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The “header image” for this blog is an example of an interesting construction in 2-dimensional conformal geometry, due to Richard Kenyon, that I learned of some time ago; I thought it might be fun to try to explain where it comes from.
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.
In a previous post, I discussed some methods for showing that a given group contains a (nonabelian) free subgroup. The methods were analytic and/or dynamical, and phrased in terms of the existence (or nonexistence) of certain functions on or on spaces derived from , or in terms of actions of on certain spaces. Dually, one can try to find a free group in by finding a homomorphism and looking for circumstances under which is injective.
For concreteness, let for some (given) space . If is a free group, a representation up to conjugation determines a homotopy class of map where is a . The most natural ‘s to consider are graphs and surfaces (with boundary). It is generally not easy to tell whether a map of a graph or a surface to a topological space is -injective at the topological level, but might be easier if one can use some geometry.
Example: Let be a complete Riemannian manifold with sectional curvature bounded above by some negative constant . Convexity of the distance function in a negatively curved space means that given any map of a graph one can flow by the negative gradient of total length until it undergoes some topology change (e.g. some edge shrinks to zero length) or it (asymptotically) achieves a local minimum (the adjective “asymptotically” here just means that the flow takes infinite time to reach the minimum, because the size of the gradient is small when the map is almost minimum; there are no analytic difficulties to overcome when taking the limit). A typical topological change might be some loop shrinking to a point, thereby certifying that a free summand of mapped trivially to and should have been discarded. Technically, one probably wants to choose to be a trivalent graph, and when some interior edge collapses (so that four points come together) to let the -valent vertex resolve itself into a pair of -valent vertices in whichever of the three combinatorial possibilities is locally most efficient. The limiting graph, if nonempty, will be trivalent, with geodesic edges, and vertices at which the three edges are all (tangentially) coplanar and meet at angles of . Such a graph can be certified as -injective provided the edges are sufficiently long (depending on the curvature ). After rescaling the metric on so that the supremum of the curvatures is , a trivalent geodesic graph with angles at the vertices and edges at least is -injective. To see this, lift to maps between universal covers, i.e. consider an equivariant map from a tree to . Let be an embedded arc in , and consider the image in . Using Toponogov’s theorem, one can compare with a piecewise isometric map from to . The worst case is when all the edges are contained in a single , and all corners “bend” the same way. Providing the image does not bend as much as a horocircle, the endpoints of the image of stay far away in . An infinite sided convex polygon in with all edges of length and all angles osculates a horocycle, so we are done.
Remark: The fundamental group of a negatively curved manifold is word-hyperbolic, and therefore contains many nonabelian free groups, which may be certified by pingpong applied to the action of the group on its Gromov boundary. The point of the previous example is therefore to certify that a certain subgroup is free in terms of local geometric data, rather than global dynamical data (so to speak). Incidentally, I would not swear to the correctness of the constants above.
Example: A given free group is the fundamental group of a surface with boundary in many different ways (this difference is one of the reasons that a group like is so much more complicated than the mapping class group of a surface). Pick a realization . Then a homomorphism up to conjugacy determines a homotopy class of map from to as above. If is negatively curved as before, each boundary loop is homotopic to a unique geodesic, and we may try to find a “good” map with boundary on these geodesics. There are many possible classes of good maps to consider:
- Fix a conformal structure on and pick a harmonic map in the homotopy class of . Such a map exists since the target is nonpositively curved, by the famous theorem of Eells-Sampson. The image is real analytic if is, and is at least as negatively curved as the target, and therefore there is an a priori upper bound on the intrinsic curvature of the image; if the supremum of the curvature on is normalized to be , then the image surface is , which just means that pointwise it is at least as negatively curved as hyperbolic space. By Gauss-Bonnet, one obtains an a priori bound on the area of the image of in terms of the Euler characteristic (which just depends on the rank of ). On the other hand, this map depends on a choice of marked conformal structure on , and the space of such structures is noncompact.
- Vary over all conformal structures on and choose a harmonic map of least energy (if one exists) or find a sequence of maps that undergo a “neck pinch” as a sequence of conformal structures on degenerates. Such a neck pinch exhibits a simple curve in that is essential in but whose image is inessential in ; such a curve can be compressed, and the topology of simplified. Since each compression increases , after finitely many steps the process terminates, and one obtains the desired map. This is Schoen-Yau‘s method to construct a stable minimal surface representative of . When the target is -dimensional, the surface may be assumed to be unbranched, by a trick due to Osserman.
- Following Thurston, pick an ideal triangulation of (i.e. a geodesic lamination of whose complementary regions are all ideal triangles); since has boundary, we may choose such a lamination by first picking a triangulation (in the ordinary sense) with all vertices on and then “spinning” the vertices to infinity. Unless factors through a cyclic group, there is some choice of lamination so that the image of can be straightened along the lamination, and then the image spanned with ideal triangles to produce a pleated surface in representing (note: if has constant negative curvature, these ideal triangles can be taken to be totally geodesic). The space of pleated surfaces in fixed (closed) of given genus is compact, so this is a reasonable class of maps to work with.
- If is merely a hyperbolic group, one can still construct pleated surfaces, not quite in , but equivariantly in Mineyev’s flow space associated to . Here we are not really thinking of the triangles themselves, but the geodesic laminations they bound (which carry the same information).
- If is complete and -dimensional but noncompact, the space of pleated surfaces of given genus is generally not compact, and it is not always easy to find a pleated surface where you want it. This can sometimes be remedied by shrinkwrapping; one looks for a minimal/pleated/harmonic surface subject to the constraint that it cannot pass through some prescribed set of geodesics in (which act as “barriers” or “obstacles”, and force the resulting surface to end up roughly where one wants it to).
Anyway, one way or another, one can usually find a map of a surface, or a space of maps of surfaces, representing a given homomorphism, with some kind of a priori control of the geometry. Usually, this control is not enough to certify that a given map is -injective, but sometimes it might be. For instance, a totally geodesic (immersed) surface in a complete manifold of constant negative curvature is always -injective, and any surface whose extrinsic curvature is small enough will also be -injective.
Geometric methods to certify injectivity of free or surface groups are very useful and flexible, as far as they go. Unfortunately, I know of very few topological methods to certify injectivity. By far the most important exception is the following:
Example: In -dimensions, one should look for properly embedded surfaces. If is a -manifold (possibly with boundary), and is a two-sided properly embedded surface, the famous Dehn’s Lemma (proved by Papakyriakopoulos) implies that either is -injective, or there is an embedded essential loop in that bounds an embedded disk in on one side of . Such a loop may be compressed (i.e. may be cut open along the loop, and two copies of the compressing disk sewn in) preserving the property of embeddedness, but increasing . After finitely many steps, either compresses away entirely, or one obtains a -injective surface. One way to ensure that does not compress away entirely is to start with a surface that is essential in (relative) homology; another way is to look for a surface dual to an action (of ) on a tree. In the latter case, one can often construct quite different free subgroups in by pingpong on the ends of the tree. Note by the way that this method produces closed surface subgroups as well as free subgroups. Note too that two-sidedness is essential to apply Dehn’s Lemma.
Remark: Modern -manifold topologists are sometimes unreasonably indifferent to the power of Dehn’s Lemma (probably because this tool has been incorporated so fully into their subconscious?); it is worth reading Ralph Fox’s review of Papakyriakopoulos’s paper (linked above). Of this paper, he writes:
. . . it has already led to renewed attack on the problem of classifying the 3-dimensional manifolds; significant results have been and are being obtained. A complete solution has suddenly become a definite possibility.
Remember this was written more than 50 years ago — before the geometrization conjecture, before the JSJ decomposition, before the Scott core theorem, before Haken manifolds. The only reasonable reaction to this is: !!!
Example: The construction of injective surfaces by Dehn’s Lemma may be abstracted in the following way. Given a target space , and a class of maps of surfaces into (in some category; e.g. homotopy classes of maps, pleated surfaces, surfaces, etc.) suppose one can find a complexity with values in some ordered set, such that if is not injective, one can find of smaller complexity. Then if is well-ordered, an injective surface may be found. If is not well-ordered, one may ask at least that is upper semi-continuous on , and hope to extend it upper semi-continuously to some suitable compactification of . Even if is not well-ordered, one can at least certify that a map is injective, by showing that it minimizes . Here are some potential examples (none of them entirely satisfactory).
- Given a (homologically trivial) homotopy class of loop in , one can look at all maps of orientable surfaces to with boundary factoring through . For such a surface, let denote the degree with which the (possibly multiple) boundary (components) of wrap homologically around , and let denote the sum of Euler characteristics of non-disk and non-sphere components of . For each surface , one considers the quantity (the factor of can be ignored if desired). The important feature of this quantity is that it does not change if is replaced by a finite cover. If is not injective, let be an essential loop on whose image in is inessential. Peter Scott showed that any essential loop on a surface lifts to an embedded loop in some finite cover. Hence, after passing to such a cover, may be compressed, and the resulting surface satisfies . In other words, a global minimizer of this quantity is injective. Such a surface is called extremal. The problem is that extremal surfaces do not always exist; but this construction motivates one to look for them.
- Given a surface with geodesic boundary in , one can retract to a geodesic spine, and encode the surface by the resulting fatgraph, with edges labelled by homotopy classes in . Since Euler characteristic is local, one does not really care precisely how the pieces of the fatgraph are assembled, but only how many pieces of what kinds are needed for a given boundary. So if only finitely many such pieces appear in some infinite family of surfaces, one can in fact construct an extremal surface as above, which is necessarily injective (more technically, one reduces the computation of Euler characteristic to a linear programming problem, finds a rational extremal solution (which corresponds to a weighted sum of pieces of fatgraph), and glues together the pieces to construct the extremal surface; one situation in which this scheme can be made to work is explained in this paper of mine). Edges can be subdivided into a finite number of possibilities, so one just needs to ensure finiteness of the number of vertex types. One condition that ensures finiteness of vertex types is the existence of a uniform constant so that for each surface in the given family, and for each point , there is an estimate . If this condition is violated, one finds pairs which converge in the geometric topology to a point in a complete (i.e. without boundary, but probably noncompact) surface.
- Given , either compress an embedded essential loop, or realize by a least area surface. If is not injective, pass to a cover, compress a loop, and realize the result by a least area surface. Repeat this process. One obtains in this way a sequence of least area surfaces in (typically of bigger and bigger genus) and there is no reason to expect the process to terminate. If is a -manifold, the curvature of a least area surface admits two-sided curvature bounds away from the boundary, by a theorem of Schoen (near the boundary, the negative curvature might blow up, but only in controlled ways — e.g. after rescaling about a sequence of points with the most negative curvature, one may obtain in the limit a helicoid). Away from the boundary, the family of surfaces one obtains vary precompactly in the topology, and one may obtain a complete locally least area lamination in the limit. If is not injective, one can continue to pass to covers (applying a version of Scott’s theorem for infinite surfaces) and compress, and by transfinite induction, eventually arrive at a locally least area lamination with injective . Of course, such a limit might well be a lamination by planes. However, the lamination one obtains is not completely arbitrary: since it is a limit of limits of . . . compact surfaces, one can choose a limit that admits a nontrivial invariant transverse measure (one must be careful here, since the lamination will typically have boundary). Or, as in bullet 2. above, one may insist that this limit lamination is complete (i.e. without boundary).
It is more tricky to find a limit lamination as in 3. without boundary and admitting an invariant transverse measure; in any case, this motivates the following:
Question: Is there a closed hyperbolic -manifold which admits a locally least area transversely measured complete immersed lamination , all of whose leaves are disks? (note that the answer is negative if one asks for the lamination to be embedded (there are several easy proofs of this fact)).
Secretly, the function that assigns to a homologically trivial loop is the stable commutator length of the conjugacy class in represented by . Extremal surfaces can sometimes be certified by constructing certain functions on called homogeneous quasimorphisms, but a discussion of such functions will have to wait for another post.