Geometry and the imagination

Groups quasi-isometric to planes

Danny Calegari

I was saddened to hear the news that Geoff Mess recently passed away, just a few days short of his 54th birthday. I first met Geoff as a beginning graduate student at Berkeley, in 1995; in fact, I believe he gave the first topology seminar I ever attended at Berkeley, on closed 3-manifolds which non-trivially cover themselves (the punchline is that there aren’t very many of them, and they could be classified without assuming the geometrization theorem, which was just a conjecture at the time). Geoff was very fast, whip-smart, with a daunting command of theory; and the impression he made on me in that seminar is still fresh in my mind. The next time I saw him might have been May 2004, at the N+1st Southern California Topology Conference, where Michael Handel was giving a talk on distortion elements in groups of diffeomorphisms of surfaces, and Geoff (who was in the audience), explained in an instant how to exhibit certain translations on a (flat) torus as exponentially distorted elements. Geoff was not well even at that stage — he had many physical problems, with his joints and his teeth; and some mental problems too. But he was perfectly pleasant and friendly, and happy to talk math with anyone. I saw him again a couple of years later when I gave a colloquium at UCLA, and his physical condition was a bit worse. But again, mentally he was razor-sharp, answering in an instant a question about (punctured) surface subgroups of free groups that I had been puzzling about for some time (and which became an ingredient in a paper I later wrote with Alden Walker).

Geoff Mess in 1996 at Kevin Scannell’s graduation (photo courtesy of Kevin Scannell)

Geoff published very few papers — maybe only one or two after finishing his PhD thesis; but one of his best and most important results is a key step in the proof of the Seifert Fibered Theorem in 3-manifold topology. Mess’s paper on this result was written but never published; it’s hard to get hold of the preprint, and harder still to digest it once you’ve got hold of it. So I thought it would be worthwhile to explain the statement of the Theorem, the state of knowledge at the time Mess wrote his paper, some of the details of Mess’s argument, and some subsequent developments (another account of the history of the Seifert Fibered Theorem by Jean-Philippe Préaux is available here).

Before stating the Seifert Fibered Theorem we must first discuss the Torus Theorem, and its place in the history of 3-manifold topology. A manifold is said to be closed if it is compact and without boundary. A closed 3-manifold is irreducible if every smoothly embedded 2-sphere bounds a 3-ball. Not all 3-manifolds are irreducible, but every closed, oriented 3-manifold admits a canonical expression as a connect sum of irreducible 3-manifolds and copies of ; these are the “prime factors” in the connect sum decomposition, and many important questions about closed oriented 3-manifolds reduce in a straightforward way to questions about their prime factors; thus in 3-manifold topology it is usual to restrict attention to irreducible 3-manifolds. We need one more definition: a closed, embedded surface S in a 3-manifold (other than a sphere) is said to be incompressible if there is no disk D properly embedded in the complement of S and bounding a homotopically essential embedded loop in S. By the Loop Theorem (proved by Papakyriakopolos), a 2-sided embedded surface (other than a sphere) is incompressible if and only if it is -injective. A closed orientable irreducible 3-manifold is said to be Haken if it contains some incompressible surface. The importance of such a surface is that once one cuts along it, one is guaranteed (for elementary homological reasons) that the resulting manifold itself contains another incompressible surface, and thus Haken manifolds may be inductively decomposed along incompressible surfaces into simple pieces. This opens up the possibility of proving theorems about Haken manifolds inductively; most famously, when Thurston formulated his Geometrization Conjecture in the late 70’s, he was able to prove it for the class of Haken 3-manifolds by an inductive argument. For the next couple of decades, the Geometrization Conjecture became the most important problem in 3-manifold topology, and it is important to view the Torus Theorem in the context of the light it sheds on this conjecture. With this understood, the statement of the Torus Theorem is as follows:

Torus Theorem (Scott): Let M be a closed orientable irreducible 3-manifold, and suppose that there is a -injective map where T is a (2-dimensional) torus. Then

  1. either M contains a 2-sided embedded incompressible torus, which is contained in any neighborhood of the image of T; or
  2. has an infinite cyclic normal subgroup.

Thus if M is a closed oriented 3-manifold whose fundamental group is known to contain a free abelian group of rank at least 2, the Torus Theorem says either that the manifold is Haken (and therefore satisfies the Geometrization Conjecture by Thurston), or its fundamental group is of a very special form (it is worth remarking that a version of the Torus Theorem was proved earlier by Waldhausen under the substantially weaker hypothesis that M is known to be Haken).

At the time Scott proved his theorem, examples were certainly known of non-Haken 3-manifolds whose fundamental group contains an infinite cyclic normal subgroup, but these examples were all of a very special kind. A 3-manifold is a Seifert Fibered space if it can be foliated by circles. Epstein showed that such foliations are always of a special form: every circle has a solid torus neighborhood which is foliated as the mapping torus of a finite order rotation of a disk (note that the very brief Math Review of this paper linked above gives an incorrect statement of the main theorem, omitting the main hypothesis that the leaves are all compact — i.e. circles!). Thus the leaf space of a Seifert-fibered 3-manifold can be thought of in a natural way as a 2-dimensional orbifold, and it makes sense to think of the 3-manifold as a circle “bundle” (in the orbifold sense) over a 2-orbifold O. This orbifold is just an ordinary surface with finitely many special singular “orbifold points”, near which the orbifold looks like the quotient of  a disk by a finite rotation; one keeps track of the kind of singularity as part of the data of the orbifold. O has a well-defined “orbifold” fundamental group, in which a small embedded loop around an orbifold point is a torsion element, of order equal to the order of the singularity. There is a reasonably well-behaved theory of bundles in the category of orbifolds, and at least in this context, there is an associated short exact sequence for . Thus the fundamental group of the fiber (which is Z) is normal in if M is a Seifert FIbered 3-manifold. If is any embedded loop in O (avoiding the orbifold singularities), the union of the circle fibers over is a torus or Klein bottle; this torus or Klein bottle is incompressible if and only if is essential in O; i.e. it does not bound a disk in O with at most one singular point in its interior. Every closed 2-orbifold admits an essential loop  except for a sphere with at most 3 singular points. Thus, every Seifert Fibered space is Haken except those which are circle bundles over a sphere with at most 3 singular points. The latter class are known as the small Seifert Fibered spaces. When the orbifold fundamental group of O is infinite, then at least we can find an immersed loop corresponding to an immersed and -injective torus in M, and thus one obtains examples showing that the second case in the Torus Theorem is unavoidable.

Seifert Fibered spaces admit homogeneous geometric structures, and thus satisfy the Geometrization Conjecture. In the case that the base orbifold O has infinite orbifold fundamental group, the orbifold can be uniformized (as the quotient of the Euclidean or hyperbolic plane by a discrete lattice) and M has a geometry which fibers over Euclidean or hyperbolic geometry. Thus the work of Scott highlighted the importance of the

Seifert Fibered Conjecture: Let M be closed, orientable and irreducible, and suppose that the fundamental group of M contains an infinite cyclic normal subgroup. Then M is Seifert Fibered.

whose resolution would complete the proof of the Geometrization Conjecture for irreducible 3-manifolds whose fundamental groups contain a free abelian group of rank 2. It is at this point that Mess’s work becomes relevant.

As near as I can tell, some version of Mess’ paper was written during 1987 and circulated in December 1987, and then a somewhat edited version was submitted to JAMS in December 1988. Although physical copies of various versions were circulated to several people, it is increasingly difficult to find a copy; I misplaced my own copy when I moved from Pasadena to Chicago. So I am indebted to Peter Scott for scanning and emailing me a copy which I am confident is very close to the final version, and to Derek Mess (Geoff’s brother) for giving me permission to post it here, for the benefit of the younger generation, and for posterity. Darryl McCullough was the referee, and he did an admirable job; Mess’ paper was written in a demanding style, with many new and unfamiliar ideas expressed sometimes in very terse language. Darryl has very kindly permitted me to attach his referee reports here, since they give some perspective on, and insight into the paper that is very valuable. Here are the links:

(note that the latter two files are stored on my department’s local computer, since wordpress does not like the suffix .tex). By carefully comparing page numbers in the preprint and in Darryl’s comments it seems that this version of the paper is probably not the final submitted version, but differs from it only very slightly, and mainly towards the end. I seem to recall in the version that I used to have Mess referred to Candel’s work on uniformization of surface laminations (which may have existed in some preprint form in 1989 or 1990, although I don’t really know). If any reader has a later version of Mess’ paper (i.e. one that is compatible with Darryl’s comments), I would be very grateful if they would send me a copy, and let me know the date their version was written, if possible.

OK, let’s begin to discuss the content of Mess’ paper. We can assume by passing to a cover if necessary that M is a closed, oriented 3-manifold whose fundamental group contains a central Z subgroup. For simplicity, let’s in fact assume that the center is actually equal to Z; it is easy (modulo facts well-known at the time) to reduce to the case that the center has rank 1, but it is subtle to deal with the possibility that the center might be infinitely generated. In any case, the first main theorem Mess proves (corresponding to Theorem 1, page 2) is:

Theorem: Let M be closed, irreducible, orientable. Suppose that center is Z. Then the covering space with fundamental group equal to this center is homeomorphic to a solid torus.

This is proved by “bare hands”, so to speak. Let’s let denote the generator of the center. Because it is central, the element is well-defined as an element of for any point p, so we can build (e.g. inductively on the skeleta of a triangulation) a homotopy such that the track of every point in M under the homotopy is in the class of . We can lift this homotopy to ; because M was compact, the length of the tracks of the homotopy have uniformly bounded length. For homological reasons, is one-ended, and the first point is that every compact set K in can be separated from this end by an embedded torus T in such a way that is still central in , where E is the noncompact region bounded by T. To see this, first observe that K can be included in a big compact set K” such that the track of under the homotopy H stays disjoint from K (this uses the fact that the tracks themselves have uniformly bounded length). The surface is essential in , and its image under H sweeps out an immersed 3-manifold whose image G in contains a central Z subgroup (the image of the tracks of the homotopy). Pass to the cover of ; this manifold has nontrivial , and is therefore Haken, so (because it has a central Z subgroup) it was known to be a Seifert fibered space. Thus the surface can be replaced by a homologically equivalent embedded torus, which necessarily bounds a solid torus in . So is an increasing union of solid tori; a further standard argument shows that these tori nest nicely in each other, and the union is a solid torus.

Now, at this stage, has two useful structures: topologically it is homeomorphic to a solid torus , while geometrically it admits a homotopy whose tracks have bounded length. The next step is to find a relationship between these two structures:

Theorem: With as above, there is a homotopy whose tracks have uniformly bounded diameter, which starts at and ends at a free circle action on witnessing its topological product structure.

In words, J is a bounded homotopy from H to the Seifert structure. In particular, because J has fibers of bounded diameter, admits a product structure for which the circle fibers have uniformly bounded length. The homotopy J is constructed inductively out of “round handles” — i.e. products of circles with ordinary (2-dimensional) handles. First, we can pick any unknotted core of the solid torus, and take this to be the image of some track of H under the homotopy J. The deck group (which is a group because is central and therefore normal) acts on by isometries, and therefore by homeomorphisms; and thus permutes the set of positively oriented unknotted cores, since these are the only unknotted circles which represent homotopically. Choose a separated net in G — a collection of elements such that no two are very close, and such that every element is not too far away from something in the net. Evidently we can choose such a net so that the translates of by elements of the net are all mutually unlinked, and collectively represent an unknotted collection of circles in . Thicken each such circle to a round 0-handle; these will be the round 0-handles in our decomposition.

Building the round 1-handles is tricky, and requires quite an ingenious argument. Because we chose a separated net, every round 0-handle is close to some, but not too many, other round 0-handles. Any two round 0-handles which are close enough can be connected by some annulus (because their cores are isotopic), and we can least area representatives. Two such least area annuli cannot intersect on their boundaries (unless they agree), by the roundoff trick. Thus, any two of them will intersect transversely in finitely many essential circles. So we pick a starting 0-handle and inductively attach least area annuli one at a time, choosing the absolute smallest area one among the finitely many (up to isotopy) which join an unattached 0-handle (which we will call ) to one of the constructed so far, and by a roundoff argument, we see that the result is embedded. By transfinite induction, all the round 0-handles can be connected up in this way after some countable ordinal stage. The annuli we attach can be thickened to become round 1-handles, and the result is a tree of round 0-handles, connected up by round 1-handles, all with uniformly bounded diameter (this is because at every stage some yet to be connected is bounded distance from the union of the handles connected so far, so the annuli which are attached have uniformly bounded diameter).

Now consider a component X of the boundary of the union of round 0- and 1-handles constructed so far. Note that X is partitioned into annuli of bounded diameter which are on the boundaries of the o-handles, and which are on the boundaries of the 1-handles. They appear in a particular order . Adding further round 1-handles splits X into components, some of which might be bounded. We would like to add new annuli, to split X up into components of uniformly bounded (combinatorial) size; to do this, we need to find pairs of which are a uniformly big combinatorial distance apart, but which can be joined by and embedded annuli of uniformly bounded diameter. It is intuitively clear that this can be done: if X is noncompact, the two “ends” of X can’t get too far away from each other, or else there would be an arbitrarily big embedded ball contained in the complement, which is incompatible with the fact that we chose a separated net’s worth of translates of our original 0-handle. A similar argument works when X is compact but sufficiently big (alternately one can suppose not and take pointed limits, since this is a purely geometric argument). Thus we can attach round 1-handles of uniformly bounded diameter so that at the end, every component X itself has bounded diameter, and can be filled in with a round 2-handle. The construction of J with this handle decomposition as the end result is routine.

This brings us to section 3 of Mess’ paper (page 11), entitled, On groups which are coarse quasi-isometric to planes. The group in question is G, i.e. . This is the group that we hope will turn out to be the orbifold fundamental group of O, if the Seifert Conjecture is true. Since it is infinite, we want to show that G is a lattice in the group of isometries of the Euclidean or hyperbolic plane; in fact, a cocompact lattice, since M is closed. In particular, this should imply at least that G is quasi-isometric either to the Euclidean or the hyperbolic plane. By the Schwarz lemma, we know that G is quasi-isometric to , and we have constructed a product structure on whose fibers have uniformly bounded length. It is therefore straightforward (e.g. by averaging over fibers) to construct a complete Riemannian metric on the plane (which we denote P) so that G is quasi-isometric to P. The next main result is Theorem 7 (page 13) which says:

Theorem: Suppose a finitely generated group G is quasi-isometric to a plane P with a complete Riemannian metric. If P is conformally equivalent to the hyperbolic plane, then G is quasi-isometric to the hyperbolic plane.

Note that P has bounded geometry (i.e. 2-sided curvature bounds, and injectivity radius bounded below). One subtlety, observed by Mess, is that the plane admits complete Riemannian metrics with bounded geometry, and in the conformal class of the hyperbolic plane, but for which 0 is the bottom of the spectrum of the Laplacian; a group quasi-isometric to such a space would be amenable, by a famous theorem of Brooks, whereas no group quasi-isometric to the hyperbolic plane can be amenable. Nevertheless, there is a short-cut to proving this theorem, by invoking Candel’s theorem, alluded to above. Candel proves that if  L is a compact Riemann surface lamination all of whose leaves are conformally hyperbolic, then the leafwise uniformization map is continuous; in particular, since L is compact, the uniformization map is bilipschitz (and in particular is a quasi-isometry). Now, a Riemannian manifold with bounded geometry can be realized as a dense leaf in a lamination by taking its closure in pointed Gromov-Hausdorff space; if we do this to P, we obtain a lamination L. A priori a lamination can have leaves of different conformal type; see e.g. this post; but in this case P is uniformly quasi-isometric to G, and therefore (since G acts cocompactly on itself) the same must be true for every leaf of L. Now apply Candel’s theorem; qed. Mess’ argument is not especially hard to follow, but I believe that invoking Candel makes the situation clearer.

Finally we must deal with the case that P is quasi-isometric to the Euclidean plane. In this case, Theorem 10 (page 20) says (paraphrasing):

Theorem: Suppose is quasi-isometric to a plane P with a complete Riemannian metric, which is conformally equivalent to the Euclidean plane. Then G is virtually rank 2 abelian, and M is Seifert fibered; thus, the Seifert Fiber Conjecture holds in this case.

The argument is a beautiful application of ideas from the theory of random walks, combined with a theorem of Varopoulos. It is a well-known fact that a simple random walk is recurrent (i.e. returns to a bounded region infinitely often) in Euclidean space of dimension 1 and 2, and transient otherwise. This is not hard to show: under random walk on Euclidean space, after n steps each coordinate function is distributed like a Gaussian with variance of order n; thus the probability that a given coordinate function will be bounded by a constant C after n steps is of order . By independence, in m-dimensional space, the probability that all coordinate functions will be bounded by the same constant C at the same time after n steps is ; thus, when m is at least 3, the total number of times this should happen in an infinite walk is bounded, by the Borel-Cantelli Lemma. Now, in the continuum limit, a simple random walk rescales to Brownian motion, and Brownian motion is conformally invariant in dimension 2; this means that if you have a complete Riemannian metric on a plane P, you can tell whether it is conformally hyperbolic or conformally Euclidean by whether Brownian motion is transient or recurrent. Using the quasi-isometry between P and G, one concludes that if P is conformally Euclidean, random walk on G is recurrent. But this is an extremely confining possibility for finitely presented groups; Varopoulos showed (when combined with Gromov’s famous theorem that groups of polynomial growth are virtually nilpotent) that it implies that G is virtually abelian of rank at most 2; this is enough to complete the proof, using the (known) classification of nilpotent 3-manifold groups.

Mess’ paper thus reduces the Seifert Fibered Conjecture to the question of whether groups quasi-isometric to the hyperbolic plane are virtually isomorphic to Fuchsian groups — i.e. to (cocompact) lattices in the group of isometries of the hyperbolic plane. Much progress on this question had already been made by Tukia, and while Mess’ paper was still under consideration at JAMS (maybe in a sense it is still under consideration there?) this question was solved in the affirmative independently (and in quite different ways) by Casson-Jungreis, and Gabai (see the comments by Darryl linked to above).

Tastes change; fashions come and go even in mathematics. After Perelman proved the Geometrization Theorem, this story and the mathematical content of these papers faded somewhat into the background, to be quoted if necessary, but rarely read. Mess’ paper in particular — and especially its beautiful and original tone, style and ideas — is in danger of disappearing from our collective consciousness. Today when borrowing some books from the Crerar Library I noticed a Latin inscription: Non est mortuus qui scientiam vivificavit (translation: “He has not died who has given life to knowledge”). But knowledge can die too, and culture, and ideas. My life has been enriched by Geoff’s beautiful ideas, and I’m happy to do my bit to see that they, and maybe some of him, live on a little longer, enriching us all.