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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 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!

I have just uploaded a paper to the arXiv, entitled “Scl, sails and surgery”. The paper discusses a connection between stable commutator length in free groups and the geometry of sails. This is an interesting example of what sometimes happens in geometry, where a complicated topological problem in low dimensions can be translated into a “simple” geometric problem in high dimensions. Other examples include the Veronese embedding in Algebraic geometry (i.e. the embedding of one projective space into another taking a point with homogeneous co-ordinates $x_i$ to the point whose homogeneous co-ordinates are the monomials of some fixed degree in the $x_i$), which lets one exhibit any projective variety as an intersection of a Veronese variety (whose geometry is understood very well) with a linear subspace.

In my paper, the fundamental problem is to compute stable commutator length in free groups, and more generally in free products of Abelian groups. Let’s focus on the case of a group $G = A*B$ where $A,B$ are free abelian of finite rank. A $K(G,1)$ is just a wedge $X:=K_A \vee K_B$ of tori of dimension equal to the ranks of $A,B$. Let $\Gamma: \coprod_i S^1 \to X$ be a free homotopy class of $1$-manifold in $X$, which is homologically trivial. Formally, we can think of $\Gamma$ as a chain $\sum_i g_i$ in $B_1^H(G)$, the vector space of group $1$-boundaries, modulo homogenization; i.e. quotiented by the subspace spanned by chains of the form $g^n - ng$ and $g-hgh^{-1}$. One wants to find the simplest surface $S$ mapping to $X$ that rationally bounds $\Gamma$. I.e. we want to find a map $f:S \to X$ such that $\partial f:\partial S \to X$ factors through $\Gamma$, and so that the boundary $\partial S$ wraps homologically $n(S)$ times around each loop of $\Gamma$, in such a way as to infimize $-\chi(S)/2n(S)$. This infimum, over all maps of all surfaces $S$ of all possible genus, is the stable commutator length of the chain $\sum_i g_i$. Computing this quantity for all such finite chains is tantamount to understanding the bounded cohomology of a free group in dimension $2$.

Given such a surface $S$, one can cut it up into simpler pieces, along the preimage of the basepoint $K_A \cap K_B$. Since $S$ is a surface with boundary, these simpler pieces are surfaces with corners. In general, understanding how a surface can be assembled from an abstract collection of surfaces with corners is a hopeless task. When one tries to glue the pieces back together, one runs into trouble at the corners — how does one decide when a collection of surfaces “closes up” around a corner? The wrong decision leads to branch points; moreover, a decision made at one corner will propogate along an edge and lead to constraints on the choices one can make at other corners. This problem arises again and again in low-dimensional topology, and has several different (and not always equivalent) formulations and guises, including -

• Given an abstract branched surface and a weight on that surface, when is there an unbranched surface carried by the abstract branched surface and realizing the weight?
• Given a triangulation of a $3$-manifold and a collection of normal surface types in each simplex satisfying the gluing constraints but *not*  necessarily satisfying the quadrilateral condition (i.e. there might be more than one quadrilateral type per simplex), when is there an immersed unbranched normal surface in the manifold realizing the weight?
• Given an immersed curve in the plane, when is there an immersion from the disk to the plane whose boundary is the given curve?
• Given a polyhedral surface (arising e.g. in computer graphics), how can one choose smooth approximations of the polygonal faces that mesh smoothly at the vertices?

I think of all these problems as examples of what I like to call the holonomy problem, since all of them can be reduced, in one way or another, to studying representations of fundamental groups of punctured surfaces into finite groups. The fortunate “accident” in this case is that every corner arises by intersecting a cut with a boundary edge of $S$. Consequently, one never wants to glue more than two pieces up at any corner, and the holonomy problem does not arise. Hence in principle, to understand the surface $S$ one just needs to understand the pieces of $S$ that can arise by cutting, and the ways in which they can be reassembled.

This is still not a complete solution of the problem, since infinitely many kinds of pieces can arise by cutting complicated surfaces $S$. The $1$-manifold $\Gamma$ decomposes into a collection of arcs in the tori $K_A$ and $K_B$ which we denote $\tau_A,\tau_B$ respectively, and the surface $S \cap K_A$ (hereafter abbreviated to $S_A$) has edges that alternate between elements of $\tau_A$, and edges mapping to $K_A \cap K_B$. Since $K_A$ is a torus, handles of $S_A$ mapping to $K_A$ can be compressed, reducing the complexity of $S_A$, and thereby $S$, so one need only consider planar surfaces $S_A$.

Let $C_2(A)$ denote the real vector space with basis the set of ordered pairs $(t,t')$ of elements of $\tau_A$ (not necessarily distinct), and $C_1(A)$ the real vector space with basis the elements of $\tau_A$. A surface $S_A$ determines a non-negative integral vector $v(S_A) \in C_2(A)$, by counting the number of times a given pair of edges $(t,t')$ appear in succession on one of the (oriented) boundary components of $S_A$. The vector $v(S_A)$ satisfies two linear constraints. First, there is a map $\partial: C_2(A) \to C_1(A)$ defined on a basis vector by $\partial(t,t') = t - t'$. The vector $v(S_A)$ satisfies $\partial v(S_A) = 0$. Second, each element $t \in \tau_A$ is a based loop in $K_A$, and therefore corresponds to an element in the free abelian group $A$. Define $h:C_2(A) \to A \otimes \mathbb{R}$ on a basis vector by $h(t,t') = t+t'$ (warning: the notation obscures the fact that $\partial$ and $h$ map to quite different vector spaces). Then $h v(S_A)=0$; moreover, a non-negative rational vector $v \in C_2(A)$ satisfying $\partial v = h v = 0$ has a multiple of the form $v(S_A)$ for some $S_A$ as above. Denote the subspace of $C_2(A)$ consisting of non-negative vectors in the kernel of $\partial$ and $h$ by $V_A$. This is a rational polyhedral cone — i.e. a cone with finitely many extremal rays, each spanned by a rational vector.

Although every integral $v \in V_A$ is equal to $v(S_A)$ for some $S_A$, many different $S_A$ correspond to a given $v$. Moreover, if we are allowed to consider formal weighted sums of surfaces, then even more possibilities. In order to compute stable commutator length, we must determine, for a given vector $v \in V_A$, an expression $v = \sum t_i v(S_i)$ where the $t_i$ are positive real numbers, which minimizes $\sum -t_i \chi_o(S_i)$. Here $\chi_o(\cdot)$ denotes orbifold Euler characteristic of a surface with corners; each corner contributes $-1/4$ to $\chi_o$. The reason one counts complexity using this modified definition is that the result is additive: $\chi(S) = \chi_o(S_A) + \chi_o(S_B)$. The contribution to $\chi_o$ from corners is a linear function on $V_A$. Moreover, a component $S_i$ with $\chi(S_i) \le 0$ can be covered by a surface of high genus and compressed (increasing $\chi$); so such a term can always be replaced by a formal sum $1/n S_i'$ for which $\chi(S_i') = \chi(S_i)$. Thus the only nonlinear contribution to $\chi_o$ comes from the surfaces $S_i$ whose underlying topological surface is a disk.

Call a vector $v \in V_A$ a disk vector if $v = v(S_A)$ where $S_A$ is topologically a disk (with corners). It turns out that the set of disk vectors $\mathcal{D}_A$ has the following simple form: it is equal to the union of the integer lattice points contained in certain of the open faces of $V_A$ (those satisfying a combinatorial criterion). Define the sail of $V_A$ to be equal to the boundary of the convex hull of the polyhedron $\mathcal{D}_A + V_A$ (where $+$ here denotes Minkowski sum). The Klein function $\kappa$ is the unique continuous function on $V_A$, linear on rays, that is equal to $1$ exactly on the sail. Then $\chi_o(v):= \max \sum t_i\chi_o(S_i)$ over expressions $v = \sum t_i v(S_i)$ satisfies $\chi_o(v) = \kappa(v) - |v|/2$ where $|\cdot|$ denotes $L^1$ norm. To calculate stable commutator length, one minimizes $-\chi_o(v) - \chi_o(v')$ over $(v,v')$ contained in a certain rational polyhedron in $V_A \times V_B$.

Sails are considered elsewhere by several authors; usually, people take $\mathcal{D}_A$ to be the set of all integer vectors except the vertex of the cone, and the sail is therefore the boundary of the convex hull of this (simpler) set. Klein introduced sails as a higher-dimensional generalization of continued fractions: if $V$ is a polyhedral cone in two dimensions (i.e. a sector in the plane, normalized so that one edge is the horizontal axis, say), the vertices of the sail are the continued fraction approximations of the boundary slope. Arnold has revived the study of such objects in recent years. They arise in many different interesting contexts, such as numerical analysis (especially diophantine approximation) and algebraic number theory. For example, let $A \in \text{SL}(n,\mathbb{Z})$ be a matrix with irreducible characteristic equation, and all eigenvalues real and positive. There is a basis for $\mathbb{R}^n$ consisting of eigenvalues, spanning a convex cone $V$. The cone — and therefore its sail — is invariant under $A$; moreover, there is a $\mathbb{Z}^{n-1}$ subgroup of $\text{SL}(n,\mathbb{Z})$ consisting of matrices with the same set of eigenvectors; this observation follows from Dirichlet’s theorem on the units in a number field, and is due to Tsuchihashi. This abelian group acts freely on the sail with quotient a (topological) torus of dimension $n-1$, together with a “canonical” cell decomposition. This connection between number theory and combinatorics is quite mysterious; for example, Arnold asks: which cell decompositions can arise? This is unknown even in the case $n=3$.

The most interesting aspect of this correspondence, between stable commutator length and sails, is that it allows one to introduce parameters. An element in a free group $F_2$ can be expressed as a word in letters $a,b,a^{-1},b^{-1}$, e.g. $aab^{-1}b^{-1}a^{-1}a^{-1}a^{-1}bbbbab^{-1}b^{-1}$, which is usually abbreviated with exponential notation, e.g. $a^2b^{-2}a^{-3}b^4ab^{-2}$. Having introduced this notation, one can think of the exponents as parameters, and study stable commutator length in families of words, e.g. $a^{2+p}b^{-2+q}a^{-3-p}b^{4-q}ab^{-2}$. Under the correspondence above, the parameters only affect the coefficients of the linear map $h$, and therefore one obtains families of polyhedral cones $V_A(p,q,\cdots)$ whose extremal rays depend linearly on the exponent parameters. This lets one prove many facts about the stable commutator length spectrum in a free group, including:

Theorem: The image of a nonabelian free group of rank at least $4$ under scl in $\mathbb{R}/\mathbb{Z}$ is precisely $\mathbb{Q}/\mathbb{Z}$.

and

Theorem: For each $n$, the image of the free group $F_n$ under scl contains a well-ordered sequence of values with ordinal type $\omega^{\lfloor n/4 \rfloor}$. The image of $F_\infty$ contains a well-ordered sequence of values with ordinal type $\omega^\omega$.

One can also say things about the precise dependence of scl on parameters in particular families. More conjecturally, one would like to use this correspondence to say something about the statistical distribution of scl in free groups. Experimentally, this distribution appears to obey power laws, in the sense that a given (reduced) fraction $p/q$ appears in certain infinite families of elements with frequency proportional to $q^{-\delta}$ for some power $\delta$ (which unfortunately depends in a rather opaque way on the family). Such power laws are reminiscent of Arnold tongues in dynamics, one of the best-known examples of phase locking of coupled nonlinear oscillators. Heuristically one expects such power laws to appear in the geometry of “random” sails — this is explained by the fact that the (affine) geometry of a sail depends only on its $\text{SL}(n,\mathbb{Z})$ orbit, and the existence of invariant measures on a natural moduli space; see e.g. Kontsevich and Suhov. The simplest example concerns the ($1$-dimensional) cone spanned by a random integral vector in $\mathbb{Z}^2$. The $\text{SL}(2,\mathbb{Z})$ orbit of such a vector depends only on the gcd of the two co-ordinates. As is easy to see, the probability distribution of the gcd of a random pair of integers $p,q$ obeys a power law: $\text{gcd}(p,q) = n$ with probability $\zeta(2)^{-1}/n^2$. The rigorous justification of the power laws observed in the scl spectrum of free groups remains the focus of current research by myself and my students.