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!

It would seem that Riemannian and Lorentzian geometry have much in common: canonical connections, geodesics, curvature tensor, etc. . . . But in fact this common part is only a common disposition at the onset: one soon enters different realms.

I will not dispute this. But it is not clear to me whether this divergence is a necessary consequence of the nature of the objects of study (in either case), or an artefact of the schism between mathematics and physics during much of the 20th century. In any case, in this blog post I have the narrow aim of describing some points of contact between Lorentzian (and more generally, causal) geometry and other geometries (hyperbolic, symplectic), which plays a significant role in some of my research.

The first point of contact is the well-known duality between geodesics in the hyperbolic plane and points in the (projectivized) “anti de-Sitter plane”. Let denote a -dimensional vector space equipped with a quadratic form

If we think of the set of rays through the origin as a copy of the real projective plane , the hyperbolic plane is the set of projective classes of vectors with , the (projectivized) anti de-Sitter plane is the set of projective classes of vectors with , and their common boundary is the set of projective classes of (nonzero) vectors with . Topologically, the hyperbolic plane is an open disk, the anti de-Sitter plane is an open Möbius band, and their boundary is the “ideal circle” (note: what people usually call the anti de-Sitter plane is actually the annulus double-covering this Möbius band; this is like the distinction between spherical geometry and elliptic geometry). Geometrically, the hyperbolic plane is a complete Riemannian surface of constant curvature , whereas the anti de-Sitter plane is a complete Lorentzian surface of constant curvature .

In this projective model, a hyperbolic geodesic is an open straight line segment which is compactified by adding an unordered pair of points in the ideal circle. The straight lines in the anti de-Sitter plane tangent to the ideal circle at these two points intersect at a point . Moreover, the set of geodesics in the hyperbolic plane passing through a point are dual to the set of points in the anti de-Sitter plane that lie on a line which does not intersect the ideal circle. In the figure, three concurrent hyperbolic geodesics are dual to three colinear anti de-Sitter points.

The anti de-Sitter geometry has a natural causal structure. There is a cone field whose extremal vectors at every point are tangent to the straight lines through that are also tangent to the ideal circle. A smooth curve is timelike if its tangent at every point is supported by this cone field, and spacelike if its tangent is everywhere not supported by the cone field. A timelike curve corresponds to a family of hyperbolic geodesics which locally intersect each other; a spacelike curve corresponds to a family of disjoint hyperbolic geodesics that foliate some region.

One can distinguish (locally) between future and past along a timelike trajectory, by (arbitrarily) identifying the “future” direction with a curve which winds positively around the ideal circle. The fact that one can distinguish in a consistent way between the positive and negative direction is equivalent to the existence of a nonzero section of timelike vectors. On the other hand, there does not exist a nonzero section of spacelike vectors, so one cannot distinguish in a consistent way between left and right (this is a manifestation of the non-orientability of the Möbius band).

The duality between the hyperbolic plane and the anti de-Sitter plane is a manifestation of the fact that (at least at the level of Lie algebras) they have the same (infinitesimal) symmetries. Let denote the group of real matrices which preserve ; i.e. matrices for which for all vectors . This contains a subgroup of index which preserves the “positive sheet” of the hyperboloid , and acts on it in an orientation-preserving way. The hyperbolic plane is the homogeneous space for this group whose point stabilizers are a copy of (which acts as an elliptic “rotation” of the tangent space to their common fixed point). The anti de-Sitter plane is the homogeneous space for this group whose point stabilizers are a copy of (which acts as a hyperbolic “translation” of the geodesic in hyperbolic space dual to the given point in anti de-Sitter space). The ideal circle is the homogeneous space whose point stabilizers are a copy of the affine group of the line. The hyperbolic plane admits a natural Riemannian metric, and the anti de-Sitter plane a Lorentz metric, which are invariant under these group actions. The causal structure on the anti de-Sitter plane limits to a causal structure on the ideal circle.

Now consider the -dimensional vector space and the quadratic form . The (-dimensional) sheets and both admit homogeneous Lorentz metrics whose point stabilizers are copies of and (which are isomorphic but sit in in different ways). These -manifolds are compactified by adding the projectivization of the cone . Topologically, this is a Clifford torus in dividing this space into two open solid tori which can be thought of as two Lorentz -manifolds. The causal structure on the pair of Lorentz manifolds limits to a pair of complementary causal structures on the Clifford torus. (edited 12/10)

Let’s go one dimension higher, to the -dimensional vector space and the quadratic form . Now only the sheet is a Lorentz manifold, whose point stabilizers are copies of , with an associated causal structure. The projectivized cone is a non-orientable twisted bundle over the circle, and it inherits a causal structure in which the sphere factors are spacelike, and the circle direction is timelike. This ideal boundary can be thought of in quite a different way, because of the exceptional isomorphism at the level of (real) Lie algebras , where denotes the Lie algebra of the symplectic group in dimension . In this manifestation, the ideal boundary is usually denoted , and can be thought of as the space of Lagrangian planes in with its usual symplectic form. One way to see this is as follows. The wedge product is a symmetric bilinear form on with values in . The associated quadratic form vanishes precisely on the “pure” -forms — i.e. those associated to planes. The condition that the wedge of a given -form with the symplectic form vanishes imposes a further linear condition. So the space of Lagrangian -planes is a quadric in , and one may verify that the signature of the underlying quadratic form is . The causal structure manifests in symplectic geometry in the following way. A choice of a Lagrangian plane lets us identify symplectic with the cotangent bundle . To each symmetric homogeneous quadratic form on (thought of as a smooth function) is associated a linear Lagrangian subspace of , namely the (linear) section . Every Lagrangian subspace transverse to the fiber over is of this form, so this gives a parameterization of an open, dense subset of containing the point . The set of positive definite quadratic forms is tangent to an open cone in ; the field of such cones as varies defines a causal structure on which agrees with the causal structure defined above.

These examples can be generalized to higher dimension, via the orthogonal groups or the symplectic groups . As well as two other infinite families (which I will not discuss) there is a beautiful “sporadic” example, connected to what Freudenthal called octonion symplectic geometry associated to the noncompact real form of the exceptional Lie group, where the ideal boundary has an invariant causal structure whose timelike curves wind around the factor; see e.g. Clerc-Neeb for a more thorough discussion of the theory of Shilov boundaries from the causal geometry point of view, or see here or here for a discussion of the relationship between the octonions and the exceptional Lie groups.

The causal structure on these ideal boundaries gives rise to certain natural -cocycles on their groups of automorphisms. Note in each case that the ideal boundary has the topological structure of a bundle over with spacelike fibers. Thus each closed timelike curve has a well-defined winding number, which is just the number of times it intersects any one of these spacelike slices. Let be an ideal boundary as above, and let denote the cyclic cover dual to a spacelike slice. If is a point in , we let denote the image of under the th power of the generator of the deck group of the covering. If is a homeomorphism of preserving the causal structure, we can lift to a homeomorphism of . For any such lift, define the rotation number of as follows: for any point and any integer , let be the the smallest integer for which there is a causal curve from to to , and then define . This function is a quasimorphism on the group of causal automorphisms of , with defect equal to the least integer such that any two points in are contained in a closed causal loop with winding number . In the case of the symplectic group with causal boundary , the defect is , and the rotation number is (sometimes) called the symplectic rotation number; it is a quasimorphism on the universal central extension of , whose coboundary descends to the Maslov class (an element of -dimensional bounded cohomology) on the symplectic group.

Causal structures in groups of symplectomorphisms or contactomorphisms are intensely studied; see for instance this paper by Eliashberg-Polterovich.

It is unfortunate in some ways that the standard way to refer to the plane emphasizes its product structure. This product structure is topologically unnatural, since it is defined in a way which breaks the natural topological symmetries of the object in question. This fact is thrown more sharply into focus when one discusses more rigid topologies.

At this point I give an example, namely that of the Zariski topology, pointing out that the product topology of two copies of the affine line with the Zariski topology is not the same as the Zariski topology on the affine plane. All well and good. I then go on to claim that part of the bias is biological in origin, citing the following example as evidence:

Example 1.2 (Primary visual cortex). The primary visual cortex of mammals (including humans), located at the posterior pole of the occipital cortex, contains neurons hardwired to fire when exposed to certain spatial and temporal patterns. Certain specific neurons are sensitive to stimulus along specific orientations, but in primates, more cortical machinery is devoted to representing vertical and horizontal than oblique orientations (see for example [58] for a discussion of this effect).

(Note: [58] is a reference to the paper “The distribution of oriented contours in the real world” by David Coppola, Harriett Purves, Allison McCoy, and Dale Purves, Proc. Natl. Acad. Sci. USA 95 (1998), no. 7, 4002–4006)

I think Wendy took this to be some kind of poetic license or conceit, and perhaps even felt that it was a bit out of place in a serious research monograph. On balance, I think I agree that it comes across as somewhat jarring and unexpected to the reader, and the tone and focus is somewhat inconsistent with that of the rest of the book. But I also think that in certain subjects in mathematics — and I would put low-dimensional geometry/topology in this category — we are often not aware of the extent to which our patterns of reasoning and imagination are shaped, limited, or (mis)directed by our psychological — and especially psychophysical — natures.

The particular question of how the mind conceives of, imagines, or perceives any mathematical object is complicated and multidimensional, and colored by historical, social, and psychological (not to mention mathematical) forces. It is generally a vain endeavor to find precise physical correlates of complicated mental objects, but in the case of the plane (or at least one cognitive surrogate, the subjective visual field) there is a natural candidate for such a correlate. Cells on the rear of the occipital lobe are arranged in a “map” in the region of the occipital lobe known as the “primary visual cortex”, or V1. There is a precise geometric relationship between the location of neurons in V1 and the points in the subjective visual field they correspond to. Further visual processing is done by other areas V2, V3, V4, V5 of the visual cortex. Information is fed forward from Vi to Vj with , but also backward from Vj to Vi regions, so that visual information is processed at several levels of abstraction simultaneously, and the results of this processing compared and refined in a complicated synthesis (this tends to make me think of the parallel terraced scan model of analogical reasoning put forward by Douglas Hofstadter and Melanie Mitchell; see Fluid concepts and creative analogies, Chapter 5).

The initial processing done by the V1 area is quite low-level; individual neurons are sensitive to certain kind of stimuli, e.g. color, spatial periodicity (on various scales),  motion, orientation, etc. As remarked earlier, more neurons are devoted to detecting horizontally or vertically aligned stimuli; in other words, our brains literally devote more hardware to perceiving or imagining vertical and horizontal lines than to lines with an oblique orientation. This is not to say that at some higher, more integrated level, our perception is not sensitive to other symmetries that our hardware does not respect, just as a random walk on a square lattice in the plane converges (after a parabolic rescaling) to Brownian motion (which is not just rotationally but conformally invariant). However the fact is that humans perform statistically better on cognitive tasks that involve the perception of figures that are aligned along the horizontal and vertical axes, than on similar tasks that differ only by a rotation of the figures.

It is perhaps interesting therefore that the earliest (?) mathematical conception of the plane, due to the Greeks, did not give a privileged place to the horizontal or vertical directions, but treats all orientations on an equal footing. In other words, in Greek (Euclidean) geometry, the definitions respect the underlying symmetries of the objects. Of course, from our modern perspective we would not say that the Greeks gave a definition of the plane at all, or at best, that the definition is woefully inadequate. According to one well-known translation, the plane is introduced as a special kind of surface as follows:

A surface is that which has length and breadth.

When a surface is such that the right line joining any two arbitrary points in it lies wholly in the surface, it is called a plane.

This definition of a surface looks as though it is introducing coordinates, but in fact one might just as well interpret it as defining a surface in terms of its dimension; having defined a surface (presumably thought of as being contained in some ambient undefined three-dimensional space) one defines a plane to be a certain kind of surface, namely one that is convex. Horizontal and vertical axes are never introduced. Perpendicularity is singled out as important, but the perpendicularity of two lines is a relative notion, whereas horizontality and verticality are absolute. In the end, Euclidean geometry is defined implicitly by its properties, most importantly isotropy (i.e. all right angles are equal to one another) and the parallel postulate, which singles it out from among several alternatives (elliptic geometry, hyperbolic geometry). In my opinion, Euclidean geometry is imprecise but natural (in the sense of category theory), because objects are defined in terms of the natural transformations they admit, and in a way that respects their underlying symmetries.

In the 15th century, the Italian artists of the Renaissance developed the precise geometric method of perspective painting (although the technique of representing more distant objects by smaller figures is extremely ancient). Its invention is typically credited to the architect and engineer Filippo Brunelleschi; one may speculate that the demands of architecture (i.e. the representation of precise 3 dimensional geometric objects in 2 dimensional diagrams) was one of the stimuli that led to this invention (perhaps this suggestion is anachronistic?). Mathematically, this gives rise to the geometry of the projective plane, i.e. the space of lines through the origin (the “eye” of the viewer of a scene). In principle, one could develop projective geometry without introducing “special” directions or families of lines. However, in one, two, or three point perspective, families of lines parallel to one or several “special” coordinate axes (along which significant objects in the painting are aligned) appear to converge to one of the vanishing points of the painting. In his treatise “De pictura” (on painting), Leon Battista Alberti (a friend of Brunelleschi) explicitly described the geometry of vision in terms of projections on to a (visual) plane. Amusingly (in the context of this blog post), he explicitly distinguishes between the mathematical and the visual plane:

In all this discussion, I beg you to consider me not as a mathematician but as a painter writing of these things.

Mathematicians measure with their minds alone the forms of things separated from all matter. Since we wish the object to be seen, we will use a more sensate wisdom.

I beg to differ: similar parts of the brain are used for imagining a triangle and for looking at a painting. Alberti’s claim sounds a bit too much like Gould’s “non-overlapping magisteria”, and in a way it is disheartening that it was made at a place and point in history at which mathematics and the visual arts were perhaps at their closest.

In the 17th century René Descartes introduced his coordinate system and thereby invented “analytic geometry”. To us it might not seem like such a big leap to go from a checkerboard floor in a perspective painting (or a grid of squares to break up the visual field) to the introduction of numerical coordinates to specify a geometrical figure, but Descartes’s ideas for the first time allowed mathematicians to prove theorems in geometry by algebraic methods. Analytic geometry is contrasted with “synthetic geometry”, in which theorems are deduced logically from primitive axioms and rules of inference. In some abstract sense, this is not a clear distinction, since algebra and analysis also rests on primitive axioms, and rules of deduction. In my opinion, this terminology reflects a psychological distinction between “analytic methods” in which one computes blindly and then thinks about what the results of the calculation mean afterwards, and “synthetic methods” in which one has a mental model of the objects one is manipulating, and directly intuits the “meaning” of the operations one performs. Philosophically speaking, the first is formal, the second is platonic. Biologically speaking, the first does not make use of the primary visual cortex, the second does.

As significant as Descartes ideas were, mathematicians were slow to take real advantage of them. Complex numbers were invented by Cardano in the mid 16th century, but the idea of representing complex numbers geometrically, by taking the real and imaginary parts as Cartesian coordinates, had to wait until Argand in the early 19th.

Incidentally, I have heard it said that the Greeks did not introduce coordinates because they drew their figures on the ground and looked at them from all sides, whereas Descartes and his contemporaries drew figures in books. Whether this has any truth to it or not, I do sometimes find it useful to rotate a mathematical figure I am looking at, in order to stimulate my imagination.

After Poincaré’s invention of topology in the late 19th century, there was a new kind of model of the plane to be (re)imagined, namely the plane as a topological space. One of the most interesting characterizations was obtained by the brilliantly original and idiosyncratic R. L. Moore in his paper, “On the foundations of plane analysis situs”. Let me first remark that the line can be characterized topologically in terms of its natural order structure; one might argue that this characterization more properly determines the oriented line, and this is a fair comment, but at least the object has been determined up to a finite ambiguity. Let me second of all remark that the characterization of the line in terms of order structures is useful; a (countable) group is abstractly isomorphic to a group of (orientation-preserving) homeomorphisms of the line if and only if admits an (abstract) left-invariant order.

Given points and the line, Moore proceeds to list a collection of axioms which serve to characterize the plane amongst topological spaces. The axioms are expressed in terms of separation properties of primitive undefined terms called points and regions (which correspond more or less to ordinary points and open sets homeomorphic to the interiors of closed disks respectively) and non-primitive objects called “simple closed curves” which are (eventually) defined in terms of simpler objects. Moore’s axioms are “natural” in the sense that they do not introduce new, unnecessary, unnatural structure (such as coordinates, a metric, special families of “straight” lines, etc.). The basic principle on which Moore’s axioms rest is that of separation — which continua separate which points from which others? If there is a psychophysical correlate of this mathematical intuition, perhaps it might be the proliferation of certain neurons in the primary visual cortex which are edge detectors — they are sensitive, not to absolute intensity, but to a spatial discontinuity in the intensity (associated with the “edge” of an object). The visual world is full of objects, and our eyes evolved to detect them, and to distinguish them from their surroundings (to distinguish figure from ground as it were). If I have an objection to Cartesian coordinates on biological grounds (I don’t, but for the sake of argument let’s suppose I do) then perhaps Moore should also be disqualified for similar reasons. Or rather, perhaps it is worth being explicitly aware, when we make use of a particular mathematical model or intellectual apparatus, of which aspects of it are necessary or useful because of their (abstract) applications to mathematics, and which are necessary or useful because we are built in such a way as to need or to be able to use them.

Question(Gromov): Let be a one-ended word-hyperbolic group. Does contain a subgroup isomorphic to the fundamental group of a closed hyperbolic surface?

Let me briefly say what “one-ended” and “word-hyperbolic” mean.

A group is said to be word-hyperbolic if it acts properly and cocompactly by isometries on a proper -hyperbolic path metric space — i.e. a path metric space in which there is a constant so that geodesic triangles in the metric space have the property that each side of the triangle is contained in the -neighborhood of the union of the other two sides (colloquially, triangles are thin). This condition distills the essence of negative curvature in the large, and was shown by Gromov to be equivalent to several other conditions (eg. that the group satisfies a linear isoperimetric inequality; that every ultralimit of the group is an -tree). Free groups are hyperbolic; fundamental groups of closed manifolds with negative sectional curvature (eg surfaces with negative Euler characteristic) are word-hyperbolic; “random” groups are hyperbolic — and so on. In fact, it is an open question whether a group that admits a finite is word hyperbolic if and only if it does not contain a copy of a Baumslag-Solitar group for (note that the group is the special case ); in any case, this is a very good heuristic for identifying the word-hyperbolic groups one typically meets in examples.

If is a finitely generated group, the ends of really means the ends (as defined by Freudenthal) of the Cayley graph of with respect to some finite generating set. Given a proper topological space , the set of compact subsets of gives rise to an inverse system of inclusions, where includes into whenever is a subset of . This inverse system defines an inverse system of maps of discrete spaces , and the inverse limit of this system is a compact, totally disconnected space , called the space of ends of . A proper topological space is canonically compactified by its set of ends; in fact, the compactification is the “biggest” compactification of by a totally disconnected space, in the sense that for any other compactification where is zero dimensional, there is a continuous map which is the identity on .

For a word-hyperbolic group , the Cayley graph can be compactified by adding the ideal boundary , but this is typically not totally disconnected. In this case, the ends of can be recovered as the components of .

A group acts on its own ends . An elementary argument shows that the cardinality of is one of (if a compact set disconnects then infinitely many translates of converging to separate from infinitely many other ends accumulating on ). A group has no ends if and only if it is finite. Stallings famously showed that a (finitely generated) group has at least ends if and only if it admits a nontrivial description as an HNN extension or amalgamated free product over a finite group. One version of the argument proceeds more or less as follows, at least when is finitely presented. Let be an -dimensional Riemannian manifold with fundamental group , and let denote the universal cover. We can identify the ends of with the ends of . Let be a least (-dimensional) area hypersurface in amongst all hypersurfaces that separate some end from some other (here the hypothesis that has at least two ends is used). Then every translate of by an element of is either equal to or disjoint from it, or else one could use the Meeks-Yau “roundoff trick” to find a new with strictly lower area than . The translates of decompose into pieces, and one can build a tree whose vertices correspond to to components of , and whose edges correspond to the translates . The group acts on this tree, with finite edge stabilizers (by the compactness of ), exhibiting either as an HNN extension or an amalgamated product over the edge stabilizers. Note that the special case occurs if and only if has a finite index subgroup which is isomorphic to .

Free groups and virtually free groups do not contain closed surface subgroups; Gromov’s question more or less asks whether these are the only examples of word-hyperbolic groups with this property.

Kim and Wilton study Gromov’s question in a very, very concrete case, namely that case that is the double of a free group along a word ; i.e. (hereafter denoted ). Such groups are known to be one-ended if and only if is not contained in a proper free factor of (it is clear that this condition is necessary), and to be hyperbolic if and only if is not a proper power, by a result of Bestvina-Feighn. To see that this condition is necessary, observe that the double is isomorphic to the fundamental group of a Seifert fiber space, with base space a disk with two orbifold points of order ; such a group contains a . One might think that such groups are too simple to give an insight into Gromov’s question. However, these groups (or perhaps the slightly larger class of graphs of free groups with cyclic edge groups) are a critical case for at least two reasons:

1. The “smaller” a group is, the less room there is inside it for a surface group; thus the “simplest” groups should have the best chance of being a counterexample to Gromov’s question.
2. If is word-hyperbolic and one-ended, one can try to find a surface subgroup by first looking for a graph of free groups in , and then looking for a surface group in . Since a closed surface group is itself a graph of free groups, one cannot “miss” any surface groups this way.

Not too long ago, I found an interesting construction of surface groups in certain graphs of free groups with cyclic edge groups. In fact, I showed that every nontrivial element of in such a group is virtually represented by a sum of surface subgroups. Such surface subgroups are obtained by finding maps of surface groups into which minimize the Gromov norm in their (projective) homology class. I think it is useful to extend Gromov’s question by making the following

Conjecture: Let be a word-hyperbolic group, and let be nonzero. Then some multiple of is represented by a norm-minimizing surface (which is necessarily -injective).

Note that this conjecture does not generalize to wider classes of groups. There are even examples of groups with nonzero homology classes with positive, rational Gromov norm, for which there are no -injective surfaces representing a multiple of at all.

It is time to define polygonal words in free groups.

Definition: Let be free. Let be a wedge of circles whose edges are free generators for . A cyclically reduced word in these generators is polygonal if there exists a van-Kampen graph on a surface such that:

1. every complementary region is a disk whose boundary is a nontrivial (possibly negative) power of ;
2. the (labelled) graph immerses in in a label preserving way;
3. the Euler characteristic of is strictly less than the number of disks.

The last condition rules out trivial examples; for example, the double of a single disk whose boundary is labeled by . Notice that it is very important to allow both positive and negative powers of as boundaries of complementary regions. In fact, if is not in the commutator subgroup, then the sum of the powers over all complementary regions is necessarily zero (and if is in the commutator subgroup, then has nontrivial , so one already knows that there is a surface subgroup).

Condition 2. means that at each vertex of , there is at most one oriented label corresponding to each generator of or its inverse. This is really the crucial geometric property. If is a van-Kampen graph as above, then a theorem of Marshall Hall implies that there is a finite cover of into which embeds (in fact, this observation underlies Stallings’s work on foldings of graphs). If we build a -complex with by attaching two ends of a cylinder to suitable loops in two copies of , then a tubular neighborhood of in (i.e. what is sometimes called a “fatgraph” ) embeds in a finite cover of , and its double — a surface of strictly negative Euler characteristic — embeds as a closed surface in , and is therefore -injective. Hence if is polygonal, contains a surface subgroup.

Not every word is polygonal. Kim-Wilton discuss some interesting examples in their paper, including:

1. suppose is a cyclically reduced product of proper powers of the generators or their inverses (e.g a word like but not a word like ); then is polygonal;
2. a word of the form is polygonal if for each ;
3. the word is not polygonal.

To see 3, suppose there were a van-Kampen diagram with more disks than Euler characteristic. Then there must be some vertex of valence at least . Since is positive, the complementary regions must have boundaries which alternate between positive and negative powers of , so the degree of the vertex must be even. On the other hand, since must immerse in a wedge of two circles, the degree of every vertex must be at most , so there is consequently some vertex of degree exactly . Since each is isolated, at least edges must be labelled ; hence exactly two. Hence exactly two edges are labelled . But one of these must be incoming and one outgoing, and therefore these are adjacent, contrary to the fact that does not contain a .

1 above is quite striking to me. When is in the commutator subgroup, one can consider van-Kampen diagrams as above without the injectivity property, but with the property that every power of on the boundary of a disk is positive; call such a van-Kampen diagram monotone. It turns out that monotone van-Kampen diagrams always exist when , and in fact that norm-minimizing surfaces representing powers of the generator of are associated to certain monotone diagrams. The construction of such surfaces is an important step in the argument that stable commutator length (a kind of relative Gromov norm) is rational in free groups. In my paper scl, sails and surgery I showed that monomorphisms of free groups that send every generator to a power of that generator induce isometries of the norm; in other words, there is a natural correspondence between certain equivalence classes of monotone surfaces for an arbitrary word in and for a word of the kind that Kim-Wilton show is polygonal (Note: Henry Wilton tells me that Brady, Forester and Martinez-Pedroza have independently shown that contains a surface group for such , but I have not seen their preprint (though I would be very grateful to get a copy!)).

In any case, if not every word is polygonal, all is not lost. To show that contains a surface subgroup is suffices to show that contains a surface subgroup, where and differ by an automorphism of . Kim-Wilton conjecture that one can always find an automorphism so that is polygonal. In fact, they make the following:

Conjecture (Kim-Wilton; tiling conjecture): A word not contained in a proper free factor of shortest length (in a given generating set) in its orbit under is polygonal.

If true, this would give a positive answer to Gromov’s question for groups of the form .

Apparently these constructions had somewhat altered the experts’s (i.e. Freedman and Kirby) feelings about whether the smooth -dimensional Poincaré conjecture is likely to be true. The Cappell-Shaneson manifolds are constructed by doing surgery on certain torus bundles over a circle — those with monodromy chosen so that the resulting torus bundles have the homology of a . A suitable surgery, killing the factor makes the manifold into homology ‘s, and also kills the subgroup of the fundamental group normally generated by the factor. On the other hand, everything else in the fundamental group “comes from” the torus, whose fundamental group is abelian, and therefore the resulting manifold is simply-connected. Since it is a homology -sphere, is it therefore a homotopy -sphere, and consequently (by Freedman), a topological -sphere.

Gompf shows these -spheres are standard by showing that a certain move — which simplifies the monodromy of the fiber — can be realized by a diffeomorphism. The move is an example of what is known to -manifold topologists as a “log transform” (and to -manifold topologists as “Dehn surgery times ”). A log transform takes as input a smooth embedded torus. A tubular neighborhood of this torus is a product whose boundary is a -torus . This tubular neighborhood is removed, and reglued by an automorphism of the factor. Usually a log transform will change the topology of the manifold, or at least the smooth structure. But in this case, the surgered torus is contained in a fiber, and the log transform can be shown to be isotopic to the identity, by using the monodromy of the fibration (technically, the monodromy of the fibration produces a once-punctured torus in the bundle with boundary on the curve along which the log transform “twists”, but after doing surgery to produce the homology ‘s, this once-punctured torus is “capped” to become a smooth disk).

The point of this blog post is to show how to construct many, many other smooth -manifolds which are topological -spheres, and for which Gompf’s method of showing they are standard does not work. Are these manifolds counterexamples to the smooth -dimensional Poincaré conjecture? I am really not the person to ask.

The construction takes as input a fibered knot — i.e. a knot in the -sphere whose complement fibers over a circle. In other words, there is a fibration , where is a (minimal genus) Seifert surface for the knot . The fibration of spaces gives rise to a short exact sequence of fundamental groups (in general, one gets a long exact sequence of homotopy groups, but the spaces are all ‘s — i.e. their homotopy groups in dimension other than all vanish). Since has boundary, the fundamental group of is free and finitely generated of rank , where is the genus. The fundamental group of is . So one exhibits as an HNN extension of a free group, where the meridian acts by conjugation on the free group by some automorphism .

Since is a knot in , the homology of is equal to in dimension . Moreover, since putting back in recovers , it follows that the fundamental group is normally generated by the meridian (which also generates the in ). For the moment everything is -dimensional, but there is a trick to promote this to dimensions. In place of the surface , consider the -manifold . In other words, is obtained by doubling a handlebody of genus . The fundamental group is free of rank . Now one builds a bundle over with monodromy ; call this -manifold . The existence of such a manifold depends on being able to realize any automorphism of a free group by a homeomorphism of a doubled handlebody; one way to see this is to observe that is generated by Nielsen moves — interchanging generators, replacing generators by their inverses, and replacing generators by . These moves are all realizable by homeomorphisms of doubled handlebodies, the last by a “handle slide”.

Now, observe that , and is normally generated by a loop representing the circle direction. Moreover, . To compute , observe that by Poincaré duality. If the action of on (a free abelian group of rank ) is represented by a matrix , then the action on is represented by the transpose . The fact that is equivalent to the fact that the first homology of dies in the bundle; i.e. ; hence , and for the same reason, the second homology of dies in the bundle, and . By (-dimensional) Poincaré duality, , and we see that is a homology .

A tubular neighborhood of the loop is a product , since is orientable. The boundary of this is . So we drill out and glue in a product to produce . Drilling out does not affect the fundamental group, by Seifert van-Kampen, and the fact that the inclusion is an isomorphism on . On the other hand, filling in a has the effect of killing the meridian , and therefore (by the discussion above), killing completely; i.e. is simply-connected. Hence . Drilling out a circle and gluing back a sphere increases Euler characteristic by ; since the rank of and have both gone down by , it follows that the rank of is still , and since the fundamental group is trivial, . So is a smooth, simply-connected homology sphere, which is to say, a smooth -manifold which is topologically .

Back in June, Freedman-Gompf-Morrison-Walker described a way to use Rasmussen’s -invariant to detect exotic ‘s, and proposed trying this invariant out on the Cappell-Shaneson examples (see Scott Morrison’s post about that here). Is it feasible to compute the invariants on these new examples?

(Corrected Update 11/10:) Some ways of doing this construction give standard ‘s, some give ‘s that are not obviously standard. And there are other variations on this construction arising from “non-geometric” automorphisms of free groups that are also not obviously standard. These examples are also not obviously the same as other known potential counterexamples to the smooth Poincare conjecture. So the conclusion seems to be that they deserve further study. (Added 11/15:) This paper by Aitchison-Silver discusses a closely related construction.

The example comes from the idea of a Riemann surface lamination. This is an object that geometrizes some ideas in 1-dimensional complex analysis. The basic idea is simple: given a noncompact infinite Riemannian -manifold , one gives it a new topology by declaring that two points on the surface are “close” in the new topology if there are balls of big radius in the surface centered at the two points which are “almost isometric”. Points that were close in the old topology are close in the new topology, but points that might have been far away in the old topology can become close in the new. For example, if is a covering space of some other Riemannian surface , then points in the orbit of the deck group are “infinitely close” in the new topology. This means that the resulting topological space is not Hausdorff; one “Hausdorffifies” by identifying pairs of points that are not contained in disjoint open sets, and the quotient recovers the surface (assuming that the metric on is sufficiently generic; otherwise, it recovers modulo its group of isometries). Morally what one is doing is mapping into the space of pointed locally compact metric spaces (which is itself a locally compact topological space), and giving it the subspace topology. In more detail, a point in is a pair where is a locally compact metric space, and is a point. A sequence converges to if there are metric balls around of diameter going to infinity, metric balls around also of diameter going to infinity, and isometric inclusions of into metric spaces in such a way that the Hausdorff distance between the images of and in goes to zero as . Any locally compact metric space has a tautological map to , where each point is sent to the point . Gromov showed (see section 6 of this paper) that the space itself is locally compact; in fact, this follows in an obvious way from the Arzela-Ascoli theorem.

If has bounded geometry — i.e. if the injectivity radius is uniformly bounded below, and the curvature is bounded above and below — then the image of in is precompact, and its closure is a compact metric space . The path components of are exactly the Riemann surfaces which are arbitrarily well approximated (in the metric sense) on every compact subset by compact subsets of . If you were wandering around on such a component , and you wandered over a compact region, and were only able to measure the geometry up to some (arbitrarily fine) definite precision, you could never rule out the possibility that you were actually wandering around on . Topologically, is a Riemann surface lamination; i.e. a locally compact topological space covered by open charts of the form where is an open two-dimensional disk, where is totally disconnected, and where the transition between charts preserves the decomposition into pieces , and is smooth (in fact, preserves the Riemann surface structure) on the slices, in the overlaps. The unions of “surface” slices — i.e. the path components of — piece together to make the leaves of the lamination, which are (complete) Riemann surfaces. In our case, the leaves have Riemannian metrics, which vary continuously in the direction transverse to the leaves. (Surface) laminations occur in other areas of mathematics, for example as inverse limits of sequences of finite covers of a fixed compact surface, or as objects obtained by inductively splitting open sheets in a branched surface (the latter can easily occur as attractors of certain kinds of partially hyperbolic dynamical systems). One well-known example is sometimes called the (punctured) solenoid; its Teichmüller theory is studied by Penner and Šarić  (question: does anyone know how to do a “\acute c” in wordpress? update 11/6: thanks Ian for the unicode hint).

A lamination is said to be minimal if every leaf is dense. In our context this means that for every compact region in and every there is a so that every ball in of radius contains a subset which is -close to in the Gromov-Hausdorff metric. In other words, every “local feature” of that appears somewhere, appears with definite density to within any desired degree of accuracy. Consequently, such features will “almost” appear, with the same definite density, in every other leaf of , and therefore is in the closure of each . Since is (in) the closure of , this implies that every leaf is dense, as claimed.

In a Riemann surface lamination, the conformal type of every leaf is well-defined. If some leaf is elliptic, then necessarily that leaf is a sphere. So if the lamination is minimal, it is equal to a single closed surface. If every leaf is hyperbolic, then each leaf admits a unique hyperbolic metric in its conformal class (i.e. each leaf can be uniformized), and Candel showed that this family of hyperbolic metrics varies continuously in . Étienne Ghys asked whether there is an example of a minimal Riemann surface lamination in which some leaves are conformally parabolic, and others are conformally hyperbolic. It turns out that the answer to this question is yes; Richard Kenyon found an example, which I will now describe.

The lamination in question has exactly one hyperbolic leaf, which is topologically a -times punctured sphere. Every other leaf is an infinite cylinder — i.e. it is conformally the punctured plane . Since the lamination is minimal, to describe the lamination, one just needs to describe one leaf. This leaf will be obtained as the boundary of a thickened neighborhood of an infinite planar graph, which is defined inductively, as follows.

Let be the planar “Greek cross” as in the following figure:

Inductively, if we have defined , define by attaching four copies of to the extremities of . The first few examples are illustrated in the following figure:

The limit is a planar tree with exactly four ends; the boundary of a thickened tubular neighborhood is conformally equivalent to a sphere with four points removed, which is hyperbolic. Every unbounded sequence of points in has a subsequence which escapes out one of the ends. Hence every other leaf in the lamination this defines has exactly two ends, and is conformally equivalent to a punctured plane, which is parabolic.

The header image is a very similar construction in -dimensional space, where the initial seed has six legs along the coordinate axes instead of four; some (quite large) approximation was then rendered in povray.

When I was in graduate school, I was very interested in the (complex) geometry of Riemann surface laminations, and wanted to understand their deformation theory, perhaps with the aim of using structures like taut foliations and essential laminations to hyperbolize -manifolds, as an intermediate step in an approach to the geometrization conjecture (now a theorem of Perelman). I know that at one point Sullivan was quite interested in such objects, as a tool in the study of Julia sets of rational functions. I have the impression that they are not studied so much these days, but I would be happy to be corrected.

Let be a hyperbolic surface with totally geodesic boundary. An orthogeodesic is a geodesic segment properly immersed in , which is perpendicular to at its endpoints. The set of orthogeodesics is countable, and their lengths are proper. Denote these lengths by (with multiplicity). The identity is:

where is the Rogers’ dilogarithm function (to be defined in a minute). Treating this function as a black box for the moment, the identity has the form a term depending only on the topology of . The proof is very, very short and elegant. By the Gauss-Bonnet theorem, the term on the right is equal to of the volume of the unit tangent bundle of . Almost every tangent vector on can be exponentiated to a geodesic on which intersects the boundary in finite forward and backward time (eg. by ergodicity of the geodesic flow on a closed hyperbolic surface obtained by doubling). If is such a tangent vector, and is the associated geodesic arc, then is homotopic keeping endpoints on to a unique orthogeodesic (which is the unique length minimizer in this relative homotopy class). The volume of the set of associated to a given orthogeodesic can be computed as follows. Lift to the universal cover, where it is the crossbar of a letter “H” whose vertical lines are lifts of the geodesics it ends on. Any lifts to a unique geodesic segment in the universal cover with endpoints on the edges of the H. So the volume of the set of such depends only on , giving rise to the explicit formula for . qed.

That’s it — that’s the whole proof! . . . modulo some calculations, which we now discuss.

The “ordinary” polylogarithms are defined by Taylor series

which converges for , and extends by analytic continuation. Taking derivatives, one sees that they satisfy , thereby giving rising to integral formulae. is the familiar geometric series , so and

The Rogers dilogarithm is then given by the formula for real . One sees that the Rogers dilogarithm is obtained by symmetrizing the integrand for the integral expression for under the involution :

Martin derives his identity by direct calculation, but in fact this calculation can be simplified a bit by some hyperbolic geometry. Consider an ideal quadrilateral (whose unit tangent bundle has area ) with one pair of opposite sides that are distance apart. Join opposite vertices in pairs to decompose the quadrilateral into four triangles, each with one non-ideal point:

In the (schematic) picture, suppose the two edges of the H are the left and right side (call them and ) and the other two edges are and . Similarly, call the four triangles depending on which edge of the quadrilateral they bound. The triangle is colored gray in the figure. We secretly identify this figure with the upper half-plane, in such a way that the ideal vertices are (in circular order) , where are the ideal vertices of the gray triangle. Call the (hyperbolic) angle of the gray triangle at its vertex, so . Moreover, it turns out that where is the distance between and . We will compute implicitly as a function of , and show that it is a multiple of the Rogers dilogarithm function, thus verifying Bridgeman’s identity.

Every vector in exponentiates to a (bi-infinite) geodesic , and we want to compute the volume of the set of vectors for which the corresponding geodesic intersects both and . The point of the decomposition is that for in (say), the geodesic intersects whenever it intersects , so we only need to compute the volume of the in for which intersects . Similarly, we only need to compute the volume of the in for which intersects . For in , we compute the volume of the which do not intersect (since these are exactly the ones that intersect both and ), and similarly for .

These volumes can be expressed in terms of integrals of harmonic functions. Let denote the harmonic function on the disk which is on the arc of the circle bounded by , and on the rest of the circle. This function at each point is equal to times the visual angle (i.e. the length in the unit tangent circle) subtended by the given arc of the circle, as seen from the given point in the hyperbolic plane. Define similarly. Then the total volume we need to compute is equal to

(here we have identified by symmetry, and similarly for the other pair of terms). Let us approach this a bit more systematically. If denotes the angle at the nonideal vertex of triangle , we denote , and . The integral we want to evaluate can be expressed easily in terms of explicit rational multiples of , and the function . These functions satisfy obvious identities:

and

where the last identity comes by observing that we are integrating a certain function over an ideal triangle, and observing that the average of this function under the symmetries of the ideal triangle is equal to the constant function . In particular, we see that we can express everything in terms of . After some elementary reorganization, we see that the contribution to the volume of the unit tangent bundle of the surface associated to this particular orthogeodesic is

To compute , it makes sense to move to the upper half-space model, and move the endpoints of the interval to and . The harmonic function is equal to on the negative real axis, and on the positive real axis. It takes the value on the line . The area form in the hyperbolic metric is proportional to the Euclidean area form, with constant . In other words, we want to integrate over the region indicated in the figure, where the nonideal angle is , and the base point is :

If we normalize so that the circular arc is part of the semicircle from to , then the real projection of the vertical lines in the figure are and . There is no elementary way to evaluate this integral, so instead we evaluate its derivative as a function of where as before, . This is the definite integral

Integrating by parts gives . This evaluates to

Thinking of as a function of , we get

Comparing values at we see that and the identity is proved.

Well, OK, this is not terribly simple, but a posteriori it gives a way to express the Rogers dilogarithm as a sum of integrals of very simple harmonic functions over hyperbolic triangles, which is a nice geometric way to think of it.

(Update 10/30): This paper by Dupont and Sah relates Rogers dilogarithm to volumes of -simplices, and discusses some interesting connections to conformal field theory and lattice model calculations. I feel like a bit of a dope, since I read this paper while I was in graduate school more than a dozen years ago, but forgot all about it until I was cleaning out my filing cabinet this morning. They cite an older paper of Dupont for the explicit calculations; these are somewhat tedious and unenlightening; however, he does manage to show that the Rogers dilogarithm is characterized by the Abel identity. In other words,

Lemma A.1 (Dupont): Let be a three times differentiable function satisfying

for all . Then there is a real constant such that where is the Rogers dilogarithm (up to an additive constant).

Nevertheless, they don’t seem to have noticed the formula in terms of integrals of harmonic functions over hyperbolic triangles. Perhaps this is also well-known. Do any readers know?

I learned about Schwarz-Christoffel transformations, Schwarzian derivatives, and Schwarz’s minimal surface as three quite separate mathematical objects, and I was very surprised to discover firstly that they had all been discovered by the same person, and secondly that they form parts of a consistent mathematical narrative, which I will try to explain in this post to the best of my ability. There is an instructive lesson in this example (for me), that we tend to mine the past for nuggets, examples, tricks, formulae etc. while forgetting the points of view and organizing principles that made their discovery possible. Another teachable example is that of Dehn’s “invention” of combinatorial (infinite) group theory, as a natural branch of geometry; several generations of followers went about the task of reformulating Dehn’s insights and ideas in the language of algebra, “generalizing” them and stripping them of their context, before geometric and topological methods were reintroduced by Milnor, Schwarz (a different one this time), Stallings, Thurston, Gromov and others to spectacular effect (note: I have the second-hand impression that the geometric point of view in group theory (and every other subject) was never abandoned in the Soviet Union).

Schwarz’s minimal surface (also called “Schwarz’s D surface”, and sometimes “Schwarz’s H surface”) is an extraordinarily beautiful triply-periodic minimal surface of infinite genus that is properly embedded in . According to Nitsche’s excellent book (p.240), this minimal surface closely resembles the separating wall between inorganic and organic materials in the skeleton of a starfish. The basic building block of the surface can be described as follows. If the vertices of a cube are -colored, the black vertices are the vertices of a regular tetrahedron. Let denote the quadrilateral formed by four edges of this tetrahedron; then a fundamental piece of Schwarz’s surface is a minimal disk spanning :

The surface may be “analytically continued” by rotating through an angle around each boundary edge. Six copies of fit smoothly around each vertex, and the resulting surface extends (triply) periodically throughout space.

The symmetries of enable us to give it several descriptions as a Riemann surface. Firstly, we could think of as a polygon in the hyperbolic plane with four edges of equal length, and angles . Twelve copies of can be assembled to make a hyperbolic surface of genus . Thinking of a surface of genus as the boundary of a genus handlebody defines a homomorphism from to , thought of as ; the cover associated to the kernel is (conformally) the triply periodic Schwarz surface, and the deck group acts on as a lattice (of index in the face-centered cubic lattice).

Another description is as follows. Since the deck group acts by translation, the Gauss map from to factors through a map . The map is injective at each point in the interior or on an edge of a copy of , but has an order branch point at each vertex. Thus, the map is a double-branched cover, with one branch point of order at each vertex of a regular inscribed cube. This leads one to think (like a late 19th century mathematician) of as the Riemann surface on which a certain multi-valued function on is single-valued. Under stereographic projection, the vertices of the cube map to the eight points where . These eight points are the roots of the polynomial , so we may think of as the hyperelliptic Riemann surface defined by the equation ; equivalently, as the surface on which the multi-valued (on ) function is single-valued.

The function is known as the Weierstrass function associated to , and an explicit formula for the co-ordinates of the embedding were found by Enneper and Weierstrass. After picking a basepoint (say ) on the sphere, the coordinates are given by integration:

The integral in each case depends on the path, and lifts to a single-valued function precisely on .

Geometrically, the three coordinate functions are harmonic functions on . This corresponds to the fact that minimal surfaces are precisely those with vanishing mean curvature, and the fact that the Laplacian of the coordinate functions (in terms of isothermal parameters on the underlying Riemann surface) can be expressed as a nonzero multiple of the mean curvature vector. A harmonic function on a Riemann surface is the real part of a holomorphic function, unique up to a constant; the holomorphic derivative of the (complexified) coordinate functions are therefore well-defined, and give holomorphic -forms which descend to (since the deck group acts by translations). These -forms satisfy the identity (this identity expresses the fact that the embedding of into via these functions is conformal). The (composition of the) Gauss map (with stereographic projection) can be read off from the , and as a meromorphic function on , it is given by the formula . Define a function on by the formula . Then are the coordinates of a rational map from into which extends to a map into , by sending each zero of to in the at infinity. Symmetry allows us to identify the image with the hyperelliptic embedding from before, and we deduce that . Solving for we obtain the integrands in the formulae above.

In fact, any holomorphic function on a domain in defines a (typically immersed with branch points) minimal surface, by the integral formulae of Enneper-Weierstrass above. Suppose we want to use this fact to produce an explicit description of a minimal surface bounded by some explicit polygonal loop in . Any minimal surface so obtained can be continued across the boundary edges by rotation; if the angles at the vertices are all of the form the resulting surface closes up smoothly around the vertices, and one obtains a compact abstract Riemann surface tiled by copies of the fundamental region, together with a holonomy representation of into . Sometimes the image of this representation in the rotational part of is finite, and one obtains an infinitely periodic minimal surface as in the case of Schwarz’s surface. A fundamental tile in can be uniformized as a hyperbolic polygon; equivalently, as a region in the upper half-plane bounded by arcs of semicircles perpendicular to the real axis. Since the edges of the loop are straight lines, the image of this hyperbolic polygon under the Gauss map is a region in also bounded by arcs of round circles; thus Schwarz’s study of minimal surfaces naturally led him to the problem of how to explicitly describe conformal maps between regions in the plane bounded by circular arcs. This problem is solved by the Schwarz-Christoffel transformation, and its generalizations, with help from the Schwarzian derivative.

Note that if and are two such regions, then a conformal map from to can be factored as the product of a map uniformizing as the upper half-plane, followed by the inverse of a map uniformizing as the upper half-plane. So it suffices to find a conformal map when the domain is the upper half plane, decomposed into intervals and rays that are mapped to the edges of a circular polygon . Near each vertex, can be moved by a fractional linear transformation to (part of) a wedge, consisting of complex numbers with argument between and , where is the angle at . The function uniformizes the upper half-plane as such a wedge; however it is not clear how to combine the contributions from each vertex, because of the complicated interaction with the fractional linear transformation. The fundamental observation is that there are certain natural holomorphic differential operators which are insensitive to the composition of a holomorphic function with groups of fractional linear transformations, and the uniformizing map can be expressed much more simply in terms of such operators.

For example, two functions that differ by addition of a constant have the same derivative: . Functions that differ by multiplication by a constant have the same logarithmic derivative: . Putting these two observations together suggest defining the nonlinearity of a function as the composition . This has the property that for any constants . Under inversion the nonlinearity transforms by . From this, and a simple calculation, one deduces that the operator is invariant under inversion, and since it is also invariant under addition and multiplication by constants, it is invariant under the full group of fractional linear transformations. This combination is called the Schwarzian derivative; explicitly, it is given by the formula . Given the Schwarzian derivative , one may recover the nonlinearity by solving the Ricatti equation . As explained in this post, solutions of the Ricatti equation preserve the projective structure on the line; in this case, it is a complex projective structure on the complex line. Equivalently, different solutions differ by an element of , acting by fractional linear transformations, as we have just deduced. Once we know the nonlinearity, we can solve for by , the usual solution to a first order linear inhomogeneous ODE. The Schwarzian of the function is . The advantage of expressing things in these terms is that the Schwarzian of a uniformizing map for a circular polygon with angles at the vertices has the form of a rational function, with principal parts , where the and the and depend (unfortunately in a very complicated way) on the edges of (for the ugly truth, see Nehari, chapter 5). To see this, observe that the map has an order two pole near finitely many points (the preimages of the vertices of under the uniformizing map) but is otherwise holomorphic. Moreover, it can be analytically continued into the lower half plane across the interval between successive , by reflecting the image across each circular edge. After reflecting twice, the image of is transformed by a fractional linear transformation, so has an analytic continuation which is single valued on the entire Riemann sphere, with finitely many isolated poles, and is therefore a rational function! When the edges of the polygon are straight, a simpler formula involving the nonlinearity specializes to the “familiar” Schwarz-Christoffel formula.

(Update 10/22): In fact, I went to the library to refresh myself on the contents of Nehari, chapter 5. The first thing I noticed — which I had forgotten — was that if is the uniformizing map from the upper half-plane to a polygon with spherical arcs, then is real-valued on the real axis. Since it is a rational function, this implies that its nonsingular part is actually a constant; i.e.

where is as above, and are real constants (which satisfy some further conditions — really see Nehari this time for more details).

The other thing that struck me was the first paragraph of the preface, which touches on some of the issues I alluded to above:

In the preface to the first edition of Courant-Hilbert’s “Methoden der mathematischen Physik”, R. Courant warned against a trend discernible in modern mathematics in which he saw a menace to the future development of mathematical analysis. He was referring to the tendency of many workers in this field to lose sight of the roots of mathematical analysis in physical and geometric intuition and to concentrate their efforts on the refinement and the extreme generalization of existing concepts.

Instead of using a word like “menace”, I would rather take this as a lesson about the value of returning to the points of view that led to the creation of the mathematical objects we study every day; which was (to some approximation) the point I was trying to illustrate in this post.