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A beautiful identity in Euclidean geometry is the Brianchon-Gram relation (also called the Gram-Sommerville formula, or Gram’s equation), which says the following: let P be a convex polytope, and for each face F of P, let \omega(F) denote the solid angle along the face, as a fraction of the volume of a linking sphere. The relation then says:

Theorem (Brianchon-Gram relation): \sum_{F \subset P} (-1)^{\text{dim}F} \omega(F)=0. In other words, the alternating sum of the (solid) angles of all dimensions of a convex polytope is zero.

Sketch of Proof: we prove the theorem in the case that P is a simplex \Delta; the more general case follows by generalizing to pyramids, and then decomposing any polytope into pyramids by coning to an interior point. This argument is due to Shephard.

Associated to each face F is a spherical polyhedron A(F) in S^{n-1}; if the span of F is the intersection of a family of half-spaces bounded by hyperplanes H_i with inward normals n_i, then A(F) is the set of unit vectors v \in S^{n-1} whose inner product with each n_i is non-negative. Note further that for each v \in S^{n-1} there is some n_i that pairs non-negatively with v; consequently to each v \in S^{n-1} one can assign a subset I(v) of indices, so that n_i pairs non-negatively with v if and only if i \in I(v). On the other hand, each subset J \subset I(v) determines a unique face F(J) of dimension n - |J|. By the inclusion-exclusion formula, we conclude that \sum_{F} (-1)^{\text{dim}F}A(F) “equals” zero, thought of as a signed union of spherical polyhedra. Since \omega(F) = \text{vol}(A(F))/\text{vol}(S^{n-1}), the formula follows. qed.

Another well-known proof starts by approximating the polytope by a rational polytope (i.e. one with rational vertices). The proof then goes via Macdonald reciprocity, using generating functions.

Example: Let T be a triangle, with angles \alpha,\beta,\gamma. The solid angle at an interior point is 1, and the solid angle at each edge is 1/2. Hence we get (\alpha + \beta + \gamma)/2\pi - 3/2 + 1 = 0 and therefore in this case Brianchon-Gram is equivalent to the familiar angle sum identity for a triangle: \alpha + \beta + \gamma = \pi.

Example: Next consider the example of a Euclidean simplex S. The contribution from the interior is -1, and the contribution from the four facets is 2. There are six edges, with angles \alpha_i, that  contribute \sum \alpha_i/2\pi. Each vertex contributes one spherical triangle, with (spherical) angles \alpha_i,\alpha_j,\alpha_k for certain i,j,k, where each \alpha_i appears as a spherical angle in exactly two spherical triangles. The Gauss-Bonnet theorem implies that the area of a spherical triangle is equal to the angle sum defect: \text{area}_{ijk} = \alpha_i + \alpha_j + \alpha_k - \pi so the vertices contribute (2\sum \alpha_i - 4 \pi)/4\pi and the identity is seen to follow in this case too.

Note in fact that the usual proof of Gauss-Bonnet for a spherical triangle is done by an inclusion-exclusion argument involving overlapping lunes, that is very similar to the proof of Brianchon-Gram given above.

The sketch of proof above just as easily proves an identity in the spherical scissors congruence group. For X^n equal to spherical, Euclidean or hyperbolic space of dimension n, the scissors congruence group \mathcal{P}(X^n) is the abelian group generated by formal symbols (x_0,x_1,\cdots,x_n,\alpha) where x_i \in X^n and \alpha is a choice of orientation, modulo certain relations, namely:

  1. (x_0,x_1,\cdots,x_n,\alpha)=0 if the x_i are contained in a hyperplane
  2. an odd permutation of the points induces multiplication by -1; changing the orientation induces multiplication by -1
  3. if g is an isometry of X^n, then (x_0,\cdots,x_n,\alpha) = (gx_0,\cdots,gx_n,g_*\alpha)
  4. \sum_i (-1)^i (x_0,\cdots,\widehat{x_i},\cdots,x_{n+1},\alpha) for any set of n+2 points, and any orientation \alpha

(Note that this definition of scissors congruence is consistent with that of Goncharov, and differs slightly from another definition consistent with Sah; this difference has to do with orientations, and has as a consequence the vanishing of spherical scissors congruence in even dimensions; whereas with Sah’s definition, \mathcal{P}(S^{2n}) = \mathcal{P}(S^{2n-1}) for each n)

The argument we gave above shows that for any Euclidean simplex \Delta, we have \sum_F(-1)^{\text{dim}F} A(F) = 0 in \mathcal{P}(S^{n-1}).

Scissors congruence satisfies several fundamental properties:

  1. S^n = 0 in \mathcal{P}(S^n). To see this, “triangulate” the sphere as a pair of degenerate simplices, whose vertices lie entirely on a hyperplane.
  2. There is a natural multiplication \mathcal{P}(S^{a-1}) \otimes \mathcal{P}(S^{b-1}) \to \mathcal{P}(S^{a+b-1}); to define it on simplices, think of S^{a+b-1} as the unit sphere in \mathbb{R}^{a+b}. A complementary pair of subspaces \mathbb{R}^a and \mathbb{R}^b intersect S^{a+b-1} in a linked pair of spheres of dimensions a-1,b-1; if \Delta_a,\Delta_b are spherical simplices in these subspaces, the image of \Delta_a \otimes \Delta_b is the join of these two simplices in S^{a+b-1}.

It follows that the polyhedra A(F)=0 in \mathcal{P}(S^{n-1}) whenever F is a face of dimension at least 1; for in this case, A(F) is the join of a spherical simplex with a sphere of some dimension, and is therefore trivial in spherical scissors congruence. Hence the identity above simplifies to \sum_v A(v)=0 in \mathcal{P}(S^{n-1}).

One nice application is to extend the definition of Dehn invariants to ideal hyperbolic simplices. We recall the definition of the usual Dehn invariant. Given a simplex P \in X^n, for each face F we let \angle(F) denote the spherical polyhedron equal to the intersection of P with the link of F. Then D(P) = \sum_F F\otimes \angle(F) \in \oplus_i \mathcal{P}(X^{n-i})\otimes \mathcal{P}(S^{i-1}). Ideal scissors congruence makes sense for ideal hyperbolic simplices, except in dimension one (where it is degenerate). For ideal hyperbolic simplices (i.e. those with some vertices at infinity), the formula above for Dehn invariant is adequate, except for the 1-dimensional faces (i.e. the edges) e. This problem is solved by the following “regularization” procedure due to Thurston: put a disjoint horoball at each ideal vertex of P, and replace each infinite edge e by the finite edge e' which is the intersection of e with the complement of the union of horoballs; hence one obtains terms of the form e' \otimes \angle(e) in D(P). This definition apparently depends on the choice of horoballs. However, if H,H' are two different horoballs, the difference is a sum of terms of the form c \otimes \angle(e) where c is constant, and e ranges over the edges sharing the common ideal vertex. The intersection of P with a horosphere is a Euclidean simplex \Delta, and the \angle(e) are exactly the spherical polyhedra A(v) as v ranges over the vertices of \Delta. By what we have shown above, the sum \sum_v A(v) is trivial in scissors congruence; it follows that D(P) is well-defined.

For more general ideal polyhedra (and finite volume complete hyperbolic manifolds) one first decomposes into ideal simplices, then computes the Dehn invariant on each piece and adds. A minor variation of the usual argument on closed manifolds shows that the Dehn invariant of any complete finite-volume hyperbolic manifold vanishes.

Update(7/29/2009): It is perhaps worth remarking that the Brianchon-Gram relation can be thought of, not merely as an identity in spherical scissors congruence, but in the “bigger” spherical polytope group, in which one does not identify simplices that differ by an isometry. Incidentally, there is an interesting paper on this subject by Peter McMullen, in which he proves generalizations of Brianchon-Gram(-Sommerville), working explicitly in the spherical polytope group. He introduces what amounts to a generalization of the Dehn invariant, with domain the Euclidean translational scissors congruence group, and range a sum of tensor products of Euclidean translational scissors congruence (in lower dimensions) with spherical polytope groups. It appears, from the paper, that McMullen was aware of the classical Dehn invariant (in any case, he was aware of Sah’s book) but he does not refer to it explicitly.

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.

The development and scope of modern biology is often held out as a fantastic opportunity for mathematicians. The accumulation of vast amounts of biological data, and the development of new tools for the manipulation of biological organisms at microscopic levels and with unprecedented accuracy, invites the development of new mathematical tools for their analysis and exploitation. I know of several examples of mathematicians who have dipped a toe, or sometimes some more substantial organ, into the water. But it has struck me that I know (personally) few mathematicians who believe they have something substantial to learn from the biologists, despite the existence of several famous historical examples.  This strikes me as odd; my instinctive feeling has always been that intellectual ruts develop so easily, so deeply, and so invisibly, that continual cross-fertilization of ideas is essential to escape ossification (if I may mix biological metaphors . . .)

It is not necessarily easy to come up with profound examples of biological ideas or principles that can be easily translated into mathematical ones, but it is sometimes possible to come up with suggestive ones. Let me try to give a tentative example.

Deoxiribonucleic acid (DNA) is a nucleic acid that contains the genetic blueprint for all known living things. This blueprint takes the form of a code — a molecule of DNA is a long polymer strand composed of simple units called nucleotides; such a molecule is typically imagined as a string in a four character alphabet \lbrace A,T,G,C \rbrace, which stand for the nucleotides Adenine, Thymine, Guanine, and Cytosine. These molecular strands like to arrange themselves in tightly bound oppositely aligned pairs, matching up nucleotides in one string with complementary nucleotides in the other, so that A matches with T, and C with G.

The geometry of a strand of DNA is very complicated — strands can be tangled, knotted, linked in complicated ways, and the fundamental interactions between strands (e.g. transcription, recombination) are facilitated or obstructed by mechanical processes depending on this geometry. Topology, especially knot theory, has been used in the study of some of these processes; the value of topological methods in this context include their robustness (fault-tolerance) and the discreteness of their invariants (similar virtues motivate some efforts to build topological quantum computers). A complete mathematical description of the salient biochemistry, mechanics, and semantic content of a configuration of DNA in a single cell is an unrealistic goal for the foreseeable future, and therefore attempts to model such systems depends on ignoring, or treating statistically, certain features of the system. One such framework ignores the ambient geometry entirely, and treats the system using symbolic, or combinatorial methods which have some of the flavor of geometric group theory.

One interesting approach is to consider a mapping from the alphabet of nucleotides to a standard generating set for F_2, the free group on two generators; for example, one can take the mapping T \to a, A \to A, C \to b, G \to B where a,b are free generators for F_2, and {}A,B denote their inverses. Then a pair of oppositely aligned strands of DNA translates into an edge of a van Kampen diagram — the “words” obtained by reading the letters along an edge on either side are inverse in F_2.

Strands of DNA in a configuration are not always paired along their lengths; sometimes junctions of three or more strands can form; certain mobile four-strand junctions, so-called “Holliday junctions”, perform important functions in the process of genetic recombination, and are found in a wide variety of organisms. A configuration of several strands with junctions of varying valences corresponds in the language of van Kampen diagrams to a fatgraph — i.e. a graph together with a choice of cyclic ordering of edges at each vertex — with edges labeled by inverse pairs of words in F_2 (note that this is quite different from the fatgraph model of proteins developed by Penner-Knudsen-Wiuf-Andersen). The energy landscape for branch migration (i.e. the process by which DNA strands separate or join along some segment) is very complicated, and it is challenging to model it thermodynamically. It is therefore not easy to predict in advance what kinds of fatgraphs are more or less likely to arise spontaneously in a prepared “soup” of free DNA strands.

As a thought experiment, consider the following “toy” model, which I do not suggest is physically realistic. We make the assumption that the energy cost of forming a junction of valence v is c(v-2) for some fixed constant c. Consequently, the energy of a configuration is proportional to -\chi, i.e. the negative of Euler characteristic of the underlying graph. Let w be a reduced word, representing an element of F_2, and imagine a soup containing some large number of copies of the strand of DNA corresponding to the string \dot{w}:=\cdots www \cdots. In thermodynamic equilibrium, the partition function has the form Z = \sum_i e^{-E_i/k_BT} where k_B is Boltzmann’s constant, T is temperature, and E_i is the energy of a configuration (which by hypothesis is proportional to -\chi). At low temperature, minimal energy configurations tend to dominate; these are those that minimize -\chi per unit “volume”. Topologically, a fatgraph corresponding to such a configuration can be thickened to a surface with boundary. The words along the edges determine a homotopy class of map from such a surface to a K(F_2,1) (e.g. a once-punctured torus) whose boundary components wrap multiply around the free homotopy class corresponding to the conjugacy class of w. The infimum of -\chi/2d where d is the winding degree on the boundary, taken over all configurations, is precisely the stable commutator length of w; see e.g. here for a definition.

Anyway, this example is perhaps a bit strained (and maybe it owes more to thermodynamics than to biology), but already it suggests a new mathematical object of study, namely the partition function Z as above, and one is already inclined to look for examples for which the partition function obeys a symmetry like that enjoyed by the Riemann zeta function, or to specialize temperature to other values, as in random matrix theory. The introduction of new methods into the study of a classical object — for example, the decision to use thermodynamic methods to organize the study of van Kampen diagrams — bends the focus of the investigation towards those examples and contexts where the methods and tools are most informative. Phenomena familiar in one context (power laws, frequency locking, phase transitions etc.) suggest new questions and modes of enquiry in another. Uninspired or predictable research programs can benefit tremendously from such infusions, whether the new methods are borrowed from other intellectual disciplines (biology, physics), or depend on new technology (computers), or new methods of indexing (google) or collaboration (polymath).

One of my intellectual heroes — Wolfgang Haken — worked for eight years in R+D for Siemens in Munich after completing his PhD. I have a conceit (unsubstantiated as far as I know by biographical facts) that his experience working for a big engineering firm colored his approach to mathematics, and made it possible for him to imagine using industrial-scale “engineering” tools (e.g. integer programming, exhaustive computer search of combinatorial possibilities) to solve two of the most significant “pure” mathematical open problems in topology at the time — the knot recognition problem, and the four-color theorem. It is an interesting exercise to try to imagine (fantastic) variations. If I sit down and decide to try to prove (for example) Cannon’s conjecture, I am liable to try minor variations on things I have tried before, appeal for my intuition to examples that I understand well, read papers by others working in similar ways on the problem, etc. If I imagine that I have been given a billion dollars to prove the conjecture, I am almost certain to prioritize the task in different ways, and to entertain (and perhaps create) much more ambitious or innovative research programs to tackle the task. This is the way in which I understand the following quote by John Dewey, which I used as the colophon of my first book:

Every great advance in science has issued from a new audacity of the imagination.

Geometric group theory is not a coherent and unified field of enquiry so much as a collection of overlapping methods, examples, and contexts. The most important examples of groups are those that arise in nature: free groups and fundamental groups of surfaces, the automorphism groups of the same, lattices, Coxeter and Artin groups, and so on; whereas the most important properties of groups are those that lend themselves to applications or can be used in certain proof templates: linearity, hyperbolicity, orderability, property (T), coherence, amenability, etc. It is natural to confront examples arising in one context with properties that arise in the other, and this is the source of a wealth of (usually very difficult) problems; e.g. do mapping class groups have property (T)? (no, by Andersen) or: is every lattice in \text{PSL}(2,\mathbb{C}) virtually orderable?

As remarked above, it is natural to formulate these questions, but not necessarily productive. Gromov, in his essay Spaces and Questions remarks that

often the mirage of naturality lures us into featureless desert with no clear perspective where the solution, even if found, does not quench our thirst for structural mathematics . . . Another approach . . . has a better chance for a successful outcome with questions following (rather than preceding) construction of new objects.

A famous question of the kind Gromov warns against is the following:

Question: Is Thompson’s group F amenable?

Recall that Thompson’s group is the group of (orientation-preserving) PL homeomorphisms of the unit interval with breakpoints at dyadic rationals (i.e. rational numbers of the form p/2^q for integers p,q) and derivatives all powers of 2. This group is a rich source of examples/counterexamples in geometric group theory: it is finitely presented (in fact FP_\infty) but “looks like” a transformation group; it contains no nonabelian free group (by Brin-Squier), but obeys no law. It is not elementary amenable (i.e. it cannot be built up from finite or abelian groups by elementary operations — subgroups, quotients, extensions, directed unions), so it is “natural” to wonder whether it is amenable at all, or whether it is one of the rare examples of nonamenable groups without nonabelian free subgroups (see this post for a discussion of amenability versus the existence of free subgroups, and von Neumann’s conjecture). This question has attracted a great deal of attention, possibly because of its long historical pedigree, rather than because of the potential applications of a positive (or negative) answer.

Recently, two papers were posted on the arXiv, promising competing resolutions of the question. In February, Azer Akhmedov posted a preprint claiming to show that the group F is not amenable. This preprint was revised, withdrawn, then revised again, and as of the end of April, Akhmedov continues to press his claim. Akhmedov’s argument depends on a new geometric criterion for nonamenability, roughly speaking, the existence of a 2-generator subgroup and a subadditive non-negative function on the group whose values grow at a definite rate on words in the subgroup whose exponents satisfy suitable parity conditions and inequalities. The non-negative function (Akhmedov calls it a “height function”) certifies the existence of a sufficiently “bushy” subset of the group to violate Folner’s criterion for amenability. Akhmedov’s paper reads like a “conventional” paper in geometric group theory, using methods from coarse geometry, careful combinatorial and counting arguments to establish the existence of a geometric object with certain large-scale properties, and an appeal to a standard geometric criterion to obtain the desired result. Akhmedov’s paper is part of a series, relating (non)amenability to certain other interesting geometric properties, some related to the so-called “traveling salesman” property, introduced earlier by Akhmedov.

On the other hand, in May, E. Shavgulidze posted a preprint claiming to show that the group F is amenable. Interestingly enough, Shavgulidze’s argument does not apply to the slightly more general class of Stein-Thompson groups in which slopes and denominators of break points can be divisible by an arbitrary (but prescribed) finite set of prime numbers. Moreover, his methods are very unlike any that one would expect to find in the typical geometric group theory paper. The argument depends on the construction, going back (in some sense) to a paper of Shavgulidze from 1978, of a measure on the space C(I) of continuous functions on the interval which is quasi-invariant under the natural action of the group of diffeomorphisms of the interval of regularity at least C^3. In more detail, let D^n denote the group of diffeomorphisms of the interval of regularity at least C^n for each n, and let C denote the Banach space of continuous functions on the interval that vanish at the origin. Define A:D^1 \to C by the formula A(f)(t) = \log(f'(t)) - \log(f'(0)). The space C can be equipped with a natural measure — the Wiener measure w_\sigma of variance \sigma, and this measure can be pulled back to D^1 by A, which is thought of as a topological space with the C^1 topology. Shavgulidze shows that the left action of D^3 on D^1 quasi-preserves this measure. Here the Wiener measure on C is the probability measure associated to Brownian motion (with given variance). A “sample” trajectory W_t from C is characterized by three properties: that it starts at the origin (i.e. W_0=0), that is it continuous almost surely (this is already implicit in the fact that the measure is supported on the space C and not some more general space), and that increments are independent, with the distribution of W_t - W_s equal to a Gaussian with mean zero and variance (t-s)\sigma. Shavgulidze’s argument depends first on an argument of Ghys-Sergiescu that shows Thompson’s group is conjugate (by a homeomorphism) to a discrete subgroup of the group of C^\infty diffeomorphisms of the interval. A bounded function f on F determines a continuous bounded function \pi_\delta(f) on D^{1+\delta} (for \delta<1/2) by a certain convolution trick, using both the group structure of F, and its discreteness in D^3. Roughly, given an element g \in D^{1+\delta}, the set of elements of F whose (group) composition with g is uniformly bounded in the C^{1+\delta} norm is finite; so the value of \pi_\delta(f) is obtained by taking a suitable average of the value of f on this finite subset of F. This reduces the problem of the amenability of F to the existence of a suitable functional on the space of bounded continuous functions on D^{1+\delta}, which is constructed via the pulled back Wiener measure as above.

There are several distinctive features of Shavgulidze’s preprint. One of the most striking is that it depends on very delicate analytic features of the Wiener measure, and the way it transforms under the action of D^3 on D^1 — a transformation law involving the Schwartzian derivative — and suggesting that certain parts of the argument could be clarified (at least from the point of view of a topologist?) by using projective geometry and Sturm-Liouville theory. Another is that the crucial analytic quality — namely differentiability of class C^{1+1/2} — is also crucial for many other natural problems in 1-dimensional analysis and geometry, from regularity estimates in the thin obstacle problem, to Navas’ work on actions of property (T) groups on the circle. At least one of the preprints by Akhmedov and Shavgulidze must be in error (in fact, a real skeptic’s skeptic such as Michael Aschbacher is not even willing to concede that much . . .) but even if wrong, it is possible that they contain things more valuable than a resolution of the question that prompted them.

Update (7/6): Azer Akhmedov sent me a construction of a (nonabelian) free subgroup of D^1 that is discrete in the C^1 topology. This is not quite enough regularity to intersect with Shavgulidze’s program, but it is interesting, and worth explaining. This is my (minor) modification of Azer’s construction (any errors are due to me):

Proposition: The group D^1 contains a discrete nonabelian free subgroup.

Sketch of Proof: First, decompose the interval [0,1] into countably many disjoint subintervals accumulating only at the endpoints. Choose a free action on two generators by doing something generic on each subinterval, in such a way that the derivative is equal to 1 at the endpoints. This can certainly be accomplished; for concreteness, choose the action so that for each subinterval I_i there is a point x_i in the interior of I_i whose stabilizer is trivial.

Second, for each pair of distinct words in the generators, choose a subinterval and modify the action there so that the derivatives of those words in that subinterval differ by at least some definite constant C at some point. In more detail: enumerate the pairs of words somehow p_1, p_2, p_3 where each p_i is a pair of words (w_{i1}, w_{i2}) in the generators, and modify the action on the subinterval I_i so the words in p_i differ by at least C in the C^1 norm on the interval I_i. Since we are modifying the generators infinitely many times, but in such a way that the support of the modification exits any compact subset of the interior, we just need to check that the modifications are C^1. Since there are only finitely many pairs of words, both of which are of bounded length (for any given bound), when i is sufficiently big, one of the words w_{i1},  w_{i2} has length at least n(i) where n(i) goes to infinity as i goes to infinity. Without loss of generality, we can order the pairs so that w_{i1} is the “long” word.

Now this is how we modify the action in I_i. Recall that the point x_i has trivial stabilizer, so the translates y_{ij} of x_i under the suffixes of w_{i1} are distinct. Take disjoint intervals about the y_{ij} and observe that each y_{ij} is taken to y_{ij+1} by one of the generators. Modify this generator inside this disjoint neighborhood so that y_{ij} is still taken to y_{ij+1}, but the derivative at that point is multiplied by 1+ C/n(i), and the derivative at nearby points is not multiplied by more than 1+C/n(i). Since the neighborhoods of the y_{ij} are disjoint, these modifications are all compatible, and the derivative of the generators does not change by more than 1+C/n(i) at any point. Since n(i) goes to infinity as i goes to infinity, we can perform such modifications for each i, and the resulting action is still C^1. But now the derivative of w_{i1} at x_i has been multiplied by 1+C, so w_{i1} and w_{i2} differ by at least C in the C^1 norm.  qed.

It is interesting to observe that this construction, while C^1, is not C^{1+\epsilon} for any \epsilon>0. For big i, we have n(i) \sim \log(i) whereas |I_i| = o(1/i). Introducing a “bump” which modifies the derivative by 1/\log(i) in a subinterval of size o(1/i) will blow up every Holder norm.

(Update 8/10): Mark Sapir has created a webpage to discuss Shavgulidze’s paper here. Also, Matt Brin has posted notes on Shavgulidze’s paper here. The notes are very nice, and go into great detail, as far as they go. Matt promises to update the notes periodically.

(Update 11/18): Matt Brin has let me know by email that a significant gap has emerged in Shavgulidze’s argument. He writes:

Lemma 5 is still unproven. It claims a property about the distributions u_n on the simplexes D_n that is needed for the second part of the paper. The main result does not need the particular distributions u_n given in the paper, but does need distributions on the D_n that satisfy the properties claimed by Lemmas 5, 6 and that cooperate with Lemma 9. Ufe Haagerup claims an argument that the u_n in the paper does not satisfy the conclusion of Lemma 5. Another distribution was said to be suggested by Shavgulidze, but at last report, it did not seem to be working out.

In light of this, it would seem to be reasonable to consider the question of whether F is amenable as wide open.

(Update 9/21/2012): Justin Moore has posted a preprint on the arXiv claiming to prove amenability of F. It is too early to suggest that there is expert consensus on the correctness of the proof, but certainly everything I have heard is promising. I have not had time to look carefully at the argument yet, but hope to get a chance to do so before too long.

(Update 10/2/2012): Justin has withdrawn his claim of a proof. A gap was found by Akhmedov.

I have struggled for a long time (and I continue to struggle) with the following question:

Question: Is the group of self-homeomorphisms of the unit disk (in the plane) that fix the boundary pointwise a left-orderable group?

Recall that a group G is left-orderable if there is a total order < on the elements satisfying g<h if and only if fg < fh for all f,g,h \in G. For a countable group, the property of being left orderable is equivalent to the property that the group admits a faithful action on the interval by orientation-preserving homeomorphisms; however, this equivalence is not “natural” in the sense that there is no natural way to extract an ordering from an action, or vice-versa. This formulation of the question suggests that one is trying to embed the group of homeomorphisms of the disk into the group of homeomorphisms of the interval, an unlikely proposition, made even more unlikely by the following famous theorem of Filipkiewicz:

Theorem: (Filipkiewicz) Let M_1,M_2 be two compact manifolds, and r_1,r_2 two non-negative integers or infinity. Suppose the connected components of the identity of \text{Diff}^{r_1}(M_1) and \text{Diff}^{r_2}(M_2) are isomorphic as abstract groups. Then r_1=r_2 and the isomorphism is induced by some diffeomorphism.

The hard(est?) part of the argument is to identify a subgroup stabilizing a point in purely algebraic terms. It is a fundamental and well-studied problem, in some ways a natural outgrowth of Klein’s Erlanger programme, to perceive the geometric structure on a space in terms of algebraic properties of its automorphism group. The book by Banyaga is the best reference I know for this material, in the context of “flexible” geometric structures, with big transformation groups (it is furthermore the only math book I know with a pink cover).

Left orderability is inherited under extensions. I.e. if K \to G \to H is a short exact sequence, and both K and H are left orderable, then so is G. Furthermore, it is a simple but useful theorem of Burns and Hale that a group G is left orderable if and only if for every finitely generated subgroup H there is a left orderable group H' and a surjective homomorphism H \to H'. The necessity of this condition is obvious: a subgroup of a left orderable group is left orderable (by restricting the order), so one can take H' to be H and the surjection to be the identity. One can exploit this strategy to show that certain transformation groups are left orderable, as follows:

Example: Suppose G is a group of homeomorphisms of some space X, with a nonempty fixed point set. If H is a finitely generated subgroup of G, then there is a point y in the frontier of \text{fix}(H) so that H has a nontrivial image in the group of germs of homeomorphisms of X at y. If this group of germs is left-orderable for all y, then so is G by Burns-Hale.

Example: (Rolfsen-Wiest) Let G be the group of PL homeomorphisms of the unit disk (thought of as a PL square in the plane) fixed on the boundary. If H is a finitely generated subgroup, there is a point p in the frontier of \text{fix}(H). Note that H has a nontrivial image in the group of piecewise linear homeomorphisms of the projective space of lines through p. Since the fixed point set of a finitely generated subgroup is equal to the intersection of the fixed point sets of a finite generating set, it is itself a polyhedron. Hence H fixes some line through p, and therefore has a nontrivial image in the group of homeomorphisms of an interval. By Burns-Hale, G is left orderable.

Example: Let G be the group of diffeomorphisms of the unit disk, fixed on the boundary. If H is a finitely generated subgroup, then at a non-isolated point p in \text{fix}(H) the group H fixes some tangent vector to p (a limit of short straight lines from p to nearby fixed points). Consequently the image of H in \text{GL}(T_p) is reducible, and is conjugate into an affine subgroup, which is left orderable. If the image is nontrivial, we are done by Burns-Hale. If it is trivial, then the linear part of H at p is trivial, and therefore by the Thurston stability theorem, there is a nontrivial homomorphism from H to the (orderable) group of translations of the plane. By Burns-Hale, we conclude that G is left orderable.

The second example does not require infinite differentiability, just C^1, the necessary hypothesis to apply the Thurston stability theorem. This is such a beautiful and powerful theorem that it is worth making an aside to discuss it. Thurston’s theorem says that if H is a finitely generated group of germs of diffeomorphisms of a manifold fixing a common point, then a suitable limit of rescaled actions of the group near the fixed point converge to a nontrivial action by translations. One way to think of this is in terms of power series: if H is a group of real analytic diffeomorphisms of the line, fixing the point 0, then every h \in H can be expanded as a power series: h(x) = c_1(h)x + c_2(h)x^2 + \cdots. The function h \to c_1(h) is a multiplicative homomorphism; however, if the logarithm of c_1 is identically zero, then if i is the first index for which some c_i(h) is nonzero, then h \to c_i(h) is an additive homomorphism. The choice of coefficient i is a “gauge”, adapted to H, that sees the most significant nontrivial part; this leading term is a character (i.e. a homomorphism to an abelian group), since the nonabelian corrections have strictly higher degree. Thurston’s insight was to realize that for a finitely generated group of germs of C^1 diffeomorphisms with trivial linear part, one can find some gauge that sees the most significant nontrivial part of the action of the group, and at this gauge, the action looks abelian. There is a subtlety, that one must restrict attention to finitely generated groups of homeomorphisms: on each scale of a sequence of finer and finer scales, one of a finite generating set differs the most from the identity; one must pass to a subsequence of scales for which this one element is constant (this is where the finite generation is used). The necessity of this condition is demonstrated by a theorem of Sergeraert: the group of germs of (C^\infty) diffeomorphisms of the unit interval, infinitely tangent to the identity at both endpoints (i.e. with trivial power series at each endpoint) is perfect, and therefore admits no nontrivial homomorphism to an abelian group.

Let us now return to the original question. The examples above suggest that it might be possible to find a left ordering on the group of homeomorphisms of the disk, fixed on the boundary. However, I think this is misleading. The construction of a left ordering in either category (PL or smooth) was ad hoc, and depended on locality in two different ways: the locality of the property of left orderability (i.e. Burns-Hale) and the tameness of groups of PL or smooth homeomorphisms blown up near a common fixed point. Rescaling an arbitrary homeomorphism about a fixed point does not make things any less complicated. Burns-Hale and Filipkiewicz together suggest that one should look for a structural dissimilarity between the group of homeomorphisms of the disk and of the interval that persists in finitely generated subgroups. The simplest way to distinguish between the two spaces algebraically is in terms of their lattices of closed (or equivalently, open) subsets. To a topological space X, one can associate the lattice \Lambda(X) of (nonempty, for the sake of argument) closed subsets of X, ordered by inclusion. One can reconstruct the space X from this lattice, since points in X correspond to minimal elements. However, any surjective map X \to Y defines an embedding \Lambda(Y) \to \Lambda(X), so there are many structure-preserving morphisms between such lattices. The lattice \Lambda(X) is an \text{Aut}(X)-space in an obvious way, and one can study algebraic maps \Lambda(Y) \to \Lambda(X) together with homomorphisms \rho:\text{Aut}(Y) \to \text{Aut}(X) for which the algebraic maps respect the induced \text{Aut}(Y)-structures. A weaker “localization” of this condition asks merely that for points (i.e. minimal elements) p,p' \in \Lambda(Y) in the same \text{Aut}(Y)-orbit, their images in \Lambda(X) are in the same \text{Aut}(X)-orbit. This motivates the following:

Proposition: There is a surjective map from the unit interval to the unit disk so that the preimages of any two points are homeomorphic.

Sketch of Proof: This proposition follows from two simpler propositions. The first is that there is a surjective map from the unit interval to itself so that every point preimage is a Cantor set. The second is that there is a surjective map from the unit interval to the unit disk so that the preimage of any point is finite. A composition of these two maps gives the desired map, since a finite union of Cantor sets is itself a Cantor set.

There are many surjective maps from the unit interval to the unit disk so that the preimage of any point is finite. For example, if M is a hyperbolic three-manifold fibering over the circle with fiber S, then the universal cover of a fiber \widetilde{S} is properly embedded in hyperbolic 3-space, and its ideal boundary (a circle) maps surjectively and finitely-to-one to the sphere at infinity of hyperbolic 3-space. Restricting to a suitable subinterval gives the desired map.

To obtain the first proposition, one builds a surjective map from the interval to itself inductively; there are many possible ways to do this, and details are left to the reader. qed.

It is not clear how much insight such a construction gives.

Another approach to the original question involves trying to construct an explicit (finitely generated) subgroup of the group of homeomorphisms of the disk that is not left orderable. There is a “cheap” method to produce finitely presented groups with no left-orderable quotients. Let G = \langle x,y \; | \; w_1, w_2 \rangle be a group defined by a presentation, where w_1 is a word in the letters x and y, and w_2 is a word in the letters x and y^{-1}. In any left-orderable quotient in which both x and y are nontrivial, after reversing the orientation if necessary, we can assume that x > \text{id}. If further y>\text{id} then w_1 >\text{id}, contrary to the fact that w_1 = \text{id}. If y^{-1} >\text{id}, then w_2 >\text{id}, contrary to the fact that w_2=\text{id}. In either case we get a contradiction. One can try to build by hand nontrivial homeomorphisms x,y of the unit disk, fixed on the boundary, that satisfy w_1,w_2 =\text{id}. Some evidence that this will be hard to do comes from the fact that the group of smooth and PL homeomorphisms of the disk are in fact left-orderable: any such x,y can be arbitrarily well-approximated by smooth x',y'; nevertheless at least one of the words w_1,w_2 evaluated on any smooth x',y' will be nontrivial. Other examples of finitely presented groups that are not left orderable include higher Q-rank lattices (e.g. subgroups of finite index in \text{SL}(n,\mathbb{Z}) when n\ge 3), by a result of Dave Witte-Morris. Suppose such a group admits a faithful action by homeomorphisms on some closed surface of genus at least 1. Since such groups do not admit homogeneous quasimorphisms, their image in the mapping class group of the surface is finite, so after passing to a subgroup of finite index, one obtains a (lifted) action on the universal cover. If the genus of the surface is at least 2, this action can be compactified to an action by homeomorphisms on the unit disk (thought of as the universal cover of a hyperbolic surface) fixed pointwise on the boundary. Fortunately or unfortunately, it is already known by Franks-Handel (see also Polterovich) that such groups admit no area-preserving actions on closed oriented surfaces (other than those factoring through a finite group), and it is consistent with the so-called “Zimmer program” that they should admit no actions even without the area-preserving hypothesis when the genus is positive (of course, \text{SL}(3,\mathbb{R}) admits a projective action on S^2). Actually, higher rank lattices are very fragile, because of Margulis’ normal subgroup theorem. Every normal subgroup of such a lattice is either finite or finite index, so to prove the results of Franks-Handel and Polterovich, it suffices to find a single element in the group of infinite order that acts trivially. Unipotent elements are exponentially distorted in the word metric (i.e. the cyclic subgroups they generate are not quasi-isometrically embedded), so one “just” needs to show that groups of area-preserving diffeomorphisms of closed surfaces (of genus at least 1) do not contain such distorted elements. Some naturally occurring non-left orderable groups include some (rare) hyperbolic 3-manifold groups, amenable but not locally indicable groups, and a few others. It is hard to construct actions of such groups on a disk, although certain flows on 3-manifolds give rise to actions of the fundamental group on a plane.

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