho_i(g)X_i,X_i\rangle| < \epsilon_i for any g \in K_i. The vectors X_i 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 L^2 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 \text{SL}(n,\mathbb{R}) has property (T) for n at least 3, and therefore the same is true for lattices in this groups, such as \text{SL}(n,\mathbb{Z}), 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 \text{SL}(n,\mathbb{R}), for instance the groups \text{SL}(n,\mathbb{Z}) (and in fact, all discrete groups with property (T)) are finitely generated. Kazhdan’s proof of this is incredibly short: let G be a discrete group and g_i and sequence of elements. For each i, let G_i be the subgroup of G generated by \lbrace g_1,g_2,\cdots,g_i\rbrace. Notice that G is finitely generated iff G_i=G for all sufficiently large i. On the other hand, consider the unitary representations of G induced by the trivial representations on the G_i. Every compact subset of G is finite, and therefore eventually fixes a vector in every one of these representations; thus there is a sequence of almost fixed vectors. If G has property (T), this sequence eventually contains a fixed vector, which can only happen if G/G_i is finite, in which case G 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 G is a group with property (T) then every action of G on a circle which is at least C^{1+1/2 + \epsilon} factors through a finite group. Navas’s argument can be generalized straightforwardly to show that if G has (FL_p) for some p>2 then every action of G on a circle which is at least C^{1+1/p+\epsilon} 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 L^p but have coboundaries which are; this gives rise to nontrivial cohomology with L^p 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 D is any elliptic operator on an n-manifold, then the commutator [D,g] is of Schatten class (n+1) whenever g 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 n-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 p) 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, \text{SL}(2,\mathbb{Z}) 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 1/3 < d < 1/2 (see Zuk or Ollivier) are both hyperbolic and have property (T). Nevertheless, these groups are not as rigid as higher rank lattices like \text{SL}(n,\mathbb{Z}) for n>2. The latter have property (FL_p) for every 1< p < \infty, whereas Yu showed that every hyperbolic group admits a proper affine isometric action on \ell^p for some p (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 \text{SL}_n(\mathbb{Z}[x_1,x_2,\cdots,x_n]) (i.e. the “universal lattice”) has property (FL_p) for every 1<p<\infty provided n is at least 4. Since property (FL_p) (like (T)) passes to quotients, this implies that \text{SL}_n(R) has (FL_p) for every unital, commutative, finitely generated ring R.

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 c:G \to B for which there is a universal constant D so that |c(gh) - c(g) -\rho(g)c(h)| < D as measured in the Banach norm on B. Any bounded map c:G \to B defines a 1-cochain; such (bounded) 1-cochains corresponds to  quasi-action with a bounded orbit. Associated to \rho: G \to U(B) 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 H^2_b(G,\rho) \to H^2(G,\rho) from bounded to ordinary cohomology. Mimura’s second main result is that when G is the universal lattice as above, and \rho has no invariant vectors, the comparison map from bounded to ordinary cohomology in dimension 2 is injective.

The fact that \rho as above is required to have no invariant vectors is a technical necessity of Mimura’s proof. When \rho is trivial, one is studying “ordinary” bounded cohomology, and there is an exact sequence

0 \to H^1(G) \to Q(G) \to H^2_b(G) \to H^2(G)

with real coefficients for any G (here Q(G) denotes the vector space of homogeneous quasimorphisms on G). In this context, one knows by Bavard duality that H^2_b \to H^2 is injective if and only if the stable commutator length is identically zero on [G,G]. By quite a different method, Mimura shows that for n at least 6, and for any Euclidean ring R (i.e. a ring for which one has a Euclidean algorithm; for example, R = \mathbb{C}[x]) the group SL_n(R) has vanishing stable commutator length, and therefore one has injectivity of bounded to ordinary cohomology in dimension 2.

(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!

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