Skinning maps are finite-to-one

Skinning maps are finite-to-one

David Dumas Department of Mathematics, Statistics, and Computer Science
University of Illinois at Chicago
ddumas@math.uic.edu
June 25, 2015 (revised). March 1, 2012 (original).
Work partially supported by the National Science Foundation.

1. Introduction

Skinning maps were introduced by William Thurston in the proof of the Geometrization Theorem for Haken -manifolds (see [Ota]). At a key step in the proof one has a compact -manifold with nonempty boundary whose interior admits a hyperbolic structure. The interplay between deformations of the hyperbolic structure and the topology of and determines a holomorphic map of Teichmüller spaces, the skinning map

where is the union of the non-torus boundary components and denotes the boundary with the opposite orientation. The problem of finding a hyperbolic structure on a related closed manifold is solved by showing that the composition of with a certain isometry has a fixed point.

Thurston’s original approach to the fixed point problem involved extending the skinning map of an acylindrical manifold to a continuous map defined on the compact space of hyperbolic structures on . (A proof of this Bounded Image Theorem can be found in [Ken, Sec. 9].) McMullen provided an alternate approach based on an analytic study of the differential of the skinning map [McM1] [McM2]. In each case there are additional complications when has essential cylinders.

More recently, Kent studied the diameter of the image of the skinning map (in cases when it is finite), producing examples where this diameter is arbitrarily large or small and relating the diameter to hyperbolic volume and the depth of an embedded collar around the boundary [Ken]. However, beyond the contraction and boundedness properties used to solve the fixed point problem—and the result of Kent and the author that skinning maps are never constant [DK3]—little is known about skinning maps in general.

Our main theorem concerns the fibers of skinning maps:

Theorem A.

Skinning maps are finite-to-one. That is, let be a compact oriented -manifold whose boundary is nonempty and not a union of tori. Suppose that the interior of admits a complete hyperbolic structure without accidental parabolics, so that has an associated skinning map . Then for each , the preimage is finite.

As a holomorphic map with finite fibers, it follows from Theorem A that skinning maps are open (answering a question in [DK3]) and locally biholomorphic away from the analytic hypersurface defined by the vanishing of the Jacobian determinant. Thus our results give strong nondegeneracy properties for all skinning maps.

Our proof of Theorem A does not bound the size of the finite set ; instead, we show that each fiber of the skinning map is both compact and discrete. In particular it is not clear if the number of preimages of a point is uniformly bounded over .

In case is acylindrical, we also show that Thurston’s extension of the skinning map to is finite-to-one; this result appears as Theorem 10.1 below.

The intersection problem

To study the fibers of the skinning map we translate the problem to one of intersections of certain subvarieties of the -character variety of . The same reduction to an intersection problem is used in [DK3]. The relevant subvarieties are:

  • The extension variety , which is the smallest closed algebraic subvariety containing the characters of all homomorphisms that can be extended to .

  • The holonomy variety , which is the analytic subvariety consisting of characters of the -lifts of holonomy representations of -structures on compatible with the complex structure .

Precise definitions of these objects are provided in Section 6, with additional details of the disconnected boundary case in Section 9.

The main theorem is derived from the following result about the intersections of holonomy and extension varieties, the proof of which occupies most of the paper.

Theorem B (Intersection Theorem).

Let be an oriented -manifold with nonempty boundary that is not a union of tori. Let be a marked Riemann surface structure on . Then the intersection is a discrete subset of the character variety.

This theorem applies in a more general setting than the specific intersection problem arising from skinning maps. For example, while skinning maps are defined for manifolds with incompressible boundary, such a hypothesis is not needed in Theorem B.

While the theorem above involves an oriented manifold , the set is independent of the orientation. Thus we also obtain discreteness of intersections where is a Riemann surface structure on that induces an orientation opposite that of the boundary orientation of .

Steps to the intersection theorem

Our study of is based on the parameterization of the irreducible components of by the vector space of holomorphic quadratic differentials; this parameterization is the holonomy map of -structures, denoted by “”. The overall strategy is to show that the preimage of , i.e. the set

is a complex analytic subvariety of that is subject to certain constraints on its behavior at infinity, and ultimately to show that only a discrete set can satisfy these.

We now sketch the main steps of the argument and state some intermediate results of independent interest. In this sketch we restrict attention to the case of a -manifold with connected boundary . Let be a marked Riemann surface structure on the boundary.

Step 1. Construction of an isotropic cone in the space of measured foliations.

The defining property of this cone in is that it determines which quadratic differentials have dual trees that admit “nice” equivariant maps into trees on which acts by isometries. Because of the way this cone is used in a later step of the argument, here we must consider actions of not only -trees but also on -trees, where is a more general ordered abelian group. And while an isometric embedding of into an -tree on which acts is the prototypical example of a nice map, we must also consider the straight maps defined in [D] and certain partially-defined maps arising from non-isometric trees with the same length function.

In the case of straight maps, we show:

Theorem C.

There is an isotropic piecewise linear cone with the following property: If acts on a -tree by isometries, and if is a quadratic differential whose dual tree admits an equivariant straight map , then the horizontal foliation of lies in .

This result is stated precisely in Theorem 4.1 below, and Theorem 4.4 presents a further refinement that is used in the proof of the main theorem.

Step 2. Limit points of have foliations in the isotropic cone.

In [D] we analyze the large-scale behavior of the holonomy map, showing that straight maps arise naturally when comparing limits in the Morgan-Shalen compactification—which are represented by actions of on -trees—to the dual trees of limit quadratic differentials. The main result can be summarized as follows:

Theorem.

If a divergent sequence in can be rescaled to have limit , then any Morgan-Shalen limit of the associated holonomy representations corresponds to an -tree that admits an equivariant straight map .

Here rescaling of the divergent sequence uses the action of on . The precise limit result we use is stated in Theorem 6.4, and other related results and discussion can be found in [D].

When we restrict attention to the subset , the associated holonomy representations lie in and so they arise as compositions of -representations with the map induced by the inclusion of as the boundary of . (Strictly speaking, this describes a Zariski open subset of .) We think of these as representations that “extend” from to the larger group .

Note that a priori the passage from a -representation to a -representation could radically change the geometry of the associated action on , as measured for example by the diameter of the orbit of a finite generating set. This possibility, combined with the need to keep track of the limiting - and -dynamics simultaneously, requires us to consider -trees more general than -trees.

Using the valuation constructions of Morgan-Shalen, we show that there is a similar extension property for the trees obtained as limits points of , or more precisely, for their length functions (Theorem 6.3). In the generic case of a non-abelian action, the combination of this construction with the holonomy limit theorem above gives an -tree (where is given the lexicographical order) on which acts by isometries and a straight map

where is the rescaled limit of a divergent sequence in . These satisfy the hypotheses of Theorem C, so the horizontal foliation of lies in . Limit points of that correspond to abelian actions introduce minor additional complications that are handled by Theorem 4.4.

Step 3. The foliation map is symplectic.

Hubbard and Masur showed that the foliation map is a homeomorphism. In order to use the isotropic cone to understand the set , we analyze the relation between the foliation map, the complex structure of , and the symplectic structure of .

We introduce a natural Kähler structure on corresponding to the Weil-Petersson-type hermitian pairing

Here we have a base point and the quadratic differentials are considered as tangent vectors. This integral pairing is not smooth, nor even well-defined for all tangent vectors, due to singularities of the integrand coming from higher-order zeros of . However we show that the pairing does give a well-defined Kähler structure relative to a stratification of .

We show that the underlying symplectic structure of this Kähler metric is the one pulled back from by the foliation map:

Theorem D.

For any , the map is a real-analytic stratified symplectomorphism, where is given the symplectic structure coming from the pairing and where has the Thurston symplectic form.

The lack of a smooth structure on means that the regularity aspect of this result must be interpreted carefully. We show that for any point there is a neighborhood in its stratum and a train track chart containing in which the local expression of the foliation map is real-analytic and symplectic. The details are given in Theorem 5.8.

Step 4. Analytic sets with totally real limits are discrete.

In a Kähler manifold, an isotropic submanifold is totally real. While the piecewise linear cone is not globally a manifold, Theorem D allows us to describe locally in a stratum of in terms of totally real, real-analytic submanifolds. Since limit points of correspond to elements of , this gives a kind of “totally real” constraint on the behavior of at infinity.

To formulate this constraint we consider the set of points in the unit sphere of that can be obtained as -rescaled limits of divergent sequences in . Projecting this set through the Hopf fibration we obtain the set of boundary points of in the complex projective compactification of . Using the results of steps 1–3 we show (in Theorem 7.2) that:

  1. In a neighborhood of some point, is contained in a totally real manifold, and

  2. The intersection of with a fiber of the Hopf map has empty interior.

Using extension and parameterization results from analytic geometry it is not hard to show that among analytic subvarieties of , only a discrete subset of can have both of these properties. Condition (i) forces any analytic curve in to extend to an analytic curve in a neighborhood of some boundary point . Within this extension there is a generically a circle of directions in which to approach the boundary point, some arc of which is realized by the original curve. Analyzing the correspondence between this circle and the Hopf fiber over , one finds that contains an open arc of this fiber, violating condition (ii).

This contradiction shows that contains no analytic curves, making it a discrete set. The intersection theorem follows.

Applications and complements

The construction of the isotropic cone in Theorem C was inspired by the work of Floyd on the space of boundary curves of incompressible, -incompressible surfaces in -manifolds [Flo]. Indeed, in the incompressible boundary case, lifting such a surface to the universal cover and considering dual trees in the boundary and in the -manifold gives rise to an isometric embedding of -trees. Using Theorem C we recover Floyd’s result in this case. This connection is explained in more detail in Section 4.5, where we also note that the same “cancellation” phenomenon is at the core of both arguments.

Since Theorem D provides an interpretation of Thurston’s symplectic form in terms of Riemannian and Kähler geometry of a (stratified) smooth manifold, we hope that it might allow new tools to be applied to problems involving the space of measured foliations. In Section 5.7 we describe work of Mirzakhani in this direction, where it is shown that a certain function connected to -orbit counting problems in Teichmüller space is constant.

As a possible extension of Theorem B one might ask whether and always intersect transversely, or equivalently, whether skinning maps are always immersions. In the slightly more general setting of manifolds with rank- cusps a negative answer was recently given by Gaster [Gas]. In addition to Gaster’s example, numerical experiments conducted jointly with Richard Kent suggest critical points for some other manifolds with rank- cusps [DK1].

Nevertheless it would be interesting to understand the discreteness of the intersection through local or differential properties rather than the compactifications and asymptotic arguments used here. The availability of rich geometric structure on the character variety and its compatibility with the subvarieties in question offers some hope in this direction; for example, both and are Lagrangian with respect to the complex symplectic structure of the character variety (see [Kaw] [Lab, Sec. 7.2] [KS, Sec. 1.7]).

Structure of the paper

Section 2 recalls some definitions and basic results related to -trees, measured foliations, and Teichmüller theory.

Sections 37 contain the proofs of the main theorems in the case of a -manifold with connected boundary. Working in this setting avoids some cumbersome notation and other issues related to disconnected spaces, while all essential features of the argument are present. Sections 34 are devoted to the isotropic cone construction, Section 5 introduces the stratified Kähler structure, and Sections 67 combine these with the results of [D] to prove Theorem A in the connected boundary case.

In Section 9 we adapt the definitions and results of the previous sections as necessary to handle a manifold with disconnected boundary, possibly including torus components, completing the proof of Theorem A.

Finally, in Section 10 we prove an analogue of Theorem A for the extended skinning map of an acylindrical -manifold.

Acknowledgments

A collaboration with Richard Kent (in [DK2] [DK3]) provided essential inspiration in the early stages of this project. The author is grateful for this and for helpful conversations with Athanase Papadopoulos, Kasra Rafi, and Peter Shalen. The author also thanks Maryam Mirzakhani for allowing the inclusion of Theorem 5.10 and the anonymous referees for many helpful comments and suggestions.

2. Preliminaries

2.1. Ordered abelian groups

An ordered abelian group is a pair consisting of an abelian group and a translation-invariant total order on . We often consider the order to be implicit and denote an ordered abelian group by alone. Note that an order on also induces an order on any subgroup of .

The positive subset of an ordered abelian group is the set . If is nonzero, then exactly one of lies in , and we denote this element by .

If , we say that is infinitely larger than if for all . If neither of is infinitely larger than the other, then and are archimedean equivalent. When the set of archimedean equivalence classes of nonzero elements is finite, the number of such classes is the rank of . In what follows we consider only ordered abelian groups of finite rank.

A subgroup is convex if whenever , , and we have . The convex subgroups of a given group are ordered by inclusion. A convex subgroup is a union of archimedean equivalence classes and is uniquely determined by the largest archimedean equivalence class that it contains (which exists, since the rank is finite). In this way the convex subgroups of a given ordered abelian group are in one-to-one order-preserving correspondence with its archimedean equivalence classes.

2.2. Embeddings and left inverses

If we equip with the lexicographical order, then the inclusion as one of the factors is order-preserving. This inclusion has a left inverse given by projecting onto the factor. This projection is of course a homomorphism but it is not order-preserving.

Similarly, the following lemma shows that an order-preserving embedding of into any ordered abelian group of finite rank has a left inverse; this is used in Section 3.6.

Lemma 2.1.

If is an ordered abelian group of finite rank and is an order-preserving homomorphism, then has a left inverse. That is, there is a homomorphism such that .

To construct a left inverse we use the following structural result for ordered abelian groups; part (i) is the Hahn embedding theorem (see e.g. [Gra] [KK, Sec. II.2]):

Theorem 2.2.

Let be an ordered abelian group of rank .

  1. There exists an order-preserving embedding where is given the lexicographical order.

  2. If , the order-preserving embedding is unique up to multiplication by a positive constant.

Proof of Lemma 2.1..

By Theorem 2.2, it suffices to consider the case of an order-preserving embedding where has the lexicographical order. Since the only order-preserving self-homomorphisms of the additive group are multiplication by positive constants, it is enough to find a homomorphism such that is order-preserving and fixes a point.

Let , and write . Since in , the first nonzero element of the tuple is positive. Let be the index of this element, i.e. .

We claim that for any , the image has the form . If not, then after possibly replacing by we have such that is infinitely larger than . The existence of a positive integer such that shows that this contradicts the order-preserving property of .

Similarly, we find that if then satisfies : The order-preserving property of implies that , so the only possibility to rule out is . But if then is infinitely larger than , yet there is a positive integer such that , a contradiction.

Now define by

This is a homomorphism satisfying , and the properties of derived above show that is order-preserving, as desired. ∎

We now consider the properties of embeddings , such as those provided by Theorem 2.2, with respect to convex subgroups. First, a proper convex subgroup maps into :

Lemma 2.3.

Let be an order-preserving embedding, where has rank . If is a proper convex subgroup, then .

Proof.

Since , the convex subgroup does not contain the largest Archimedean equivalence class of . Thus there exists a positive element such that for all .

Suppose that there exists such that with . Then we have for some . This contradicts the order-preserving property of , so no such exists and has the desired form. ∎

Building on the previous result, the following lemma shows that in some cases the embeddings given by Hahn’s theorem behave functorially with respect to rank- subgroups. This result is used in Section 6.4.

Lemma 2.4.

Let be an ordered abelian group of finite rank and a subgroup contained in the minimal nontrivial convex subgroup of . Then there is a commutative diagram of order-preserving embeddings

where and is the rank of .

Proof.

Let be the convex subgroups of . We can assume that since all other cases are handled by restricting the maps from this one.

We are given the inclusion and the Hahn embedding theorem provides an order-preserving embedding . Applying Lemma 2.3 to each step in the chain of convex subgroups of , we find that for all we have

and the induced map is order-preserving. Since by construction, these maps complete the commutative diagram. ∎

2.3. -metric spaces and -trees

We refer to the book [Chi3] for general background on -metric spaces and -trees. Here we recall the essential definitions and fix notation.

As before let denote an ordered abelian group. A -metric space is a pair where is a set and is a function which satisfies the usual axioms for the distance function of a metric space. In particular an -metric space (where has the standard order) is the usual notion of a metric space.

An isometric embedding of one -metric space into another is defined in the natural way. Generalizing this, let be a -metric space and a -metric space. An isometric embedding of into is a pair consisting of a map and an order-preserving homomorphism such that

More generally we say is an isometric embedding if there exists an order-preserving homomorphism such that the pair satisfy this condition.

An ordered abelian group is an example of a -metric space, with metric . An isometric embedding of the subspace into a -metric space is a segment. A -metric space is geodesic if any pair of points can be joined by a segment.

A -tree is a -metric space satisfying three conditions:

  • is geodesic,

  • If two segments in share an endpoint but have no other intersection points, then their union is a segment, and

  • If two segments in share an endpoint, then their intersection is a segment (or a point).

The notion of -hyperbolicity for metric spaces generalizes naturally to -metric spaces, where now , . In terms of this generalization, any -tree is -hyperbolic. (The converse holds under mild additional assumptions on the space.) The -hyperbolicity condition has various equivalent characterizations, but the one we will use in the sequel is the following condition on -tuples of points:

Lemma 2.5 (-hyperbolicity of -trees).

If is a -tree, then for all we have

For a proof and further discussion see [Chi3, Lem. 1.2.6 and Lem. 2.1.6]. By permuting a given -tuple and considering the inequality of this lemma, we obtain the following corollary (see [Chi3, p. 35]):

Lemma 2.6 (Four points in a -tree).

Let be a -tree and . Then among the three sums

two are equal, and these two are not less than the third. ∎

Given a -tree, there are natural constructions that associate trees to certain subgroups or extensions of ; in what follows we require two such operations. First, let be a -tree and a convex subgroup. For any we can consider the subset . Then the restriction of to takes values in , and this gives the structure of a -tree [MoSh1, Prop. II.1.14].

Second, suppose that is an order-preserving homomorphism and that is a -tree. Then there is a natural base change construction that produces a -tree and an isometric embedding with respect to (see [Chi3, Thm. 4.7] for details). Roughly speaking, if one views as a union of segments, each identified with some interval , then is obtained by replacing each such segment with .

2.4. Group actions on -trees and length functions

Every isometry of a -tree is either elliptic, hyperbolic, or an inversion; see [Chi3, Sec. 3.1] for a detailed discussion of this classification. Elliptic isometries are those with fixed points, while hyperbolic isometries have an invariant axis (identified with a subgroup of ) on which they act as a translation. An inversion is an isometry that has no fixed point but which induces an elliptic isometry after an index- base change; permitting such base change allows us to make the standing assumption that isometric group actions on -trees that we consider are without inversions.

The translation length of an isometry of a -tree is defined as

Note that . It can be shown that the translation length is also given by .

When a group acts on a -tree by isometries, taking the translation length of each element of defines a function , the translation length function (or briefly, the length function) of the action.

When the translation length function takes values in a convex subgroup, one can extract a subtree whose distance function takes values in the same subgroup:

Lemma 2.7.

Let act on a -tree with length function . If is a convex subgroup and then there is a -tree that is invariant under and such that is also the length function of the induced action of on .

This lemma is implicit in the proof of Theorem 3.7 in [Mor1], which uses the structure theory of actions developed in [MoSh1]. For the convenience of the reader, we reproduce the argument here.

Proof.

Because is convex, there is an induced order on the quotient group . Define an equivalence relation on where if . Then the quotient is a -tree, and each fiber of the projection is a -tree. The action of on induces an action on whose length function is the composition of with the map , which is identically zero since . It follows that the action of on has a global fixed point [MoSh1, Prop. II.2.15], and thus acts on the fiber of over , which is a -tree . By [MoSh1, Prop. II.2.12] the length function of the action of on is . ∎

2.5. Measured foliations and train tracks

Let denote the space of measured foliations on a compact oriented surface of genus . Then is a piecewise linear manifold which is homeomorphic to . A point is an equivalence class up to Whitehead moves of a singular foliation on equipped with a transverse measure of full support. For detailed discussion of measured foliations and of the space see [FLP].

Piecewise linear charts of correspond to sets of measured foliations that are carried by a train track; we will now discuss the construction of these charts in some detail. While this material is well-known to experts, most standard references that discuss train track charts use the equivalent language of measured laminations, whereas our primary interest in foliations arising from quadratic differentials makes the direct consideration of foliations preferable. Additional details of the carrying construction from this perspective can be found in [Pap] [Mos].

A train track on is a embedded graph in which all edges incident on a given vertex share a tangent line at that point. Vertices of the train track are called switches and its edges are branches. We consider only generic train tracks in which each switch is has three incident edges, two incoming and one outgoing, such that the union of any incoming edge and the outgoing edge forms a curve.

The complement of a train track is a finite union of subsurfaces with cusps on their boundaries. In order to give a piecewise linear chart of , each complementary disk must have at least three cusps on its boundary and each complementary annulus must have at least one cusp. We will always require this of the train tracks we consider.

If is such a train track, let denote the vector space of real-valued functions on its set of edges that obey the relation for any switch with incoming edges and outgoing edge . This switch relation ensures that determines a signed transverse measure, or weight, on the embedded train track. Within there is the finite-sided convex cone of nonnegative weight functions, denoted . It is this cone which forms a chart for .

A measured foliation is carried by the train track if the foliation can be cut open near singularities and along saddle connections and then moved by an isotopy so that all of the leaves lie in an arbitrarily small open neighborhood of and are nearly parallel to its branches, as depicted in Figure 1. Here “cutting open” refers to the procedure of replacing a union of leaf segments and saddle connections coming out of singularities with a subsurface with cusps on its boundary. The result of cutting open a measured foliation is a partial measured foliation in which there are non-foliated regions, each of which has a union of leaf segments of the original foliation as a spine.

Figure 1. Local picture of carrying: Cutting a measured foliation along saddle connections and singular leaf segments and inflating these to cusped subsurfaces allows an isotopy into a small neighborhood of the train track.

A measured foliation determines a weight on any train track that carries it, as follows: For each branch choose a tie , a short closed arc that crosses transversely at an interior point and which is otherwise disjoint from . Now select an open neighborhood of of that intersects each tie in a connected open interval. Let be a partial measured foliation associated to that has been isotoped to lie in and to be transverse to each tie. Note that each tie then has endpoints in non-foliated regions of , since the endpoints of lie outside . Let be the total transverse measure of with respect to .

The resulting function lies in and regarding this construction as a map gives a one-to-one correspondence between equivalence classes of measured foliations that are carried by and the convex cone . Furthermore, these cones in train track weight spaces form the charts of a piecewise linear atlas on .

2.6. The symplectic structure of

The orientation of induces a natural antisymmetric bilinear map on the space of weights on a train track . This Thurston form can be defined as follows (compare [PH, Sec. 3.2] [Bon, Sec. 3]): For each switch , let be its incoming edges and its outgoing edge, where are ordered so that intersecting with a small circle around gives a positively oriented triple. Then we define

(2.1)

If defines a chart of then this form is nondegenerate, and the induced symplectic forms on train track charts are compatible. This gives the structure of a piecewise linear symplectic manifold.

The Thurston form can also be interpreted as a homological intersection number. If can be consistently oriented then each weight function defines a -cycle , where denotes the singular -simplex defined by the oriented edge of . In terms of these cycles, we have . For a general train track, there is a branched double cover (with branching locus disjoint from ) such that the preimage is orientable. Lifting weight functions we obtain cycles such that

(2.2)

Note that if denotes the opposite orientation of the surface , then there is a natural identification between measured foliation spaces , but this identification does not respect the Thurston symplectic forms. Rather, in corresponding local charts we have .

2.7. Dual trees

Let be a measured foliation on and its lift to the universal cover . There is a pseudo-metric on where is the minimum -transverse measure of a path connecting to . The quotient metric space is an -tree (see [Bow] [MS2]). The action of on by deck transformations determines an action on by isometries. The dual tree of the zero foliation is a point.

This pseudo-metric construction can be applied to the partial measured foliation obtained by cutting open along leaf segments from singularities, as when is carried by a train track . The result is a tree naturally isometric to , which we identify with from now on. Non-foliated regions of are collapsed to points in this quotient, so in particular each complementary region of the lift has a well-defined image point .

Similarly, the lift of a tie of to the universal cover projects to a geodesic segment in of length ; the endpoints of this segment are the projections of the two complementary regions adjacent to the lift of the edge .

To summarize, we have the following relation between carrying and dual trees:

Proposition 2.8.

Let be a measured foliation carried by the train track with associated weight function . Let denote the lift of to the universal cover. If are complementary regions of that are adjacent along an edge of , and if are the associated points in , then we have

where is the distance function of . ∎

2.8. Teichmüller space and quadratic differentials

Let be the Teichmüller space of marked isomorphism classes of complex structures on compatible with its orientation. For any we denote by the set of holomorphic quadratic differentials on , a complex vector space of dimension .

Associated to we have the following structures on :

  • The flat metric , which has cone singularities at the zeros of ,

  • The measured foliation whose leaves integrate the distribution , with transverse measure given by , and

  • The dual tree and the -equivariant map that collapses leaves of the lifted foliation to points of .

The dual tree construction is homogeneous with respect to the action of on in the sense that for any we have

where the right hand side represents the metric space obtained from by multiplying its distance function by .

Note that the point is a degenerate case in which there is no corresponding flat metric, and by convention is the empty measured foliation whose dual tree is a point.

We say that a -geodesic is nonsingular if its interior is disjoint from the zeros of . Choosing a local coordinate in which (a natural coordinate for ), a nonsingular -geodesic segment becomes a line segment in the -plane. The vertical variation of this segment in (i.e.  , where are the endpoints) is the height of .

Note that leaves of the foliation are geodesics of the -metric. Conversely, a nonsingular -geodesic is either a leaf of or it is transverse to . In the latter case, the height of is equal to its -transverse measure, and any lift of to projects homeomorphically to a geodesic segment in of length .

3. The isotropic cone: Embeddings

The goal of this section is to establish the following result relating -manifold actions on -trees and measured foliations:

Theorem 3.1.

Let be a -manifold with connected boundary . There exists an isotropic piecewise linear cone with the following property: If is a measured foliation on whose dual tree embeds isometrically and -equivariantly into a -tree equipped with an isometric action of , then .

Here a piecewise linear cone refers to a closed -invariant subset of whose intersection with any train track chart is a finite union of finite-sided convex cones in linear subspaces of . Such a cone is isotropic if the linear spaces can be chosen to be isotropic with respect to the Thurston symplectic form. Since transition maps between these charts are piecewise linear and symplectic, it suffices to check these conditions in any covering of the set by train track charts.

The first step in the proof of Theorem 3.1 will be to use the foliation and embedding to construct a weight function on the -skeleton of a triangulation of . We begin with some generalities about train tracks, triangulations, and weight functions.

3.1. Complexes and weight functions

Let be a simplicial complex, and let denote the its set of -simplices. Given an abelian group , define the space of -valued weights on as the -module consisting of functions ; we denote this space by

The case will be of primary interest and so we abbreviate . If and we say that is the weight of with respect to .

A homomorphism of groups induces a homomorphism of weight spaces , and an inclusion of simplicial complexes induces a -linear restriction map . These functorial operations commute, i.e. .

A map from to a -metric space induces a -valued weight on in a natural way: For each we define the weight to be the distance between the -images of its endpoints. We write for the weight function defined in this way.

This construction has a natural extension to equivariant maps on regular covers. Suppose that are simplicial complexes such that there is a regular covering which is also a simplicial map. Suppose also that the deck group of this covering acts isometrically on a -metric space . Then if is a -equivariant map, the resulting weight function is also -invariant, hence it descends to a weight function on the base of the covering.

3.2. Extending triangulations and maps

We will now consider the space of weight functions as defined above in cases where the complex is a triangulation of a - or -manifold, possibly with boundary.

For example, let be a maximal, generic train track on a surface . Then there is a triangulation of dual to the embedded trivalent graph underlying . Each triangle of contains one switch of the train track, each edge of corresponds to an edge of , and each vertex of corresponds to a complementary region of . The correspondence between edges gives a natural (linear) embedding

Now suppose that is a measured foliation on that is carried by the train track , so we consider the class as an element of . Let denote the lift of to the universal cover . As in Section 2.7, the carrying relationship between and gives a map from complementary regions of to the dual tree . In terms of the dual triangulation , this is a map

and it is immediate from the definitions above and Proposition 2.8 that the associated weight function is the image of under the embedding .

Let us further assume that, as in the hypotheses of Theorem 3.1, there is an equivariant isometric embedding of into a -tree equipped with an action of , where is a -manifold with . Using this embedding we can consider the map constructed above as taking values in . We extend the triangulation of to a triangulation of , and the map to a -equivariant map

Such an extension can be constructed by choosing a fundamental domain for the -action on and mapping elements of to arbitrary points in . Combining these with the values of on and the free action of gives a unique equivariant extension to all of .

Associated to the map is the weight function . By construction, its values on the edges of are the coordinates of relative to the train track chart of , considered as elements of using the embedding that is implicit in the isometric map .

We record the constructions of this paragraph in the following proposition.

Proposition 3.2.

Let be a measured foliation on carried by a maximal generic train track , and let be a triangulation of extending the dual triangulation of . Suppose that there exists a -tree equipped with an isometric action of and an equivariant isometric embedding

relative to an order-preserving embedding . Then there exists a weight function with the following properties:

  1. The weight is induced by an equivariant map

  2. The restriction of to is the image of under the natural inclusion . ∎

3.3. Triangle forms and the symplectic structure

In [PP], Penner and Papadopoulos relate the Thurston symplectic structure of for a punctured surface to a certain linear -form on the space of weights on a “null-gon track” dual to an ideal triangulation of . In this section we discuss a related construction for a triangulation of a compact surface dual to a train track.

Let be an oriented triangle with edges (cyclically ordered according to the orientation). Let denote the corresponding linear functionals on , which evaluate a function on the given edge. We call the alternating -form

the triangle form associated with . Note that if represents the triangle with the opposite orientation, then .

Given a triangulation of a compact oriented -manifold , the triangle form corresponding to any (with its induced orientation) is naturally an element of . Denote the sum of these by

(3.1)

This -form on is an analogue of the Thurston symplectic form, in a manner made precise by the following:

Lemma 3.3.

If is a maximal generic train track on with dual triangulation , then the Thurston form on is the pullback of by the natural inclusion .

Proof.

By direct calculation: Both the Thurston form and are given as a sum of -forms, one for each triangle of (equivalently, switch of ). The image of in is cut out by imposing a switch condition for each triangle , which for an appropriate labeling of the sides as can be written as

On the subspace defined by this constraint the triangle form pulls back to

which is the associated summand in the Thurston form (2.1). ∎

3.4. Tetrahedron forms

Figure 2. Labeled edges of an oriented -simplex and an oriented -simplex . The -forms and are defined in terms of these labels.

Let be an oriented -simplex. Call a pair of edges of opposite if they do not share a vertex. Label the edges of as so that the following conditions are satisfied:

  1. The pairs , , and are opposite.

  2. The ordering gives the oriented boundary of one of the faces of .

An example of such a labeling is shown in Figure 2.

Define the tetrahedron form as

Here we abbreviate and similarly for the other edges. It is easy to check that this -form does not depend on the labeling (as long as it satisfies the conditions above). As in the case of triangle forms, is naturally a -form on the space of weights for any oriented simplicial complex containing .

A simple calculation using the definition above gives the following:

Lemma 3.4.

The tetrahedron form is equal to the sum of the triangle forms of its oriented boundary faces, i.e.

Now consider a triangulation of an oriented -manifold with boundary , and let denote the induced triangulation of the boundary. Denote the sum of the tetrahedron forms by

In fact, due to cancellation in this sum, the -form defined above “lives” on the boundary:

Lemma 3.5.

The form is equal to the pullback of under the restriction map .

Proof.

By Lemma 3.4 we have

In this sum, each interior triangle of appears twice (once with each orientation) and so these terms cancel. The remaining terms are the elements of with the boundary orientation, so we are left with the sum (3.1) defining . The result is the pullback of by the restriction map because in the formula above, we are considering as an element of rather than . ∎

3.5. The four-point condition

Given four points in a -tree, Lemma 2.6 implies that there is always a labeling of these points such that the distance function satisfies

(3.2)

We call this the weak four-point condition to distinguish it from the stronger four-point condition of Lemma 2.6 which also involves an inequality.

If we think of as labeling the vertices of a -simplex , then the pairwise distances give a weight function . Condition (3.2) is equivalent to the existence of opposite edge pairs such that

(3.3)

Given a simplicial complex , let denote the set of -valued weights such that in each -simplex of there exist opposite edge pairs so that (3.3) is satisfied.

The following basic properties of follow immediately from the definition of this set (and the relation between the four-point condition and -tuples in -trees):

Lemma 3.6.

  1. The set is a finite union of subspaces (i.e. -submodules) of ; each subspace corresponds to choosing opposite edge pairs in each of the -simplices of .

  2. If is a homomorphism, then we have .

  3. If is an equivariant map to a -tree, then

Ultimately, the isotropic condition in Theorem 3.1 arises from the following property of the set :

Lemma 3.7.

Let be an oriented -manifold and a triangulation. Then is a finite union of -isotropic subspaces of .

Proof.

Let be one of the subspaces comprising (as in Lemma 3.6.(i)). Then for each we have opposite edge pairs and such that (3.3) holds, or equivalently, on the subspace the equation

is satisfied. Substituting this into the definition of the tetrahedron form gives zero. Since is the sum of these forms, the subspace is isotropic. ∎

3.6. Construction of the isotropic cone

We now combine the results on triangulations, weight functions, and the symplectic structure of with the constructions of Proposition 3.2 to prove Theorem 3.1.

Proof of Theorem 3.1.

Let be a finite set of maximal, generic train tracks such that any measured foliation on is carried by one of them. For each , let be an extension of to a triangulation of .

Define

where