Centroids and the rapid decay property
in mapping class groups
We study a notion of an equivariant, Lipschitz, permutation-invariant centroid for triples of points in mapping class groups , which satisfies a certain polynomial growth bound. A consequence (via work of Druţu-Sapir or Chatterji-Ruane) is the Rapid Decay Property for .
A finitely generated group has the Rapid Decay property111This property is also sometimes called property RD or the Haagerup inequality if the space of rapidly decreasing functions on (with respect to every word metric) is inside the reduced –algebra of (see the end of section 2 for a more detailed definition). Rapid Decay was first introduced for the free group by Haagerup [Haagerup]. Jolissaint then formulated this property in its modern form and established it for several classes of groups, including groups of polynomial growth and discrete cocompact subgroups of isometries of hyperbolic space [Jolissaint:RD]. Jolissaint also showed that many groups, for instance , fail to have the Rapid Decay property [Jolissaint:RD]. Rapid Decay was established for Gromov-hyperbolic groups by de la Harpe [delaHarpe:hyperbolicRD].
Throughout this paper will denote a compact orientable surface with genus and punctures. The mapping class group of , denoted , is the group of isotopy classes of orientation preserving homeomorphisms of . We will prove:
has the Rapid Decay property for every compact orientable surface .
The only previously known cases of this theorem were in low complexity when the mapping class group is hyperbolic (these are tori with at most one puncture, or spheres with at most 4 punctures) and for the braid group on four strands, which was recently established by Barré and Pichot [BarrePichot:4strandbraid]. The results in this paper also hold for the braid group on any number of strands. (The case of braid groups follows from the above theorem, since braid groups are subgroups of mapping class groups of surfaces [Ivanov:mcg, Theorem 2.7.I] and RD is inherited by subgroups [Jolissaint:RD, Proposition 2.1.1].)
The Rapid Decay property has several interesting applications. For instance, in order to prove the Novikov Conjecture for hyperbolic groups, Connes-Moscovici [ConnesMoscovici:hyperbolic, Theorem 6.8] showed that if a finitely generated group has the Rapid Decay property and has group cohomology of polynomial growth (property PC), then it satisfies Kasparov’s Strong Novikov Conjecture [Kasparov:Novikov]. Accordingly, since any automatic group has property PC [Meyer:combable], Kasparov’s Strong Novikov Conjecture follows from the above Theorem 1.1 and Mosher’s result that mapping class groups are automatic [mosher:automatic]. The strong Novikov conjecture for has been previously established by both Hamenstädt [Hamenstadt:boundaryamenability] and Kida [Kida:MCGmeasureequivviewpoint].
We prove the Rapid Decay property by appealing to a reduction by Druţu-Sapir (alternatively Chatterji-Ruane) to a geometric condition. Namely, we introduce a notion of centroids for unordered triples in the mapping class group which satisfies a certain polynomial growth property. Despite the presence of large quasi-isometrically embedded flat subspaces in the mapping class group, these centroids behave much like centers of triangles in hyperbolic space. Our notion of centroid is provided by the following result which to each unordered triple in the mapping class group gives a Lipschitz assignment of a point, which has the property that it is a centroid in every curve complex projection. We obtain the following:
For each with there exists a map with the following properties:
is invariant under permutation of the arguments.
For any and we have the following cardinality bound:
where depends only on .
These properties, and especially the count provided by part (4), are essentially Druţu and Sapir’s condition of (**)-relative hyperbolicity with respect to the trivial subgroup [DrutuSapir:RD], and the main theorem of [DrutuSapir:RD] states that this condition implies the Rapid Decay property. Thus to obtain Theorem 1.1 from Theorem 1.2 we appeal to [DrutuSapir:RD], without dealing directly with the Rapid Decay property itself.
Outline of the proof
Let us first recall the situation for a hyperbolic group, . In this setting, for a triple of points , one defines a centroid for the triangle with vertices to be a point with the property that is in the –neighborhood of any geodesics , where is the hyperbolicity constant for considered with some fixed word metric. Thus, if one fixes and and allows to vary in the ball of radius around , the corresponding centroid must lie in a –neighborhood of the length initial segment of . It follows that the number of such centers is linear in .
When , then is not hyperbolic. Nonetheless, it has a closely associated space, the complex of curves, , which is hyperbolic [MasurMinsky:complex1]. Moreover, given any subsurface , there is a geometrically defined projection map, , from the mapping class group of to the curve complex of .
For any , in Theorem LABEL:centroid_from_projections, we construct a centroid with the property that for each , in the hyperbolic space the point is a centroid of the triangle with vertices , , and .
Due to the lack of hyperbolicity in , if one were to fix ahead of time a geodesic , it need not be the case that the center is close to . For this reason, we do not fix a geodesic between and , but rather we use the notion of a –hull, as introduced in [BKMM]. The –hull of a finite set is a way of taking the convex hull of these points, in particular, the convex hull of a pair of points is roughly the union of all geodesics between those points.
In analogy to the fact that for any triangle in a Gromov-hyperbolic space any centroid is uniformly close to each of the three geodesics, in Section LABEL:polynomial_bounds we show that in , any centroid is contained in each –hull between a pair of vertices. This reduces the problem of counting centroids to counting subsets of the –hull, which we also do in this section.
In Section 2, we will review the relevant properties of surfaces, curve complexes, and mapping class groups. In Section LABEL:centroid_construction, we will use properties of curve complexes and –hulls, as developed in [BKMM], to construct the Lipschitz, permutation-invariant centroid map. In Section LABEL:polynomial_bounds we will prove the polynomial bound (3), thus completing the proof of Theorem 1.2.
The authors would like to thank Indira Chatterji, Cornelia Druţu, and Mark Sapir for useful conversations and for raising our interest in the Rapid Decay property. We would like to thank Ken Shackleton for comments on an earlier draft. Behrstock would also like to thank the Columbia University Mathematics Department for their support.
We recall first some notation and results that were developed in [MasurMinsky:complex1], [MasurMinsky:complex2] and [BKMM].
Surfaces and subsurfaces
As above, is an oriented connected surface with genus and punctures (or boundary components) and we measure the complexity of this surface by . An essential subsurface is one whose inclusion is –injective, and which is not peripheral, i.e., not homotopic to the boundary or punctures of . We also consider disconnected essential subsurfaces, in which each component is essential and no two are isotopic. For such a subsurface we define another notion of complexity as follows: if is connected and , if is an annulus, and is additive over components of a disconnected surface. (In [BehrstockMinsky:rankconj], was denoted ). It is not hard to check that is monotonic, i.e., if is an essential subsurface. (From now on we implicitly understand subsurfaces to be essential, and defined up to isotopy.)
If is a subsurface and a curve in we say that and overlap, or , if cannot be isotoped outside of . We say that two surfaces and overlap, or , if neither can be isotoped into the other or into its complement. Equivalently, iff and .
See [BKMM] for a careful discussion of these and related notions.
Curves and markings
The curve complex, , is a complex whose vertices are essential simple closed curves up to homotopy, and whose -simplices correspond to -tuples of disjoint curves. Endow the 1-skeleton with a path metric giving each edge length 1. With this metric is a –hyperbolic metric space [MasurMinsky:complex1].
The definition of is slightly different for : If is a torus with at most one puncture then edges correspond to pairs of curves intersecting once, and if is a sphere with 4 punctures then edges correspond to pairs of vertices intersecting twice. In all these cases is isomorphic to the Farey graph. If is an annulus embedded in a larger surface then admits a natural compactification as an annulus with boundary and is the set of homotopy classes of essential arcs in rel endpoints, with edges corresponding to arcs with disjoint interior. In this case admits a quasi-isometry to which takes Dehn twists to translation by 1. See [MasurMinsky:complex2] for details.
The marking graph is a locally finite, connected graph whose vertices are complete markings on and whose edges are elementary moves. A complete marking is a system of closed curves consisting of a base, which is a maximal simplex in , together with a choice of transversal curve for each element of the base, satisfying certain intersection properties. For a detailed discussion, and proofs of the properties we will list below, see [MasurMinsky:complex2] or [BKMM].
We make into a path metric space by again assigning length 1 to edges. We will denote distance in as , or sometimes just . We will need to use the fact that acts on , and any orbit map induces a quasi-isometry from to .
For an annulus , we identify with , and map this to via the twist-equivariant quasi-isometry mentioned above.
Given a curve in that intersects essentially a subsurface , we can apply a surgery to the intersection to obtain a curve in . This gives a partially-defined map from to which we call a subsurface projection, and in fact this construction extends to a system of maps of both curve and marking complexes that fit into coarsely commutative diagrams: