The universal Cannon–Thurston map and the boundary of the curve complex
In genus two and higher, the fundamental group of a closed surface acts naturally on the curve complex of the surface with one puncture. Combining ideas from previous work of Kent–Leininger–Schleimer and Mitra, we construct a universal Cannon–Thurston map from a subset of the circle at infinity for the closed surface group onto the boundary of the curve complex of the once-punctured surface. Using the techniques we have developed, we also show that the boundary of this curve complex is locally path-connected.
AMS subject classification = 20F67(Primary), 22E40 57M50
1.1 Statement of results.
Fix a hyperbolic metric on a closed surface of genus at least two. This identifies the universal cover with the hyperbolic plane . Fix a basepoint and a point . This defines an isomorphism between the group of homotopy classes of loops based at and the group of covering transformations of .
We will also regard the basepoint as a marked point on . As such, we write for the surface with the marked point . We could also work with the punctured surface ; however a marked point is more convenient for us.
Let and denote the curve complexes of and respectively and let denote the forgetful projection. From [kls], the fiber over is –equivariantly isomorphic to the Bass-Serre tree determined by . The action of on comes from the inclusion into the mapping class group via the Birman exact sequence (see Section 1.2.2). We define a map
by sending to in a –equivariant way and then extending over simplices using barycentric coordinates (see Section LABEL:S:construct_Phi).
Given , let denote the restriction to
Suppose that is a geodesic ray that eventually lies in the preimage of some proper essential subsurface of S. We will prove in Section 3 that has finite diameter. The remaining rays define a subset (of full Lebesgue measure). Our first result is the following.
Theorem 1.1 (Universal Cannon–Thurston map).
For any , the map has a unique continuous –equivariant extension
The map does not depend on and is a quotient map onto . Given distinct points , if and only if and are ideal endpoints of a leaf (or ideal vertices of a complementary polygon) of the lift of an ending lamination on .
We recall that a Cannon–Thurston map was constructed by Cannon and Thurston [CT] for the fiber subgroup of the fundamental group of a closed hyperbolic –manifold fibering over the circle. The construction was then extended to simply degenerate, bounded geometry Kleinian closed surface groups by Minsky [minsky-top], and in the general simply degenerate case by the second author [mahan-amalgeo],[mahan-split]. In all these cases, one produces a quotient map from the circle onto the limit set of the Kleinian group . In the quotient, distinct points are identified if and only if they are ideal endpoints of a leaf (or ideal vertices of a complementary polygon) of the lift of an ending lamination for . This is either one or two ending laminations depending on whether the group is singly or doubly degenerate; see [mahan-elct].
In a similar fashion, the second author [mitra-endlam] has constructed a Cannon–Thurston map for any –hyperbolic extension of a group by ,
(for a discussion of such groups see [mosher-hbh, FMcc]). This is a –equivariant quotient map from onto the Gromov boundary of . As above, the quotient identifies distinct points if and only if they are ideal endpoints of a leaf (or ideal vertices of a complementary polygon) of the lift of an ending lamination for .
The map is universal in that distinct points are identified if and only if they are the ideal endpoints of a leaf (or ideal vertices of a complementary polygon) of the lift of any ending lamination on . We remark that the restriction to is necessary to get a reasonable quotient: the same quotient applied to the entire circle is a non-Hausdorff space.
It follows from the above description of the various Cannon–Thurston maps that the universal property of can also be rephrased as follows. If is any Cannon–Thurston map as above—so, is either the limit set of a Kleinian group, or the Gromov boundary of a hyperbolic extension —then there exists a map
so that . Moreover, because identifies precisely the
required points to make this valid, one sees that any –equivariant quotient of with this property is
actually a –equivariant quotient of .
It is a classical fact, due to Nielsen, that the action of on extends to the entire mapping class group . It will become apparent from the description of given below that this action restricts to an action on . In fact, we have
The quotient map
is equivariant with respect to the action of .
As an application of the techniques we have developed, we also prove the following.
The Gromov boundary is path-connected and locally path-connected.
This strengthens the work of the first and third authors in [leinsch] in a special case: in [leinsch] it was shown that the boundary of the curve complex is connected for surfaces of genus at least with any positive number of punctures and closed surfaces of genus at least . The boundary of the complex of curves describes the space of simply degenerate Kleinian groups as explained in [leinsch]. These results seem to be the first ones providing some information about the topology of the boundary of the curve complex. The question of connectivity of the boundary was posed by Storm, and the general problem of understanding its topology by Minsky in his 2006 ICM address. Gabai [gabai] has now given a proof of Theorem 1.3 for all surfaces for which is nontrivial, except the torus, –punctured torus and –punctured sphere, where it is known to be false.
Acknowledgements. The authors wish to thank the Mathematical Sciences Research Institute for its hospitality during the Fall of 2007 where this work was begun. We would also like to thank the other participants of the two programs, Kleinian Groups and Teichmüller Theory and Geometric Group Theory, for providing a mathematically stimulating and lively atmosphere.
1.2 Notation and conventions.
For a discussion of laminations, we refer the reader to [harer-penner], [CEG], [bonahon-curr-teich], [thurstonnotes], [CB].
A measured lamination on is a lamination with a transverse measure of full support. A measured lamination on will be denoted with the support—the underlying lamination—written . We require that all of our laminations be essential, meaning that the leaves lift to quasigeodesics in the universal cover.
If is an arc or curve in and a measured lamination, we write for the total variation of along . We say that is transverse to if is transverse to every leaf of . If is the isotopy class of a simple closed curve, then we write
for the intersection number of with , where varies over all representatives of the isotopy class .
Two measured laminations and are measure equivalent if for every isotopy class of simple closed curve , . Every measured lamination is equivalent to a unique measured geodesic lamination (with respect to the fixed hyperbolic structure on ). This is a measured lamination for which is a geodesic lamination. Given a measured lamination , we let denote the measure equivalent measured geodesic lamination. We will describe a preferred choice of representative of the measure class of a measured lamination in Section 2 below.
We similarly define measured laminations on as compactly supported measured laminations on . In the situations that we will be considering, these will generally not arise as geodesic laminations for a hyperbolic metric on , though any one is measure equivalent to a measured geodesic lamination for a complete hyperbolic metric on .
The spaces of (measure classes of) measured laminations will be denoted by and . The topology on is the weakest topology for which is continuous for every simple closed curve . Scaling the measures we obtain an action of on and , and we denote the quotient spaces and , respectively.
A particularly important subspace is the space of filling laminations which we denote . These are the measure classes of measured laminations for which all complementary regions of its support are disks (for , there is also a single punctured disk). The quotient of by forgetting the measures will be denoted and is the space of ending laminations. For notational simplicity, we will denote the element of associated to the measure class of in by its support .
Train tracks provide another useful tool for describing measured laminations. See [thurstonnotes] and [harer-penner] for a detailed discussion of train tracks and their relation to laminations. We recall some of the most relevant information.
A lamination is carried by a train track if there is a map homotopic to the identity with so that for every leaf of the restriction of to is an immersion. If is a measured lamination carried by a train track , then the transverse measure defines weights on the branches of satisfying the switch condition—the sum of the weights on the incoming branches equals the sum on the outgoing branches. Conversely, any assignment of nonnegative weights to the branches of a train track satisfying the switch condition uniquely determines an element of . Given a train track carrying , we write to denote the train track together with the weights defined by .
Suppose that are all carried by the train track . Then if and only if the weights on each branch of defined by converge to those defined by .
This is an immediate consequence of [harer-penner, Theorem 2.7.4]. ∎
There is a well-known construction of train tracks carrying a given lamination which will be useful for us. For a careful discussion, see [harer-penner, Theorem 1.6.5], or [brock-length-cont, Section 4]. Starting with a geodesic lamination one chooses very small and constructs a foliation, transverse to , of the –neighborhood . The leaves of this foliation are arcs called ties. Taking the quotient by collapsing each tie to a point produces a train track on ; see Figure 1.
We can view as being built from finitely many rectangles, each foliated by ties, glued together along arcs of ties in the boundary of the rectangle. In the collapse each rectangle projects to a branch of . When is trivalent we may assume that is contained in , transverse to the foliation by ties, and the branch is contained in the rectangle .
Suppose now that is any measured lamination with , and transverse to the ties. If is a rectangle and a tie in , then the weight on the branch , defined by , is given by ; see Figure 1.
1.2.2 Mapping class groups.
Recall that we have fixed a hyperbolic structure on as well as a locally isometric universal covering . We also have a basepoint determining an isomorphism from , the covering group of , to , the group of homotopy classes of based loops. All of this is considered fixed for the remainder of the paper.
The mapping class group of is the group , where is the group of orientation preserving diffeomorphisms of . We define to be , where is the group of orientation preserving diffeomorphisms of that fix .
The evaluation map
given by defines a locally trivial principal fiber bundle
A theorem of Earle and Eells [earleeells] says that , the component containing the identity, is contractible. So the long exact sequence of a fibration gives rise to Birman’s exact sequence [birmansequence, birmanbook]
We elaborate on the injection in Birman’s exact sequence. Let
The long exact sequence of homotopy groups identifies . This isomorphism is induced by a homomorphism
given by where , , is an isotopy from to , and is the based homotopy class of , . To see that this is a homomorphism, suppose and and are paths from and respectively to . Write and . There is a path from to given as
Then is the path obtained by first traversing then , while and . So, , and is the required homomorphism.
Given , we will write for a loop (or the homotopy class) representing . Similarly, we will let denote the mapping class (or a representative homeomorphism) determined by . When convenient, we will simply identify with a subgroup of .
1.2.3 Curve complexes.
A closed curve in is essential if it is homotopically nontrivial in . We will refer to a closed curve in simply as a closed curve in , and will say it is essential if it is homotopically nontrivial and nonperipheral in . Essential simple closed curves in are isotopic if and only if they are isotopic in .
Let and denote the curve complexes of and , respectively; see [harvey] and [masur-minsky]. These are geodesic metric spaces obtained by isometrically gluing regular Euclidean simplices with all edge lengths equal to one. The following is proven in [masur-minsky].
Theorem 1.5 (Masur-Minsky).
The spaces and are -hyperbolic for some .
We will refer to a simplex or and not distinguish between this simplex and the isotopy class of multicurve it determines. Any simple closed curve in can be viewed as a curve in which we denote . This gives a well-defined “forgetful” map
which is simplicial.
Given a multicurve , unless otherwise stated, we assume that is realized by its geodesic representative in . Associated to there is an action of on a tree , namely, the Bass–Serre tree for the splitting of determined by . We will make use of the following theorem of [kls].
Theorem 1.6 (Kent-Leininger-Schleimer).
The fiber of over a point is –equivariantly homeomorphic to the tree , where is the unique simplex containing in its interior.∎
1.2.4 Measured laminations and the curve complex.
The curve complex naturally injects into sending a simplex to the simplex of measures supported on . We denote the image subspace . We note that this bijection is not continuous in either direction. We will use the same notation for a point of and its image in .
In [klarreich-el] Klarreich proved that . Therefore, if we define
then there is a natural surjective map
extending . The following is a consequence of Klarreich’s work [klarreich-el], stated using our terminology.
Proposition 1.7 (Klarreich).
The natural map is continuous at every point of . Moreover, a sequence converges to if and only if every accumulation point of in has as its support.
Theorem 1.4 of [klarreich-el] implies that if a sequence converges in to , then every accumulation point of in has as its support. We need only verify that if and every accumulation point in of a sequence has then converges to in .
To see this, let be any sequence in the Teichmüller space for which is the shortest curve in . In particular is uniformly bounded. Since every accumulation point of is in , it follows that exits every compact set and so accumulates only on in the Thurston compactification of . Moreover, if is any accumulation point of in , then , and so since is filling.
Now according to Theorem 1.1 of [klarreich-el], the map
sending to any shortest curve in extends to
continuously at every point of . It follows that
in and we are done. ∎
1.2.5 Cannon–Thurston maps.
Fix and hyperbolic metric spaces, a continuous map, and a subset of the Gromov boundary. A –Cannon–Thurston map is a continuous extension of . That is, . We will simply call a Cannon–Thurston map when the set is clear from the context. We sometimes refer to the restriction as a Cannon–Thurston map.
This definition is more general than that in [mitra-ct] in the sense that here we require only to be continuous, whereas in [mitra-ct] it was demanded that be an embedding. Also, we do not require to be defined on all of .
To prove the existence of such a Cannon–Thurston map, we shall use the following obvious criterion:
Fix and hyperbolic metric spaces, a continuous map and a subset. Fix a basepoint . Then there is a –Cannon–Thurston map if and only if for every there is a neighborhood basis of and a collection of uniformly quasiconvex sets with and as . Moreover,
determines uniquely. ∎
2 Point position.
We now describe in more detail the map
as promised in the introduction, and explain how this can be extended continuously to .
2.1 A bundle over .
The bundle determining the Birman exact sequence has a subbundle obtained by restricting to :