<![CDATA[Mapping class groups: the next generation]]>

Nothing stands still except in our memory.

– Phillipa Pearce, Tom’s Midnight Garden

In mathematics we are always putting new wine in old bottles. No mathematical object, no matter how simple or familiar, does not have some surprises in store. My office-mate in graduate school, Jason Horowitz, described the experience this way. He said learning to use a mathematical object was like learning to play a musical instrument (let’s say the piano). Over years of painstaking study, you familiarize yourself with the instrument, its strengths and capabilities; you hone your craft, your knowledge and sensitivity deepens. Then one day you discover a little button on the side, and you realize that there is a whole new row of green keys under the black and white ones.

In this blog post I would like to talk a bit about a beautiful new paper by Juliette Bavard which opens up a dramatic new range of applications of ideas from the classical theory of mapping class groups to 2-dimensional dynamics, geometric group theory, and other subjects.

1. Surfaces.

Surfaces and their symmetries are ubiquitous throughout geometry, and even more broadly throughout mathematics. In the first place, surfaces arise as Riemann surfaces, and can be found wherever one finds the complex numbers (which is to say, everywhere). Moreover, surfaces frequently arise in families, and the global study of these families is governed by mapping class groups. And for this reason, mapping class groups are among the most widely-studied objects in topology/dynamics/complex analysis/geometric group theory.

But: when surfaces arise in nature, one does not always know in advance the genus or the topology (think of Seifert surfaces for knots, or Heegaard surfaces for 3-manifolds); moreover, families (especially the kinds of families – think Lefschetz pencils – that arise in complex or symplectic geometry) are typically singular, and depend on choices (a meromorphic function on a complex surface, an integral symplectic form in the symplectic cone etc.). So for a proper appreciation of the role of surfaces in mathematics, one must often consider the totality of all possible surfaces at once; here I explicitly mean to emphasize the importance of considering surfaces of different topological types all at once.

Let’s say we are interested in (oriented) surfaces up to homeomorphism, and homotopy classes of maps between them. We should also be interested in localizing our objects of interest; hence we should also pay attention to surfaces with boundary and maps between them. Connected surfaces with boundary are classified (up to finite ambiguity) by Euler characteristic. Stabilizing the domain of a map (by adding handles which map homotopically trivially) decreases Euler characteristic, so the most interesting information is obtained by trying to minimize $-\chi(S)$ . For closed surfaces, this leads directly to the Gromov-Thurston norm on 2-dimensional homology. For surfaces with boundary, this leads directly to the stable commutator length norm on (homogeneous) 1-boundaries. When the target is a surface with boundary itself, we are discussing stable commutator length in free groups, a topic initiated by Christoph Bavard, and extensively studied by me and my coauthors; see e.g. my monograph scl for an introduction.

Christoph Bavard with some guy

2. Inverse limits and big mapping class groups.

A different route to understanding maps between surfaces of different topological types is to take inverse limits; this naturally leads to the study of homotopy classes of homeomorphisms between surfaces of infinite type; i.e. to the study of big mapping class groups. One natural way in which such things turn up is in the theory of 1-dimensional dynamics. Suppose X is a tree, and $\phi:X \to X$ is an expanding endomorphism. Under many circumstances (e.g. if X is an interval) it is possible to embed X in the plane, and take a topological neighborhood U of X which deformation retracts to X, in such a way that there is an embedding $\tilde{\phi}:U \to U$ such that the composition of inclusion $X \to U$ with $\tilde{\phi}$ with the retraction $U \to X$ is $\phi$ . In this case the intersection $\tilde{X}$ of the forward iterates of U is a continuum, homeomorphic to the inverse limit of $\phi:X \to X$ , in such a way that the homeomorphism $\tilde{\phi}:\tilde{X} \to \tilde{X}$ is the inverse limit of $\phi$ .

Geometrically, $\tilde{\phi}$ looks like a (pseudo-) Anosov map: the contracting directions are the fibers of $U \to X$ , and the expanding directions are the 1-dimensional leaves of $\tilde{X}$ . This idea has been vigorously pursued by Andre de Carvalho and Toby Hall, in several papers beginning (I believe) with this one, focussed on the example of interval endomorphisms. Let me not try to add to the brilliant and incisive math review of the linked paper; instead I will summarize the main points. de Carvalho and Hall develop a train track theory for such endomorphisms, with finitely many “big” edges, but infinitely many “infinitesimal” edges, which are organized in a well-ordered way. The dynamics can be complexified to an honest pseudo-Anosov homeomorphism of the Riemann sphere (with a suitable complex structure), with 1-pronged singularities accumulating only at finitely many limit points. Away from these limit points, the dynamics looks like a pseudo-Anosov on a finitely punctured surface, except that some of the dynamics is carried out to the limit “ends” by eventually periodic homeomorphisms. The suspension of this infinitely-punctured sphere by the dynamics gives rise to an open 3-manifold, which can be (partially) compactified to a sutured manifold by adding finitely many surfaces of finite type — the quotient of the germs near the ends by the eventually periodic maps. Topologically, this sutured manifold has the structure of a finite depth foliation (of depth 1), whose depth 0 leaves are the end quotients, and whose depth 1 leaves are the fibers of the fibration of the open manifold (generalizations to generalized pseudo-Anosovs with one-pronged singularities of different order type, and finite depth foliations of higher depth, should be straightforward).

If the “top” and “bottom” surfaces of the sutured manifold are homeomorphic, they can be glued up to give a closed (well, cusped) foliated manifold, in which the depth 0 leaves are Thurston norm minimizing. By Agol’s recent resolution of the virtual fibered conjecture, there is a finite cover of this manifold which fibers over the circle, and in which the class of the depth 0 leaves is in the boundary of a fibered face. Perturbing the depth 1 foliation to a nearby fibration gives a way of approximating the dynamics of the generalized pseudo-Anosov on a surface of infinite type by a sequence of (ordinary) pseudo-Anosovs on surfaces of finite type.

This is part of a general story: the theory of taut foliations of 3-manifolds is the study of mapping classes of surfaces of infinite type. The best results in the theory are concerned with developing a “pseudo-Anosov package” for a taut foliation which synthesizes the geometric, topological and dynamical avatars of the object in a way which generalizes Thurston’s classical picture for 3-manifolds fibering over the circle. For an introduction to this story, see my monograph Foliations and the geometry of 3-manifolds, especially the first chapter.

3. Artinization of automorphism groups of trees.

Another route to big mapping class groups, or subgroups of them, has a more algebraic flavor, as certain familiar groups from geometric group theory are seen to be close cousins of mapping class groups of infinite type.

The first main example is Thompson’s group V of “dyadic homeomorphisms of the Cantor set”. Here one thinks of the Cantor set C as being made up of smaller Cantor sets $C_s$ for each finite binary string $s$ ; if we identify C with the “middle third” Cantor set, then $C_0$ is the left third, and $C_1$ is the right third, and so on. An automorphism is in V if it breaks up $C$ into some finite disjoint set of $C_{s_i}$ , and shrinks or grows each $C_{s_i}$ , possibly rotating it, and rearranging them in some order so they make a new copy of C. The group V is a beautiful example of a finitely presented infinite simple group (one with no nontrivial proper normal subgroups). For more on this group (and Thompson’s other groups T and F), one can hardly beat Cannon-Floyd-Parry’s introductory paper.

But Cantor sets sit comfortably in the plane; e.g. the middle third Cantor set. Shrinking or growing a sub(Cantor)set can be accomplished by a planar isotopy, as can a rotation of a subset (although one needs to choose a direction of rotation; this ambiguity is resolved by choosing both!) and a rearrangement (by a choice of braid lifting the given permutation). One thus obtains in an obvious way a subgroup $\tilde{V}$ of the mapping class group of the plane minus a Cantor set, which comes with a natural surjective homomorphism to V. Similar “Artinizations” of T and of V were considered by Neretin, Kapoudjian, Funar, Sergiescu and others; they can all be shown to be finitely presented by uniform methods (similar to the methods that work for F,T and V); see e.g. this survey paper.

Similar methods let one Artinize other groups of automorphisms of trees, for example the famous Grigorchuk groups of intermediate growth. One partial Artinization procedure lifts these groups to groups of homeomorphisms of the line – which are necessarily torsion-free – while still being of intermediate growth. Navas studied these groups and found a deep connection between growth rate and analytic quality of the group action. Navas’s groups (actually, any countable left-ordered group) embed in the mapping class group of the plane minus a Cantor set. Another way that self-similar groups give rise to mapping class groups of infinite type is by taking iterated monodromy groups of post-critically finite branched self-coverings of the (Riemann) sphere; this can also be viewed as an example of the inverse limit construction, of course, and realized in the language of taut foliations.

Apropos of nothing, here’s one of the authors of the modern theory of iterated monodromy groups with some guy:

4. Bavard, the next generation.

Let me now come to discuss Juliette Bavard‘s exciting new preprint (note that Juliette is the daughter of Christoph, mentioned earlier). A few years ago I wrote a blog post about the mapping class group of the plane minus a Cantor set. This is a very interesting group; like many mapping class groups, it is circularly orderable; this is a key step in my proof that a group of $C^1$ diffeomorphisms of the plane with a bounded orbit is circularly orderable. In my post I was very curious about the extent to which this group resembles “ordinary” mapping class groups when seen through the lens of bounded cohomology (and scl, as above). Bounded cohomology (in the form of quasimorphisms) arises on mapping class groups through their action on natural hyperbolic spaces – e.g. the complex of curves and its cousins. One can define a complex of curves for the mapping class group of the plane minus a Cantor set, but it is easily seen to be of bounded diameter, and therefore essentially useless. So one needs some substitute. One discouraging fact is that there is a very natural (surjective)map from the mapping class group of the plane minus a Cantor set to the mapping class group of the sphere minus a Cantor set. But this latter group is uniformly perfect, so that it admits no nontrivial quasimorphisms at all, and certainly can’t act in the way one would like on a hyperbolic space. So I proposed a substitute in my blog post: one can consider instead the ray graph (or complex), whose vertices are isotopy classes of proper rays from a point in the Cantor set to infinity, and whose edges (or simplices) are pairs (collections) of isotopy classes that can be realized disjointly. Without having any clear idea one way or the other, I asked (in increasing order of greediness) whether this graph is hyperbolic, whether it has infinite diameter, and whether the action of the mapping class group of the plane minus a Cantor set on this graph gives rise to lots of interesting quasimorphisms.

Juliette’s preprint exceeded my wildest expectations, answering all three questions positively, and doing so in a way which connects up the geometry of this ray graph to the geometry of classical curve complexes. Significantly, it builds on a recent result, proved independently by three separate groups, that the curve and arc graphs associated to surfaces of finite type are uniformly hyperbolic (in the version of this result I understand best, the constant of hyperbolicity is 7). Experts that I know who discussed this result all agreed that it was beautiful and worth knowing, but perhaps that it lacked immediate “killer applications”. I would like to suggest that the adaptation of these arguments to mapping class groups of infinite type by Bavard (which is how hyperbolicity is established) is the killer application: uniform theorems for all surfaces of finite type translate into theorems for surfaces of infinite type (and their mapping class groups).

The action of the mapping class groups on this complex satisfies the necessary conditions to construct lots of interesting (nontrivial) quasimorphisms, and Juliette spells out some specific examples. This gives an enormous range of new quantitative tools with which to attack problems in 2-dimensional (group) dynamics, where the existence of proper invariant closed sets for an action gives rise to homomorphisms to mapping class groups. I think it would also be very interesting to try to construct other hyperbolic graphs on which such mapping class groups of infinite type act, with asymmetric metrics, by combining distances tuned to left-veering maps with subsurface projection; one might be able to use such methods to construct interesting new chiral invariants for area-preserving homeomorphisms of surfaces; see this post for the sort of thing I mean in the finite case.

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