2 Realizations of graphs in surfaces

In this section, we discuss certain spaces of maps of graphs into a surface. These spaces play a key rôle in this paper. Although long, the material here should be nice and pleasant to read – only the proof of Proposition 2.3 takes some amount of work.

Realizations

We will be interested in connected graphs living inside our hyperbolic surface $\Sigma $ , or more precisely in continuous maps

(2.1) $$ \begin{align} \phi:X\to\Sigma \end{align} $$

of graphs X into $\Sigma $ which when restricted to each edge are geodesic. We will say that such a map (2.1) is a realization of X in $\Sigma $ . We stress that realizations do not need to be injective and that in fact the map $\phi $ could be constant on certain edges or even on larger pieces of the graph. A regular realization is one whose restriction to every edge is nonconstant. If $\phi :X\to \Sigma $ is a regular realization and if $\vec e\in \operatorname {\mathrm {\mathbf {half}}}(X)$ is a half-edge incident to a vertex $v\in \operatorname {\mathrm {\mathbf {vert}}}(X)$ , then we denote by $\phi (\vec e)\in T^1_{\phi (v)}\Sigma $ the unit tangent vector at $\phi (v)$ pointing in the direction of the image of $\vec e$ .

We endow the set $\mathcal G^X$ of all realizations of the (always connected) graph X with the compact-open topology and note that unless X itself is contractible, the space $\mathcal G^X$ is not connected: The connected components of $\mathcal G^X$ correspond to the different possible free homotopy classes of maps of X into $\Sigma $ . Indeed, pulling segments tight relative to their endpoints we get that any homotopy

$$ \begin{align*}[0,1]\times X\to\Sigma,\ \ (t,x)\mapsto \varphi_t(x)\end{align*} $$

between two realizations is homotopic, relative to $\{0,1\}\times X$ , to a homotopy, which we are still denoting by the same symbol, such that

1. 1. $t\mapsto \varphi _t(v)$ is geodesic for every vertex $v\in \operatorname {\mathrm {\mathbf {vert}}}(X)$ , and

2. 2. $\varphi _t:X\to \Sigma $ is a realization for all t.

A homotopy $[0,1]\times X\to \Sigma $ satisfying (1) and (2) is said to be a geodesic homotopy.

Geodesic homotopies admit a different intrinsic description. Indeed, uniqueness of geodesic representatives in each homotopy class of arcs implies that each realization $\phi \in \mathcal G^X$ has a neighborhood which is parametrized by the image of the vertices. This implies that the map

(2.2) $$ \begin{align} \Pi:\mathcal G^X\to\Sigma^{\operatorname{\mathrm{\mathbf{vert}}}(X)},\ \ \phi\mapsto(\phi(v))_{v\in\operatorname{\mathrm{\mathbf{vert}}}(X)} \end{align} $$

is a cover. Pulling back the product of hyperbolic metrics, we think of it as a manifold locally modeled on the product $({\mathbb {H}}^2)^{\operatorname {\mathrm {\mathbf {vert}}}(X)}={\mathbb {H}}^2\times \dots \times {\mathbb {H}}^2$ of $\operatorname {\mathrm {\mathbf {vert}}}(X)$ worth of copies of the hyperbolic plane. Geodesic homotopies are, from this point of view, nothing other than geodesics in $\mathcal G^X$ .

Since all of this will be quite important, we record it here as a proposition:

Proposition 2.1. Let X be a graph. The map (2.2) is a cover and geodesic homotopies are geodesics with respect to the pull-back metric.

Length function

On the space $\mathcal G^X$ of realizations of the graph X in $\Sigma $ , we have the length function

$$ \begin{align*}\ell_{\Sigma}:\mathcal G^X\to\mathbb R_{\geqslant 0},\ \ \ell_{\Sigma}(\phi)=\sum_{e\in\operatorname{\mathrm{\mathbf{edge}}}(X)}\ell_{\Sigma}(\phi(e)).\end{align*} $$

First, note that Arzela–Ascoli implies that $\ell _{\Sigma }$ is a proper function. It follows thus that the restriction of $\ell _{\Sigma }$ to any and every connected component of $\mathcal G^X$ has a minimum. Now, convexity of the distance function $d_{{\mathbb {H}}^2}(\cdot ,\cdot )$ implies that $\ell _{\Sigma }$ is convex. More precisely, if $(\varphi _t)$ is a geodesic homotopy between two realizations in $\Sigma $ , then the function $t\mapsto \ell _{\Sigma }(\varphi _t)$ is convex. Indeed, it is strictly convex unless the image of $[0,1]\times X$ is contained in a geodesic in $\Sigma $ , or rather if the image of some (and hence any) lift to the universal cover is contained in a geodesic in ${\mathbb {H}}^2$ .

Note now also that the length function is smooth when restricted to the set of regular realizations – its derivative is given by the first variation formula

$$ \begin{align*}\frac d{dt}\ell_{\Sigma}(\phi_t)=\sum_{v\in\operatorname{\mathrm{\mathbf{vert}}}(X)}\sum_{\vec e\in\operatorname{\mathrm{\mathbf{half}}}_v(X)} \left\langle-\phi(\vec e),\frac d{dt}\phi_t(v)\right\rangle,\end{align*} $$

where $\phi (\vec e)\in T_v^1\Sigma $ is the unit tangent vector based at v and pointing as the image of the half-edge $\vec e$ . It follows that a regular realization $\phi $ is a critical point for the length function $\ell _{\Sigma }:\mathcal G^X\to \mathbb R_{\geqslant 0}$ if and only if for every $v\in \operatorname {\mathrm {\mathbf {vert}}}(X)$ we have $\sum _{\vec e\in \operatorname {\mathrm {\mathbf {half}}}_v}\phi (\vec e)=0$ . Note that this implies, in the for us relevant case that X is trivalent, that the (unsigned) angle $\angle (\phi '(\vec e_1),\phi '(\vec e_2))$ between the images of any two half-edges incident to the same vertex is equal to $\frac {2\pi }3$ .

Definition. A regular realization $\phi :X\to \Sigma $ of a trivalent graph X into $\Sigma $ is critical if we have

$$ \begin{align*}\angle(\phi(\vec e_1),\phi(\vec e_2))=\frac{2\pi}3\end{align*} $$

for every vertex $v\in \operatorname {\mathrm {\mathbf {vert}}}(X)$ and for any two distinct $\vec e_1,\vec e_2\in \operatorname {\mathrm {\mathbf {half}}}_v(X)$ .

We collect in the next lemma a few of the properties of critical realizations:

Lemma 2.2. Let X be a trivalent graph. A regular realization $\phi \in \mathcal G^X$ is a critical point for the length function if and only if $\phi $ is a critical realization. Moreover, if $\phi \in \mathcal G^X$ is a critical realization and $\mathcal G^{\phi }$ is the connected component of $\mathcal G^X$ containing $\phi $ , then the following holds:

1. 1. $\phi $ is the unique critical realization in $\mathcal G^{\phi }$ .

2. 2. $\phi $ is the global minimum of the length function on $\mathcal G^{\phi }$ .

3. 3. Besides $\phi $ , there are no other local minima in $\mathcal G^{\phi }$ of the length function.

4. 4. The connected component $\mathcal G^{\phi }$ is isometric to the product ${\mathbb {H}}^2\times \dots \times {\mathbb {H}}^2$ of $\operatorname {\mathrm {\mathbf {vert}}}(X)$ many copies of the hyperbolic plane.

Note that lack of global smoothness of the length function means that we cannot directly derive (2) and (3) from (1). This is why they appear as independent statements.

Proof. Statement (1) was actually discussed in the paragraph preceding the definition of critical realization. Let us focus in the subsequent ones. Recall that if

$$ \begin{align*}[0,1]\to\mathcal G^{\phi},\ t\mapsto\phi_t\end{align*} $$

is a (nonconstant) geodesic homotopy with $\phi _0=\phi $ , then the length function

(2.3) $$ \begin{align} t\mapsto\ell_{\Sigma}(\phi_t(X)) \end{align} $$

is convex. In our situation it is strictly convex: Since $\phi $ is critical, we get that its image, or rather the image of its lifts to the universal cover, are not contained in a geodesic because if they were, then the angles between any two half-edges starting at the same vertex could only take the values $0$ or $\pi $ . Now, strict convexity and the fact that $\phi =\phi _0$ is a critical point of the length function implies that $t=0$ is the minimum and only critical point of the function (2.3). From here, we get directly (2) and (3). To prove (4), note that if $\mathcal G^{\phi }$ were not simply connected, then there would be a nontrivial geodesic homotopy starting and ending at $\phi $ , contradicting the strict convexity of the length function. Note now that the restricting the cover (2.2) to $\mathcal G^{\phi }$ we get a locally isometric cover $\mathcal G^{\phi }\to \Sigma ^{\operatorname {\mathrm {\mathbf {vert}}}(X)}$ . Since the domain of this cover is connected and simply connected, we get that it is nothing other than the universal cover of $\Sigma ^{\operatorname {\mathrm {\mathbf {vert}}}(X)}$ . This proves (4) and concludes the proof of the lemma.

Remark. Although we will not need it here, let us comment briefly on the topology of the connected components of $\mathcal G^X$ . The fundamental group of the connected component $\mathcal G^{\phi }$ containing a realization $\phi \in \mathcal G^X$ is isomorphic to the centralizer $\mathcal Z_{\pi _1(\Sigma )}(\phi _*(\pi _1(X)))$ in $\pi _1(\Sigma )$ of the image of $\pi _1(X)$ under $\phi _*:\pi _1(X)\to \pi _1(\Sigma )$ . It follows that $\mathcal G^{\phi }$ is isometric to the quotient under the diagonal action of $\mathcal Z_{\pi _1(\Sigma )}(\phi _*(\pi _1(X)))$ of ${\mathbb {H}}^2\times \dots \times {\mathbb {H}}^2$ of $\operatorname {\mathrm {\mathbf {vert}}}(X)$ -copies of the hyperbolic plane. In particular, we have:

• ○ If $\phi $ is homotopically trivial, then $\mathcal G^{\phi }\simeq \pi _1(\Sigma )\backslash ({\mathbb {H}}^2\times \dots \times {\mathbb {H}}^2)$ .

• ○ If $\phi _*(\pi _1(X))\neq \operatorname {\mathrm {Id}}_{\pi _1(\Sigma )}$ is abelian, then $\mathcal G^{\phi }\simeq \mathbb Z\backslash ({\mathbb {H}}^2\times \dots \times {\mathbb {H}}^2)$ .

• ○ $\phi _*(\pi _1(X))$ is non-abelian, then $\mathcal G^{\phi }\simeq {\mathbb {H}}^2\times \dots \times {\mathbb {H}}^2.$

In the appendix, we will give a concrete description of the components of $\mathcal G^X$ for the case that X is a loop, that is, the graph with a single vertex and a single edge.

Lemma 2.2, with all its beauty, does not say anything about the existence of critical realizations. Our next goal is to prove that any realization that from far away kind of looks like a critical realization is actually homotopic to a critical realization.

Quasicritical realizations

Recall that a regular realization $\phi :X\to \Sigma $ of a trivalent graph X is critical if the angles between the images of any two half-edges incident to the same vertex are equal to $\frac {2\pi }3$ . We will say that a regular realization is quasicritical if those angles are bounded from below by $\frac {1}{2}\pi $ and that the realization is $\ell _0$ -long if $\ell (\phi (e))>\ell _0$ for all edges e of X. Recall that we measure all angles in $[0,\pi ]$ .

Our next goal here is to prove that every sufficiently long quasicritical realization is homotopic to a critical realization. This will follow easily from the following technical result:

Proposition 2.3. For any trivalent graph X and constant $C\geq 0$ , there exist constants $\ell _0, D>0$ such that given an $\ell _0$ -long quasicritical realization $\phi : X\to \Sigma $ , a trivalent graph Y and realization $\psi : Y\to \Sigma $ satisfying

1. 1. $\ell (\psi )\leq \ell (\phi )+C$ , and

2. 2. there is a homotopy equivalence $\sigma : X\to Y$ with $\phi $ and $\sigma \circ \psi $ homotopic,

then there exists a homeomorphism $F: Y\to X$ mapping each edge with constant speed such that $\sigma \circ F$ is homotopic to the identity and such that the geodesic homotopy $X\times [0,1]\to \sigma $ , $(x, t)\mapsto \phi _t(x)$ joining $\phi _0(\cdot ) = \phi \circ F(\cdot )$ to $\phi _1(\cdot ) = \psi (\cdot )$ has tracks bounded by D.

Since the proof of Proposition 2.3 is technical and pretty long, it may be a good idea to skip it in a first reading. Nevertheless, before proving it, we demonstrate that it might be useful:

Corollary 2.4. Let X be a trivalent graph. There are positive constants $\ell _0$ and D such that every component of $\mathcal G^X$ which contains an $\ell _0$ -long quasicritical realization $\phi : X\to \Sigma $ also contains a critical realization $\psi $ , which moreover is unique and homotopic to $\phi $ by a homotopy whose tracks have length bounded by D.

Proof. Let $\ell _0$ and D be given by Proposition 2.3 for $C=0$ and if needed increase $\ell _0$ so that it is larger than $2D$ . Let $\phi : X\to \Sigma $ be an $\ell _0$ -long quasicritical realization. Let $\psi : X\to \Sigma $ be the minimizer for the length function in the component $\mathcal G^{\phi }$ – it exists because the length function is proper. We claim that $\psi $ is critical. In the light of Lemma 2.2, it suffices to prove that $\psi $ is regular.

Being a minimizer we have that $\ell _{\Sigma }(\psi )\leqslant \ell _{\Sigma }(\phi )$ . We thus get from Proposition 2.3 that there exists a homeomorphism $F: X\to X$ homotopic to the identity, mapping edges with constant speed and such that the geodesic homotopy from $\phi \circ F$ to $\psi $ has tracks bounded by D. Now, since X is trivalent we get that the homeomorphism F, being homotopic to the identity and mapping edges with constant speed, is actually equal to the identity. What we thus have is a homotopy from $\phi $ and $\psi $ with tracks bounded by D. Now, since each edge of $\phi (X)$ has at least length $\ell _0>2D$ and since the tracks of the homotopy are bounded by D, we get that $\psi $ is regular and hence critical by Lemma 2.2, as we needed to prove. Lemma 2.2 also yields that $\psi $ is unique.

Let us next prove the proposition:

Proof of Proposition 2.3

Let $\phi : X\to \Sigma $ be a quasicritical realization, and assume it is $\ell _0$ -long for an $\ell _0$ large enough to satisfy some conditions we will give in the course of the proof. Let $H: X\times [0,1]\to \Sigma $ be the homotopy between $\phi $ and $\psi \circ \sigma $ .

To be able to consistently choose lifts of $\phi $ , $\psi $ and $\sigma $ to the universal covers $\widetilde X, \widetilde Y$ and ${\mathbb {H}}^2$ of $X,Y$ and $\Sigma $ , let us start by picking base points. Fixing $x_0\in X$ , consider the base points $\sigma (x_0)\in Y$ and $\phi (x_0)\in \Sigma $ , and pick lifts $\widetilde {x}_0\in \widetilde X$ , $\widetilde {\sigma (x_0)}\in \widetilde Y$ and $\widetilde {\phi (x_0)}\in \mathbb {H}^2$ of each one of those endpoints. Having chosen those base points we have uniquely determined lifts $\widetilde {\phi }: \widetilde {X}\to \mathbb {H}^2$ and $\widetilde {\sigma }: \widetilde {X}\to \widetilde {Y}$ of $\phi $ and $\sigma $ satisfying $\widetilde {\phi }(\widetilde x_0) = \widetilde {\phi (x_0)}$ and $\widetilde {\sigma }(\widetilde {x}_0) = \widetilde {\sigma (x_0)}$ . We can also lift to ${\mathbb {H}}^2$ , starting at $\widetilde \phi (\widetilde {x_0})$ , the path in $\Sigma $ given by $t\mapsto H(x_0, t)$ . The endpoint of this path is a lift of $\psi \circ \sigma (x_0)$ , and we take the lift $\widetilde {\psi }: \widetilde {Y}\to \mathbb {H}^2$ which maps $\widetilde {\sigma }(\widetilde x_0)$ to this point. All those lifts are related by the following equivariance property:

(2.4) $$ \begin{align} \widetilde{\phi}(g(x)) = \phi_*(g)(\widetilde{\phi}(x)) \text{ and } (\widetilde{\psi}\circ\widetilde{\sigma})(g(x)) = \phi_*(g)\left((\widetilde{\psi}\circ\widetilde{\sigma})(x)\right) \end{align} $$

for all $x\in \widetilde {X}$ and for all $g\in \pi _1(X, x_0)$ , where $\phi _*: \pi _1(X, x_0)\to \pi _1(\Sigma , \phi (x_0))$ is the homomorphism induced by $\phi $ and the chosen base points.

Note that, since $\phi $ is almost critical and $\ell _0$ -long we get, as long as $\ell _0$ is large enough, that the lift $\widetilde \phi :\widetilde X\to {\mathbb {H}}^2$ is an injective quasi-isometric embedding. For ease of notation, denote by $T_X=\widetilde {\phi }(\widetilde {X})\subset \mathbb {H}^2$ the image of $\widetilde \phi $ and let us rephrase what we just said: If $\ell _0$ is sufficiently large, then $T_X$ is a quasiconvex tree, with quasiconvexity constants only depending on a lower bound for $\ell _0$ . We denote by $\hat \pi :{\mathbb {H}}^2\to T_X$ a nearest point retraction which is equivariant under $\phi _*(\pi _1(X, x_0))$ . We would like to define

$$ \begin{align*}\pi: \widetilde{Y}\to T_X\end{align*} $$

as $\hat \pi \circ \widetilde \psi $ , but since $\hat \pi $ is definitively not continuous, we have to be slightly careful. We define $\pi $ as follows: First, set $\pi (v) = \hat \pi (\tilde {\psi }(v))$ for all vertices $v\in \operatorname {\mathrm {\mathbf {vert}}}(\widetilde Y)$ of $\widetilde Y$ , and then extend (at constant speed) over the edges of Y. Note that Equation (2.4), together with the equivariance of $\pi $ , implies that $\pi \circ \widetilde {\sigma }: \tilde {X}\to T_X$ satisfies

$$ \begin{align*}(\pi\circ\widetilde{\sigma})(g(x)) = \phi_*(g)(\pi\circ\widetilde{\sigma}(x))\end{align*} $$

for all $x\in \widetilde {X}$ and $g\in \pi _1(X, x_0)$ .

Claim 1. As long as $\ell _0$ is large enough, we have that $\pi $ maps each vertex of $\widetilde Y$ within $\operatorname {\mathrm {\mathbf {const}}}$ of one and only one vertex of $T_X$ .

Starting with the proof of the claim, note that there is a constant $A\geqslant 0$ depending only on the quasiconvexity constants for $T_X$ with

(2.5) $$ \begin{align} d_{{\mathbb{H}}^2}(x,y)\geqslant -A+d_{\mathbb{H}^2}(\hat\pi(x),\hat\pi(y)) \end{align} $$

for all $x,y\in {\mathbb {H}}^2$ . In fact, there exists constants $K, A'>0$ which once again depend only on the quasiconvexity constant of $T_X$ such that whenever $d_{\mathbb {H}^2}(\hat \pi (x),\hat \pi (y))>K$ we have the better bound

(2.6) $$ \begin{align} d_{{\mathbb{H}}^2}(x,y)\geqslant -A'+d_{{\mathbb{H}}^2}(x,\hat\pi(x))+d_{{\mathbb{H}}^2}(\hat\pi(x),\hat\pi(y))+d_{{\mathbb{H}}^2}(\hat\pi(y),y). \end{align} $$

For $a, b\in T_X$ , let $d_{T_X}(a,b)$ denote the interior distance in $T_X$ between them. We will care about a modified version of this distance: Let B be the collection of balls in ${\mathbb {H}}^2$ of radius 1 centered at each vertex of $T_X$ , and define $d_{T_X\text {rel}B}(a, b)$ to be the length of the part of the path in $T_X$ between them that lies outside of B. For all $\ell _0$ large enough (depending only on hyperbolicity of ${\mathbb {H}}^2$ ), we have from the choice of the radiusFootnote ¹ that

(2.7) $$ \begin{align} d_{T\text{rel}B}(a, b) \leq d_{{\mathbb{H}}^2}(a, b) \end{align} $$

for all $a, b \in T_X$ .

Given two edges $e, e'$ of $\widetilde Y$ adjacent to the same vertex, we define

$$ \begin{align*}\operatorname{\mathrm{Fold}}(e, e') = \ell_{T\text{rel}B}(\pi(e)\cap\pi(e'))\end{align*} $$

and let then

$$ \begin{align*}\operatorname{\mathrm{Fold}}(\pi)=\max\operatorname{\mathrm{Fold}}(e,e')\end{align*} $$

where the maximum is taken over all such pairs of edges. The reason to introduce this quantity is that we have

(2.8) $$ \begin{align} \sum_{[v, v']\in\operatorname{\mathrm{\mathbf{edge}}}(Y)} d_{T_X\text{rel}B}(\pi(v), \pi(v')) \geq \ell(\phi) + \operatorname{\mathrm{Fold}}(\pi) - A" \end{align} $$

for some positive constant $A"$ depending only on that number of vertices (and the fact that we chose the balls B to have radius $1$ ) – here, we have identified each edge of Y with one of its representatives in $\widetilde Y$ .

Now, using Equations (2.5), (2.7) and (2.8), we get that

$$ \begin{align*} \ell(\psi) &= \sum_{[v, v']\in\operatorname{\mathrm{\mathbf{edge}}}(Y)}d_{{\mathbb{H}}^2}(\tilde\psi(v), \tilde\psi(v')) \\ & \geq -\operatorname{\mathrm{\mathbf{const}}} + \sum_{[v, v']\in\operatorname{\mathrm{\mathbf{edge}}}(Y)} d_{{\mathbb{H}}^2}(\pi(v), \pi(v'))\\ & \geq -\operatorname{\mathrm{\mathbf{const}}} + \sum_{[v, v']\in\operatorname{\mathrm{\mathbf{edge}}}(Y)} d_{T_X\text{rel}B}(\pi(v), \pi(v'))\\ & \geq -\operatorname{\mathrm{\mathbf{const}}} + \ell(\phi) + \operatorname{\mathrm{Fold}}(\pi), \end{align*} $$

where each $\operatorname {\mathrm {\mathbf {const}}}$ is a positive constant (but not necessarily the same) depending on the quasiconvexity constant and combinatorics of Y.

Since $\ell (\psi )\leq \ell (\phi )+C$ , it follows that $\operatorname {\mathrm {Fold}}(\pi ) \leq \operatorname {\mathrm {\mathbf {const}}}$ , where the constant depends on C and on the lower bound for $\ell _0$ and on combinatorics of Y. In particular, for each $v\in \operatorname {\mathrm {\mathbf {vert}}}$ there is a vertex $v'\in T_X$ such that $d_{T_X\text {rel}B}(\pi (v), v') \leq \operatorname {\mathrm {\mathbf {const}}}$ and we can choose $\ell _0$ large such that $v'$ is unique. We have proved Claim 1.

Armed with Claim 1, we can define a map $\widetilde {F}: \widetilde Y\to T_X$ by first mapping $v\in \operatorname {\mathrm {\mathbf {vert}}}(\widetilde Y)$ , to the unique vertex in $T_X$ closest to $\pi (v)$ , extending it so that it maps each $e\in \operatorname {\mathrm {\mathbf {edge}}}(\widetilde Y)$ with constant speed. Note that equivariance of $\pi $ implies that $\widetilde {F}$ satisfies $\widetilde F(\sigma _*(g)(y)) = g(\widetilde F(y))$ for all $y\in \widetilde Y$ and $g\in \pi _1(X, x_0)$ , where $\sigma _*: \pi _1(X, x_0)\to \pi _1(Y, \sigma (x_0))$ is the isomorphism induced by $\sigma $ and the chosen base points. It follows that $\widetilde F$ descends to a map $F: Y\to X$ .

Claim 2. $F:Y\to X$ is a homeomorphism with $F\circ \sigma $ homotopic to the identity.

The fact that $F\circ \sigma $ is homotopic to the identity follows directly from the fact $\widetilde F\circ \widetilde \sigma : \widetilde X\to \widetilde X$ satisfies that $(\widetilde F\circ \sigma )(gx)=g((\widetilde F\circ \widetilde \sigma )(x))$ for $g\in \pi _1(X,x_0)$ . What we really have to prove is that F is a homeomorphism. To see that this is the case, note that since the maps $\widetilde F,\pi :\widetilde Y\to \widetilde X$ send points within $\operatorname {\mathrm {\mathbf {const}}}$ of each other, and since in our way to proving Claim 1 we proved that $\operatorname {\mathrm {Fold}}(\pi )<\operatorname {\mathrm {\mathbf {const}}}$ , we get that $\operatorname {\mathrm {Fold}}(\widetilde F)<\operatorname {\mathrm {\mathbf {const}}}$ . On the other hand, since $\widetilde F$ maps vertices to vertices and has constant speed on the edges we get that if $\operatorname {\mathrm {Fold}}(\widetilde F)\neq 0$ , then $\operatorname {\mathrm {Fold}}(F)$ has to be at least as large as the shortest edge of X. This implies, using the fact that $\operatorname {\mathrm {Fold}}(\widetilde F)$ is uniformly bounded, that as long as $\ell _0$ is large enough, then $\operatorname {\mathrm {Fold}}(\widetilde F)=0$ . This implies in turn that $\widetilde F$ is locally injective. Local injectivity of $\widetilde F$ , together with the fact that F, being a homotopy inverse to $\sigma $ , is a homotopy equivalence, implies that F is a homeomorphism. We have proved Claim 2.

What is left is to bound the tracks of the geodesic homotopy from $\phi \circ F$ to $\psi $ – the argument is similar to the proof of the existence of F. First, note that we know that for every vertex $v\in \widetilde Y$ , the nearest point projection $\hat \pi $ maps $\widetilde \psi (v)$ close to the vertex $\tilde \phi (\widetilde F(v))$ of $T_X$ . It follows that if we choose $\ell _0$ large enough we can also assume that $d_{\mathbb {H}^2}(\hat \pi (\tilde \psi (v)),\hat \pi (\tilde \psi ((v'))))> K$ for any two distinct vertices $v,v'\in \operatorname {\mathrm {\mathbf {vert}}}(\widetilde Y)$ . This means that Equation (2.6) applies, and hence that we have

$$ \begin{align*} \ell(\psi(Y)) &= \sum_{[v, v']\in\operatorname{\mathrm{\mathbf{edge}}}}d_{{\mathbb{H}}^2}(\tilde\psi(v), \tilde\psi(v')) \\ & \geq -\operatorname{\mathrm{\mathbf{const}}} + \sum_{[v, v']\in\operatorname{\mathrm{\mathbf{edge}}}} d_{\mathbb{H}^2}(\hat\pi(\tilde\psi(v)),\hat\pi(\tilde\psi(v')))+ \\ & \hspace{1cm}+ 3\sum_{v\in\operatorname{\mathrm{\mathbf{vert}}}} d_{{\mathbb{H}}^2}(\tilde\psi(v),\hat\pi(\tilde\psi(v)))\\ & \geq -\operatorname{\mathrm{\mathbf{const}}} + \ell(\phi(X)) + 3\sum_{v\in\operatorname{\mathrm{\mathbf{vert}}}} d_{\mathbb{H}^2}(\tilde\psi(v), \hat\pi(\tilde\psi(v))), \end{align*} $$

where as before each $\operatorname {\mathrm {\mathbf {const}}}$ is a positive constant, the first inequality follows by Equation (2.6), the second by Equations (2.7) and (2.8). Hence, again by assumption (1) in the lemma, we must have that $d_{\mathbb {H}^2}(\tilde \psi (v), \hat \pi (\tilde \psi (v)))<\operatorname {\mathrm {\mathbf {const}}}$ for all $v\in \operatorname {\mathrm {\mathbf {vert}}}(\widetilde Y)$ . Since $\widetilde {F}(v)$ and $\hat \pi (\tilde \psi (v))$ are near each other, we have that $d_{\mathbb {H}^2}(\tilde \psi (v), \widetilde F(v))<\operatorname {\mathrm {\mathbf {const}}}$ for all $v\in \operatorname {\mathrm {\mathbf {vert}}}(\widetilde Y)$ . Convexity of the length function implies that the geodesic homotopy between $\tilde \psi $ and $\tilde \phi \circ \widetilde F$ has then tracks bounded by the same constant $\operatorname {\mathrm {\mathbf {const}}}$ , as we had claimed.

$\varepsilon $ -critical realizations

Corollary 2.4 asserts that whenever we have a realization with very long edges and which looks vaguely critical, then it is not far from a critical realization. Our next goal is to give a more precise description of the situation in the case that our realization looks even more as if it were critical. To be precise, suppose that X is a trivalent graph and $\varepsilon $ positive and small. A regular realization $\phi :X\to \Sigma $ is $\varepsilon $ -critical if

$$ \begin{align*}\angle(\phi(\vec e_1),\phi(\vec e_2))\in\left[\frac{2\pi}3-\varepsilon,\frac{2\pi}3+\varepsilon\right]\end{align*} $$

for every two half-edges $\vec e_1,\vec e_2\in \operatorname {\mathrm {\mathbf {half}}}_v$ incident to any vertex $v\in \operatorname {\mathrm {\mathbf {vert}}}(X)$ .

The goal of this section is to determine how the set $\mathcal G^X_{\varepsilon -\operatorname {\mathrm {crit}}}$ of $\varepsilon $ -critical realizations of X looks. To do that, we need a bit of of notation and preparatory work. First, we understand a tripod to be a tuple $\tau =(p,\{v,v',v"\})$ , where $p\in {\mathbb {H}}^2$ is a point and where $\{v,v',v"\}\subset T^1_p{\mathbb {H}}^2$ is an unordered collection consisting of three distinct unit vectors based at p. A tripod $\tau =(p,\{v,v',v"\})$ is critical if the three vectors $v,v',v,"$ have pairwise angle equal to $\frac {2\pi }3$ . Any tripod $\tau $ determines three geodesic rays, that is, three points in $\partial _{\infty }{\mathbb {H}}^2$ , that is an ideal triangle $T_{\tau }\subset {\mathbb {H}}^2$ (see the left-hand side of Figure 1). Conversely, every ideal triangle $T\subset {\mathbb {H}}^2$ determines a tripod $\tau _p^T=(p,\{v,v',v"\})$ for every $p\in {\mathbb {H}}^2$ : The vectors $v,v',v,"$ are the unit vectors based at p and pointing to the (ideal) vertices of the ideal triangle T. Note that T consists of exactly those points p such that the vectors in the tripod $\tau _p^T$ have pairwise (always unoriented) angles adding up to $2\pi $ . For a given $\varepsilon $ , we let

$$ \begin{align*}T(\varepsilon)=\left\{p\in T\ \middle\vert\begin{array}{l}\text{the angles between the vectors}\\ \text{in }\tau_p^T\text{ lie in }[\frac{2\pi}3-\varepsilon,\frac{2\pi}3+\varepsilon]\end{array}\right\}\end{align*} $$

be the set of points such that the angles of the tripod $\tau _p^T$ are within $\varepsilon $ of $\frac {2\pi }3$ . The set $T(\varepsilon )$ is, at least for $\varepsilon \in (0,\frac \pi 6)$ , a hexagon centered at the center of the triangle – the sides of the hexagon are not geodesic but rather subsegments of curves of constant curvature; see Figure 1. However, if we scale everything by $\varepsilon ^{-1}$ , then $\varepsilon ^{-1}\cdot T(\varepsilon )$ converges to the regular euclidean hexagon of side length $\frac 23$ . This means in particular that

(2.9) $$ \begin{align} \operatorname{\mathrm{diam}}(T(\varepsilon))\sim\frac 43\varepsilon\text{ and }\operatorname{\mathrm{vol}}(T(\varepsilon))\sim \frac 2{\sqrt 3}\varepsilon^2\text{ as }\varepsilon\to 0. \end{align} $$

Figure 1

Left: A critical tripod and (part of) the ideal triangle it determines. Right: The hexagon $T(\varepsilon )$ . Each of the lines making up the boundary of the hexagon corresponds to points p having an angle between vectors in $\tau ^T_P$ of measure $\frac {2\pi }{3}-\varepsilon $ (dotted lines) and $\frac {2\pi }{3}+\varepsilon $ (solid lines).

Remark. To get the shape of $T(\varepsilon )$ , one can use elementary synthetic hyperbolic geometry. But one can also resort to hyperbolic trigonometry. For example, evoking formula 2.2.2 (iv) in the back of Buser’s book [Reference Buser3] one gets

$$ \begin{align*}T(\varepsilon)=\left\{p\in T\,\middle\vert\, \cot\left(\frac{\pi}{3}-\frac{\varepsilon}{2}\right)\leq \sinh(d(p, \partial T))\leq\cot\left(\frac{\pi}{3}+\frac{\varepsilon}{2}\right)\right\}.\end{align*} $$

Suppose now that $\phi :X\to \Sigma $ is a critical realization of a trivalent graph X in our closed hyperbolic surface, and recall that by Lemma 2.2 we have that the connected component $\mathcal G^{\phi }$ of geodesic realizations homotopic to $\phi $ is isometric to a product of hyperbolic planes ${\mathbb {H}}^2\times \dots \times {\mathbb {H}}^2$ , one factor for each vertex of X. To each vertex x of X, we can associate first the tripod $\tau _x^{\phi }=(\phi (x),\{\phi (\vec e)\}_{\vec e\in \operatorname {\mathrm {\mathbf {half}}}_x(X)})$ consisting of the image under $\phi $ of the vertex x and of the unit vectors $\phi (\vec e)$ tangent to the images of the half-edges incident to x, and then $T_{\tau ^{\phi }_x}$ the ideal triangle associated to the critical tripod $\tau ^{\phi }_x$ . The assumption that $\phi $ is critical implies that the point $\phi (x)$ is the center of $T_{\tau ^{\phi }_x}$ for every vertex $x\in \operatorname {\mathrm {\mathbf {vert}}}$ . We let

$$ \begin{align*}\mathcal T^{\phi}=\prod_{x\in\operatorname{\mathrm{\mathbf{vert}}}(X)}T_{\tau_x^{\phi}}\subset\prod_{x\in\operatorname{\mathrm{\mathbf{vert}}}(X)}{\mathbb{H}}^2=\mathcal G^{\phi}\end{align*} $$

be the subset of $\mathcal G^{\phi }$ consisting of geodesic realizations homotopic to $\phi $ via a homotopy that maps each vertex x within $T_{\tau _x^{\phi }}$ . Accordingly, we set

$$ \begin{align*}\mathcal T^{\phi}(\varepsilon)=\prod_{x\in\operatorname{\mathrm{\mathbf{vert}}}(X)}T_{\tau_x^{\phi}}(\varepsilon)\subset\mathcal T^{\phi}\end{align*} $$

and we note that

$$ \begin{align*}\operatorname{\mathrm{vol}}(\mathcal T^{\phi}(\varepsilon))\sim \left(\frac 2{\sqrt 3}\varepsilon^2\right)^{\vert\operatorname{\mathrm{\mathbf{vert}}}(X)\vert}.\end{align*} $$

The reason we care about all these sets is that, asymptotically, $\mathcal T^{\phi }(\varepsilon )$ agrees with the set $\mathcal G^{\phi }_{\varepsilon -\operatorname {\mathrm {crit}}}$ of $\varepsilon $ -critical realizations $\psi $ homotopic to $\phi $ . Indeed, we get from Corollary 2.4 that the geodesic homotopy from $\psi $ to $\phi $ has lengths bounded by a uniform constant. This means that for all $e\in \operatorname {\mathrm {\mathbf {edge}}}(X)$ the geodesic segments $\phi (e)$ and $\psi (e)$ are very long but have endpoints relatively close to each other. This means that, when looked from (each) one of its vertices x, the segment $\psi (e)$ is very close to being asymptotic to $\phi (e)$ . In particular, the angle between $\psi (e)$ and the geodesic ray starting at $\psi (x)$ and asymptotic to the ray starting in $\phi (x)$ in direction $\phi (e)$ is smaller than $\delta =\delta (\min _{e\in \operatorname {\mathrm {\mathbf {edge}}}(X)}\ell (\psi (e)))$ for some positive function with $\delta (t)\to 0$ as $t\to \infty $ . With the rather clumsy notation, we find ourselves working with we have that $\delta $ bounds from above the angles between corresponding edges of the tripods $\tau ^{\psi }_x$ and $\tau ^{T^{\phi }_x}_{\psi (x)}$ . Since by assumption the angles of $\tau ^{\psi }_x$ belong to $[\frac {2\pi }3-\varepsilon ,\frac {2\pi }3+\varepsilon ]$ , we get that the angles of $\tau ^{T^{\phi }_x}_{\psi (x)}$ belong to $[\frac {2\pi }3-\varepsilon -2\delta ,\frac {2\pi }3+\varepsilon +2\delta ]$ , meaning that $\psi (x)\in T_{\tau _x^{\phi }}(\varepsilon +2\delta )$ .

To summarize, what we have proved is that for all $\varepsilon _1>\varepsilon $ there is $\ell _1$ such that if $\phi :X\to \Sigma $ is critical with $\ell (\phi (e))\geqslant \ell _1$ for all $e\in \operatorname {\mathrm {\mathbf {edge}}}$ , then

$$ \begin{align*}\mathcal G^{\phi}_{\varepsilon-\operatorname{\mathrm{crit}}}\subset\mathcal T^{\phi}(\varepsilon_1).\end{align*} $$

The same argument proves that for all $\varepsilon _2<\varepsilon $ there is $\ell _2$ such that if $\phi :X\to \Sigma $ is critical with $\ell (\phi (e))\geqslant \ell _2$ for all $e\in \operatorname {\mathrm {\mathbf {edge}}}$ , then

$$ \begin{align*}\mathcal T^{\phi}(\varepsilon_2)\subset\mathcal G^{\phi}_{\varepsilon-\operatorname{\mathrm{crit}}}\end{align*} $$

We record these facts:

Lemma 2.5. There are functions $h:(0,\frac \pi 6)\to \mathbb R_{>0}$ and $\delta :(0,\varepsilon _0)\to \mathbb R_{>0}$ with $\lim _{t\to 0}h(t)=0$ and $\lim _{t\to \infty }\delta (t)=0$ such that for every critical realizations $\phi :X\to \Sigma $ we have

$$ \begin{align*}\mathcal T^{\phi}(\varepsilon-r(\varepsilon,\phi))\subset\mathcal G^{\phi}_{\varepsilon-\operatorname{\mathrm{crit}}}\subset\mathcal T^{\phi}(\varepsilon+r(\varepsilon,\phi)),\end{align*} $$

where $r(\varepsilon ,\phi )=h(\varepsilon )+\delta (\min _{e\in \operatorname {\mathrm {\mathbf {edge}}}(X)}\ell (\phi (e)))$ .

To conclude this section, we collect what we will actually need about the set of $\varepsilon $ -critical realizations in a single statement.

Proposition 2.6. Let X be a trivalent graph. There are $\ell>0$ and functions $\rho _0,\rho _1:\mathbb R_{>0}\to \mathbb R_{>0}$ with $\lim _{t\to 0}\rho _0(t)=0$ and $\lim _{t\to \infty }\rho _1(t)=0$ and such that the following holds for all $\varepsilon \in [0,\frac \pi 6]$ :

If $\phi \in \mathcal G_{\varepsilon -\operatorname {\mathrm {crit}}}(X)$ is an $\ell $ -long $\varepsilon $ -critical realization, then

(2.10) $$ \begin{align} \left\vert\operatorname{\mathrm{vol}}(\mathcal G_{\varepsilon-\operatorname{\mathrm{crit}}}^{\phi})-\left(\frac 2{\sqrt 3}\varepsilon^2\right)^V\right\vert\leqslant \rho_0(\varepsilon)\cdot\rho_1\left(\min_{e\in\operatorname{\mathrm{\mathbf{edge}}}}\ell(\phi(e))\right)\cdot\varepsilon^{2V}. \end{align} $$

Moreover, $\mathcal G^{\phi }_{\varepsilon -\operatorname {\mathrm {crit}}}$ contains a unique critical realization $\psi $ and we have

(2.11) $$ \begin{align} \max_{e\in\operatorname{\mathrm{\mathbf{edge}}}}\vert\ell_{\Sigma}(\phi(e))-\ell_{\Sigma}(\psi(e))\vert\leqslant\rho_0(\varepsilon)+\rho_1\left(\min_{e\in\operatorname{\mathrm{\mathbf{edge}}}}\ell(\phi(e))\right)\cdot\varepsilon. \end{align} $$

Here, as before, V is the number of vertices of X and $\operatorname {\mathrm {\mathbf {edge}}}=\operatorname {\mathrm {\mathbf {edge}}}(X)$ is its set of edges.

Proof. Suppose to begin with that $\ell $ is at least as large as the $\ell _0$ in Corollary 2.4. We thus get that $\mathcal G^{\phi }$ contains a unique critical realization $\psi $ . Now, we get from Lemma 2.5 that

(2.12) $$ \begin{align} \mathcal T^{\phi}(\varepsilon-r(\varepsilon,\phi))\subset\mathcal G^{\phi}_{\varepsilon-\operatorname{\mathrm{crit}}}\subset\mathcal T^{\phi}(\varepsilon+r(\varepsilon,\phi)), \end{align} $$

where $r(\varepsilon ,\phi )=h(\varepsilon )+\delta (\min _{e\in \operatorname {\mathrm {\mathbf {edge}}}}\ell (\phi (e)))$ for some functions $h:(0,\frac \pi 6)\to \mathbb R_{>0}\text { with }$ and $\delta :(0,\varepsilon _0)\to \mathbb R_{>0}$ with $\lim _{t\to \infty }h(t)=0$ and $\lim _{t\to \infty }\delta (t)=0$ . The bounds (2.10) and (2.11) follow now from Equations (2.12) and (2.9).

3 Variations of Delsarte’s theorem

In this section, we establish the asymptotics of the number of realizations $\phi :X\to \Sigma $ satisfying some length condition, mapping each vertex to some predetermined point in $\Sigma $ , and so that the tuple of directions of images of half-edges belongs to some given open set of such tuples – see Theorem 3.2 for details. Other than some bookkeeping, what is needed is a slight extension of Delsarte’s lattice counting theorem [Reference Delsarte7], namely Theorem 3.1 below. This result is known to experts, and much more sophisticated versions than what we need here can be found in the literature – see, for example, [Reference Roblin17, Théorème 4.1.1]. However, for the sake of completeness, we will explain how to derive Theorem 3.1 from the fact that the geodesic flow is mixing. We refer to the very nice books [Reference Bekka and Mayer2, Reference Roblin17] for the needed background.

We start by recalling a few well-known facts about dynamics of geodesic flows on hyperbolic surfaces. First, recall that we can identify $T^1{\mathbb {H}}^2$ with $\operatorname {\mathrm {PSL}}_2\mathbb R$ , and $T^1\Sigma =T^1(\Gamma \backslash {\mathbb {H}}^2)$ with $\Gamma \backslash \operatorname {\mathrm {PSL}}_2\mathbb R$ . More specifically, when using the identification

$$ \begin{align*}\operatorname{\mathrm{PSL}}_2\mathbb R\to T^1{\mathbb{H}}^2\ \ g\mapsto \frac d{dt}g(e^ti)\vert_{t=0}\end{align*} $$

where the computation happens in the upper half-plane model and $i\in {\mathbb {H}}^2$ is the imaginary unit, then the geodesic and horocyclic flows amount to right multiplication by the matrices

$$ \begin{align*}\rho_t=\left(\begin{array}{cc}e^{\frac t2} & 0 \\ 0 & e^{-\frac t2}\end{array}\right)\text{ and }h_s=\left(\begin{array}{cc}1 & s \\ 0 & 1\end{array}\right),\end{align*} $$

respectively.

Note that $K=\operatorname {\mathrm {SO}}_2\subset \operatorname {\mathrm {PSL}}_2\mathbb R$ , the stabilizer of i under the action $\operatorname {\mathrm {PSL}}_2\mathbb R\curvearrowright {\mathbb {H}}^2$ , is a maximal compact subgroup – K corresponds under the identification $\operatorname {\mathrm {PSL}}_2\mathbb R\simeq T^1{\mathbb {H}}^2$ to the unit tangent space $T^1_i{\mathbb {H}}^2$ of the base point $i\in {\mathbb {H}}^2$ . The KAN decomposition (basically the output of the Gramm–Schmidt process from linear algebra) asserts that every element in $g\in \operatorname {\mathrm {PSL}}_2\mathbb R$ can be written in a unique way as

(3.1) $$ \begin{align} g=k\rho_th_s \end{align} $$

for $k\in K$ and $t,s\in \mathbb R$ . In those coordinates, the Haar measure of $\operatorname {\mathrm {PSL}}_2\mathbb R$ is given by

$$ \begin{align*}\int f(g)\,\operatorname{\mathrm{vol}}_{T^1{\mathbb{H}}^2}(g)=\iiint f(k\rho_th_s)\cdot e^{-t}\, dkdtds\end{align*} $$

where $dk$ stands for integrating over the arc length in $K\simeq T^1_i{\mathbb {H}}^2$ , normalized to have total measure $2\pi $ .

The basic fact we will need is that the geodesic flow $\rho _t$ on $T^1\Sigma $ is mixing [Reference Bekka and Mayer2, III.2.3], or rather one of its direct consequences, namely the fact that for each $x_0\in {\mathbb {H}}^2$ the projection to $\Sigma $ of the sphere $S(x_0,L)$ centered at $x_0$ and with radius L gets equidistributed in $\Sigma $ when $L\to \infty $ [Reference Bekka and Mayer2, III.3.3]. To be more precise, note that we might assume without loss of generality that $x_0=i$ , meaning that we are identifying $T_{x_0}^1{\mathbb {H}}^2=K$ . The fact that the geodesic flow is mixing implies then that for every nondegenerate interval $I\subset K$ the spherical arcs $I\rho _t$ equidistribute in $\Gamma \backslash \operatorname {\mathrm {PSL}}_2\mathbb R$ in the sense that for any continuous function $f\in C^0(T^1\Sigma )=C^0(\Gamma \backslash \operatorname {\mathrm {PSL}}_2\mathbb R)$ we have

(3.2) $$ \begin{align} \lim_{L\to\infty}\frac 1{\ell(I)}\int_{I}f(\Gamma k\rho_L)\,dk=\frac 1{\operatorname{\mathrm{vol}}_{T^1\Sigma}(T^1\Sigma)}\int_{T^1\Sigma} f(\Gamma g)\cdot dg, \end{align} $$

where $\ell (I)$ is the arc length of I and where the second integral is with respect to $\operatorname {\mathrm {vol}}_{T^1\Sigma }$ . Anyways, we care about equidistribution of the spheres for the following reason. If $f\in C^0(\Sigma )$ is a continuous function and if we let $\tilde f$ be the composition of f with the cover ${\mathbb {H}}^2\to \Sigma $ , then Equation (3.2) implies, with $I=K$ , that

$$ \begin{align*}\lim_{L\to\infty}\frac 1{\operatorname{\mathrm{vol}}_{{\mathbb{H}}^2}(B(x_0,L))}\cdot\int_{B(x_0,L)}\tilde f(x)\, dx=\frac 1{\operatorname{\mathrm{vol}}(\Sigma)}\int_{\Sigma} f(x)dx.\end{align*} $$

Applying this to a nonnegative function $f_{y_0,\varepsilon }\in C^0(\Sigma )$ with total integral $1$ and supported by $B(y_0,\varepsilon )$ , we get that

$$ \begin{align*}\int_{B(x_0,L)}\tilde f_{y_0,\varepsilon}(x)\, dx\sim\frac{\operatorname{\mathrm{vol}}_{{\mathbb{H}}^2}(B(x_0,L))}{\operatorname{\mathrm{vol}}(\Sigma)}.\end{align*} $$

Now, from the properties of $f_{y_0,\varepsilon }$ we get that

(3.3) $$ \begin{align} \int_{B(x,L-\varepsilon)}\tilde f_{y_0,\varepsilon}(\cdot)d\operatorname{\mathrm{vol}}_{{\mathbb{H}}^2}\leqslant\vert\Gamma\cdot y_0\cap B(x_0,L)\vert\leqslant\int_{B(x,L+\varepsilon)}\tilde f_{y_0,\varepsilon}(\cdot)d\operatorname{\mathrm{vol}}_{{\mathbb{H}}^2}. \end{align} $$

When taken together, the last two displayed equations imply that for all $\varepsilon $ one has

$$ \begin{align*}\frac{\pi\cdot e^{L-\varepsilon}}{\operatorname{\mathrm{vol}}(\Sigma)}\leqslant\vert\Gamma\cdot y_0\cap B(x_0,L)\vert\leqslant\frac{\pi\cdot e^{L+\varepsilon}}{\operatorname{\mathrm{vol}}(\Sigma)}\end{align*} $$

and hence that

(3.4) $$ \begin{align} \vert\Gamma\cdot y_0\cap B(x_0,L)\vert\sim\frac{\pi\cdot e^L}{\operatorname{\mathrm{vol}}(\Sigma)}\text{ as }L\to\infty. \end{align} $$

This is Delsarte’s lattice point counting theorem [Reference Delsarte7] – we refer to [Reference Bekka and Mayer2, III.3.5] for more details.

An observation: Note that counting elements of $\Gamma \cdot y_0$ contained in the ball $B(x_0,L)$ is exactly the same thing as counting geodesic arcs in $\Sigma $ of length at most L going from $x_0$ to $y_0$ , or to be precise, from the projection of $x_0$ to the projection of $y_0$ . In this way, if we are given a segment $I\subset K=T_{x_0}^1\Sigma $ and we use the equidistribution of the spherical segments $I\rho _t$ , then when we run the argument above we get the cardinality of the set ${\mathbf {A}}_{I,y_0}(L)$ of geodesic arcs of length at most L, starting in $x_0$ with initial speed in I, and ending in $y_0$ . As expected, the result is that

$$ \begin{align*}\vert{\mathbf{A}}_{I,y_0}(L)\vert\sim \frac{\ell(I)}{2\pi}\cdot\frac{\pi\cdot e^L}{\operatorname{\mathrm{vol}}(\Sigma)}\ \ \ \text{ when }L\to\infty.\end{align*} $$

Suppose now that we dial it up a bit giving ourselves a second sector $J\subset T^1_{y_0}\Sigma $ and care, for some fixed h, about the cardinality of the set ${\mathbf {A}}_{I,J}(L,h)$ of geodesic arcs with length in $[L,L+h]$ , joining $x_0$ to $y_0$ , and with initial and terminal velocities in I and J, respectively. We can obtain the asymptotics of $\vert {\mathbf {A}}_{I,J}(L,h)\vert $ following the same basic idea as in the proof of Delsarte’s theorem above. We need, however, to replace the bump function $f_{y_0,\varepsilon }$ by something else.

Using the coordinates (3.1), consider for h and $\delta $ positive and small the set

$$ \begin{align*}{\mathcal{J}}(J,h,\delta)=\{J\rho_t h_s\text{ with }s\in[-\delta,\delta]\text{ and }t\in[0,h]\}\subset T^1\Sigma\end{align*} $$

and note that it has volume $\ell (J)\cdot 2\delta \cdot (1-e^{-h})$ . Note also that the intersection of ${\mathcal {J}}(J,h,\delta )$ with the outer normal vector field of any horosphere consists of segments of the form $H_{v\rho _t}=\{v\rho _t h_s\text { with }s\in [-\delta ,\delta ]\}$ , where $v\in J$ and $t\in [0,h]$ and that each such segment has length $2\delta $ . Finally, observe that each one of the sets $H_{v\rho _t}$ contains exactly one vector, namely $v\rho _t$ , which lands in J when we geodesic flow it for some time in $[-h,0]$ . It follows that for all intervals $\mathbb I\subset \mathbb R$ and $w\in T^1\Sigma $ we have

$$ \begin{align*}\left\vert\left\vert\left\{r\in\mathbb I\,\middle\vert\begin{array}{c}\exists s\in[-\delta,\delta],t\in[-h,0]\\ \text{ with }wh_s\rho_t\in J\end{array}\right\}\right\vert-\frac{\ell(\{s\in\mathbb I\text{ with }wh_s\in{\mathcal{J}}\})}{2\delta}\right\vert\leqslant 2,\end{align*} $$

where ${\mathcal {J}}={\mathcal {J}}(J,h,\delta )$ and where the number $2$ is there to take into account possible over counting near the ends of the horospherical segment.

Using then the fact that when $L\to \infty $ the sphere $S(x_0,L+h)\subset {\mathbb {H}}^2$ looks more and more like a horosphere, one gets the following analogue of Equation (3.3): For any $\delta '<\delta <\delta "$ and $h'<h<h"$ and $J'\Subset J\Subset J"$ , we have for all sufficiently large L

$$ \begin{align*}\int_{I\rho_{L+h}}\chi_{{\mathcal{J}}(J',h',\delta')}\leqslant 2\delta\cdot\vert{\mathbf{A}}_{I,J}(L,h)\vert\leqslant\int_{I\rho_{L+h}}\chi_{{\mathcal{J}}(J",h",\delta")},\end{align*} $$

where $\chi _{\mathcal {J}}$ is the characteristic function of ${\mathcal {J}}$ . From here, we get that

$$ \begin{align*} \vert{\mathbf{A}}_{I,J}(L,h)\vert &\sim \frac 1{2\delta}\int_{I\rho_{L+h}}\chi_{{\mathcal{J}}(J,h,\delta)}\sim\frac 1{2\delta}\cdot\frac{e^{L+h}}2\cdot\int_I\chi_{{\mathcal{J}}(J,h,\delta)}(k\rho_{L+h})\ dk\\ &\stackrel{\text{(3.2)}}\sim\frac 1{2\delta}\cdot\frac{e^{L+h}}2\cdot\ell(I)\cdot\frac{\operatorname{\mathrm{vol}}_{T^1\Sigma}({\mathcal{J}}(J,h,\delta))}{\operatorname{\mathrm{vol}}_{T^1\Sigma}(T^1\Sigma)}\\ &\sim \frac{\ell(I)\cdot\ell(J)}{4\pi}\cdot\frac{e^{L+h}-e^L}{\operatorname{\mathrm{vol}}(\Sigma)}. \end{align*} $$

We record this slightly generalized version of Delsarte’s theorem for later reference:

Theorem 3.1 (Delarte’s theorem for in-out sectors)

Let $\Sigma $ be a closed hyperbolic surface, let h be positive and let $I\subset T_{x_0}\Sigma $ and $J\subset T_{y_0}\Sigma $ be nondegenerate segments. Let then ${\mathbf {A}}_{I,J}(L,h)$ be the set of geodesic arcs $\alpha :[0,r]\to \Sigma $ with length $r\in [L,L+h]$ , with endpoints $\alpha (0)=x_0$ and $\alpha (r)=y_0$ , and with initial and terminal speeds satisfying $\alpha '(0)\in I$ and $\alpha '(r)\in J$ . Then we have

$$ \begin{align*}\vert{\mathbf{A}}_{I,J}(L,h)\vert\sim \frac{\ell(I)\cdot\ell(J)}{4\pi}\cdot\frac{e^{L+h}-e^L}{\operatorname{\mathrm{vol}}(\Sigma)}\end{align*} $$

when $L\to \infty $ .

As we mentioned earlier, Theorem 3.1 is a very special case of the much more general [Reference Roblin17, Théorème 4.1.1]. We would not be surprised if there were also other references we are not aware of.

Now, let X be a trivalent graph with vertex set $\operatorname {\mathrm {\mathbf {vert}}}(X)$ and let $\vec x=(x_v)_{v\in \operatorname {\mathrm {\mathbf {vert}}}(X)}\in \Sigma ^{\operatorname {\mathrm {\mathbf {vert}}}(X)}$ be a $\operatorname {\mathrm {\mathbf {vert}}}(X)$ -tuple of points in the surface. Let also

$$ \begin{align*}U\subset\prod_{v\in\operatorname{\mathrm{\mathbf{vert}}}(X)}\left(\bigoplus_{\bar e\in\operatorname{\mathrm{\mathbf{half}}}_v(X)} T^1_{x_v}\Sigma\right)\stackrel{\text{def}}=\mathbb T_{\vec x}\end{align*} $$

be an open set where, as always, $\operatorname {\mathrm {\mathbf {half}}}_v(X)$ is the set of all half-edges of X starting at the vertex v.

Given for each edge $e\in \operatorname {\mathrm {\mathbf {edge}}}(X)$ a positive number $L_e$ and writing $\vec L=(L_e)_{e\in \operatorname {\mathrm {\mathbf {edge}}}(X)}$ , we are going to be interested in the set ${\mathbf {G}}^X_{U}(\vec L_e,h)$ of realizations $\phi :X\to \Sigma $

1. (a) mapping each vertex $v\in \operatorname {\mathrm {\mathbf {vert}}}(X)$ to the point $x_v$ ,

2. (b) with $(\phi (\vec e))_{\bar e\in \operatorname {\mathrm {\mathbf {half}}}(X)}\in U$ and

3. (c) with $\ell (e)\in [L_e,L_e+h]$ for all $e\in \operatorname {\mathrm {\mathbf {edge}}}(X)$ .

In this setting, we have the following version of Delsarte’s theorem relating the cardinality of ${\mathbf {G}}^X_{U}(\vec L_e,h)$ with the volume $\operatorname {\mathrm {vol}}_{\vec x}(U)$ of U, where the volume is measured in the flat torus $\mathbb T_{\vec x}$ .

Theorem 3.2 (Delarte’s theorem for graph realizations)

Let $\Sigma $ be a closed hyperbolic surface, let X be a finite graph, fix $\vec x\in \Sigma ^{\operatorname {\mathrm {\mathbf {vert}}}(X)}$ and an open set $U_{\vec x}\subset \mathbb T_{\vec x}$ . If $\operatorname {\mathrm {vol}}_{\vec x}(\bar U_{\vec x}\setminus U_{\vec x})=0$ , then for every $h>0$ we have

$$ \begin{align*}\vert{\mathbf{G}}^X_{U_{\vec x}}(\vec L,h)\vert\sim \frac{\operatorname{\mathrm{vol}}_{\vec x}(U_{\vec x})}{(4\pi)^{E}}\cdot\frac{(e^h-1)^{E}\cdot e^{\Vert\vec L\Vert}}{\operatorname{\mathrm{vol}}(\Sigma)^{E}}\end{align*} $$

when $\min _{e\in \operatorname {\mathrm {\mathbf {edge}}}(X)}L_e\to \infty $ . Here, $\bar U_{\vec x}$ is the closure of $U_{\vec x}$ in $\mathbb T_{\vec x}$ and $\operatorname {\mathrm {vol}}_{\vec x}(\cdot )$ stands for the volume therein. Also, $E=\vert \operatorname {\mathrm {\mathbf {edge}}}(X)\vert $ is the number of edges in X, and $\Vert \vec L\Vert =\sum _{e\in \operatorname {\mathrm {\mathbf {edge}}}(X)}L_e$ .

We stress that $\operatorname {\mathrm {vol}}_{\vec x}$ is normalized in such a way that $\operatorname {\mathrm {vol}}_{\vec x}(\mathbb T_{\vec x})=(2\pi )^{2 E}$ . We also stress that this is consistent with the fact that we use the interval $[0,\pi ]$ to measure unoriented angles.

Proof. Let us say that a closed set of the form

$$ \begin{align*}\prod_{v\in\operatorname{\mathrm{\mathbf{vert}}}(X)}\left(\prod_{\bar e\in\operatorname{\mathrm{\mathbf{half}}}_v(X)} I_{\bar e}\right)\subset\mathbb T_{\vec x}=\prod_{v\in\operatorname{\mathrm{\mathbf{vert}}}(X)}\left(\bigoplus_{\bar e\in\operatorname{\mathrm{\mathbf{half}}}_v(X)} T^1_{x_v}\Sigma\right),\end{align*} $$

where each $I_{\bar e}$ is a segment in $T^1_{x_v}\Sigma $ is a cube. We say that a closed subset of $\prod _{v\in \operatorname {\mathrm {\mathbf {vert}}}(X)}\left (\oplus _{\bar e\in \operatorname {\mathrm {\mathbf {half}}}_v(X)} T^1_{x_v}\Sigma \right )$ it cubical if it can be given as the union of finitely many cubes with disjoint interiors. The assumption that the open set $U=U_{\vec x}$ in the statement is such that $\bar U\setminus U$ is a null-set implies that U can be approximated from inside and outside by cubical sets $U'\subset U\subset U"$ with $\operatorname {\mathrm {vol}}(U")-\operatorname {\mathrm {vol}}(U')$ as small as we want. It follows that it suffices to prove the theorem if U is (the interior of) a cubical set.

Note now that if $U=U_1\cup U_2$ is the disjoint union of two sets $U_1$ and $U_2$ and if the statement of the theorem holds true for $U_1$ and $U_2$ , then it also holds true for U. Since every cubical set is made out of finitely many cubes with disjoint interior, we deduce that it really suffices to prove the theorem for individual cubes

$$ \begin{align*}U=\prod_{v\in\operatorname{\mathrm{\mathbf{vert}}}(X)}\left(\prod_{\bar e\in\operatorname{\mathrm{\mathbf{half}}}_v(X)} I_{\bar e}\right).\end{align*} $$

Note that, up to shuffling the factors we can see U as

$$ \begin{align*}U=\prod_{e\in\operatorname{\mathrm{\mathbf{edge}}}(X)}\left(I_{e^+}\times I_{e^-}\right).\end{align*} $$

Here, we are denoting by $e^+$ and $e^-$ the two half-edges of the edge $e\in \operatorname {\mathrm {\mathbf {edge}}}(X)$ . Now, unpacking the notation one sees that a realization $\phi $ belongs to ${\mathbf {G}}_{U}(\vec L_e,h)$ if and only if for all $e\in \operatorname {\mathrm {\mathbf {edge}}}(X)$ the arc $\phi (e)$ belongs to ${\mathbf {A}}_{I_{e^+},I_{e^-}}(L_e,h)$ . It follows thus from Delsarte’s theorem for in-out sectors that

$$ \begin{align*} \vert{\mathbf{G}}_{U}^X(\vec L,h)\vert &=\prod_{e\in\operatorname{\mathrm{\mathbf{edge}}}(X)}\left\vert{\mathbf{A}}_{I_{e^+},I_{e^-}}(L_e,h)\right\vert\\ &\sim\prod_{e\in\operatorname{\mathrm{\mathbf{edge}}}(X)}\left(\frac{\ell(I_{e^+})\cdot\ell(I_{e^-})}{4\pi}\cdot \frac{(e^h-1)\cdot e^{L_e}}{\operatorname{\mathrm{vol}}(\Sigma)}\right)\\ &=\frac{\prod_{e\in\operatorname{\mathrm{\mathbf{edge}}}(X)}(\ell(I_{e^+})\cdot\ell(I_{e^-}))}{(4\pi)^{E}}\cdot \frac{(e^h-1)^{E}\cdot e^{\sum_{e\in\operatorname{\mathrm{\mathbf{edge}}}(X)}L_e}}{\operatorname{\mathrm{vol}}(\Sigma)^{E}}\\ &=\frac{\operatorname{\mathrm{vol}}(U)}{(4\pi)^{E}}\cdot \frac{(e^h-1)^{E}\cdot e^{\Vert\vec L\Vert}}{\operatorname{\mathrm{vol}}(\Sigma)^{E}}, \end{align*} $$

where, as in the statement we have set $E=\vert \operatorname {\mathrm {\mathbf {edge}}}(X)\vert $ . We are done.

Let us consider now the concrete case we will care mostly about. Suppose namely that X is a trivalent graph and that we want all the angles to be between $\frac 23\pi -\varepsilon $ and $\frac 23\pi +\varepsilon $ . In other words, we are interested in the set of $\varepsilon $ -critical realizations $\phi :X\to \Sigma $ which map the vertices to our prescribed tuple $\vec x\in \Sigma ^{\operatorname {\mathrm {\mathbf {vert}}}(X)}$ . Then we have to consider the set

$$ \begin{align*}U_{\vec x,\varepsilon-\operatorname{\mathrm{crit}}}^X\subset\prod_{v\in\operatorname{\mathrm{\mathbf{vert}}}(X)}\left(\bigoplus_{\bar e\in\operatorname{\mathrm{\mathbf{half}}}_v(X)} T^1_{x_v}\Sigma\right)\end{align*} $$

of those tuples $(v_{\vec e})_{\vec e\in \operatorname {\mathrm {\mathbf {half}}}(X)}$ with $\angle (v_{\vec e_1},v_{\vec e_2})\in [\frac {2\pi }3-\varepsilon ,\frac {2\pi }3+\varepsilon ]$ for all distinct $\vec e_1,\vec e_2\in \operatorname {\mathrm {\mathbf {half}}}(X)$ incident to the same vertex.

Let us compute the volume of $U_{\vec x,\varepsilon -\operatorname {\mathrm {crit}}}^X$ . Noting that the conditions on $U_{\vec x,\varepsilon -\operatorname {\mathrm {crit}}}^X$ associated to different vertices are independent, we get that it suffices to think vertex-by-vertex and then multiply all the numbers obtained for all vertices. For each vertex v, label arbitrarily the half-edges incident to v by $\vec e_1,\vec e_2$ and $\vec e_3$ . We have no restriction for the position of $v_{\vec e_1}\in T_{x_v}\Sigma $ . Once we have fixed $v_{\vec e_1}\in T_{x_v}^1(\Sigma )$ , we get that $v_{\vec e_2}$ can belong to two segments, each one of length $2\varepsilon $ , in $T^1_{x_v}(\Sigma )$ – recall that all our angles are unoriented. Then, once we have fixed $v_{\vec e_1}$ and $v_{\vec e_2}$ , we have to choose $v_{\vec e_3}$ in an interval of length $2\varepsilon -\vert \angle (v_{\vec e_1},v_{\vec e_2})-\frac {2\pi }3\vert $ . This means that when we have chosen $v_{\vec e_1}$ and which segment $v_{\vec e_2}$ is in, then we have $3\varepsilon ^2$ worth of choices of $(v_{\vec e_2},v_{\vec e_3})$ . This means that the set of possible choices in $T_{x_v}^1\Sigma \times T_{x_v}^1\Sigma \times T_{x_v}^1\Sigma $ has volume equal to $12\pi \varepsilon ^2$ . Since, as we already mentioned earlier, the conditions at all vertices of X are independent, we get that

$$ \begin{align*}\operatorname{\mathrm{vol}}_{\vec x}(U_{\vec x,\varepsilon-\operatorname{\mathrm{crit}}}^X)=(12\pi\varepsilon^2)^{\vert\operatorname{\mathrm{\mathbf{vert}}}(X)\vert}.\end{align*} $$

From Theorem 3.2, we get thus that for all $h>0$ and all $\vec x\in \Sigma ^{\operatorname {\mathrm {\mathbf {vert}}}(X)}$ we have

$$ \begin{align*}\vert{\mathbf{G}}^X_{\vec x,\varepsilon-\operatorname{\mathrm{crit}}}(\vec L,h)\vert\sim \frac{(12\pi\varepsilon^2)^{V}}{(4\pi)^{E}}\cdot\frac{(e^h-1)^{E}\cdot e^{\Vert\vec L\Vert}}{\operatorname{\mathrm{vol}}(\Sigma)^{E}}\text{ as }\min_{e\in\operatorname{\mathrm{\mathbf{edge}}}(X)}L_e\to\infty,\end{align*} $$

where we are writing V and E for the number of vertices and edges of X, and ${\mathbf {G}}^X_{\vec x,\varepsilon -\operatorname {\mathrm {crit}}}$ instead of ${\mathbf {G}}^X_{U_{\vec x,\varepsilon -\operatorname {\mathrm {crit}}}}$ . Taking into account that X is trivalent and hence satisfies $V=-2\chi (X)$ and $E=-3\chi (X)$ , we can clean this up to

$$ \begin{align*}\vert{\mathbf{G}}^X_{\vec x,\varepsilon-\operatorname{\mathrm{crit}}}(\vec L,h)\vert\sim \varepsilon^{4\vert\chi(X)\vert}\cdot\left(\frac{2}{3}\right)^{2\chi(X)}\cdot\pi^{\chi(X)}\cdot\frac{(e^h-1)^{-3\chi(X)}\cdot e^{\Vert\vec L\Vert}}{\operatorname{\mathrm{vol}}(\Sigma)^{-3\chi(X)}}\end{align*} $$

as $\min _{e\in \operatorname {\mathrm {\mathbf {edge}}}(X)}L_e\to \infty $ . Moreover, since the geometry of the set $U_{\vec x}^X(\varepsilon -\operatorname {\mathrm {crit}})$ is independent of the point $\vec x$ , we get that the speed of convergence to this asymptotic is independent of $\vec x$ . Altogether, we have the following result:

Corollary 3.3. Let $\Sigma $ be a closed hyperbolic surface and X be a trivalent graph, fix $\varepsilon>0$ and $h>0$ and for $\vec x\in \Sigma ^{\operatorname {\mathrm {\mathbf {vert}}}(X)}$ , let ${\mathbf {G}}^X_{\vec x,\varepsilon -\operatorname {\mathrm {crit}}}(\vec L,h)$ be the set of $\varepsilon $ -critical realizations $\phi :X\to \Sigma $ mapping the vertex v to the point $x_v=\phi (v)$ . Then we have

(3.5) $$ \begin{align} \vert{\mathbf{G}}_{\vec x,\varepsilon-\operatorname{\mathrm{crit}}}^X(\vec L,h)\vert \sim \varepsilon^{4\vert\chi(X)\vert}\cdot\left(\frac{2}{3}\right)^{2\chi(X)}\cdot\pi^{\chi(X)}\cdot\frac{(e^h-1)^{-3\chi(X)}\cdot e^{\Vert\vec L\Vert}}{\operatorname{\mathrm{vol}}(\Sigma)^{-3\chi(X)}} \end{align} $$

as $\min _{e\in \operatorname {\mathrm {\mathbf {edge}}}(X)}L_e\to \infty $ .

Note that the set $U_{\vec x,\varepsilon -\operatorname {\mathrm {crit}}}^X$ needed to establish Corollary 3.3 is basically the same for all choices of $\vec x$ . For example, if we take another $\vec y\in \Sigma ^{\operatorname {\mathrm {\mathbf {vert}}}(X)}$ and we identity $\mathbb T_{\vec x}$ with $\mathbb T_{\vec y}$ isometrically by parallel transport (along any collection of curves whatsoever), then $U_{\vec x,\varepsilon -\operatorname {\mathrm {crit}}}^X$ is sent to $U_{\vec y,\varepsilon -\operatorname {\mathrm {crit}}}^X$ . It follows that we can approximate $U_{\vec x,\varepsilon -\operatorname {\mathrm {crit}}}^X$ and $U_{\vec y,\varepsilon -\operatorname {\mathrm {crit}}}^X$ at the same time by cubical sets consisting of the same number of cubes with the same side lengths. It thus follows from the comment following Theorem 3.1 that the asymptotics in Corollary 3.3 is uniform in $\vec x$ . We record this fact:

Addendum to Corollary 3.3. The asymptotics in Equation (3.5) is uniform in $\vec x\in \Sigma ^{\operatorname {\mathrm {\mathbf {vert}}}(X)}$ .

4 Counting critical realizations

In this section, we prove Theorem 1.3 from the introduction. Before restating the theorem, recall that if X is a trivalent graph then we denote by

(4.1) $$ \begin{align} {\mathbf{G}}^X(L)=\left\{\begin{array}{c}\phi:X\to\Sigma\text{ critical realization}\\ \text{with length }\ell_{\Sigma}(\phi)\leqslant L\end{array}\right\} \end{align} $$

the set of all critical realizations of length $\ell _{\Sigma }(\phi )$ at most L.

Theorem 1.3. Let $\Sigma $ be a closed, connected and oriented hyperbolic surface. For every connected trivalent graph X, we have

$$ \begin{align*}\vert{\mathbf{G}}^X(L)\vert\sim \left(\frac 23\right)^{3\chi(X)}\cdot\frac{\operatorname{\mathrm{vol}}(T^1\Sigma)^{\chi(X)}}{(-3\chi(X)-1)!}\cdot L^{-3\chi(X)-1}\cdot e^{L}\end{align*} $$

as $L\to \infty $ .

Fixing for the remaining of this section the trivalent graph X, we will write $\operatorname {\mathrm {\mathbf {vert}}}=\operatorname {\mathrm {\mathbf {vert}}}(X)$ and $\operatorname {\mathrm {\mathbf {edge}}}=\operatorname {\mathrm {\mathbf {edge}}}(X)$ for the sets of vertices and edges and $V=\vert \operatorname {\mathrm {\mathbf {vert}}}\vert $ and $E=\vert \operatorname {\mathrm {\mathbf {edge}}}\vert $ for their cardinalities. Similarly, we will denote by $\mathcal G=\mathcal G^X$ the manifold of realizations of X in $\Sigma $ , and by ${\mathbf {G}}(L)={\mathbf {G}}^X(L)$ the set of critical realizations with length at most L.

The main step in the proof of Theorem 1.3 is to count critical realizations of X such that the corresponding vectors of lengths $(\ell (\phi (e)))_{e\in \operatorname {\mathrm {\mathbf {edge}}}}$ belong to a box of size $h>0$ . More concretely, we want to count how many elements there are in the set

(4.2) $$ \begin{align} {\mathbf{G}}(\vec L,h)=\left\{\begin{array}{c}\phi:X\to\Sigma\text{ critical realization with}\\ \ell(\phi(e))\in(L_e,L_e+h]\text{ for all }e\in\operatorname{\mathrm{\mathbf{edge}}}\end{array}\right\}, \end{align} $$

where $\vec L=(L_e)_{e\in \operatorname {\mathrm {\mathbf {edge}}}}$ is a positive vector. We start by establishing some form of an upper bound for the number of homotopy classes of realizations when we bound the length of each individual edge – recall that by Lemma 2.2 any two homotopic critical realizations are identical.

It is worth pointing out that Lemma 4.1 fails if $\Sigma $ is allowed to have cusps – see Section 9.

Proof. Let us fix a point $x_0\in \Sigma $ , and note that every point in $\Sigma $ – think of the images under a realization of the vertices of X – can be moved to $x_0$ along a path of length at most $\operatorname {\mathrm {diam}}(\Sigma )$ . It follows that every realization $\phi :X\to \Sigma $ is homotopic to a new realization $\psi :X\to \Sigma $ mapping all vertices of X to $x_0$ and with

(4.3) $$ \begin{align} \ell(\psi(e))\leqslant\ell(\phi(e))+2\cdot\operatorname{\mathrm{diam}}(\Sigma)\leqslant L_e+2\cdot\operatorname{\mathrm{diam}}(\Sigma) \end{align} $$

for every edge $e\in \operatorname {\mathrm {\mathbf {edge}}}(X)$ . Note that the homotopy class of $\psi $ is determined by the homotopy classes of the loops $\psi (e)$ when e ranges over the edges of X. Now, Equation (4.3) implies that, up to homotopy, we have at most $\operatorname {\mathrm {\mathbf {const}}}\cdot e^{L_e}$ choices for the geodesic segment $\psi (e)$ . This implies that there are at most $\operatorname {\mathrm {\mathbf {const}}}\cdot e^{\Vert \vec L\Vert }$ choices for the homotopy class of $\psi $ , and hence for the homotopy class of $\phi $ . We are done.

Although it is evidently pretty coarse, Lemma 4.1 will play a key role in the proof of Theorem 1.3. However, the main tool in the proof of the theorem is the following:

Proposition 4.2. For all $h>0$ , we have

$$ \begin{align*}\vert{\mathbf{G}}(\vec L,h)\vert\sim\frac {2^{4\chi(X)}}{3^{3\chi(X)}}\cdot\pi^{\chi(X)}\cdot \frac{(e^{h}-1)^{-3\chi(X)}\cdot e^{\Vert\vec L\Vert}}{\operatorname{\mathrm{vol}}(\Sigma)^{-\chi(X)}}\end{align*} $$

as $\min _{e\in \operatorname {\mathrm {\mathbf {edge}}}}L_e\to \infty $ . Here, ${\mathbf {G}}(\vec L,h)$ is as in Equation (4.2).

Proof. Denote by $\mathcal G(\vec L,h)\subset \mathcal G$ the set of all realizations $\phi :X\to \Sigma $ with $\ell _{\Sigma }(\phi )\in [L_e,L_e+h]$ for all $e\in \operatorname {\mathrm {\mathbf {edge}}}$ and then let

$$ \begin{align*} G_{\varepsilon}(\vec L,h)&=\left\{ \mathcal G^{\phi}\in\pi_0(\mathcal G)\text{ with }\mathcal G^{\phi}_{\varepsilon-\operatorname{\mathrm{crit}}}\subset \mathcal G(\vec L,h)\right\}\\ \hat G_{\varepsilon}(\vec L,h)&=\left\{ \mathcal G^{\phi}\in\pi_0(\mathcal G)\text{ with }\mathcal G^{\phi}_{\varepsilon-\operatorname{\mathrm{crit}}}\cap\mathcal G(\vec L,h)\neq\varnothing\right\} \end{align*} $$

be the sets of connected components of $\mathcal G$ whose set of $\varepsilon $ -critical realizations is fully contained in (resp. which meet) $\mathcal G(\vec L,h)$ . It follows from Corollary 2.4 that there is some $\ell _0$ such that as long as $\vec L$ satisfies that $\min L_e\geqslant \ell _0$ , then each component listed in $\hat G_{\varepsilon }(\vec L,h)$ contains exactly one critical realization of the graph X. Assuming from now on that we are in this situation, we get that

$$ \begin{align*}\vert G_{\varepsilon}(\vec L,h)\vert\leqslant\vert{\mathbf{G}}(\vec L, h)\vert\leqslant\vert\hat G_{\varepsilon}(\vec L,h)\vert.\end{align*} $$

Now, from Equation (2.10) in Proposition 2.6 we get that for all $\delta>0$ there is $\ell _1>\ell _0$ with

$$ \begin{align*}(1-\delta)\cdot\operatorname{\mathrm{vol}}\left(\bigcup_{\mathcal G^{\phi}\in G_{\varepsilon}(\vec L,h)}\mathcal G^{\phi}_{\varepsilon-\operatorname{\mathrm{crit}}}\right)<\left(\frac 2{\sqrt 3}\varepsilon^2\right)^V\cdot\vert G_{\varepsilon}(\vec L,h)\vert\end{align*} $$

$$ \begin{align*}(1+\delta)\cdot\operatorname{\mathrm{vol}}\left(\bigcup_{\mathcal G^{\phi}\in\hat G_{\varepsilon}(\vec L,h)}\mathcal G^{\phi}_{\varepsilon-\operatorname{\mathrm{crit}}}\right)>\left(\frac 2{\sqrt 3}\varepsilon^2\right)^V\cdot \vert \hat G_{\varepsilon}(\vec L,h)\vert\end{align*} $$

whenever $\varepsilon $ is small enough and $\min L_e\geqslant \ell _1$ . Altogether, we get that for all $\varepsilon $ positive and small we have

$$ \begin{align*} (1-\delta)\cdot\left(\frac 2{\sqrt 3}\varepsilon^2\right)^{-V}\cdot\operatorname{\mathrm{vol}}\left(\bigcup_{\mathcal G^{\phi}\in G_{\varepsilon}(\vec L,h)}\mathcal G^{\phi}_{\varepsilon-\operatorname{\mathrm{crit}}}\right)&<\vert{\mathbf{G}}(\vec L, h)\vert\\[5pt] (1+\delta)\cdot\left(\frac 2{\sqrt 3}\varepsilon^2\right)^{-V}\cdot\operatorname{\mathrm{vol}}\left(\bigcup_{\mathcal G^{\phi}\in\hat G_{\varepsilon}(\vec L,h)}\mathcal G^{\phi}_{\varepsilon-\operatorname{\mathrm{crit}}}\right)&>\vert{\mathbf{G}}(\vec L, h)\vert \end{align*} $$

for all $\vec L$ with $\min L_e\geqslant \ell _1$ .

We get now from Equation (2.11) in Proposition 2.6 that there is $\ell _2>\ell _1$ such that, as long as $\varepsilon $ is under some threshold, we have that whenever $\phi ,\psi \in \mathcal G$ are homotopic $\varepsilon $ -critical realizations with $\ell (\phi (e)),\ell (\psi (e))\geqslant \ell _2$ for all $e\in \operatorname {\mathrm {\mathbf {edge}}}$ , then we have that the lengths $\ell (\phi (e))$ and $\ell (\psi (e))$ differ by at most $2\varepsilon $ for each edge $e\in \operatorname {\mathrm {\mathbf {edge}}}$ . This implies that for any such $\vec L$ and $\varepsilon $ we have

$$ \begin{align*} \mathcal G_{\varepsilon-\operatorname{\mathrm{crit}}}(\vec L+[2\varepsilon],h-4\varepsilon)&\subset \bigcup_{\mathcal G^{\phi}\in G_{\varepsilon}(\vec L,h)}\mathcal G^{\phi}_{\varepsilon-\operatorname{\mathrm{crit}}}\\ \mathcal G_{\varepsilon-\operatorname{\mathrm{crit}}}(\vec L-[2\varepsilon],h+4\varepsilon)&\supset \bigcup_{\mathcal G^{\phi}\in\hat G_{\varepsilon}(\vec L,h)}\mathcal G^{\phi}_{\varepsilon-\operatorname{\mathrm{crit}}}, \end{align*} $$

where $\vec L+[t]\in \mathbb R^{\operatorname {\mathrm {\mathbf {edge}}}}$ is the vector with entries $(\vec L+[t])_e=\vec L_e+t$ .

Let us summarize what we have obtained so far:

$$ \begin{align*} (1-\delta)\cdot\left(\frac 2{\sqrt 3}\varepsilon^2\right)^{-V}\cdot\operatorname{\mathrm{vol}}\left(\mathcal G_{\varepsilon-\operatorname{\mathrm{crit}}}(\vec L+[2\varepsilon],h-4\varepsilon)\right)&<\vert{\mathbf{G}}(\vec L, h)\vert\\ (1+\delta)\cdot\left(\frac 2{\sqrt 3}\varepsilon^2\right)^{-V}\cdot\operatorname{\mathrm{vol}}\left(\mathcal G_{\varepsilon-\operatorname{\mathrm{crit}}}(\vec L-[2\varepsilon],h+4\varepsilon)\right)&>\vert{\mathbf{G}}(\vec L, h)\vert. \end{align*} $$

Our next goal is to compute the volumes on the left. Using the cover (2.2), we can compute volumes $\operatorname {\mathrm {vol}}(\mathcal G_{\varepsilon -\operatorname {\mathrm {crit}}}(\vec L,h))$ by integrating over $\Sigma ^{\operatorname {\mathrm {\mathbf {vert}}}}$ the cardinality of the intersection

$$ \begin{align*}{\mathbf{G}}_{\vec x,\varepsilon-\operatorname{\mathrm{crit}}}^X(\vec L,h)=\Pi^{-1}(\vec x)\cap\mathcal G_{\varepsilon-\operatorname{\mathrm{crit}}}(\vec L,h)\end{align*} $$

of the fiber $\Pi ^{-1}(\vec x)$ with the set we care about. In light of Corollary 3.3, we get in this way that

$$ \begin{align*} \operatorname{\mathrm{vol}}(\mathcal G_{\varepsilon-\operatorname{\mathrm{crit}}}(\vec L,h)) &=\int_{\Sigma^{\operatorname{\mathrm{\mathbf{vert}}}}} \left\vert{\mathbf{G}}_{\vec x,\varepsilon-\operatorname{\mathrm{crit}}}^X(\vec L,h)\right\vert\, d\vec x\\[4pt] &\stackrel{\text{Cor. 3.3}}\sim\int_{\Sigma^{\operatorname{\mathrm{\mathbf{vert}}}}} \varepsilon^{4\vert\chi(X)\vert}\cdot\left(\frac{2}{3}\right)^{2\chi(X)}\cdot\pi^{\chi(X)}\cdot\frac{(e^h-1)^{-3\chi(X)}\cdot e^{\Vert\vec L\Vert}}{\operatorname{\mathrm{vol}}(\Sigma)^{-3\chi(X)}}\, d\vec x\\[4pt] &=\varepsilon^{4\vert\chi(X)\vert}\cdot\left(\frac{2}{3}\right)^{2\chi(X)}\cdot\pi^{\chi(X)}\cdot\frac{(e^h-1)^{-3\chi(X)}\cdot e^{\Vert\vec L\Vert}}{\operatorname{\mathrm{vol}}(\Sigma)^{-3\chi(X)}}\operatorname{\mathrm{vol}}(\Sigma)^{V}\\[4pt] &=\varepsilon^{4\vert\chi(X)\vert}\cdot\left(\frac{2}{3}\right)^{2\chi(X)}\cdot\pi^{\chi(X)}\cdot\frac{(e^h-1)^{-3\chi(X)}\cdot e^{\Vert\vec L\Vert}}{\operatorname{\mathrm{vol}}(\Sigma)^{-\chi(X)}}, \end{align*} $$

where we have used that $V=-2\chi (X)$ and where the asymptotics hold true when $\min _eL_e\to \infty $ . This means that whenever $\min _eL_e$ is large enough we have

$$ \begin{align*} (1-\delta)\cdot\frac {2^{4\chi(X)}}{3^{3\chi(X)}}\cdot\pi^{\chi(X)}\cdot \frac{(e^{h-4\varepsilon}-1)^{-3\chi(X)}\cdot e^{\sum_{e\in\operatorname{\mathrm{\mathbf{edge}}}}(L_e+2\varepsilon)}}{\operatorname{\mathrm{vol}}(\Sigma)^{-\chi(X)}}&<\vert{\mathbf{G}}(\vec L, h)\vert,\\[7pt] (1+\delta)\cdot\frac {2^{4\chi(X)}}{3^{3\chi(X)}}\cdot\pi^{\chi(X)}\cdot \frac{(e^{h+4\varepsilon}-1)^{-3\chi(X)}\cdot e^{\sum_{e\in\operatorname{\mathrm{\mathbf{edge}}}}(L_e-2\varepsilon)}}{\operatorname{\mathrm{vol}}(\Sigma)^{-\chi(X)}}&>\vert{\mathbf{G}}(\vec L, h)\vert. \end{align*} $$

Since this is true for all $\varepsilon>0$ , we get that

$$ \begin{align*}(1-\delta)\cdot\frac {2^{4\chi(X)}}{3^{3\chi(X)}}\cdot\pi^{\chi(X)}\cdot \frac{(e^{h}-1)^{-3\chi(X)}\cdot e^{\Vert\vec L\Vert}}{\operatorname{\mathrm{vol}}(\Sigma)^{-\chi(X)}}\leqslant\vert{\mathbf{G}}(\vec L, h)\vert\end{align*} $$

$$ \begin{align*}(1+\delta)\cdot\frac {2^{4\chi(X)}}{3^{3\chi(X)}}\cdot\pi^{\chi(X)}\cdot \frac{(e^{h}-1)^{-3\chi(X)}\cdot e^{\Vert\vec L\Vert}}{\operatorname{\mathrm{vol}}(\Sigma)^{-\chi(X)}}\geqslant\vert{\mathbf{G}}(\vec L, h)\vert\end{align*} $$

and hence, since for all $\delta>0$ we can choose L large enough so that the above bounds hold, we have

$$ \begin{align*}\frac {2^{4\chi(X)}}{3^{3\chi(X)}}\cdot\pi^{\chi(X)}\cdot \frac{(e^{h}-1)^{-3\chi(X)}\cdot e^{\Vert\vec L\Vert}}{\operatorname{\mathrm{vol}}(\Sigma)^{-\chi(X)}}\sim\vert{\mathbf{G}}(\vec L, h)\vert\end{align*} $$

as we wanted to prove.

Armed with Lemma 4.1 and Proposition 4.2, we can now prove the theorem:

Proof of Theorem 1.3

Let $h>0$ be small, and for $\vec n\in \mathbb {N}^{\operatorname {\mathrm {\mathbf {edge}}}}$ consider, with the same notation as in Equation (4.2), the set ${\mathbf {G}}(h\cdot \vec n,h)$ . Setting

$$ \begin{align*}\Delta(N)=\{\vec n\in\mathbb N^{\operatorname{\mathrm{\mathbf{edge}}}}\text{ with }\Vert\vec n\Vert\leqslant N\},\end{align*} $$

where $\Vert \vec n\Vert =\sum _e n_e$ , note that

(4.4) $$ \begin{align} \sum_{\vec n\in\Delta(N-E)}\vert{\mathbf{G}}(h\cdot\vec n,h)\vert\leqslant\vert{\mathbf{G}}(h\cdot N)\vert\leqslant\sum_{\vec n\in\Delta(N)}\vert{\mathbf{G}}(h\cdot\vec n,h)\vert, \end{align} $$

where ${\mathbf {G}}(h\cdot N)={\mathbf {G}}^X(h\cdot N)$ is as in Equation (4.1) and where, once again, $E=\vert \operatorname {\mathrm {\mathbf {edge}}}\vert $ is the number of edges of the graph X. Finally, write

$$ \begin{align*}\kappa=\frac {2^{4\chi(X)}}{3^{3\chi(X)}}\cdot(\pi\cdot\operatorname{\mathrm{vol}}(\Sigma))^{\chi(X)}\end{align*} $$

Proposition 4.2 now reads as

$$ \begin{align*}\vert{\mathbf{G}}(h\cdot\vec n,h)\vert\sim\kappa\cdot(e^{h}-1)^{-3\chi(X)}\cdot e^{h\cdot\Vert\vec n\Vert},\end{align*} $$

where the asymptotic holds for fixed h when $\min \vec n_e\to \infty $ . This means that for all h and $\delta $ there is $n(h,\delta )$ with

$$ \begin{align*}\vert{\mathbf{G}}(h\cdot\vec n,h)\vert> (\kappa-\delta)\cdot(e^{h}-1)^{-3\chi(X)}\cdot e^{h\cdot\Vert\vec n\Vert}\end{align*} $$

and

$$ \begin{align*}\vert{\mathbf{G}}(h\cdot\vec n,h)\vert< (\kappa+\delta)\cdot(e^{h}-1)^{-3\chi(X)}\cdot e^{h\cdot\Vert\vec n\Vert}\end{align*} $$

for all $\vec n$ with $\min \vec n_e\geqslant n(h,\delta )$ . It follows thus from the left side of Equation (4.4) that

$$ \begin{align*} \vert{\mathbf{G}}(h\cdot N)\vert &\geqslant\sum_{\tiny\begin{array}{c}\vec n\in\Delta(N-E),\\ \min\vec n_e\geqslant n(h,\delta)\end{array}}\vert{\mathbf{G}}(h\cdot\vec n,h)\vert\\ &> (\kappa-\delta)(e^h-1)^{-3\chi(X)}\sum_{\tiny\begin{array}{c}\vec n\in\Delta(N-E),\\ \min\vec n_e\geqslant n(h,\delta)\end{array}}e^{h\cdot\Vert\vec n\Vert}\\ &= (\kappa-\delta)(e^h-1)^{-3\chi(X)}\sum_{K=0}^{N-E}P(K)\cdot e^{h\cdot K}, \end{align*} $$

where $P(K)$ is the number of those $\vec n\in \mathbb N^{\operatorname {\mathrm {\mathbf {edge}}}}$ with $\Vert n\Vert =K$ and $\min \vec n_e\geqslant n(h,\delta )$ . As K tends to $\infty $ , we have $P(K)\sim \frac 1{(E-1)!}K^{E-1}$ . Taking into account that $E=-3\chi (X)$ , we get that for all N large enough we have

$$ \begin{align*} \vert{\mathbf{G}}(h\cdot N)\vert &\succeq\frac{\kappa-\delta}{(-3\chi(X)-1)!}(e^h-1)^{-3\chi(X)}\sum_{K=0}^{N-E}K^{-3\chi(X)-1}\cdot e^{h\cdot K}\\ &=\frac{\kappa-\delta}{(-3\chi(X)-1)!}\left(\frac{e^h-1}h\right)^{-3\chi(X)}\sum_{K=0}^{N-E}(hK)^{-3\chi(X)-1}\cdot e^{h\cdot K}\cdot h\\ &\succeq \frac{\kappa-\delta}{(-3\chi(X)-1)!}\left(\frac{e^h-1}h\right)^{-3\chi(X)}\int_0^{(N-E)h} x^{-3\chi(X)-1}e^xdx, \end{align*} $$

where the symbol $\succeq $ means that asymptotically the ratio between the left side and the right side is at least $1$ . When $N\to \infty $ then the value of the integral is asymptotic to $((N-E)\cdot h)^{-3\chi (X)-1}\cdot e^{(N-E)h}$ , and this means that for all N large enough we have

$$ \begin{align*}\vert{\mathbf{G}}(h\cdot N)\vert\succeq \frac{\kappa-\delta}{(-3\chi(X)-1)!}\left(\frac{e^h-1}h\right)^{-3\chi(X)}(Nh-Eh)^{-3\chi(X)-1}\cdot e^{Nh-Eh}.\end{align*} $$

This being true for all $\delta $ and all h and replacing $Nh$ by L, we have

$$ \begin{align*}\vert{\mathbf{G}}(L)\vert\succeq \frac{\kappa}{(-3\chi(X)-1)!}L^{-3\chi(X)-1}\cdot e^{L}\end{align*} $$

as $L\to \infty $ . In other words, we have established the desired asymptotic lower bound.

Starting with the upper bound we get, again for h positive and small, from the right side in Equation (4.4) that

$$ \begin{align*}\vert{\mathbf{G}}(h\cdot N)\vert\leqslant\sum_{\tiny\begin{array}{c}\vec n\in\Delta(N),\\ \min\vec n_e\geqslant n(h,\delta)\end{array}}\vert{\mathbf{G}}(h\cdot\vec n,h)\vert+\sum_{\tiny\begin{array}{c}\vec n\in\Delta(N),\\ \min\vec n_e\leqslant n(h,\delta)\end{array}}\vert{\mathbf{G}}(h\cdot\vec n,h)\vert.\end{align*} $$

The same calculation as above yields that

(4.5) $$ \begin{align} \begin{aligned} \sum_{\tiny\begin{array}{c}\vec n\in\Delta(N),\\ \min\vec n_e\geqslant n(h,\delta)\end{array}}&\vert{\mathbf{G}}(h\cdot\vec n,h)\vert\preceq \\ &\preceq\frac{\kappa+\delta}{(-3\chi(X)-1)!}\left(\frac{e^h-1}h\right)^{-3\chi(X)}(h(N+E))^{-3\chi(X)-1}\cdot e^{h(N + E)} \end{aligned} \end{align} $$

as $N\to \infty $ . On the other hand, we get from Lemma 4.1 that there is $C>0$ with $\vert {\mathbf {G}}(h\cdot \vec n,h)\vert \leqslant C\cdot e^{h\cdot \vert \vec n\vert }$ for all $\vec n$ . This means thus that

$$ \begin{align*}\sum_{\tiny\begin{array}{c}\vec n\in\Delta(N),\\ \min\vec n_e\leqslant n(h,\delta)\end{array}}\vert{\mathbf{G}}(h\cdot\vec n,h)\vert\leqslant C\cdot\sum_{K=1}^NQ(K)\cdot e^{h\cdot K},\end{align*} $$

where $Q(K)$ is the number of those $\vec n\in \mathbb N^{\operatorname {\mathrm {\mathbf {edge}}}}$ with $\min \vec n_e< n(h,\delta )$ and $\vert \vec n\vert =K$ . When $K\to \infty $ the function $Q(K)$ is asymptotic to $C'\cdot K^{E-2}$ for some positive constant $C'$ , meaning that we have

$$ \begin{align*}\sum_{\tiny\begin{array}{c}\vec n\in\Delta(N),\\ \min\vec n_e\leqslant n(h,\delta)\end{array}}\vert{\mathbf{G}}(h\cdot\vec n,h)\vert\leqslant \frac{(1+\delta)\cdot C\cdot C'}{h^{E-1}}\cdot\sum_{K=1}^N h\cdot(hK)^{E-2}\cdot e^{h\cdot K}\end{align*} $$

for all L large enough. A similar estimation as the one above yields thus that there is another positive constant $C"$ with

(4.6) $$ \begin{align} \sum_{\tiny\begin{array}{c}\vec n\in\Delta(N),\\ \min\vec n_e\leqslant n(h,\delta)\end{array}}\vert{\mathbf{G}}(h\cdot\vec n,h)\vert\leqslant C"\cdot (N\cdot h)^{-3\chi(X)-2}\cdot e^{N\cdot H}. \end{align} $$

The quantity in the right-hand side of Equation (4.6) is negligible when compared to the right-hand side of Equation (4.5), and this means that we have

$$ \begin{align*}\vert{\mathbf{G}}(h\cdot N)\vert\preceq \frac{\kappa+2\delta}{(-3\chi(X)-1)!}\left(\frac{e^h-1}h\right)^{-3\chi(X)}(h(N+E))^{-3\chi(X)-1}\cdot e^{h(N+E)}\end{align*} $$

for all large N. Since this holds true for all $\delta $ and all h, replacing $hN$ by L we deduce that

$$ \begin{align*}\vert{\mathbf{G}}(L)\vert\preceq \frac{\kappa}{(-3\chi(X)-1)!}L^{-3\chi(X)-1}\cdot e^{L}.\end{align*} $$

Having now also established the upper asymptotic bound, we are done with the proof of the theorem.

Before moving on to other matters, we include an observation that we will use later on. The basic strategy of the proof of Theorem 1.3 was to decompose the problem of counting all critical realizations of at most some given length into the problem of counting those whose edge lengths are in a given box and then adding over all boxes. For most boxes, Proposition 4.2 gives a pretty precise estimation for the number of critical realizations in the box, and from Lemma 4.1 we get an upper bound for all boxes. We used these two to get the desired upper bound in the theorem, deducing that we could ignore the boxes where Proposition 4.2 does not apply. A very similar argument implies that the set of critical realizations where some edge is shorter than $\ell _0$ is also negligible. We state this observation as a lemma:

It is probably clear from the context what negligible means here, but to be precise, we mean that the set is negligible inside the set of all critical realizations in the sense of Equation (7.2) below.

5 Fillings

Let $S_g$ be a compact, connected and oriented surface of genus g and with one boundary component. Below, we will be interested in continuous maps

(5.1) $$ \begin{align} \beta:S_g\to\Sigma \end{align} $$

which send $\partial S_g$ to a closed geodesic $\gamma =\beta (S_g)$ . We will refer to such a map as a filling of genus g of $\gamma $ , a genus g filling of $\gamma $ , or just simply as a filling of $\gamma $ when the genus g is either undetermined or understood from the context. The genus of a curve $\gamma \subset \Sigma $ is the infimum of all g’s for which there is a genus g filling Equation (5.1) with $\gamma =\beta (\partial S_g)$ . Note that the genus of a curve is infinite unless $\gamma $ is homologically trivial, that is, unless $\gamma $ is represented by elements in the commutator subgroup of $\pi _1(\Sigma )$ . Indeed, as an element of $\pi _1(S_g)$ the boundary $\partial S_g$ is a product of g commutators. It follows that if a curve $\gamma $ in $\Sigma $ has genus g, then it is, when considered as an element in $\pi _1(\Sigma )$ , a product of g commutators. Conversely, if $\gamma $ is a product of g commutators, then there is a map as in Equation (5.1) with $S_g$ of genus g. In a nutshell, what we have is that the genus and the commutator length of $\gamma $ agree. We record this fact for ease of reference:

Continuing with the same notation and terminology, suppose that a curve $\gamma $ in $\Sigma $ has genus g. We then refer to any $\beta :S\to \Sigma $ as in Equation (5.1) with S of genus g as a minimal genus filling. Minimal genus fillings have very nice topological properties. Indeed, suppose that $\beta :S_g\to \Sigma $ is a filling as in Equation (5.1) and suppose that there is an essential simple curve $m\subset S_g$ with $\beta (m)$ homotopically trivial. Then, performing surgery on the surface $S_g$ and the map $\beta $ we get a smaller genus filling $\beta ':S_{g'}\to \Sigma $ with $\beta '(\partial S_{g'})=\beta (\partial S_g)$ and $g'<g$ . It follows that if $\beta :S_g\to \Sigma $ is a minimal genus filling for $\gamma =\beta (\partial S_g)$ , then $\beta $ is geometrically incompressible in the sense that there are no elements in $\ker (\beta _*:\pi _1(S_g)\to \pi _1(\Sigma ))$ which are represented by simple curves. We record this fact:

From now on, we will be working exclusively with minimal genus fillings and the reader can safely add the words ‘minimal genus’ every time they see the word ‘filling’.

We will not be that much interested in individual fillings, but rather in homotopy classes of fillings. In particular, we will allow ourselves to select particularly nice fillings. More precisely, we will be working with hyperbolic fillings, by which we will understand a particular kind of pleated surface. We remind the reader that according to Thurston [Reference Thurston18] (see also [Reference Canary, Epstein and Green5]) a pleated surface is a map from a hyperbolic surface to a hyperbolic manifold with the property that every point in the domain is contained in a geodesic segment which gets mapped isometrically.

An important observation is that if $\beta :S\to \Sigma $ is a hyperbolic filling, if $x\in \partial S$ and if $I\subset S$ is a geodesic segment with x in its interior, then $I\subset \partial S$ . It follows that hyperbolic fillings map the boundary isometrically.

We should not delay making sure that hyperbolic fillings exist:

To prove this proposition, we will first show that if $\beta :S\to \Sigma $ is any filling and if the surface S admits a triangulation with certain properties, then $\beta $ is homotopic to a hyperbolic filling. For lack of a better name, we will say that a triangulation $\mathcal T$ of S is useful if it satisfies the following three conditions:

1. (i) $\mathcal T$ has exactly $g+1$ vertices $v_0,\dots ,v_g$ , with $v_0\in \partial S$ and the others in the interior of S.

2. (ii) There is a collection of edges $I_0,\dots ,I_g$ of $\mathcal T$ such that both endpoints of $I_i$ are attached to $v_i$ for all $i=0,\dots ,g$ . Moreover, $I_0=\partial S$ .

These two conditions are evidently pretty soft. It is the condition we will state next what makes useful triangulations actually useful. We first need a bit of notation: If I is any edge of $\mathcal T$ other than $I_0,\dots ,I_g$ , then let $G_I$ be the connected component of $I\cup I_0\cup \dots \cup I_g$ containing I.

1. (iii) For any edge I other than $I_0,\dots ,I_g$ , we have that the image under $\beta _*:\pi _1(G_I)\to \pi _1(\Sigma )$ is not abelian.

Note that $\pi _1(G_I)$ is a a free group of rank $2$ . This means that its image $\beta _*(\pi _1(G_I)))$ has at most rank $2$ . Since we are assuming that it is not abelian, we actually get that it is a rank $2$ free group. Free groups being Hopfian we deduce that $\beta _*:\pi _1(G_I)\to \pi _1(\Sigma )$ is an isomorphism onto its image. In other words, $\beta _*$ is injective on $\pi _1(G_I)$ .

The following result makes clear why we care about such triangulations:

We can now prove Proposition 5.4.

Proof. Let $\beta :S\to \Sigma $ be a minimal genus filling of a geodesic $\gamma =\beta (\partial S)$ , say of genus g. In light of Lemma 5.5, to prove that $\beta $ is homotopic to a hyperbolic filling it suffices to show that S admits a useful triangulation. Let us start by taking g disjoint compact one-holed tori $T_1,\ldots ,T_g\subset S$ . We claim that the restriction of $\beta $ to each $T_i$ is $\pi _1$ -injective. Indeed, since we know that $\beta $ is geometrically incompressible we deduce that $\partial T_i$ is not in the kernel of the induced homomorphism at the fundamental group level. It follows that the image of $\beta _*(\pi _1(T_i))$ cannot be abelian. Now, this implies that $\beta _*(\pi _1(T_i))$ is free and evidently of rank 2. Hence, again since free groups are Hopfian, we get that the restriction of $\beta _*$ to $\pi _1(T_i)$ is injective for all i.

Figure 2

Construction of the useful triangulation in Proposition 5.4: The $2g+1$ holed sphere obtained from S by cutting along $\gamma _1, \ldots , \gamma _g$ . The vertices $v_1,\dots ,v_g$ and their copies $v_1',\dots ,v_g'$ are joined by the sequence of edges $[v_1,v_2],[v_2,v_3],\dots ,[v_g,v_1'],[v_1',v_2'],\dots ,[v_{g-1}',v_g']$ . To complete the triangulation, add as many edges as needed joining $v_0$ to the other vertices.

Now, why do we care about that? Knowing that the restriction of $\beta _*$ to $\pi _1(T_i)$ is injective for any i implies that when the images under $\beta $ of the nonboundary parallel simple closed curves in $T_i$ determine infinitely many conjugacy classes of maximal abelian subgroups of $\pi _1(\Sigma )$ . We can thus choose for each $i=1,\dots ,g$ a nonboundary parallel simple closed curve $\gamma _i\subset T_i$ such that if we also set $\gamma _0=\partial S$ , then we have that

• (*) no two of the the maximal abelian subgroups of $\pi _1(\Sigma )$ containing $\beta _*(\gamma _0),\dots ,\beta _*(\gamma _g)$ are conjugate to each other.

Choosing now a vertex $v_i$ in each one of the curves $\gamma _i$ , we get from $(*)$ that if I is any simple path joining $v_i$ to $v_j$ for $i\neq j$ , then the image of $\beta _*(\pi _1(\gamma _i\cup I\cup \gamma _j))$ is not abelian.

The upshot of all of this is that any triangulation $\mathcal T$ with

1. 1. vertex set $v_0,\dots ,v_g$ ,

2. 2. such that there are edges $I_0,\dots ,I_g$ incident on both ends to $v_i$ and with image $\gamma _i$ , and

3. 3. such that all other edges connect distinct endpoints,

is useful. To see that such a triangulation exists, cut S along $\gamma _1,\dots ,\gamma _g$ . When doing this, we get a $2g+1$ holed sphere $\Delta $ and each vertex $v_1,\dots ,v_g$ arises twice – denote the two copies of $v_i$ by $v_i$ and $v_i'$ as in Figure 2. Now, any triangulation of $\Delta $ with vertex set $v_0,v_1,v_1',v_2,v_2',\dots ,v_g,v_g'$ and which

• ○ contains a sequence of $2g-1$ edges yielding a path

  $$ \begin{align*}[v_1,v_2],[v_2,v_3],\dots,[v_g,v_1'],[v_1',v_2'],\dots,[v_{g-1}',v_g']\end{align*} $$
  and

• ○ such that all other edges are incident to $v_0$

yields a triangulation of S satisfying (1)–(3) above. Having proved that there is a useful triangulation, we get from Lemma 5.5 that $\beta $ is homotopic to a hyperbolic filling, as we needed to prove.

Being able to work with hyperbolic fillings is going to be key in the next section, where we bound the number of closed geodesics $\gamma $ in $\Sigma $ with length $\ell _{\Sigma }(\gamma )\leqslant L$ and which admit at least two homotopically distinct fillings.

Evidently, to bound the number of pairs of nonhomotopic fillings with the same boundary we will need some criterion to decide when two fillings are homotopic. To be able to state it note that if $\beta _1:S_1\to \Sigma $ and $\beta _2:S_2\to \Sigma $ are homotopic hyperbolic fillings, then there is

For lack of a better name, we refer to $\Theta :S\to \Sigma $ as the pseudo-double associated to $\beta _1$ and $\beta _2$ , and to the curve $\partial S_1=\partial S_2\subset S$ as the crease of the pseudo-double.

Our criterion to decide if the two hyperbolic fillings $\beta _1$ and $\beta _2$ of a geodesic $\gamma $ are homotopic will be in terms of the structure of the $\varepsilon _0$ thin part of the domain of the associated pseudo-double, where we choose once and forever the constant

(5.2) $$ \begin{align} \varepsilon_0=\frac 1{10}\cdot\min\left\{\text{Margulis constant of }{\mathbb{H}}^2,\ \text{systole of }\Sigma\right\}. \end{align} $$

The following is our criterion.

6 Bounding the number of multifillings

The goal of this section is to prove Theorem 1.4:

The proof of Theorem 1.4 turns out to be kind of involved. We hope that the reader will not despair with all the weird objects and rather opaque statements they will find below.

Wired surfaces

Under a wired surface, we will understand a compact connected simplicial complex $\Delta $ obtained as follows: Start with a compact, possibly disconnected, triangulated surface $\operatorname {\mathrm {Surf}}(\Delta )$ and an, evidently finite, subset $P_{\Delta }$ of the set of vertices of the triangulation of $\operatorname {\mathrm {Surf}}(\Delta )$ such that every connected component of $\operatorname {\mathrm {Surf}}(\Delta )\setminus P_{\Delta }$ has negative Euler characteristic. We think of the elements in $P_{\Delta }$ as plugs. Now, attach 1-simplices, to which we will refer as wires, by attaching both endpoints to plugs and do so in such a way that each plug arises exactly once as the end-point of a wire – we denote the set of wires by $\operatorname {\mathrm {\mathbf {wire}}}(\Delta )$ . A wired surface $\Delta $ without wires, that is, one with $\Delta =\operatorname {\mathrm {Surf}}(\Delta )$ , is said to be degenerate. Otherwise, it is nondegenerate.

How will wired surfaces arise? We will say that a pair $({\mathbb {F}},\mathbb T)$ is a decoration of a surface S if

• ○ ${\mathbb {F}}$ is a partial foliation of S supported by the union of a finite collection of disjoint, essential and nonparallel, cylinders, and

• ○ $\mathbb T$ is a triangulation of the complement of the interior of those cylinders.

Now, if $({\mathbb {F}},\mathbb T)$ is a decoration of S, then we get an associated wired surface $\Delta =S/\sim _{\mathbb {F}}$ by collapsing each leaf of ${\mathbb {F}}$ to a point and dividing each one of the arising bigons into two triangles by adding a vertex in the interior of the bigon. We will say that the quotient map $\pi :S\to \Delta $ is a resolution of $\Delta $ with associated foliation ${\mathbb {F}}$ . Every wired surface admits an essentially unique resolution, unique in the sense that any two differ by a PL-homeomorphism mapping one of the foliations to the other one.

Suppose now that $\Delta $ is a wired surface. A simple curve on $\Delta $ is a map $\eta :\mathbb S^1\to \Delta $ such that there are a resolution $\pi :S\to \Delta $ with associated foliation ${\mathbb {F}}$ and an essential simple curve $\eta ':\mathbb S^1\to S$ which is transversal to ${\mathbb {F}}$ and with $\eta =\pi \circ \eta '$ .

Note that transversality to the associated foliation implies that if $\eta $ is a simple curve of a wired surface $\Delta $ then $\eta \cap \pi ^{-1}(\Delta \setminus \operatorname {\mathrm {Surf}}(\Delta ))$ consists of a collection of segments, each one of them mapped homeomorphically to a wire. If I is such a wire, then we denote by $n_I(\eta )$ the weight of $\eta $ in I, that is, the number of connected components of $\eta \cap \pi ^{-1}(\Delta \setminus \operatorname {\mathrm {Surf}}(\Delta ))$ which are mapped homeomorphically to I, or in other words, the number of times that $\eta $ crosses I. We refer to the vector

(6.1) $$ \begin{align} \vec n_{\Delta}(\eta)=(n_I(\eta))_{I\in\operatorname{\mathrm{\mathbf{wire}}}(\Delta)} \end{align} $$

as the weight vector for $\eta $ in $\Delta $ . The intersection of the image of $\eta $ with the surface part $\operatorname {\mathrm {Surf}}(\Delta )$ is a simple arc system with endpoints in the set $P_{\Delta }$ . Note that up to homotopying $\eta $ to another simple curve in $\Delta $ we might assume that all the components of the arc system $\eta \cap \operatorname {\mathrm {Surf}}(\Delta )$ are essential. This is equivalent to asking that for each wire I we have $n_I(\eta )\leqslant n_I(\eta ')$ for any other simple curve $\eta '$ in $\Delta $ homotopic to $\eta $ . We will suppose from now on, without further mention, that all simple curves in $\Delta $ satisfy these minimality requirements.

So far, wired surfaces are just topological objects. Let us change this. Under a hyperbolic wired surface we understand a wired surface $\Delta $ whose surface part $\operatorname {\mathrm {Surf}}(\Delta )$ is endowed with a piece-wise hyperbolic metric, that is, one with respect to which the simplexes in the triangulation of $\operatorname {\mathrm {Surf}}(\Delta )$ are isometric to hyperbolic triangles.

Let $\Delta $ be a hyperbolic wired surface, and as always let $\Sigma $ be our fixed hyperbolic surface. We will say that a map $\Xi :\Delta \to \Sigma $ is tight if the following holds:

• ○ $\Xi $ maps every wire to a geodesic segment, and

• ○ $\Xi $ is an isometry when restricted to each one of the simplexes in the triangulation of $\operatorname {\mathrm {Surf}}(\Delta )$ .

We will be interested in counting pairs $(\Xi :\Delta \to \Sigma ,\gamma )$ consisting of tight maps and simple curves. Evidently, without further restrictions, there could be infinitely many such pairs. What we are going to count is pairs were the curve has bounded length. We are, however, going to use a pretty strange notion of length. Consider namely for some given small but positive $\varepsilon $ , the following quantity

(6.2) $$ \begin{align} \ell_{\Xi}^{\varepsilon}(\gamma)=\varepsilon\cdot\ell_{\operatorname{\mathrm{Surf}}(\Delta)}(\gamma\cap\operatorname{\mathrm{Surf}}(\Delta))+\sum_{I\in\operatorname{\mathrm{\mathbf{wire}}}}n_I(\gamma)\cdot\max\left\{\ell_{\Sigma}(\Xi(I))-\frac 1\varepsilon,0\right\}. \end{align} $$

It is evident that this notion of length is exactly tailored to what we will need later on, but let us try to parse what Equation (6.2) actually means. What is the role of $\varepsilon $ ? If we think of the length as a measure of the cost of a journey, then the first $\varepsilon $ just makes traveling along the surface part pretty cheap, meaning that for the same price we can cruise longer over there. Along the same lines, when traveling through the wires, we only pay when the wires are very long.

Lemma 6.1. Let $\Delta $ be a nondegenerate hyperbolic wired surface with set of wires $\operatorname {\mathrm {\mathbf {wire}}}=\operatorname {\mathrm {\mathbf {wire}}}(\Delta )$ . Fix a tight map $f:\Delta \to \Sigma $ and a positive integer vector $\vec n=(n_I)_{I\in \operatorname {\mathrm {\mathbf {wire}}}}\in \mathbb N_+^{\operatorname {\mathrm {\mathbf {wire}}}}$ , and denote by

$$ \begin{align*}\min=\min_{I\in\operatorname{\mathrm{\mathbf{wire}}}}n_I\geqslant 1\text{ and }d=\vert\{I\in \operatorname{\mathrm{\mathbf{wire}}}\text{ with }n_I=\min\}\vert\end{align*} $$

the smallest entry of $\vec n$ and the number of times that this value is taken.

For any $\varepsilon>0$ , there are at most $\operatorname {\mathrm {\mathbf {const}}}\cdot L^{d-1}\cdot e^{\frac {L}{\min }}$ homotopy classes of pairs $(\Xi :\Delta \to \Sigma ,\gamma )$ , where $\Xi $ is a tight map with $\Xi \vert _{\operatorname {\mathrm {Surf}}(\Delta )}=f\vert _{\operatorname {\mathrm {Surf}}(\Delta )}$ and where $\gamma $ is a simple multicurve in $\Delta $ with $n_I(\gamma )\geqslant n_I$ for every wire I and with $\ell _{\Xi }^{\varepsilon }(\gamma )\leqslant L$ .

Note that the fact that the obtained bound, or rather its rate of growth, does not depend on $\varepsilon $ implies that actually the only way to get many homotopy classes is to play with the wires. In fact, since the given bound only depends on d and $\min $ , the only wires that matter are those which the curve crosses as little as possible.

Another comment before launching the proof. Namely, what happens if the wired surface $\Delta $ in Lemma 6.1 is degenerate? If there are no wires, then $\Delta $ is nothing other than a surface with a (piece-wise hyperbolic) metric. In such a surface, there are at most $\operatorname {\mathrm {\mathbf {const}}}\cdot L^{3\cdot \vert \chi (\Delta )\vert }$ simple multicurves of length at most L – see, for example, [Reference Erlandsson and Souto8]. This means that for a degenerate wired surface one actually gets a polynomial bound instead of an exponential one.

We are now ready to launch the proof of Lemma 6.1:

Proof. Note that, since the map $\Xi $ is fixed on $\operatorname {\mathrm {Surf}}(\Delta )$ , we get that the homotopy type of the map, or even the map itself, is determined by what happens to the wires. In particular, as in the proof of Lemma 4.1 we get that if we give ourselves a positive vector $\vec \lambda =(\lambda _I)_{I\in \operatorname {\mathrm {\mathbf {wire}}}}\in \mathbb R_+^{\operatorname {\mathrm {\mathbf {wire}}}}$ , then there are at most $\operatorname {\mathrm {\mathbf {const}}}\cdot e^{\Vert \vec \lambda \Vert }$ homotopy classes of tight maps $\Xi :\Delta \to \Sigma $ with $\Xi \vert _{\operatorname {\mathrm {Surf}}(\Delta )}=f$ and such that

(6.3) $$ \begin{align} \lambda_I\leqslant \ell_{\Sigma}(\Xi(I))\leqslant\lambda_I+1\text{ for all }I\in\operatorname{\mathrm{\mathbf{wire}}}. \end{align} $$

As always, we have set $\Vert \vec \lambda \Vert =\lambda _1+\dots +\lambda _r$ .

Now, we are not counting homotopy classes of maps but, rather, of pairs $(\Xi :\Delta \to \Sigma ,\gamma )$ , where the multicurve $\gamma $ satisfies $\ell _{\Xi }^{\varepsilon }(\gamma )\leqslant L$ . Note that, if $\Xi $ satisfies Equation (6.3), then our given length bound $\ell _{\Xi }^{\varepsilon }(\gamma )\leqslant L$ implies that

$$ \begin{align*} \ell_{\operatorname{\mathrm{Surf}}(\Delta)}(\gamma\cap\operatorname{\mathrm{Surf}}(\Delta)) &= \frac 1\varepsilon\left(\ell_{\Xi}^{\varepsilon}(\gamma)-\sum_{I\in\operatorname{\mathrm{\mathbf{wire}}}}n_I(\gamma)\cdot\max\left\{\ell_{\Sigma}(\Xi(I))-\frac 1\varepsilon,0\right\}\right)\\ &\leqslant \frac 1\varepsilon\left(L-\sum_{I\in\operatorname{\mathrm{\mathbf{wire}}}}n_I\cdot\max\left\{\lambda_I-\frac 1\varepsilon,0\right\}\right)\\ &\leqslant \frac 1\varepsilon\left(L-\langle\vec n,\vec\lambda\rangle+\frac 1\varepsilon\cdot\Vert\vec n\Vert\right). \end{align*} $$

Now, since for any given length there are only polynomially many simple arc systems of bounded length we deduce that for each $\Xi $ satisfying Equation (6.3) there are at most $\operatorname {\mathrm {\mathbf {const}}}\cdot (L-\langle \vec n,\vec \lambda \rangle )^{\operatorname {\mathrm {\mathbf {const}}}}$ homotopy classes of simple multicurves $\gamma $ in $\Delta $ with $\ell _{\Xi }^{\varepsilon }(\gamma )\leqslant L$ and satisfying $n_{I}(\gamma )\geqslant n_{I}$ for each wire I. Putting all of this together, we get the following:

Fact. There are at most $\operatorname {\mathrm {\mathbf {const}}}\cdot (L-\langle \vec n,\vec \lambda \rangle )^{\operatorname {\mathrm {\mathbf {const}}}}\cdot e^{\Vert \lambda \Vert }$ homotopy classes of pairs $(\Xi :\Delta \to \Sigma ,\gamma )$ , where $\Xi $ is a tight map with $\Xi \vert _{\operatorname {\mathrm {Surf}}(\Delta )}=f$ , satisfying Equation (6.3) and where $\gamma $ is a simple curve in $\Delta $ with $n_I(\gamma )\geqslant n_I$ for every wire I and with $\ell _{\Xi }^{\varepsilon }(\gamma )\leqslant L$ .

Now, we get that the quantity we want to bound, that is, the number of homotopy classes of pairs $(\Xi :\Delta \to \Sigma ,\gamma )$ , where $\Xi $ is a tight map with $\Xi \vert _{\operatorname {\mathrm {Surf}}(\Delta )}=f$ and where $\gamma $ is a simple curve in $\Delta $ with $n_I(\gamma )\geqslant n_I$ for every wire I and with $\ell _{\Xi }^{\varepsilon }(\gamma )\leqslant L$ is bounded from above by

$$ \begin{align*}\sum_{\lambda\in\mathbb N^{\operatorname{\mathrm{\mathbf{wire}}}},\ \Vert\lambda\Vert\leqslant L}\operatorname{\mathrm{\mathbf{const}}}\cdot (L-\langle\vec n,\vec\lambda\rangle)^{\operatorname{\mathrm{\mathbf{const}}}}\cdot e^{\Vert\lambda\Vert}.\end{align*} $$

This quantity is then bounded from above by the value of the integral

$$ \begin{align*}\operatorname{\mathrm{\mathbf{const}}}\int_{\{\vec x\in\mathbb R_+^{\operatorname{\mathrm{\mathbf{wire}}}},\ \langle\vec n,\vec x\rangle\}\leqslant L}(L-\langle \vec n,\vec x\rangle)^{\operatorname{\mathrm{\mathbf{const}}}}\cdot e^{\Vert\vec x\Vert}dx,\end{align*} $$

and now it is a calculus problem that we leave to the reader to check that this integral is bounded by $\operatorname {\mathrm {\mathbf {const}}}\cdot L^{d-1}\cdot e^{\frac {L}{n_{\min }}}$ , where $n_{\min }=\min _In_I\geqslant 1$ and where $d=\vert \{I\text { wire with }n_I=n_{\min }\}\vert $ . We are done.

At this point, we know how to bound the number of homotopy classes of tight maps of wired surfaces. It is time to explain why we care about being able to do so.

Pseudo-doubles

Earlier, just before the statement of Lemma 5.6, we introduced the pseudo-double associated to two fillings. Let us extend that terminology a bit: Under a pseudo-double, we understand a pair $(\Theta :S\to \Sigma ,\gamma )$ where

• ○ $\Theta :S\to \Sigma $ is a pleated surface with S closed,

• ○ $\gamma \subset S$ , the crease, is a simple curve cutting S into two connected components and

• ○ $\Theta $ maps $\gamma $ to a geodesic in $\Sigma $ and its restriction $\Theta \vert _{S\setminus \gamma }$ to the complement of $\gamma $ is geometrically incompressible.

Two pseudo-doubles $(\Theta :S\to \Sigma ,\gamma )$ and $(\Theta ':S'\to \Sigma ,\gamma ')$ are homotopic if there is a homeomorphism $f:S\to S'$ with $\gamma '$ homotopic to $f(\gamma )$ and with $\Theta $ homotopic to $\Theta '\circ f$ .

Note that this terminology is consistent with the use of the word pseudo-double in the previous section.

Recall now that we fixed earlier some $\varepsilon _0$ satisfying Equation (5.2) and note that if $(\Theta :S\to \Sigma ,\gamma )$ is a pseudo-double then the crease $\gamma $ , being separating, crosses every component U of the thin part $S^{\leqslant \varepsilon _0}$ an even number of times $\iota (\gamma ,U)\in 2\mathbb N$ . In fact, since by the choice of $\varepsilon _0$ we get that $\Theta $ maps the core of every component of the thin part to a homotopically trivial curve and since the restriction of $\Theta $ to $S\setminus \gamma $ is geometrically incompressible we get that actually

(6.4) $$ \begin{align} \iota(\gamma,U)\geqslant 2\text{ for all components }U\text{ of the thin part of }S, \end{align} $$

by which we mean that $\gamma $ traverses each such U at least twice.

Our next goal is to bound, for growing L, the number of homotopy classes of pseudo-doubles $(\Theta :S\to \Sigma ,\gamma )$ where S has given topological type, where there are precisely d components U of the thin part with $\iota (\gamma ,U)=2$ and where $\ell _S(\gamma )\leqslant L$ :

Proposition 6.2. Let $\varepsilon _0$ be as in Equation (5.2), and suppose that $S_0$ is a closed orientable surface.

1. 1. For every $d\geqslant 1$ , there are at most $\operatorname {\mathrm {\mathbf {const}}}\cdot L^{d-1}\cdot e^{\frac L2}$ homotopy classes of pseudo-doubles $(\Theta :S\to \Sigma ,\gamma )$ , where S is homeomorphic to $S_0$ where the thin part $S^{\leqslant \varepsilon _0}$ of S has exactly d components U with $\iota (\gamma ,U)=2$ and where $\gamma $ has length $\ell _S(\gamma )\leqslant L$ .

2. 2. There are at most $\operatorname {\mathrm {\mathbf {const}}}\cdot e^{\frac L3}$ homotopy classes of pseudo-doubles $(\Theta :S\to \Sigma ,\gamma )$ where S is homeomorphic to $S_0$ , where there is no component U of the thin part $S^{\leqslant \varepsilon _0}$ with $\iota (\gamma ,U)=2$ , and where $\gamma $ has length $\ell _S(\gamma )\leqslant L$ .

Recall that we declared a decoration $({\mathbb {F}},\mathbb T)$ of a surface S to be a pair consisting of partial foliation ${\mathbb {F}}$ supported by the union of disjoint essential and nonparallel cylinders and of a triangulation $\mathbb T$ of the complement of the interior of those cylinders. To see where these decorations come from, assume that S is a hyperbolic surface.

• ○ The $\varepsilon _0$ -thick part of S is metrically bounded in the sense that it has a triangulation $\mathbb T$ whose vertices are $\frac 1{10}\varepsilon _0$ -separated and whose edges have length at most $\frac 13\varepsilon _0$ and are geodesic unless contained in the boundary $\partial (S^{\geqslant \varepsilon _0})$ of the $\varepsilon _0$ -thick part – in that case, they have just constant curvature.

• ○ The components of the $\varepsilon _0$ -thin part $S^{\leqslant \varepsilon _0}$ are not metrically bounded but still have a very simple structure: They are cylinders foliated by constant curvature circles, namely the curves at constant distance from the geodesic at the core of the cylinder.

Putting these things together, that is, the triangulation $\mathbb T$ of the thick part and the foliation ${\mathbb {F}}$ of the thin part, we get what we will refer as a thin-thick decoration $({\mathbb {F}},\mathbb T)$ of S – note that the triangulation $\mathbb T$ is not unique, and this is why we use an undetermined article. More importantly, note also that the number of components of the thin part is bounded just in terms of the topology of S and that the number of vertices in the triangulation $\mathbb T$ is bounded by some number depending on the chosen $\varepsilon _0$ and again the topological type of S. Since we have fixed $\varepsilon _0$ , it follows that every compact surface $S_0$ admits finitely many decorations $({\mathbb {F}}_1,\mathbb T_1),\dots ,({\mathbb {F}}_r,\mathbb T_r)$ such that if S is any hyperbolic surface homeomorphic to $S_0$ , then there is a homeomorphism $\sigma :S_0\to S$ and some i such that $(\sigma ({\mathbb {F}}_i),\sigma (\mathbb T_i))$ is a $\varepsilon _0$ -thin-thick decoration of S. We state this fact for later reference:

Lemma 6.3. For every closed surface $S_0$ , there are finitely many decorations $({\mathbb {F}}_1,\mathbb T_1),\dots ,({\mathbb {F}}_r,\mathbb T_r)$ such that for any hyperbolic surface S homeomorphic to $S_0$ there are $i\in \{1,\dots ,r\}$ and a homeomorphism $\sigma :S_0\to S$ such that $\sigma ({\mathbb {F}}_i,\mathbb T_i)$ is a $\varepsilon _0$ -thin-thick decoration of S.

After these comments, we can finally launch the proof of Proposition 6.2:

Proof of Proposition 6.2

Starting with the proof of (1), note that from Lemma 6.3 we get that it suffices to prove for each fixed decoration $({\mathbb {F}},\mathbb T)$ of $S_0$ that

• (*) there are at most $\operatorname {\mathrm {\mathbf {const}}}\cdot L^{d-1}\cdot e^{\frac L2}$ homotopy classes of pseudo-doubles $(\Theta :S\to \Sigma ,\gamma )$ where there is a homeomorphism $\sigma :S_0\to S$ such that $(\sigma ({\mathbb {F}}),\sigma (\mathbb T))$ is a decoration of the $\varepsilon _0$ -thin-thick decomposition of S, where the thin part $S^{\leqslant \varepsilon _0}$ of S has exactly d components U with $\iota (\gamma ,U)=2$ and where $\gamma $ has length $\ell _S(\gamma )\leqslant L$ .

Assuming from now on that we have an $\varepsilon _0$ -thin-thick decoration $({\mathbb {F}}, \mathbb T)$ , let $\Delta =S_0/\sim _{\mathbb {F}}$ be the wired surface obtained from S by collapsing each leaf of ${\mathbb {F}}$ to a point and let $\pi :S_0\to \Delta $ be the corresponding quotient map.

The reason we consider $\Delta $ is that, as we already mentioned earlier, we have that by the choice of $\varepsilon _0$ all the leaves of the foliation $\sigma ({\mathbb {F}})$ are mapped by $\Theta $ to homotopically trivial curves. This implies that there is a map

$$ \begin{align*}\Xi:\Delta\to\Sigma\end{align*} $$

mapping the wires of $\Delta $ to geodesic segments and with $\Theta \circ \sigma $ homotopic to $\Xi \circ \pi $ by a homotopy whose tracks are bounded by $\operatorname {\mathrm {\mathbf {const}}}$ . In particular, the edges in $\mathbb T$ are mapped to paths homotopic to geodesic paths of at most length $\operatorname {\mathrm {\mathbf {const}}}$ . Pulling tight relative to the vertices, we can assume that $\Xi $ maps two-dimensional simplices in the triangulation of $\Delta $ to hyperbolic triangles. Note that the bound on the lengths of the images of the edges of $\mathbb T$ imply that the restriction of $\Xi $ to $\operatorname {\mathrm {Surf}}(\Delta )$ is $\operatorname {\mathrm {\mathbf {const}}}$ -Lipschitz.

This uniform Lipschitz bound implies that $\Xi \vert _{\operatorname {\mathrm {Surf}}(\Delta _0)}$ belongs to finitely many homotopy classes and that the tracks of the homotopy to any chosen representative of the correct homotopy class are bounded by $\operatorname {\mathrm {\mathbf {const}}}$ . Choosing the representatives to be tight maps with respect to some hyperbolic structure on $\Delta _0$ , we get:

Fact. There are finitely many hyperbolic structures $\Delta _1,\dots ,\Delta _r$ on $\Delta $ and finitely many tight maps $f_1,\dots ,f_r:\Delta _i\to \Sigma $ such that for any $\Theta :S\to \Sigma $ and $\sigma :S_0\to S$ as in ( $*$ ) there are $i\in \{1,\dots ,r\}$ and a tight map $\Xi :\Delta _i\to \Sigma $ with $\Xi \vert _{\operatorname {\mathrm {Surf}}(\Delta _i)}=f_i\vert _{\operatorname {\mathrm {Surf}}(\Delta _i)}$ and such that $\Theta \circ \sigma :S_0\to \Sigma $ is homotopic to $\Xi \circ \pi :S_0\to \Sigma $ by a homotopy whose tracks have at most length $\operatorname {\mathrm {\mathbf {const}}}$ .

Continuing with the same notation, note that the bound on the tracks of the homotopy between $\Theta \circ \sigma $ and $\Xi \circ \pi $ means that when we compare the length of the geodesic $\gamma $ in S with that of the curve $\eta =(\pi \circ \sigma ^{-1})(\gamma )$ in the hyperbolic wired surface $\Delta _i$ then there is at most an increase by an additive amount every time $\gamma $ crosses a simplex of the wired surface. It means that lengths

• ○ increase at most by an additive amount every time we cross a component of the thin part, and

• ○ increase by a multiplicative amount while we are in the thick part.

Said in other words: There is some R with

$$ \begin{align*} \ell_{\Sigma}(\Xi(\eta\cap\operatorname{\mathrm{Surf}}(\Delta)))&\leqslant R\cdot \ell_S(\gamma\cap S^{\geqslant\varepsilon_0})\\ \ell_{\Sigma}(\Xi(\eta\setminus\operatorname{\mathrm{Surf}}(\Delta)))&\leqslant \sum_{\kappa\in\pi_0(\gamma\cap S^{\leqslant\varepsilon_0})}(\ell_S(\kappa)+R). \end{align*} $$

Recalling that the wires I of $\Delta _0$ correspond to the thin parts of $S^{\leqslant \varepsilon _0}$ and that the weight $n_I(\eta )$ is nothing other than the number of times that $\eta $ crosses the wire I, we get from the last two inequalities that

(6.5) $$ \begin{align} \frac 1R\cdot\ell_{\Xi}(\eta\cap\operatorname{\mathrm{Surf}}(\Delta))+\sum_{I\in\operatorname{\mathrm{\mathbf{wire}}}(\Delta_0)}n_I(\eta)\cdot\max\left\{\ell_{\Xi}(I)-R,0\right\}\leqslant\ell_S(\gamma), \end{align} $$

where the ‘max’ arises because a length is always nonnegative. Anyways, note that with the notation introduced in Equation (6.2) we can rewrite Equation (6.5) as

$$ \begin{align*}\ell_{\Xi}^{\frac 1R}(\gamma)\leqslant\ell_S(\gamma).\end{align*} $$

Note also that from Equation (6.4) we get that $n_I(\eta )\geqslant 2$ for all $I\in \operatorname {\mathrm {\mathbf {wire}}}(\Delta _0)$ . This means that using the notation of Lemma 6.1 we can restate the assumptions in Proposition 6.2 (1) as follows: $\min =2$ and this value is achieved $d\geqslant 1$ times. Lemma 6.1 implies thus that there are at most $\operatorname {\mathrm {\mathbf {const}}}\cdot L^{d-1}\cdot e^{\frac L2}$ homotopy classes of pairs $(\Xi ,\eta )$ arising from pseudo-doubles $(\Theta :S\to \Sigma ,\gamma )$ , where $\Theta $ is as in the fact and where $\ell _S(\gamma )\leqslant L$ . This implies a fortiori that we have at most that many choices for the homotopy class of $\Xi $ . Since the homotopy class of $\Xi $ determines that of $\Theta $ , we are done with the proof of (1). The proof of (2) is pretty much identical; the only difference is that now $\min \geqslant 4$ . Plugging this in the argument above, we get the bound $\operatorname {\mathrm {\mathbf {const}}}\cdot L^{k-1}\cdot e^{\frac L4}$ for some k. This is evidently a stronger bound that $\operatorname {\mathrm {\mathbf {const}}}\cdot e^{\frac L3}$ , and we are done.