Peripheral birationality for 3-dimensional convex co-compact ${\rm PSL}_{2}\mathbb{C}$ varieties

Let $S$ be a genus $g$ surface. Let us fix an auxiliary hyperbolic metric $m_{0}$ and $\Gamma=\{\gamma_{i}\}_{1\leq i\leq 3g-3}$ a maximal collection of oriented, disjoint, essential, pairwise non-isotopic, simple closed curves (i.e. an oriented pants decomposition). Take $\lambda_{0}$ to be the maximal lamination extension of $\Gamma$ so that any leaf of $\lambda_{0}$ that accumulates at $\gamma_{i}\in\Gamma$ does it in the direction of the orientation (see Figure 1). Equivalently, the orientation for each $\gamma_{i}\in\Gamma$ gives a preferred endpoint for any lift $\tilde{\gamma_{i}}$ of $\gamma_{i}$ in the universal cover. Then any ideal triangle in the lift $(\tilde{S},\tilde{\lambda_{0}})$ uses only preferred endpoints as ideal vertices.

Take representations $\rho_{t}:\pi_{1}(S)\rightarrow PSL(2,\mathbb{C})$ so that $\rho(\gamma_{i})$ is neither parabolic or the identity (so there is a well-defined axis), and the endpoints of $\rho(\gamma_{i}),\rho(\gamma_{j})$ are distinct for $i\neq j$. Assume as well that we have a given equivariant orientation/endpoint for axis of each lift of $\gamma_{i}$, which we denote by $\zeta$. Then we define the (generalized) pleated surface $(\tilde{f}_{\zeta}=\tilde{f}:\tilde{S}\rightarrow\overline{\mathbb{H}^{3}},\rho)$ as follows

1. (1)

   For any lift $\tilde{\gamma_{i}}$ of $\gamma_{i}$, map its preferred endpoint to the corresponding endpoint given by $\zeta$.

2. (2)

   For any lift of a component of $\lambda_{0}\setminus\Gamma$, send it to the geodesic in $\mathbb{H}^{3}$ joining the corresponding preferred endpoints. This is possible since by assumption the preferred endpoints in $\partial_{\infty}\mathbb{H}^{3}$ are distinct.

3. (3)

   For any lift of an ideal triangle in $\tilde{S}\setminus\tilde{\lambda_{0}}$, send it to the ideal triangle spanned by the distinct corresponding endpoints.

4. (4)

   Finally, extend continuously to $\tilde{\Gamma}$. This makes the map $\tilde{f}:\tilde{S}\rightarrow\overline{\mathbb{H}^{3}}$ equivariant by $\rho$.

Note that if every $\rho(\gamma_{i})$ was loxodromic and the orientation $\zeta$ agrees with the orientation of $\Gamma$, this will be the classical notion of pleated surface. On the other hand if for some $\rho(\gamma_{i})$ the orientations of $\zeta$ and $\Gamma$ disagree, then $\tilde{f}$ is the classical notion of pleated surface for the lamination $\lambda_{1}$, where $\lambda_{1}$ is obtained from $\Gamma$ after changing the orientations to have full agreement with $\zeta$.

If $\rho(\gamma_{i})$ is elliptic then the situation is a bit more delicate. The continuous extension of step (4) will send any lift $\tilde{\gamma_{i}}$ entirely to the preferred endpoint given by $\zeta$. This means that the pull-back metric on $S$ given by $\tilde{f}$ is of finite type with $\ell(\gamma_{i})=0$, which is consistent with the fact that the complex length of $\rho(\gamma_{i})$ is purely imaginary.

In [Bon96], Bonahon defines the bending transverse cocycle of a pleated surface. This means that for each arc $\alpha$ transverse to a lamination $\lambda$ we have a number $\beta(\alpha)\in\mathbb{R}/2\pi\mathbb{Z}$ called the bending, which represents the amount of turning made by the pleated surface between the geodesic faces containing the endpoints of $\alpha$. The cocycle $\beta$ is additive under finite subdivision of a transverse arcs and can be defined as follows. Consider all the geodesics of the lamination that $\alpha$ crosses, and the geodesic faces (or rather plaques, as denoted by Bonahon) of the pleated surface going between those geodesics. The boundary of the plaques define two curves $\eta_{1,2}$ in $\partial_{\infty}\mathbb{H}^{3}$. Then the bending of $\alpha$ is defined as the difference of angles between the end plaques minus the integral of the signed curvature of either $\eta_{1,2}$.

For our fixed lamination $(\tilde{S},\tilde{\lambda_{0}})$ we can finitely decompose any transverse arc into smaller transverse arcs so that each smaller arc intersects $\tilde{\Gamma}$ at most once. This simplifies the description of the bending cocycle as follows.

If an arc $\alpha$ does not intersect $\tilde{\Gamma}$ then it only intersects finitely many geodesic lines of $\tilde{\lambda_{0}}$. Then the bending is just given by the sum of the finitely many angles involved.

Now say that $\alpha$ intersects exactly one component $\tilde{\gamma_{i}}$ of $\tilde{\Gamma}$. Furthermore, assume that $\alpha$ only intersects leaves of $\lambda_{0}$ that accumulate in $\gamma_{i}$. Then all plaques involved contain the preferred endpoint of $\tilde{\gamma_{i}}$. This means that one of the two curves $\eta_{1,2}$ degenerates to a point, so we are only left with the angle of the two intersecting end plaques.

Now it is easy to see that for our definition of (generalized) pleated surface, the two above definitions are well-defined and are additive under finite subdivision. Then for a general transverse arc $\alpha$ we can take any finite subdivision so every arc is of one of the two cases analyzed above, and define the bending of $\alpha$ as the sum of bending. Since the cocycle was already additive between the two types, this shows that the bending cocycle is well-defined and additive under finite subdivision. Moreover, this implies that for a smooth 1-parameter family of $\Gamma$-adapted representations $\rho_{t}$ where the orientation/endpoint choice $\zeta$ varies continuously, the bending varies smoothly. This is because the family of preferred endpoints will vary smoothly, and the bending of any arc is decomposed as the finite sum of finitely many angles of planes defined by preferred endpoints.

Let $\rho_{t}$ be a smooth 1-parameter family of $\Gamma$-adapted representations. Let $f_{t}$ be the $\lambda_{0}$ generalized pleated surface, $m_{t}$ the metric induced from $\mathbb{H}^{3}$ in $S$ by $f_{t}$, and let $V_{t}$ be the volume of a $3$-chain bounded by $f_{t}$. Then

where $b_{t}$ is the bending cocycle of $f_{t}$, and $\ell_{t}$ is the length of the (real-valued) transverse cocycle $b^{\prime}_{t}$ with respect to the induced metric $m_{t}$. This is known for classical pleated surfaces as the Bonahon-Schalfli formula, as proved by Bonahon in [Bon98]. We will explain the outline of Bonahon’s proof while justifying that the method holds for our case. This in particular means that we need to explain how to make sense of the formula when elliptic transformations are involved.

Cover $\lambda_{0}$ by geodesic rectangles $R^{0}_{1},\ldots R^{0}_{m}$ with disjoint interior, so that the components of $\lambda_{0}\cap R^{0}_{i}$ are parallel to opposite sides of the rectangle. Each rectangle $R^{0}_{i}$ can be collapsed to an edge in order to obtain an embedded graph in $S$, and we can complete that graph to a triangulation $T_{0}$ of $S$, which lifts to a triangulation $\tilde{T_{0}}$ of $\tilde{S}$. Take then a $\rho_{0}$ equivariant map $g:\tilde{S}\rightarrow\mathbb{H}^{3}$ that is polyhedral with respect to $\tilde{T_{0}}$ and homotopic to $f_{0}$ through $\rho_{0}$-equivariant maps. For $t$ small, take $m_{t}$-geodesic rectangles $R^{t}_{i}$ so that the analogous statement holds for $\lambda_{t}$, $R^{t}_{i}$ is smooth on $t$, and up to isotopy on $S$ we have that $T_{t}=T_{0}$. Consider then $\rho_{t}$-equivariant maps $g_{t}:\tilde{S}\rightarrow\mathbb{H}^{3}$ that are polyhedral with respect to $\tilde{T_{0}}$ and homotopic to $f_{t}$ through $\rho_{t}$-equivariant maps. By the Schläfli Formula for polyhedral maps, we can calculate the variation of volume for the quotient of a $3$-chain bounded by $g_{t}$. Hence we can reduce the problem to calculate the variation of volume for the quotient of a $\rho_{t}$-equivariant homotopy between $f_{t},g_{t}$. Since we can take maps $h_{t}:S\rightarrow S$ homotopic to the identity so that $h_{t}(R^{t}_{i})$ is an arc of $T$, the volume of the quotient of the homotopy between $g_{t}$ and $g_{t}\circ\tilde{h_{t}}$ is equal to $0$. Then we can further reduce to calculate the variation of volume for the quotient of a $\rho_{t}$-equivariant homotopy $H_{t}$ between $f_{t},g_{t}\circ\tilde{h_{t}}$.

The next step is to divide the homotopy $H_{t}$ into a family of $\rho_{t}$ equivariant polyhedral pieces. Fixing a rectangle $R^{t}_{i}$, and given $g_{t}\circ\tilde{h_{t}}=H_{t}(.,0)$ sends $R^{t}_{i}$ to a geodesic segment, we want extend $H$ to $R^{t}_{i}\times[0,1]$ by geodesic segments so that $f_{t}=H_{t}(.,1)$. In order to do so, for each component $R$ of $R^{t}_{i}\setminus\lambda_{t}$, we define $H_{t}(R\times[0,1])$ so that decomposes into the union of a pyramid with square basis given by $R$, and a tetrahedra that shares a side with the pyramid (see Figure 6). Because $f_{t},g_{t}\circ\tilde{h_{t}}$ are $\rho_{t}$-equivariant, this family of polyhedra $P_{t}$ is $\rho_{t}$-equivariant. Since $(\lambda_{t}\cap R^{t}_{i})\times[0,1]$ has $3$-dimensional Lebesgue measure $0$, we can focus solely in the family of polyhedra $P_{t}$ in order to calculate volumes.

Recall that the variation of volume of a polyhedra is given by the sum of half lengths of edges times the variation of the dihedral angle. Then Bonahon [Bon98] argues that for any interior edge and for any edge share with $g_{t}\circ\tilde{h_{t}}$, the different contributions cancel out. As for edges appearing in $f_{t}$, their sum can be reinterpreted as the half-length of the variation of the bending cocycle. This is the delicate part of the argument, because as Bonahon points out, edges are not locally finite, so appropriate sumability and convergence should be proved. Bonahon’s statement covers the case when all $\gamma\in\Gamma$ are loxodromic, so we are left to justify when some $\gamma$ is elliptic. Hence we concentrate on this case.

For $\lambda_{0}$ the polyhedral subdivision can be simplified in such a way that the questions of sumability and convergence are easier to conclude.

The rectangles $R^{t}_{1},\ldots,R^{t}_{m}$ are taken in such a way that each closed curve $\gamma$ in $\Gamma$ is covered by the closure of a single rectangle $R^{t}_{i}$, which we label by $R^{t}_{\gamma}$ (see Figure 3). In the case that $\gamma$ is elliptic the vertical sides are not well-defined segments, but we have a well-defined pair of cusped cylinders which union we still denoted by $R^{0}_{\gamma}$. Observe that it is only in these rectangles where $\lambda_{0}\cap R^{t}_{i}$ is not a finite union of geodesic segments (see Figure 4 for how to cover these finitely many segments by rectangles). The horizontal collapse of such $R^{t}_{\gamma}$ is a closed edge homotopic to the curve $\gamma$ covered by $R^{t}_{\gamma}$ (see Figure 5 for how to complete to a triangulation on each pair of pants), from where we will choose that the $\rho_{t}$-equivariant polyhedral maps $g_{t}:\tilde{S}\rightarrow\mathbb{H}^{3}$ send these closed edges to the axis of $\rho_{t}(\gamma)$. The potential issues of choosing $g_{t}$ in such a way are that (1) the polyhedral decomposition of the homotopy $H_{t}$ might include degenerate polyhedra and (2) how to make sense of the length of $b^{\prime}$ in the Bonahon-Schläfli formula.

For (1) the only $3$-chains that could degenerate are the ones concerning $\gamma$. In preparation, take $R^{0}_{\gamma}$ such that each horizontal side extends to a geodesic ray with the same endpoint as $\gamma$. In the axis $\tilde{\gamma}$ take a point $A$ so that any $\gamma$ translate of $A$ does not belong to either $\tilde{\lambda_{0}}$ or to any lifting of the horizontal sides of $R^{0}_{\gamma}$. This is possible because we are in the open set where $\tilde{\gamma}$ is not included in $\tilde{\lambda_{0}}$. Use $A$ as the vertex in $\gamma$ of the triangulation of Figure 5. Since the horizontal sides of $R^{0}_{\gamma}$, $\tilde{\gamma}$ and $\lambda_{0}\cap R^{0}_{\gamma}$ have all a point at infinity in common, the associated $3$-chains to $R^{0}_{\gamma}$ are in fact prisms. This is the combinatorial type we will take for these $3$-chains. Such prisms are not necessarily non-degenerate, as if $\gamma$ is elliptic then a side of the prism will collapse to a point. Regardless, since by choice the translates of $A$ are disjoint from the boundary of the opposite rectangle, the planes containing a face of the prism are all well-defined. Similarly, each edge of the prism is contained in a well-defined geodesic line (in Figure 7 the collapsed segment belongs to $\tilde{\gamma}$). Hence, even if the prism degenerates, we have a well-defined notion of angles between adjacent faces, where we allowed the values of $0,\pi$ in $\mathbb{R}/2\pi\mathbb{Z}$ for angles and $0$ for lengths. Same follows for the $3$-chains bounded by $g_{0}\circ\tilde{h}_{0}$, and we extend this configuration for $t$ small.

For (2) Bonahon observes that $\frac{1}{2}\ell(b^{\prime}_{t})$ in $R^{t}_{\gamma}$ is equal to the variations of volume of the $3$-chains obtained from $R^{t}_{\gamma}$ minus the edge contributions of edges that do not belong to the lamination. More specifically, Bonahon proves that the contribution to the Schläfli formula of the edges in the lamination add up to $\frac{1}{2}\ell(b^{\prime}_{t})$, while the sum of contributions of all other edges cancel out while adding all $R^{t}_{i}$ together. So for our particular construction we will rearrange the rectangles of $R^{t}_{\gamma}\setminus\lambda^{t}_{0}$ for the top to the bottom diagram of Figure 9. Doing so will stack all prisms to form finitely many tetrahedra with an ideal vertex such as in Figure 8. Polyhedra with ideal vertices have as well a Schläfli formula, where one take an horoball at each ideal vertex and computes the (signed) length of an edge by taking the finite segment in the edge that goes between vertices/horoballs. Then the sum of (edge length)(derivative of dihedral angle) is well defined and independent of the family of horoballs at ideal vertices.

Now we have to subtract the contribution of edges that are not in the lamination $\tilde{\lambda_{t}}$. Edges joining $\tilde{\gamma}$ to the rest of the prism already cancel out once we assembled the ideal tetrahedra, with the exception of the finite face of the tetrahedra, which we exclude. Hence we are left with the contributions of the edges at the ideal vertex. Observe that from the change of orientation of the tetrahedra from each cuff side of $\gamma$, the dihedral angles at $\tilde{\gamma}$ cancel out, as along as we consider the same horoball for all ideal tetrahedra. Hence we are left with the edges coming from the bending.

On the other hand, $\ell(b^{\prime}_{t})$ over a region is obtained by integrating the horizontal length agains the measure induced by the change of bending cocycle. When $\gamma$ is elliptic the pleated surface is of finite type, which makes all the lengths infinite (they correspond to geodesic rays towards the fixed endpoint of $\tilde{\gamma}$). But the horoball of the previous paragraph corresponds to an horocycle in the pleated surface, where we now we can compute length of a segment from the horocycle, and multiply that by the corresponding bending. This sum will be well-defined by the same reasons as the Schläfli formula for ideal polyhedra, as long as we take matching horocycles at each side of $\gamma$, namely horocycles coming from the same horoball at the fixed endpoint of $\hat{\gamma}$.

Putting all together, both $V^{\prime}_{t}$ and $\frac{1}{2}\ell(b^{\prime}_{t})$ have the same contribution on the rectangle $R^{t}_{\gamma}$, namely both contributions are equal to

where $\ell$ denotes length of the segment of the edge $e$ in $R^{t}_{\gamma}$ outside a fixed horoball based at the endpoint of $\tilde{\gamma}$, and $\theta(e)$ is the exterior dihedral angle. With these conventions we have justified

for all cases, which is sufficient for the purpose of this article, since what we need is to characterize the derivative of volume as depending only on the boundary information.