Hyperbolic structures from Sol on pseudo-Anosov mapping tori
The invariant measured foliations of a pseudo-Anosov homeomorphism induce a natural (singular) Sol structure on mapping tori of surfaces with pseudo-Anosov monodromy. We show that when the pseudo-Anosov has orientable foliations and does not have 1 as an eigenvalue of the induced cohomology action on the closed surface, then the Sol structure can be deformed to nearby cone hyperbolic structures, in the sense of projective structures. The cone angles can be chosen to be decreasing from multiples of .
Let be a surface of genus with punctures such that . Given a homeomorphism , we can define the mapping torus . The hyperbolization theorem by Thurston  states that is hyperbolic if and only if is pseudo-Anosov. A pseudo-Anosov homeomorphism has two transverse (possibly singular) foliations and with transverse measures and , respectively, and a constant such that preserves and and scales the measures by and . When is not closed, the map induces a pseudo-Anosov map on the closed surface of genus , where the punctures have been filled in. We will also call this map .
The measured foliations and endow with a singular Euclidean metric. The corresponding suspension flow on , expanding the leaves of by a factor of and contracting the leaves of by , has period , so that . One model for Sol geometry is to take with the metric , so the suspension flow can be viewed as an isometry of Sol translating the surface in the direction. The identification then defines a singular Sol structure on , with singular locus given by the orbits of the singular points and punctures of and .
In the case where is a punctured torus, Hodgson  studied how to deform representations of near a representation corresponding to a projection of the Sol structure. Sol space contains embedded hyperbolic planes, and the representations studied in  correspond to projecting the 3-manifold onto a hyperbolic plane inside Sol, resulting in a reducible representation that gives the structure of a transversely hyperbolic foliation (recall that a representation is irreducible if the only subspaces of that are invariant under are trivial). Further results about deforming reducible representations to irreducible representations can be found in , , and . Heusener, Porti, and Suárez  have also shown that hyperbolic structures can be regenerated from Sol, constructing a path of nearby hyperbolic structures that collapse onto a circle, and rescaling the metric as it collapses to obtain the Sol metric on .
In the case where is not the punctured torus, such a regeneration theorem is not known. In this paper, we utilize half-pipe (HP) geometry, studied by Danciger , to regenerate hyperbolic structures in a more general setting. In particular, we will prove the following result.
Let be a pseudo-Anosov homeomorphism whose stable and unstable foliations, and , are orientable and does not have 1 as an eigenvalue. Then, there exists a family of singular hyperbolic structures on , smooth on the complement of and with cone singularities along , that degenerate to a transversely hyperbolic foliation. The degeneration can be rescaled so that the path of rescaled structures limit to the singular Sol structure on , as projective structures. Moreover, the cone angles can be chosen to be decreasing.
The proof of Theorem 3 uses HP structures as an intermediate. We find a family of HP structures that collapse, such that rescaling the collapse in an appropriate manner yields Sol. The HP structures involved are built from a representation arising from projecting the 3-dimensional Sol space to one of its embedded hyperbolic planes, along with a first order deformation of the representation. The following is an application of the Ehresmann–Thurston principle:
Theorem (, Proposition 3.6).
Let be a compact -manifold with boundary and let be a thickening of so that is a collar neighborhood of . Suppose has an HP structure defined by the developing map , and holonomy representation . Let be either or and let be a family of representations compatible to first order at time with . Then we can construct a family of structures on with holonomy for short time.
As noted in , given an HP structure, the regeneration of a hyperbolic structure only requires that it exists on the level of representations. In Theorem 3, the conditions that the invariant foliations and are orientable and that does not have as an eigenvalue guarantee smoothness of the representation variety at , so we can find a nearby family of representations . We also do a simple computation to generalize Danciger’s notion of infinitesimal cone angle to multiple components. This allows us to adapt the HP machinery to show that there are singular hyperbolic structures near the HP structures, which are themselves collapsing to the Sol structure. We will then show that the singular locus can be controlled so that the family of structures are cone manifolds.
In Section 2, we present an overview of geometric structures and infinitesimal deformations. Section 3 describes the collapsed structure as a metabelian representation and establishes the notation used in the following section. Section 4 proves smoothness of the representation variety at the metabelian representation, which is used in Section 5 to show that we can find nearby three dimensional hyperbolic structures via HP geometry. Section 6 analyzes the behavior of the singular locus to show that the singularities can be realized as cone singularities, providing the final step to Theorem 3.
The author would like to thank Steven Kerckhoff for advising much of this work at Stanford University and Jeffrey Danciger for many useful conversations about HP structures. The author would also like to thank the reviewer for helpful comments and references.
An structure on a manifold is a collection of charts , where the are an open cover of and the transition maps are restrictions of elements .
In the context of this paper, we will take to be (a subset of) and to be (a subgroup of) , with and Sol being described as projective structures. An structure on defines a developing map that is equivariant under the holonomy representation .
A smooth family of -structures on a manifold can be described by a family of developing maps and corresponding holonomy representations . Two families of -structures and such that are equivalent if there exists a smooth family of elements in and a smooth family of diffeomorphisms defined on all but a neighborhood of such that where is the lift of , , and is the identity. Such a deformation is trivial if is equivalent to the family of structures . In this case, the holonomy representations also differ by conjugation by a smooth family , i.e. .
We will study deformations of geometric structures through their representations. Let be the variety of representations of into , be the character variety, where the quotient is the GIT quotient as acts by conjugation, and let be the space of -structures on up to the equivalence defined. The Ehresmann–Thurston principle states that locally, deformations of geometric structures can be studied by their holonomy representations (see  for a proof of the theorem).
The map taking an structure to its holonomy representation is a local homeomorphism on , where is the subset of consisting of stable orbits.
Given a smooth family of representations , we can study the infinitesimal change in at , as in . The derivative of the homomorphism condition yields
In order to normalize the derivative, we multiply on the right by to translate back to the identity element to obtain
The second term is defined to be
The Lie algebra of , denoted by , turns into a module, with acting via . Then a cocycle of with coefficients in twisted by is defined as a map , where and is the derivative evaluated at , such that the map satisfies the cocycle condition
The group of all maps satisfying the cocycle condition in Equation (1) is defined to be . Differentiating the triviality condition for representations yields the coboundary condition
for some . The set of cocycles satisfying Equation (2) are defined to be , the set of coboundaries of with coefficients in twisted by . Weil [23, 16] has noted that contains the tangent space to at as a subspace. Provided that we can show that the representation variety at is smooth, we can study the space of cocycles to determine the first order behavior of deformations of a representation .
2.2. Hyperbolic geometry
The hyperboloid model for is described as a subspace of . Topologically, is the space , but it is endowed with the Lorentzian metric . Then,
with the metric induced by is isometric to . The isometry group of in the hyperboloid model is the identity component of . Each point in the hyperboloid model intersects exactly 1 line through the origin in . Hence, we can also identify the hyperboloid with a subset of , given by
There is a well-known method for taking an isometry of from the upper half-space model (i.e. an element ) to the corresponding isometry in the hyperboloid model (see for instance [1, p. 66]). First, a point from the hyperboloid model is identified with the matrix
Then, acts on the point by
where denotes the Hermitian transpose of . This operation preserves , so it sends points of the hyperboloid in to points of the hyperboloid. The corresponding isometry in the hyperboloid model is the element so that
2.3. Sol geometry
Topologically, Sol is , with the metric . In this model for Sol, one can see that by restricting to any plane , we obtain a 2-dimensional space that is isometric to the hyperbolic plane via the upper half-plane model. Restricting to the plane also yields a space isometric to the hyperbolic plane as the lower half-plane model.
Sol also has an embedding into by
The image of this map gives Sol as the subspace
The group contains the identity component of the isometry group of Sol inside as elements of the form
where . Other components can be found by multiplying the diagonal blocks by or the upper left block by . A further treatment of Sol geometry can be found in .
2.4. HP geometry
There are also multiple copies of lying inside . For each , we can take the hyperboloid
and the subgroup of preserving the form
to obtain a space isometric to . The isometry to the usual hyperboloid model of is given by the rescaling map
Geometrically, we can think of the family of hyperboloids, , as flattening out to in . Taking the limit as yields a model for half-pipe geometry.
Danciger  studies degenerations of singular hyperbolic structures using the projective models. An appropriate rescaling of the degeneration yields half-pipe (HP) geometry, a transition geometry between hyperbolic geometry and anti-de Sitter (AdS) geometry.
Three-dimensional HP geometry, , topologically is . In terms of representations, it can be described as a rescaling of the collapse of the structure group from to . Begin with a representation of into , and describe the collapse of the manifold in the coordinate by a family of representations , so that we end with a representation into of matrices of the form
Conjugate the path of representations degenerating in this matter by
and take the limit as . This will yield a representation whose image lies in the set of matrices of of the form
where is the transpose of a vector in . The vector can be interpreted as an infinitesimal deformation of into . A path of representations satisfying Equation (3) is said to be compatible to first order with . The map takes the standard copy of inside to the isometric copy . As we take the limit , we obtain as
As a subset of , we can think of as
The structure group is the set of matrices of the form in Equation (3).
A concrete description of can be found by generalizing the isomorphism . Let be a non-zero element such that , and define an algebra generated over by and . Furthermore, define a conjugation by
Then let be the conjugate transpose of .
We can define a map by
where is the set of matrices with entries in such that . Then define the map by where is the matrix that satisfies
When , this is the usual isometry from to . Danciger proves the following:
Theorem (, Propositions 4.15, 4.19).
For , the map is an isomorphism. When , the map is an isomorphism onto the group of HP matrices.
Moreover, in the case , we obtain a geometric interpretation for the vector in Equation (3). If we have a matrix in , we can write it as , where is symmetric and is skew-symmetric. Similarly, we can write where
Then . In the map , the symmetric part determines the first three rows of the HP matrix, and the skew-symmetric part determines the bottom row of the HP matrix.
Lemma (, Lemma 4.20).
Let have determinant . Then and . In other words is in the tangent space at of matrices of constant determinant .
Hence, when mapped into , the symmetric part is the usual map , and the bottom row of an HP matrix comes from the skew-symmetric part. The vector in the HP matrix of Equation (3) is an infinitesimal deformation of the matrix from the collapsed structure.
The key result about HP structures is that we can recover hyperbolic structures from them [4, Proposition 3.6]. Thus, if we can find an HP structure for and construct a transition at the level of representations, then we can deform it to nearby hyperbolic and AdS structures.
3. The metabelian representation
Let be a pseudo-Anosov homeomorphism with orientable invariant foliations with singular set and transverse measures and . If has a puncture , then we can fill in the puncture by taking . Either the measured foliations extend smoothly to , or is a singular point of the foliation. In either case, we simply include in the set , so we can simplify our analysis to the case where is closed. The orientability assumption gives us some control over the eigenvalues of . It also implies that the cone angles at the singular points in the singular Euclidean metric induced by the measured foliations are multiples of – in particular, they are larger than .
Lemma 1 (c.f. McMullen , Theorem 5.3).
Let be a pseudo-Anosov homeomorphism with dilatation factor . Suppose also that has orientable unstable and stable foliations, and . Then and are simple eigenvalues of .
If and are orientable, then their transverse measures represent cohomology classes . The fact that scales the invariant measures by implies that , so that are eigenvalues of .
Let be any cohomology class dual to a simple closed curve . Since is pseudo-Anosov, limits to the either or . In particular,
for some . Since the classes dual to simple closed curves span , the eigenspaces for are 1-dimensional. In fact, must be simple eigenvalues by considering the Jordan canonical form. If there existed a generalized eigenvector such that , we would have , so that the condition in Equation (4) is not satisfied. ∎
Note that in addition to and being simple eigenvalues, we also have that the corresponding eigenvectors come from the measures and . In particular, if we take to be a basis for , then the eigenvector is given by
where the transverse measure is taken to be a signed measure, i.e. , if is the closed curve taken with the orientation opposite that of . The eigenvector corresponding to is given by
Choose a disk that contains all of the points in , and fix a point on as the base point for . Let be generators of , so that each encircles exactly one singularity , each lies entirely inside , and the product is homotopic to the boundary .
Choose standard generators and of such that for each , (a representative of) and do not intersect for , except at the basepoint for . We will also refer to these curves as , , . When convenient, we will use and to refer to their respective homology classes.
On the dual generators of , has a block upper triangular action: the first block on the diagonal corresponding to the action on the closed surface , and the second block a permutation of the generators coming from the curves around the singular points. Strictly speaking, this matrix is a square matrix with dimensions one greater than the dimension of . There is one redundancy in the generators by the relation in homology. However, using the additional generator from the singularities makes the lower right block for easier to understand. When discussing (or ) in this section, it will mean with this additional generator (resp. the action on with the additional generator).
Using these generators for , we can describe by the following presentation.
where are words in the s, s, and s.
We start with the metabelian representation with
where is the signed length of in . Note that for . We also set
where is the generator in the direction of , and is the pseudo-Anosov dilatation factor of . There is a singular Sol structure on coming from the pseudo-Anosov action on and , where and provide a singular Euclidean structure on the fibers of . Recall from Section 2.3 that Sol contains embedded hyperbolic planes as “vertical” planes. In the singular Sol structure on , these can be seen as products of a leaf of with the direction. The metabelian representation is a projection of the singular Sol structure along the leaves of onto one of these hyperbolic planes inside of Sol. Such a projection yields a transversely hyperbolic foliation – locally, can be viewed as an open subset of , and the pseudometric is given by the metric on the factor and ignoring the second factor.
4. Smoothness of the representation variety
The goal is to deform to a representation into , and to realize the representation as the holonomy representation of a -structure on . We consider as the metabelian representation from the previous section. We begin by computing the dimension of the space of classes of twisted cocycles .
Let be pseudo-Anosov with stable and unstable foliations which are orientable. Suppose also that does not have 1 as an eigenvalue. Then where is the number of components of the boundary of .
Let . Then is determined by its values on , and , subject to the cocycle condition imposed by the relations in . These can be computed via the Fox calculus [16, Chapter 3]. Differentiating the relations
Choosing the basis,
for , the values can be expressed in coordinates , where is the matrix
and we similarly let be given in the coordinates . We note that by using the coboundary condition from Equation (2), we can compute the set of coboundaries as the set of cocycle satisfying,
where parametrize . In particular, adding the appropriate coboundary to , we can set . To simplify the calculation somewhat, we will assume that has this form
We first note that if is a word in the , then for some real number . Then, under the chosen basis for , acts by
We obtain one term from for each instance of in (with a negative sign if appears), and each term is a word in the ’s.
Similarly, we can compute that acts on via
We see that is determined, as in , by a subset of vectors such that , where decomposes into blocks
Here, is the matrix describing the cohomology action induced by , which can be written as a block matrix
where is a permutation matrix denoting the permutation of the singularities in by . In particular, if