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I recently uploaded a paper to the arXiv entitled Knots with small rational genus, joint with Cameron Gordon. The genesis of this paper was a couple of nice (and related) talks at Caltech by Matthew Hedden and Jake Rasmussen in 2007. They both talked about potential applications of the theory of knot Floer homology to the Berge conjecture. A Berge knot is a (tame) knot in the 3-sphere which lies on a genus two Heegaard surface, and with the property that on each side of the Heegaard surface there is a meridian disk that the knot intersects exactly once. Equivalently, the inclusion of the knot into each (closed) handlebody sends the generator of to a generator of . Note that since the 3-sphere admits a unique (up to isotopy) Heegaard splitting of any genus, one may think of such a knot as lying on a specific genus 2 surface in . Such knots were classified by Berge; they admit (Dehn) surgeries which result in (nontrivial) Lens spaces. The Berge conjecture is the converse; i.e.:
Berge Conjecture: Let be a knot in which admits a nontrivial Lens space surgery; i.e. there is a Lens space and a knot in for which is homeomorphic to . Then is a Berge knot.
An equivalent formulation (of course) is to try to classify knots in Lens spaces which admit an surgery, i.e. to identify the knots as in the formulation of the conjecture above. The equivalent formulation says that these knots should be 1-bridge. The strategy of Hedden-Rasmussen (building on work of Ken Baker and Eli Grigsby) to approach the Berge conjecture depends on characterizing such knots by properties which can be detected by topological invariants that behave well under surgery. An example of such a topological invariant is the Casson invariant , a -valued invariant of integer homology spheres which satisfies the surgery formula where denotes the result of surgery on some integral homology sphere along a fixed knot , and is the Arf invariant. For more sophisticated invariants like knot Floer homology, the surgery formula is replaced by an exact triangle. One important piece of topological information that is detected by knot Floer homology is the genus of a knot. The approach to the Berge conjecture thus rests on Ken Baker’s impressive paper showing that small genus knots (in a sense to be made precise) in Lens spaces have small bridge number.
Hedden remarked in his talk that his work, and that of his collaborators “gave the first examples of an infinite family of knots that were characterized by their knot Floer homology”. Though technically true, I think this overstates the role of knot Floer homology in this case, since the knots (1-bridge knots in Lens spaces) are entirely characterized (up to isotopy) by their genus (and therefore by any topological invariant which detects genus). My immediate instinct was to think that knots with small genus in any 3-manifold should always be quite special, and that a complete classification might even be feasible. My paper with Cameron confirms this suspicion, and gives such a classification. Let me admit at this point that I am not especially interested in the Berge conjecture per se, although I find it interesting that new ideas in 3-manifold topology are starting to have something meaningful to say about it. In any case, I shall not have anything else to say about it (meaningful or otherwise) in this post.
First I should say that I have been using the word “genus” in a somewhat sloppy manner. For an oriented knot in , a Seifert surface is a compact oriented embedded surface whose boundary is . The genus of such a surface is a non-negative integer, and the least such genus over all Seifert surfaces is (said to be) the genus of , denoted . Such a surface represents the generator in the relative homology group which equals since has vanishing homology in dimensions 1 and 2. This relative homology group is dual to , which is parameterized by homotopy classes of maps from to a circle (which is a ). The preimage of a regular value under a smooth map dual to the homology class is a smooth proper surface in whose closure is a Seifert surface. It is immediate that if and only if is an unknot; in other words, the unknot is “characterized” by its genus. There are infinitely many knots of any positive genus in ; on the other hand, there are only two fibered genus 1 knots — the trefoil and the figure 8 knot (three if you distinguish the left-handed from the right-handed trefoil), and it is worth remarking (from the point of view of the motivation of characterizing knots by topological invariants) that a theorem of Yi Ni says that fiberedness of knots can be detected by knot Floer homology.
For knots in integral homology -spheres, the situation is very similar: every knot admits a Seifert surface, and the least genus of such a surface is the genus of a knot. The unknot is (always) characterized by the fact that it has genus , but there are infinitely many knots of every positive genus. For a knot in a general -manifold it is not so easy to define genus. A necessary and sufficient condition for to bound an embedded surface in its complement is that in . However, if has finite order, one can find an open properly embedded surface in the complement of whose “boundary” wraps some number of times around . Technically, let be a compact oriented surface, and a map which restricts to an embedding from the interior of into , and which restricts to an oriented covering map from to (note that we allow to have multiple boundary components). If is the degree of the covering map , we call a -Seifert surface, and define the rational genus of to be , where denotes Euler characteristic, and (for a connected surface ). The reason to use Euler characteristic instead of genus is that Euler characteristic is multiplicative under coverings (unlike genus), and behaves well with respect to “local” operations on surfaces like cut-and-paste. Moreover, (negative) Euler characteristic, unlike genus, is a good measure of complexity for surfaces with possibly many boundary components. The coefficient of in the denominator reflects the fact that genus is “almost” times Euler characteristic. With this definition, we say that the rational genus of , for any knot with of finite order in , is the infimum of over all -Seifert surfaces for and all . The purpose of our paper is to give a complete classification of knots with sufficiently small rational genus, and to show that such knots are always “geometric” — i.e. they can be isotoped into a normal form which is sensitive to the geometric decomposition of the ambient -manifold . Thus the concept of rational genus makes contact between the homological world of the Thurston norm, knot Floer homology and such invariants, and the geometric world of hyperbolic structures, JSJ decompositions and so on.
It is worth pointing out at this point that knots with small rational genus are not special by virtue of being rare: if is any knot in (for instance) of genus , and in is obtained by Dehn surgery on , then the knot has order in , and . Since for “most” coprime the integer is arbitrarily large, it follows that “most” knots obtained in this way have arbitrarily small rational genus.
There is a precise connection between rational genus and the Thurston norm. There is an exact sequence in homology, which contains the fragment . Since , the kernel of is generated by some class , and one can define the affine subspace . By excision, we identify with where is a tubular neighborhood of . Under this identification, the rational genus of is equal to where denotes the (relative) Thurston norm, and the infimum is taken over classes in in the affine subspace corresponding to . Since the Thurston norm is a convex piecewise rational function, this infimum is realized at some rational point. In other words, rational genus of any knot is rational, and is realized by some -Seifert surface, where as above divides (note: if is a rational homology sphere, then necessarily , but if the rank of is positive, this is not necessarily true, and might be arbitrarily large). This relationship to the Thurston norm also gives a straightforward algorithm to compute rational genus, since one can compute Thurston norm e.g. by linear programming in normal surface space relative to any triangulation.
The precise statement of results depends on the geometric decomposition of the ambient manifold . By the geometrization theorem (of Perelman), a closed, orientable -manifold is either reducible (i.e. contains an embedded sphere that does not bound a ball), or is a Lens space, or is hyperbolic, or is a small Seifert fiber space, or is toroidal (i.e. contains an essential (-injective) embedded torus). For the record, the complete “classification” is as follows:
Reducible Theorem: Let be a knot in a reducible manifold . Then either
- ; or
- there is a decomposition , and either
- is irreducible, or
Lens Theorem: Let be a knot in a lens space . Then either
- ; or
- lies on a Heegaard torus in ; or
- is of the form and lies on a Klein bottle in as a non-separating orientation-preserving curve.
Hyperbolic Theorem: Let be a knot in a closed hyperbolic -manifold . Then either
- ; or
- is trivial; or
- is isotopic to a cable of the core of a Margulis tube.
Small SFS Theorem: Let be an atoroidal Seifert fiber space over with three exceptional fibers and let be a knot in . Then either
- ; or
- is trivial; or
- is a cable of an exceptional Seifert fiber of ; or
- is a prism manifold and is a fiber in the Seifert fiber structure of over with at most one exceptional fiber.
Toroidal Theorem: Let be a closed, irreducible, toroidal 3-manifold, and let be a knot in . Then either
- ; or
- is trivial; or
- is contained in a hyperbolic piece of the JSJ decomposition of and is isotopic either to a cable of a core of a Margulis tube or into a component of ; or
- is contained in a Seifert fiber piece of the JSJ decomposition of and either
- is isotopic to an ordinary fiber or a cable of an exceptional fiber or into , or
- contains a copy of the twisted bundle over the Möbius band and is contained in as a fiber in this bundle structure;
- is a -bundle over with Anosov monodromy and is contained in a fiber.
The constant is presumably not optimal, but reflects the coarseness of certain geometric estimates at a particular step in the argument. Broadly speaking, there are two cases to consider: when the knot complement is hyperbolic, and when it is not. The complement is hyperbolic unless it contains an essential subsurface of non-negative Euler characteristic.
The case that is hyperbolic is conceptually easiest to analyze. Let be a surface, embedded in and with boundary wrapping some number of times around , realizing the rational genus of . The complete hyperbolic structure on may be deformed, adding back as a cone geodesic. Just as a cone can be obtained from a wedge of paper by gluing the two edges together, the geometry of a cone geodesic is locally modeled on the quotient space obtained from a (3-dimensional hyperbolic) wedge by gluing the two flat faces together. The thinner the wedge, the smaller the cone angle along the geodesic. For all sufficiently small angles , Thurston proved that there exists a unique hyperbolic metric on which is singular along a cone geodesic, isotopic to , with cone angle . Call this metric space . The cone angle can be increased, deforming the geometry in a family of spaces, until one of the following three things happens:
- The cone angle is increased all the way to , resulting in the complete hyperbolic structure on , in which is isotopic to an embedded geodesic; or
- The volume of the family of manifolds goes to zero (and either converges after rescaling to a Euclidean cone manifold, or converges after rescaling to have fixed diameter and injectivity radius going to zero everywhere); or
- The cone locus bumps into itself (this can only happen for ).
As the cone angle along increases, so does the length of the cone geodesic. Simultaneously, the diameter of an embedded tube about this diameter decreases. While the diameter of the tube is big, the deformation can continue. Hodgson-Kerckhoff analyzed the kinds of degenerations that can occur, and obtained universal geometric control on how fast the tube diameter can shrink, or the length of the cone geodesic grow. They showed that the cone angle can be increased (giving rise to a family of singular hyperbolic structures ) either until , or until the product , where is the length of the cone geodesic, is at least , at which point the diameter of an embedded tube about this cone geodesic is at least . Since in the latter case, one obtains a lower bound on both the length of the cone geodesic and the diameter of an embedded tube, independent of or .
Now, one would like to use this big tube to conclude that is large. This is accomplished as follows. Geometrically, one constructs a -form which agrees with the length form on the cone geodesic, which is supported in the tube, and which satisfies pointwise for some (universal) constant . Then one uses this -form to control the topology of . By Stokes theorem, for any surface homotopic to in one has an estimate
In particular, the area of divided by can’t be too small. However, it turns out that one can find a surface as above with ; such an estimate is enough to obtain a universal lower bound on . Such a surface can be constructed either by the shrinkwrapping method of Calegari-Gabai, or the (related) PL-wrapping method of Soma. Roughly speaking, one uses the cone geodesic as an “obstacle”, and finds a surface of least area homotopic to (rel. boundary) subject to the constraint that it cannot cross the geodesic. Away from the cone geodesic, looks like an ordinary minimal surface. In particular, its intrinsic curvature is no more than the extrinsic curvature of hyperbolic space, which is everywhere. Along the geodesic, looks like a bedsheet hanging on a clothesline; in particular, it does not accumulate any corners or atoms of positive curvature along this singularity, so the Gauss-Bonnet theorem gives the desired bound on .
This leaves the case that is not hyperbolic to analyze. As remarked above, this only occurs when contains an essential surface (which might be closed or proper) of non-negative Euler characteristic, i.e. a sphere, a disk, an annulus or a torus. In this case, one tries to make the intersection of with this essential surface as simple as possible; if one arranges this just right, every intersection contributes a definite amount to the topology of , and one can conclude either that is complicated (in which case is large), or that the intersection is simple, and therefore draw some topological conclusion.
To actually do this in practice is quite complicated, but fortunately it relies on (largely combinatorial) methods developed at length by Gabai, Scharlemann, Gordon and others over the last 30 years to analyze (so-called) “exceptional surgeries”. Of course, the argument is still complicated, and this analysis takes up most of the length of the paper. It is also worth pointing out that every case provided for by the classification above actually occurs, with examples of arbitrarily small rational genus.
This paper raises several natural questions, the most obvious of which is whether the explicit (but quite small) constants can be improved in any way. The constant in the statement of the Toroidal Theorem is really only there to take care of a knot sitting inside a hyperbolic piece in the decomposition; a knot that interacts in a meaningful way with an essential torus necessarily has rational genus at least (for a precise statement, see the paper). As remarked above, knots of (ordinary) genus are very plentiful, even in , and do not “see” any of the ambient geometry, so the wildest and most optimistic guess might be that there is a chance of classifying knots of rational genus at most . There are some (very weak) reasons to think that this fraction is critical, at least in some cases, not least of which is the papers of Hedden and Ni mentioned above. But in the hyperbolic case, it is probably not easy to get a better estimate using purely geometric arguments.
Another approach might be to try to substitute another conclusion (again in the hyperbolic case) than that be isotopic to the cable of a core of a Margulis tube. For instance, one might ask for to admit an insulator family (of the kind Gabai used here), or one might merely ask that be unknotted in the universal cover, or satisfy some other condition. This goes to the heart of a very, very difficult and important question, namely how to identify geometric features of codimension 2 objects in (especially hyperbolic) geometric 3-manifolds from purely topological properties. If I am optimistic, then I can imagine that this paper makes a contribution, however small, to this ongoing project.
I am (update: was) currently (update: but am no longer) in Brisbane for the “New directions in geometric group theory” conference, which has been an entirely enjoyable and educational experience. I got to eat fish and chips, to watch Australia make 520 for 7 (declared) against the West Indies at the WACA, and to hear Masato Mimura give a very nice talk about his recent results on rigidity of the “universal lattice”.
His talk included a quick and beautiful survey of some geometric aspects of the theory of rigidity for infinite groups, which I will attempt to partially reproduce (despite the limitations of the wordpress format). In this context, rigidity is expressed in terms of isometric affine actions of groups on Banach spaces. This means the following. Suppose is a Banach space (i.e. a complete, normed vector space) and is a group. A linear isometric action is a representation from to the group of linear isometries of — i.e. linear norm-preserving automorphisms. An affine action is a representation from to the group of affine isometries of — i.e. isometries as a metric space that do not necessarily fix the zero element. The group of isometries of a Banach space is a semi-direct product where is the group of linear isometries, and is the Banach space, thought of as an Abelian group, acting on itself by (isometric) translations. Such an action is usually encoded by a pair which records the “linear” part of the action, and a 1-cocycle with coefficients in , i.e. a function satisfying for every . This formula might look strange if you don’t know where it comes from: it is just the way that factors transform in semi-direct products. The affine action is given by sending to the transformation that sends each to . Consequently, is sent to the transformation that sends to and the fact that this is a group action becomes the formula
Equating the left and right hand sides gives the cocycle condition. Given one affine isometric action, one can obtain another in a silly way by conjugating by an isometry for some . Under conjugation by such an isometry, a cocycle transforms by . A function of the form is called a 1-coboundary, and the quotient of the space of 1-cocycles by the space of 1-coboundaries is the 1 dimensional cohomology of with coefficients in . This is usually denoted , where is suppressed in the notation. In particular, an affine isometric action of on with linear part has a global fixed point iff it represents in . Contrapositively, admits an affine isometric action on without a global fixed point iff for some .
A group is said to satisfy Serre’s Property (FH) if every affine isometric action of on a Hilbert space has a global fixed point. In 2007, Bader-Furman-Gelander-Monod introduced a property (FB) for a group to mean that every affine isometric action of on some (out of a class of) Banach space(s) has a global fixed point. Mimura used the notation property (FL_p) for the case that is allowed to range over the class of spaces (for some fixed ).
Intimately related is Kazhdan’s Property (T), introduced by Kazhdan in this paper. Let be a locally compact topological group (for example, a discrete group). The set of irreducible unitary representations of is called its dual, and denoted . This dual is topologized in the following way. Associated to a representation , a unit vector , a positive number and a compact subset there is an open neighborhood of consisting of representations for which there is a unit vector such that whenever . With this topology (called the Fell topology), one says that a group has property (T) if the trivial representation is isolated in . Note that this topology is very far from being Hausdorff: the trivial representation fails to be isolated exactly when there are a sequence of representations , unit vectors , numbers and compact sets exhausting so that for any . The vectors are said to be (a sequence of) almost invariant vectors. Hence (informally) a group has property (T) if some compact subset must move some unit vector a definite amount in every irreducible nontrivial unitary representation. If a group fails to have property (T), one can rescale a sequence of irreducible actions near a sequence of almost invariant vectors in such a way that one obtains in the geometric limit a nontrivial isometric action on without a global fixed point. A famous theorem of Delorme-Guichardet says that property (T) and property (FH) are equivalent for (locally compact second countable) groups. Property (T) passes to quotients, and to lattices (i.e. finite covolume discrete subgroups of a topological group). Kazhdan already showed in his paper that has property (T) for at least , and therefore the same is true for lattices in this groups, such as , a fact which is not easy to see directly from the definition. One beautiful application, already pointed out by Kazhdan, is that this means that all lattices in , for instance the groups (and in fact, all discrete groups with property (T)) are finitely generated. Kazhdan’s proof of this is incredibly short: let be a discrete group and and sequence of elements. For each , let be the subgroup of generated by . Notice that is finitely generated iff for all sufficiently large . On the other hand, consider the unitary representations of induced by the trivial representations on the . Every compact subset of is finite, and therefore eventually fixes a vector in every one of these representations; thus there is a sequence of almost fixed vectors. If has property (T), this sequence eventually contains a fixed vector, which can only happen if is finite, in which case is finitely generated, as claimed.
Property (FL_p) generalizes (FH) (equivalently (T)) in many significant ways, with interesting applications to dynamics. For example, Navas showed that if is a group with property (T) then every action of on a circle which is at least factors through a finite group. Navas’s argument can be generalized straightforwardly to show that if has (FL_p) for some then every action of on a circle which is at least factors through a finite group. The proof rests on a beautiful construction due to Reznikov (although a similar construction can be found in Pressley-Segal) of certain functions on a configuration space of the circle which are not in but have coboundaries which are; this gives rise to nontrivial cohomology with coefficients for groups acting on the circle in a sufficiently interesting way.
(Update: Nicolas Monod points out in an email that the “function on a configuration space” is morally just the derivative. In fact, he made the nice remark that if is any elliptic operator on an -manifold, then the commutator is of Schatten class whenever is a sufficiently smooth function; morally this should give rise to nontrivial cohomology with suitable coefficients for groups acting with enough regularity on any given -manifold, and one would like to use this e.g. to approach Zimmer’s conjecture, but nobody seems to know how to make this work as yet; in fact the work of Monod et. al. on (FL_p) is at least partly motivated by this general picture.)
Mimura discussed a spectrum of rigid behaviour for infinite groups, ranging from most rigid (property (FL_p) for every ) to least rigid (amenable) (note: every finite group is both amenable and has property (T), so this only really makes sense for infinite groups; moreover, every reasonable measure of rigidity for infinite groups is usually invariant under passing to subgroups of finite index). Free groups, and so on are very non-rigid. However, it is well-known that certain infinite families of (word) hyperbolic groups, including lattices in groups of isometries of quaternion-hyperbolic symmetric spaces, and “random” groups with relations having density parameter (see Zuk or Ollivier) are both hyperbolic and have property (T). Nevertheless, these groups are not as rigid as higher rank lattices like for . The latter have property (FL_p) for every , whereas Yu showed that every hyperbolic group admits a proper affine isometric action on for some (the existence of a proper affine isometric action on a Hilbert space is called “a-T-menability” by Gromov, and the “Haagerup property” by some. Groups satisfying this property, or even Yu’s weaker property, are known to satisfy some version of the Baum-Connes conjecture, the subject of a very nice minicourse by Graham Niblo at the same conference).
It is in this context that one can appreciate Mimura’s results. His first main result is that the group (i.e. the “universal lattice”) has property (FL_p) for every provided is at least 4. Since property (FL_p) (like (T)) passes to quotients, this implies that has (FL_p) for every unital, commutative, finitely generated ring .
His second main result concerns a “quasification” of FL_p, to a property called (FFL_p). Without getting too technical, this property concerns “quasi-actions” of a group on a Banach space by affine isometries; algebraically these are encoded by 1-cochains for which there is a universal constant so that as measured in the Banach norm on . Any bounded map defines a 1-cochain; such (bounded) 1-cochains corresponds to quasi-action with a bounded orbit. Associated to one defines in a similar way a complex of bounded cochains; quasi-actions modulo bounded quasi-actions are parameterized by the kernel of the comparison map from bounded to ordinary cohomology. Mimura’s second main result is that when is the universal lattice as above, and has no invariant vectors, the comparison map from bounded to ordinary cohomology in dimension 2 is injective.
The fact that as above is required to have no invariant vectors is a technical necessity of Mimura’s proof. When is trivial, one is studying “ordinary” bounded cohomology, and there is an exact sequence
with real coefficients for any (here denotes the vector space of homogeneous quasimorphisms on ). In this context, one knows by Bavard duality that is injective if and only if the stable commutator length is identically zero on . By quite a different method, Mimura shows that for at least , and for any Euclidean ring (i.e. a ring for which one has a Euclidean algorithm; for example, ) the group has vanishing stable commutator length, and therefore one has injectivity of bounded to ordinary cohomology in dimension .
(Update 1/9/2010): Nicholas Monod sent me a nice email commenting on a couple of points in this blog entry, and I have consequently modified the language a bit in a few places. Ta much!
On page 10 of Besse’s famous book on Einstein manifolds one finds the following quote:
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.