$\Xi\circ\pi:S_{0}\to\Sigma$ by a homotopy whose tracks have at most length
$\operatorname{\mathbf{const}}$.∎

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

$\displaystyle\ell_{\Sigma}(\Xi(\eta\cap\operatorname{Surf}(\Delta)))$
$\displaystyle\leqslant R\cdot\ell_{S}(\gamma\cap
S^{\geqslant\varepsilon_{0}})$
$\displaystyle\ell_{\Sigma}(\Xi(\eta\setminus\operatorname{Surf}(\Delta)))$
$\displaystyle\leqslant\sum_{\kappa\in\pi_{0}(\gamma\cap
S^{\leqslant\varepsilon_{0}})}(\ell_{S}(\kappa)+R)$

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 number of times that $\eta$ crosses the wire $I$, we get from the
last two inequalities that

(6.34)
$\frac{1}{R}\cdot\ell_{\Xi}(\eta\cap\operatorname{Surf}(\Delta))+\sum_{I\in\operatorname{\mathbf{wire}}(\Delta_{0})}n_{I}(\eta)\cdot\max\left\\{\ell_{\Xi}(I)-R,0\right\\}\leqslant\ell_{S}(\gamma)$

where the “max” arises because a length is always non-negative. Anyways, note
that with the notation introduced in (6.31) we can rewrite (6.34) as

$\ell_{\Xi}^{\frac{1}{R}}(\gamma)\leqslant\ell_{S}(\gamma)$

Note also that from (6.33) we get that $n_{I}(\eta)\geqslant 2$ for all
$I\in\operatorname{\mathbf{wire}}(\Delta_{0})$. 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 $\operatorname{\mathbf{const}}\cdot
L^{d-1}\cdot e^{\frac{L}{2}}$ 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 $\operatorname{\mathbf{const}}\cdot L^{k-1}\cdot e^{\frac{L}{4}}$ for
some $k$. This is evidently a stronger bound that
$\operatorname{\mathbf{const}}\cdot e^{\frac{L}{3}}$, and we are done. ∎

### The fruits of our labor

After all this work we are now ready to prove Theorem 1.4.

###### Proof of Theorem 1.4.

Suppose that $\gamma$ is a closed geodesic with length
$\ell_{\Sigma}(\gamma)\leqslant L$ and such that there are two non-homotopic
minimal genus fillings $\beta_{1}:S_{1}\to\Sigma$ and
$\beta_{2}:S_{2}\to\Sigma$ with $\beta_{i}(\partial S_{i})=\gamma$. From
Proposition 5.4 we get that, without loss of generality we might assume that
both $\beta_{1}:S_{1}\to\Sigma$ and $\beta_{2}:S_{2}\to\Sigma$ are hyperbolic
fillings.

Let then $S=S_{i}\cup_{\partial S_{1}=\partial S_{2}}S_{2}$ be the hyperbolic
surface obtained by gluing both surfaces $S_{1}$ and $S_{2}$ along the
boundary in such a way that there is a pleated surface $\Theta:S\to\Sigma$
with $\Theta|_{S_{i}}=\beta_{i}$. Note once again that the map $\Theta$ maps
the crease $\hat{\gamma}=\partial S_{1}=\partial S_{2}$ geodesically to
$\gamma$. Moreover, Lemma 5.2 implies that the restriction of $\Theta$ to
$S\setminus\hat{\gamma}$ is geometrically incompressible. Taken together, all
of this means that the pair $(\Theta:S\to\Sigma,\hat{\gamma})$ is a pseudo-
double.

Now, Lemma 5.6 implies, together with the assumption that $\beta_{1}$ and
$\beta_{2}$ are not homotopic, that the $\varepsilon_{0}$-thin part of $S$ has
at most $6g-4$ connected components which are traversed twice by the crease
$\hat{\gamma}$. We thus get from Proposition 6.2 that there are at most
$\operatorname{\mathbf{const}}\cdot L^{6g-5}\cdot e^{\frac{L}{2}}$ choices for
the homotopy class of $(\Theta:S\to\Sigma,\hat{\gamma})$. Since the geodesic
$\gamma$ is determined by the homotopy class of $\Theta(\hat{\gamma})$, we
have proved that there are $\operatorname{\mathbf{const}}\cdot L^{6g-5}\cdot
e^{\frac{L}{2}}$ choices for $\gamma$, as we had claimed. ∎

##  7\. Proof of the main theorem

In this section we prove Theorem 1.1 from the introduction.

######  Theorem 1.1.

Let $\Sigma$ be a closed, connected, and oriented hyperbolic surface and for
$g\geqslant 1$ and $L>0$ let $\mathbf{B}_{g}(L)$ be as in (1.3). We have

$|\mathbf{B}_{g}(L)|\sim\frac{2}{12^{g}\cdot
g!\cdot(3g-2)!\cdot\operatorname{vol}(T^{1}\Sigma)^{2g-1}}\cdot L^{6g-4}\cdot
e^{\frac{L}{2}}$

as $L\to\infty$.

Before we can even explain the idea of the proof of Theorem 1.1 we need to
recall what fat graphs are, and a few of their properties:

### Fat graphs

A fat graph $X$ is a graph endowed with a cyclic ordering of the set
$\operatorname{\mathbf{half}}_{v}$ for each $v$—fat graphs are also sometimes
called ribbon graphs. Every fat graph is endowed with a canonically built
thickening $\operatorname{\mathbf{neigh}}(X)$, the thickening of $X$. For the
sake of concreteness let us discuss this in the particular case that $X$ is
trivalent. Well, we start by taking an oriented filled-in hexagon $G_{v}$ for
every vertex $v$, see Figure 3. If we label by $a,b,c$ the three elements in
$\operatorname{\mathbf{half}}_{v}$, given in the correct cyclic order, then we
label the boundary components of $G_{v}$ by $a,ab,b,bc,c,ca$, also given in
the correct cyclic order. Now, for every edge
$e\in\operatorname{\mathbf{edge}}(X)$ let
$v_{1},v_{2}\in\operatorname{\mathbf{vert}}(X)$ be the two (possibly
identical) vertices at which the two half-edges
$\vec{e}_{1}\in\operatorname{\mathbf{half}}_{v_{1}}$ and
$\vec{e}_{2}\in\operatorname{\mathbf{half}}_{v_{2}}$ corresponding to $e$ are
based, and identify in an orientation reversing way the $\vec{e}_{1}$-edge of
$\partial G_{v_{1}}$ with the $\vec{e}_{2}$-edge of $\partial G_{v_{2}}$.
Proceeding like this for all edges we end up with the thickening
$\operatorname{\mathbf{neigh}}(X)$ of $X$.

$\circlearrowleft$ $\circlearrowleft$

Figure 3. Constructing the thickening of $X$. Pictured are two oriented
hexagons corresponding to two vertices connected by an edge. The lighter lines
indicate the gluing of those two hexagons.

######  Definition.

A trivalent fat graph $X$ has genus $g$ if $\operatorname{\mathbf{neigh}}(X)$
is homeomorphic to a surface of genus $g$ with one boundary component.

One of the quantities that appear in the right side of Theorem 1.1 is the
number of genus $g$ fat graphs, or rather, that number weighted by the number
of automorphims of each such fat graph, where a map $F:X\to X^{\prime}$
between two fat graphs is a fat graph homeomorphism if first it is a
homeomorphism between two the underlying graphs, and then if it sends one fat
structure to the other one. Anyways, what is really lucky for us is that
Bacher and Vdovina [1] have computed that

$\sum\frac{1}{|\operatorname{Aut}(X)|}=\frac{2}{12^{g}}\cdot\frac{(6g-5)!}{g!\cdot(3g-3)!}$

where the sum takes place over all homeomorphism classes of genus $g$ fat
graphs.

######  Remark.

Bacher and Vdovina’s result is phrased in terms of 1-vertex triangulations of
the closed surface of genus $g$ up to orientation preserving homeomorphism.
Let us explain briefly how one goes from such triangulations to genus $g$ fat
graphs and back. The dual graph of a triangulation of a surface is a trivalent
fat graph—its thickening is the surface minus the vertices of the
triangulation. It follows that if the triangulation has a single vertex and
the surface is closed of genus $g$ then the fat graph has genus $g$.
Conversely, the thickening of a fat graph is equipped with a natural arc
system (one arc dual to each edge). When collapsing each boundary component of
the thickening to a point one gets a closed surface together with a
triangulation with as many vertices as connected components of the boundary.
It follows that if $X$ is a genus $g$ fat graph then one gets a 1-vertex
triangulation of a genus $g$ surface.

Let $X$ now be a fat graph and, for the sake of concreteness assume that it
has genus $g$. By construction we have a canonical embedding of $X$ into
$\operatorname{\mathbf{neigh}}(X)$ whose image is a spine. In particular there
is a retraction

$\operatorname{spine}:\operatorname{\mathbf{neigh}}(X)\to X$

such that the pre-image of every vertex is a tripod and the pre-image of every
point in the interior of and edge in $X$ is a segment. The image of
$\partial\operatorname{\mathbf{neigh}}(X)$ under $\operatorname{spine}$ runs
twice over every edge of $X$. We will refer to this parametrized curve in $X$
as $\partial X$.

######  Remark.

Note that reversing the orientation of $X$, that is reversing the cyclic order
at each vertex, has the effect of reversing the orientation of $\partial X$.

### The map

We will reduce the proof of Theorem 1.1 to the fact that we know how to count
critical realizations of graphs, that is to Theorem 1.3. Let us explain the
basic idea. For given $g$ consider the set

$\mathbf{X}_{g}=\left\\{(X,\phi)\middle|\begin{array}[]{c}X\text{ is a fat
graph of genus }g\text{ and}\\\ \phi:X\to\Sigma\text{ is a criticial
realization}\\\ \text{of the underlying
graph}\end{array}\middle\\}\right/_{\text{equiv}}$

of equivalence classes of realizations, where two realizations
$\phi:X\to\Sigma$ and $\phi^{\prime}:X^{\prime}\to\Sigma$ are equivalent if
there is a fat graph homeomorphism $\sigma:X\to X^{\prime}$ such that
$\phi=\phi^{\prime}\circ\sigma$.

> We stress something very important: the critical realization $\phi$ for
> $(X,\phi)\in\mathbf{X}_{g}$ is explicitly not assumed to respect the fat
> structure. On the other hand, the homeomorphism $\sigma$ in the definition
> of equivalence definitively has to preserve the fat structure.

Note that if $(X,\phi)$ is a equivalent to $(X^{\prime},\phi^{\prime})$ then
the curves $\phi(\partial X)$ and $\phi^{\prime}(\partial X^{\prime})$ are
freely homotopic to each other. In particular we have a well-defined map

(7.35) $\Lambda:\mathbf{X}_{g}\to\mathbf{C},\ \ (X,\phi)\mapsto\text{ geodesic
homotopic to }\phi(\partial X)$

where $\mathbf{C}$ is, as it was earlier, the collection of all oriented
geodesics in $\Sigma$.

The basic idea of the proof of Theorem 1.1 is that the map (7.35) has the
following informally stated properties:

  1. (1)

$\Lambda$ is basically injective with image basically contained in
$\mathbf{B}_{g}$,

  2. (2)

$\Lambda$ is basically surjective onto $\mathbf{B}_{g}$, and

  3. (3)

generically, the geodesic $\Lambda(X,\phi)$ has length almost exactly equal to
$2\cdot\ell(\phi)-C$ for some explicit constant $C$.

Let us start by clarifying the final point. Suppose that
$(X,\phi)\in\mathbf{X}_{g}$ is such that $\phi$ is $\ell_{0}$-long for some
large $\ell_{0}$. Well, since the curve $\partial X$ runs exactly twice over
each edge $X$ we get that it its image $\phi(\partial X)$ consists of
$2\operatorname{\mathbf{edge}}(X)$ geodesic segments of length at least
$\ell_{0}$ and with $3\cdot\operatorname{\mathbf{vert}}(X)=-6\cdot\chi(X)$
corners where it makes angle equal to $\frac{2\pi}{3}$. We get now from a
standard hyperbolic geometry computation (or, if you so wish, from a limiting
argument) that when $\ell_{0}$ is large then, up to an small error depending
on $\ell_{0}$, when we pull tight $\phi(\partial X)$ to get $\Lambda(X,\phi)$
we save $\log\frac{4}{3}$ at each one of those corners. Taking into account
that $\chi(X)=1-2g$ when $X$ has genus $g$, this is what we have proved:

######  Lemma 7.1.

For every $\delta>0$ there is $\ell_{\delta}$ with

$\left|\ell_{\Sigma}(\Lambda(X,\phi))-\left(2\ell_{\Sigma}(\phi)-6\cdot(2g-1)\cdot\log\frac{4}{3}\right)\right|\leqslant\delta$

for every $(X,\phi)\in\mathbf{X}_{g}$ such that $\phi$ is
$\ell_{\delta}$-long. ∎

######  Remark.

Where does $\log\frac{4}{3}$ come from? Well, if $\Delta\subset\mathbb{H}^{2}$
is an ideal triangle with vertices $\theta_{1},\theta_{2}$ and $\theta_{3}$
and center $p$, and if $p^{\prime}$ is the projection of $p$ to the side say
$(\theta_{1},\theta_{2})$ then $\log\frac{2}{\sqrt{3}}$ is the difference
between the values at $p$ and $p^{\prime}$ of the Buseman function centered at
$\theta_{1}$, and every time we pass by a vertex we basically save twice that
amount (see Figure 4).

$p$ $p^{\prime}$ $\theta_{1}$ $\theta_{2}$

Figure 4. The ideal triangle $\Delta$ with vertices
$\theta_{1},\theta_{2},\theta_{3}=\infty$ and center $p$. Each one of the legs
of the bold printed tripod has length $\log\frac{2}{\sqrt{3}}$.

Our next goal is to prove that the map $\Lambda$ is basically bijective onto
$\mathbf{B}_{g}$, but we should first formalize what we mean by “basically”.
Well, suppose that we have a set $Z$ consisting of either realizations, or
curves, or of anything else consisting of elements $\alpha\in Z$ whose length
$\ell_{\Sigma}(\alpha)$ can be measured, and set $Z(L)=\\{\alpha\in Z\text{
with }\ell_{\Sigma}(\alpha)\leqslant L\\}$. We will say that a subset

(7.36) $W\subset Z\text{ is {\em negligible} in }Z\text{ if
}\limsup_{L\to\infty}\frac{|W(L)|}{|Z(L)|}=0$

If the ambient set $Z$ is understood from the context, then we might just say
that $W$ is negligible. The complement of a negligible set is said to be
generic and the elements of a generic set are themselves generic. For example
we get from Lemma 4.3 and Lemma 7.1 that for all $\delta$ we have that

$\left|\ell_{\Sigma}(\Lambda(X,\phi))-\left(2\ell_{\Sigma}(\phi)-2\cdot(6g-3)\cdot\log\frac{4}{3}\right)\right|\leqslant\delta$

for $(X,\phi)\in\mathbf{X}_{g}$ generic.

### Basic bijectivity of $\Lambda$

Above we used the word “basically” as meaning that something was true up to
negligible sets. Let us start by proving that the map $\Lambda$ is basically
injective and that its image is contained in the set $\mathbf{B}_{g}$ of
closed geodesics of genus $g$.

######  Lemma 7.2.

There is a generic subset $W\subset\mathbf{X}_{g}$ such that the restriction
of $\Lambda$ to $W$ is injective and that its image is contained in
$\mathbf{B}_{g}$.

###### Proof.

Let $\ell_{0}$ and $C$ be such that for every $(X,\phi)\in\mathbf{X}_{g}$ so
that $\phi$ is $\ell_{0}$-long we have

$2\cdot\ell_{\Sigma}(\phi)-C\leqslant\ell_{\Sigma}(\Lambda(X,\phi))\leqslant
2\ell_{\Sigma}(\phi)$

and let $\mathbf{X}_{g,\ell_{0}}$ be the set of those pairs. We get from Lemma
4.3 that $\mathbf{X}_{g,\ell_{0}}$ is generic in $\mathbf{X}_{g}$. It follows
hence from Theorem 1.3 that

(7.37) $|\\{(X,\phi)\in\mathbf{X}_{g,\ell_{0}}\text{ with
}\ell_{\Sigma}(\Lambda(X,\phi))\leqslant
L\\}|\geqslant\operatorname{\mathbf{const}}\cdot L^{6g-4}\cdot
e^{\frac{L}{2}}$

Note now that each element $(X,\phi)$ of $\mathbf{X}_{g}$, and thus of
$\mathbf{X}_{g,\ell_{0}}$, determines not only the curve $\Lambda(X,\phi)$ but
also a homotopy class of fillings for this curve, namely
$\phi\circ\operatorname{spine}:\operatorname{\mathbf{neigh}}(X)\to\Sigma$. Let
now $Z\subset\mathbf{X}_{g,\ell_{0}}$ be the set of pairs $(X,\phi)$ so that
the filling
$\phi\circ\operatorname{spine}:\operatorname{\mathbf{neigh}}(X)\to\Sigma$ is
not unique in the sense that the curve $\Lambda(X,\phi)$ admits another non-
homotopic genus $g$ filling and let

$W=\mathbf{X}_{g,\ell_{0}}\setminus Z$

be its complement. From Theorem 1.4 we get that $Z$ consists of at most
$\operatorname{\mathbf{const}}\cdot L^{6g-5}\cdot e^{\frac{L}{2}}$ many
elements and hence that $W$ is generic.

######  Claim.

If $\ell_{0}$ is over some threshold we have that
$\Lambda(W)\subset\mathbf{B}_{g}$.

###### Proof.

First we have by construction that the curve $\Lambda(X,\phi)$ admits the
genus $g$ filling
$\phi\circ\operatorname{spine}_{X}:\operatorname{\mathbf{neigh}}(X)\to\Sigma$.
Suppose that it admits a smaller genus filling. Then, adding handles and
mapping them to points we get that $\Lambda(X,\phi)$ admits a
non-$\pi_{1}$-injective genus $g$ filling. Since on the other hand the filling
$\phi\circ\operatorname{spine}_{X}:\operatorname{\mathbf{neigh}}(X)\to\Sigma$
is $\pi_{1}$-injective as long as $\ell_{0}$ is over some threshold, we get
that $\Lambda(X,\phi)$ admits two non-homotopic genus $g$ fillings,
contradicting the assumption that $(X,\phi)\in W$. ∎

It remains to prove that the restriction of $\Lambda$ to the generic set $W$
is injective.

Well, suppose that we have $(X,\phi),(X^{\prime},\phi^{\prime})\in W$ with
$\Lambda(X,\phi)=\Lambda(X^{\prime},\phi^{\prime})$. Since $(X,\phi)$ and
$(X^{\prime},\phi^{\prime})$ belong to $W$ we know that the two fillings

$\phi\circ\operatorname{spine}_{X}:\operatorname{\mathbf{neigh}}(X)\to\Sigma\text{
and
}\phi^{\prime}\circ\operatorname{spine}_{X^{\prime}}:\operatorname{\mathbf{neigh}}(X^{\prime})\to\Sigma$

are homotopic. Recall that this means that there is a homeomorphism

$\sigma:\operatorname{\mathbf{neigh}}(X)\to\operatorname{\mathbf{neigh}}(X^{\prime})$

with $\phi^{\prime}\circ\operatorname{spine}_{X^{\prime}}\circ\sigma$
homotopic to $\phi\circ\operatorname{spine}_{X}$. Since $X$ and $X^{\prime}$
are spines of $\operatorname{\mathbf{neigh}}(X)$ and
$\operatorname{\mathbf{neigh}}(X^{\prime})$ we deduce that there is a homotopy
equivalence $\bar{\sigma}:X\to X^{\prime}$ such that
$\phi^{\prime}\circ\bar{\sigma}$ is homotopic to $\phi$. Now, since the
lengths of both $\phi(X)$ and $\phi^{\prime}(X^{\prime})$ are, up to a
constant, basically half the length of
$\Lambda(X,\phi)=\Lambda(X^{\prime},\phi^{\prime})$ we see that $\phi$ and
$\phi^{\prime}$ satisfy the conditions in Proposition 2.3. It thus follows
that there is a homeomorphism $F:X^{\prime}\to X$ mapping edges at constant
velocity, with $F\circ\bar{\sigma}$ homotopic to the identity, and with
$\phi\circ F$ homotopic to $\phi^{\prime}$. Now, both

$\phi\circ F,\phi^{\prime}:X\to\Sigma$

are critical realizations and both are homotpic to each other. We get then
from (1) in Lemma 2.2 that $\phi^{\prime}=\phi\circ F$. To conclude, note that
since $F$ is a homotopy inverse of $\bar{\sigma}$ and since $\bar{\sigma}$ is
induced by the homeomorphism
$\sigma:\operatorname{\mathbf{neigh}}(X^{\prime})\to\operatorname{\mathbf{neigh}}(X)$
we get that $F$ is a fat graph homeomorphism. This proves that $(X,\phi)$ and
$(X^{\prime},\phi^{\prime})$ are equivalent, and hence that the restriction of
$\Lambda$ to $W$ is injective. We are done with Lemma 7.2. ∎

######  Remark.

Note that (7.37) and Lemma 7.2 imply that

(7.38) $|\mathbf{B}_{g}(L)|>\operatorname{\mathbf{const}}\cdot L^{6g-4}\cdot
e^{\frac{L}{2}}$

for all $L$ large enough.

Our next goal is to show that the image of $\Lambda$ contains a large subset
of $\mathbf{B}_{g}$.

######  Lemma 7.3.

There is a generic subset of $\mathbf{B}_{g}$ which is contained in the image
of $\Lambda$.

###### Proof.

Let $Z\subset\mathbf{B}_{g}$ be the set of those geodesics which admits a
hyperbolic genus $g$ filling $\beta:S\to\Sigma$ such that the
$\varepsilon_{0}$-thin part of the double $DS$ of $S$ has at most $6g-4$
connected components $U$ through with $\iota(U,\partial S)=2$. It follows from
Proposition 6.2 that $Z$ has at most $\operatorname{\mathbf{const}}\cdot
L^{6g-5}\cdot e^{\frac{L}{2}}$ elements with length $\leqslant L$. It follows
from (7.38) that $Z$ is negligible and hence that its complement
$W=\mathbf{B}_{g}\setminus Z$ is generic. We claim that $W$ is contained in
the image of $\Lambda$, at least if $\varepsilon_{0}$ is chosen small enough.

Well, each $\gamma\in W$ admits a hyperbolic filling $\beta:S\to\Sigma$ such
that the $\varepsilon_{0}$-thin part of the double $DS$ has at least $6g-3$
connected components $U$ with $\iota(\partial S,U)=2$. Since $DS$ has genus
$2g$ we have that $6g-3$ is actually the maximal number of connected
components that its thin part can have. It follows that the double $DS$ of $S$
admits a pants decomposition consisting of very short curves and that moreover
$\partial S$ cuts each one of them exactly twice. It follows that $S$ has
$6g-3$ orthogeodesics cutting $S$ into a union of $4g-2$ right-angle hexagons.
These hexagons have 3 alternating sides which are extremely short. It follows
that each one of the hexagons contains a pretty large compact set which is
almost isometric to a large neighborhood of the center of an ideal triangle in
$\mathbb{H}^{2}$. Declare the center of the hexagon to be the image of the
center of the ideal triangle by this almost isometric map. Now, we can
represent the dual graph of the decomposition of $S$ by our short
orthogeodesic segments as a geodesic subgraph $X$ of $S$ with vertices in the
centers of the hexagons, see Figure 5.

The graph $X$ is a spine of $S$ and hence it inherits from $S$ a fat graph
structure with $S$ as its neighborhood. Let now $\psi:X\to\Sigma$ be the
realization obtained from the restriction of $\beta$ to $X$ by pulling the
edges tight.

Figure 5. Getting an almost critical realization out of a filling whose domain
can be cut into hexagons by very short orthogeodesics.

######  Claim.

For all $\delta$ there is $\varepsilon$ such that if
$\varepsilon_{0}<\varepsilon$ then the realization $\psi:X\to\Sigma$ is
$\delta$-critical.

###### Proof.

Suppose that the claim fails to be true. This means that for some $\delta$
there are a sequence of counter examples with $\varepsilon_{0}\to 0$. Since
$\varepsilon_{0}\to 0$ we get that the images of the hexagons converge to a
1-Lipschitz map of a geodesic triangle into $\Sigma$ which however maps the
boundary geodesics to geodesics. Such a map is an isometric embedding of the
triangle in question. Now, in a geodesic triangle, the geodesic rays starting
in the center and pointing into the cusps make angle $\frac{2\pi}{3}$. This
means that the angles in the approximates converge to $\frac{2\pi}{3}$
contradicting our assumption that some of them were at least $\delta$ off
$\frac{2\pi}{3}$. We have proved the claim. ∎

It follows thus from the claim and Corollary 2.4 that, as long as
$\varepsilon_{0}$ is under some threshold, the realization $\psi:X\to\Sigma$
is homotopic to a critical realization $\phi:X\to\Sigma$. This implies
$(X,\phi)$ belongs to the domain of $\Lambda$ and that
$\gamma=\Lambda(X,\phi)$. We have proved that the generic set
$W\subset\mathbf{B}_{g}$ is contained in the image of $\Lambda$. ∎

We are now ready to wrap all of this up.

### Proof of Theorem 1.1

Combining Lemma 7.1, Lemma 7.2 and Lemma 7.3 we get that for all $\delta>0$
there is a generic subset $W\subset\mathbf{X}_{g}$ which is mapped injectively
under $\Lambda$ to a generic subset of $\mathbf{B}_{g}$ in such a way that

$\left|\ell_{\Sigma}(\Lambda(X,\phi))-2\ell_{\Sigma}(\phi)+2\kappa\right|\leqslant\delta$

where

(7.39)
$\kappa=-3\chi(X)\cdot\log\frac{4}{3}=\log\left(\left(\frac{4}{3}\right)^{6g-3}\right)$

It follows that for all $\delta>0$ our set $\mathbf{B}_{g}(L)$ has, for
$L\to\infty$, at least as many elements as
$\mathbf{X}_{g}(\frac{L}{2}+\kappa-\delta)$ and at most as many as
$\mathbf{X}_{g}(\frac{L}{2}+\kappa+\delta)$. In symbols this just means that

(7.40)
$\left|\mathbf{X}_{g}\middle(\frac{L}{2}+\kappa-\delta\middle)\right|\preceq|\mathbf{B}_{g}(L)|\preceq\left|\mathbf{X}_{g}\middle(\frac{L}{2}+\kappa+\delta\middle)\right|$

for large $L$. It thus remains to estimate the cardinality of
$\mathbf{X}_{g}(L)$.

######  Lemma 7.4.

We have

$|\mathbf{X}_{g}(L)|\sim\frac{1}{12^{g}}\cdot\left(\frac{3}{2}\right)^{6g-3}\cdot\frac{1}{g!\cdot(3g-2)!\cdot\operatorname{vol}(T^{1}\Sigma)^{2g-1}}\cdot
L^{6g-4}\cdot e^{L}$

as $L\to\infty$.

###### Proof.

From Theorem 1.3 we get that every trivalent graph $X$ has

$|\mathbf{G}^{X}(L)|\sim\left(\frac{2}{3}\right)^{3\chi(X)}\cdot\frac{\operatorname{vol}(T^{1}\Sigma)^{\chi(X)}}{(-3\chi(X)-1)!}\cdot
L^{-3\chi(X)-1}\cdot e^{L}$

critical realizations of length at most $L$ in $\Sigma$. This implies that
whenever $X$ is fat graph of genus $g$ then asymptotically there are

$\frac{1}{|\operatorname{Aut}(X)|}\left(\frac{2}{3}\right)^{3\chi(X)}\cdot\frac{\operatorname{vol}(T^{1}\Sigma)^{\chi(X)}}{(-3\chi(X)-1)!}\cdot
L^{-3\chi(X)-1}\cdot e^{L}$

many elements in $\mathbf{X}_{g}(L)$ represented by $(X,\phi)$ for some
critical $\phi:X\to\Sigma$ of length $\ell(\phi)\leqslant L$. Adding over all
possible types of genus $g$ fat graphs we get that

$|\mathbf{X}_{g}(L)|\sim\sum_{X}\frac{1}{|\operatorname{Aut}(X)|}\left(\frac{2}{3}\right)^{3\chi(X)}\cdot\frac{\operatorname{vol}(T^{1}\Sigma)^{\chi(X)}}{(-3\chi(X)-1)!}\cdot
L^{-3\chi(X)-1}\cdot e^{L}$

From the Bacher-Vdovina [1] result mentioned earlier and taking into
consideration that $\chi(X)=1-2g$ we get

$|\mathbf{X}_{g}(L)|\sim\frac{2}{12^{g}}\cdot\frac{(6g-5)!}{g!\cdot(3g-3)!}\cdot\left(\frac{3}{2}\right)^{6g-3}\cdot\frac{\operatorname{vol}(T^{1}\Sigma)^{1-2g}}{(6g-4)!}\cdot
L^{6g-4}\cdot e^{L}$

The claim follows now from elementary algebra. ∎

Now, from Lemma 7.4 we get that

$\left|\mathbf{X}_{g}\left(\frac{L}{2}\right)\right|\sim\frac{1}{12^{g}}\cdot\left(\frac{3}{2}\right)^{6g-3}\cdot\frac{1}{g!\cdot(3g-2)!\cdot\operatorname{vol}(T^{1}\Sigma)^{2g-1}}\cdot\frac{L^{6g-4}}{2^{6g-4}}\cdot
e^{\frac{L}{2}}.$

Taking into account that

$\left|\mathbf{X}_{g}\left(\frac{L}{2}+\kappa\right)\right|\sim\left|\mathbf{X}_{g}\left(\frac{L}{2}\right)\right|\cdot
e^{\kappa}=\left|\mathbf{X}_{g}\left(\frac{L}{2}\right)\right|\cdot\left(\frac{4}{3}\right)^{6g-3}$

we get that

$\left|\mathbf{X}_{g}\left(\frac{L}{2}+\kappa\right)\right|\sim\frac{2}{12^{g}\cdot
g!\cdot(3g-2)!\cdot\operatorname{vol}(T^{1}\Sigma)^{2g-1}}\cdot L^{6g-4}\cdot
e^{\frac{L}{2}}$

It follows thus from (7.40) that

$|\mathbf{B}_{g}(L)|\sim\frac{2}{12^{g}\cdot
g!\cdot(3g-2)!\cdot\operatorname{vol}(T^{1}\Sigma)^{2g-1}}\cdot L^{6g-4}\cdot
e^{\frac{L}{2}}$

as we wanted to show. This concludes the proof of Theorem 1.1.∎

##  8\. Curves bounding immersed surfaces

We now turn our attention to Theorem 1.2:

######  Theorem 1.2.

Let $\Sigma$ be a closed, connected, and oriented hyperbolic surface and for
$g\geqslant 1$ and $L>0$ let $\mathbf{B}_{g}(L)$ be as in (1.3). We have

$|\\{\gamma\in\mathbf{B}_{g}(L)\text{ bounds immersed surface of genus
}g\\}|\sim\frac{1}{2^{4g-2}}|\mathbf{B}_{g}(L)|$

as $L\to\infty$.

Denote by

$\mathbf{B}_{g}^{\operatorname{imm}}(L)=\\{\gamma\in\mathbf{B}_{g}(L)\text{
bounds immersed surface of genus }g\\}$

the set we want to count. From the proof of Theorem 1.1 we get that

$|\mathbf{B}_{g}^{\operatorname{imm}}(L)|\sim\left|\mathbf{X}_{g}^{\operatorname{imm}}\left(\frac{L}{2}+\kappa\right)\right|$

where $\kappa$ is as in (7.39) and where

$\mathbf{X}_{g}^{\operatorname{imm}}(L)\stackrel{{\scriptstyle\text{def}}}{{=}}\left\\{(X,\phi)\in\mathbf{X}_{g}(L)\,\middle|\begin{array}[]{c}\text{the
realization }\phi:X\to\Sigma\text{ extends}\\\ \text{to an immersion of the
thickening}\\\ \operatorname{\mathbf{neigh}}(X)\text{ of
}X\end{array}\right\\}.$

Equivalently, $(X,\phi)\in\mathbf{X}_{g}$ belongs to
$\mathbf{X}_{g}^{\operatorname{imm}}$ if the cyclic ordering at each vertex of
$X$ agrees with the one pulled back from $\Sigma$ via $\phi$, that is the one
coming from the orientation of $\Sigma$. We can refer to such realizations of
a fat graph as fat realizations. With this language we have that

$\mathbf{X}_{g}^{\operatorname{imm}}(L)=\left\\{(X,\phi)\in\mathbf{X}_{g}(L)\,\middle|\,\phi\text{
is a fat realization of the fat graph }X\right\\}.$

For a given trivalent fat graph $X$ let

$\mathbf{G}^{X}_{\operatorname{imm}}(L)=\\{\phi\in\mathbf{G}^{X}(L)\,|\,\phi\text{
is a fat realization of }X\\}$

be the set of fat critical realizations of $X$ of total length at most $L$.
Note that

$|\mathbf{X}_{g}^{\operatorname{imm}}(L)|=\sum_{X}\frac{1}{|\operatorname{Aut}(X)|}|\mathbf{G}^{X}_{\operatorname{imm}}(L)|.$

What is still missing to be able to run the proof as in that of Theorem 1.1 is
a version of Theorem 1.3 for $\mathbf{G}^{X}_{\operatorname{imm}}$. Well, here
it is:

######  Theorem 8.1.

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

$|\mathbf{G}^{X}_{\operatorname{imm}}(L)|\sim
2^{2\chi(X)}\cdot\left(\frac{2}{3}\right)^{3\chi(X)}\cdot\frac{\operatorname{vol}(T^{1}\Sigma)^{\chi(X)}}{(-3\chi(X)-1)!}\cdot
L^{-3\chi(X)-1}\cdot e^{L}$

as $L\to\infty$.

Assuming Theorem 8.1 for the moment we get from $\chi(X)=1-2g$ and from the
Bacher-Vdovina theorem that

$|\mathbf{X}^{\operatorname{imm}}_{g}(L)|\sim\frac{1}{2^{4g-2}}\cdot\frac{2}{12^{g}}\cdot\frac{(6g-5)!}{g!\cdot(3g-3)!}\cdot\left(\frac{3}{2}\right)^{6g-3}\cdot\frac{\operatorname{vol}(T^{1}\Sigma)^{1-2g}}{(6g-4)!}\cdot
L^{6g-4}\cdot e^{L}$

from where we get, as in the proof of Theorem 1.1, that

$|\mathbf{B}_{g}^{\operatorname{imm}}(L)|\sim\frac{1}{2^{4g-2}}\frac{2}{12^{g}\cdot
g!\cdot(3g-2)!\cdot\operatorname{vol}(T^{1}\Sigma)^{2g-1}}\cdot L^{6g-4}\cdot
e^{\frac{L}{2}}$

The claim of Theorem 1.2 follows now from this statement combined with Theorem
1.1.

All that is left to do is to prove Theorem 8.1. Since it is basically
identical to the proof of Theorem 1.3 we just point out the differences. The
key is to obtain a fat graph version of Proposition 4.2. In a nutshell, the
idea of the proof of this proposition was that

  1. (1)

we could compute the volume
$\operatorname{vol}(\mathcal{G}_{\varepsilon-\operatorname{crit}}^{X}(\vec{L},h))$
of the set of $\varepsilon$-critical realizations of $X$ whose edge lengths
were in a box, and

  2. (2)

we knew that every connected component contributes the same amount, and how
much.

Let thus
$\mathcal{G}_{\varepsilon-\operatorname{crit}-\operatorname{imm}}^{X}(\vec{L},h)\subset\mathcal{G}_{\varepsilon-\operatorname{crit}}^{X}(\vec{L},h)$
be those $\varepsilon$-critical realizations in our box which preserve the fat
structure. Recalling now that by Proposition 2.6 the connected components of
$\mathcal{G}_{\varepsilon-\operatorname{crit}}^{X}(\vec{L},h)$ have small
diameter, we deduce that the induced fat structure is constant over each such
connected component. It follows that
$\mathcal{G}_{\varepsilon-\operatorname{crit}-\operatorname{imm}}^{X}(\vec{L},h)$
is a union of connected components of
$\mathcal{G}_{\varepsilon-\operatorname{crit}}^{X}(\vec{L},h)$. In particular,
to be able to obtain a fat graph version of Proposition 4.2 we just need to be
able to compute
$\operatorname{vol}(\mathcal{G}_{\varepsilon-\operatorname{crit}-\operatorname{imm}}^{X}(\vec{L},h))$.

Now, in the proof of Proposition 4.2, the key ingredient of the computation of
the volume of $\mathcal{G}_{\varepsilon-\operatorname{crit}}(\vec{L},h)$ was
Corollary 3.3—we remind the reader that the statement of the said corollary
was that for any $\vec{x}\in\Sigma^{\operatorname{\mathbf{vert}}X}$ we have

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

as $\min_{e\in\operatorname{\mathbf{edge}}(X)}L_{e}\to\infty$, where
$\mathbf{G}^{X}_{\vec{x},\varepsilon-\operatorname{crit}}(\vec{L},h)$ is the
set of $\varepsilon$-critical realizations $\phi:X\to\Sigma$ mapping the
vertex $v$ to the point $x_{v}=\phi(v)$. As we see, if we want to just copy
line-by-line the computation of the volume of
$\mathcal{G}_{\varepsilon-\operatorname{crit}}(\vec{L},h)$ to get the volume
of
$\mathcal{G}_{\varepsilon-\operatorname{crit}-\operatorname{imm}}(\vec{L},h)$
what we need to know is the number of elements in the set
$\mathbf{G}^{X}_{\vec{x},\varepsilon-\operatorname{crit}-\operatorname{imm}}(\vec{L},h)$
of $\varepsilon$-critical realizations $\phi:X\to\Sigma$ mapping the vertex
$v$ to the point $x_{v}=\phi(v)$ and preserving the fat structure.

Now, to obtain the number of elements in
$\mathbf{G}_{\vec{x},\varepsilon-\operatorname{crit}}^{X}(\vec{L},h)$ what we
did was to invoke Theorem 3.2 and compute the volume of the set

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

of those tuples $(v_{\vec{e}})_{\vec{e}\in\operatorname{\mathbf{half}}(X)}$
with
$\angle(v_{\vec{e}_{1}},v_{\vec{e}_{2}})\in[\frac{2\pi}{3}-\varepsilon,\frac{2\pi}{3}+\varepsilon]$
for all distinct $\vec{e}_{1},\vec{e}_{2}\in\operatorname{\mathbf{half}}(X)$
incident to the same vertex. Accordingly, to compute the number of elements in
$\mathbf{G}_{\vec{x},\varepsilon-\operatorname{crit}\operatorname{imm}}^{X}(\vec{L},h)$
we need to compute the volume of the set

$U_{\vec{x},\varepsilon-\operatorname{crit}-\operatorname{imm}}^{X}\subset
U_{\vec{x},\varepsilon-\operatorname{crit}}^{X}$

consisting of tuples such that for any vertex
$v\in\operatorname{\mathbf{vert}}(X)$ the cyclic order of the half-edges
incident to $v$ agrees with the one of the corresponding unit tangent vectors.
Following the computation of
$\operatorname{vol}(U_{\vec{x},\varepsilon=\operatorname{crit}})$ we get that

$\operatorname{vol}(U_{\vec{x},\varepsilon-\operatorname{crit}-\operatorname{imm}}^{X})=\frac{1}{2^{|\operatorname{\mathbf{vert}}(X)|}}\operatorname{vol}(U_{\vec{x},\varepsilon-\operatorname{crit}}^{X})=2^{2\chi(X)}\cdot\operatorname{vol}(U_{\vec{x},\varepsilon-\operatorname{crit}}^{X})$

As we just discussed, this implies that

$|\mathbf{G}_{\vec{x},\varepsilon-\operatorname{crit}-\operatorname{imm}}^{X}(\vec{L},h)|\sim
2^{2\chi(X)}\cdot|\mathbf{G}_{\vec{x},\varepsilon-\operatorname{crit}}^{X}(\vec{L},h)|$

and hence that

$\operatorname{vol}(\mathcal{G}_{\varepsilon-\operatorname{crit}-\operatorname{imm}}^{X}(\vec{L},h))\sim
2^{2\chi(X)}\cdot\operatorname{vol}(\mathcal{G}_{\varepsilon-\operatorname{crit}}^{X}(\vec{L},h))$

and thus that

$|\mathbf{G}^{X}_{\operatorname{imm}}(L)|\sim
2^{2\chi(X)}\cdot|\mathbf{G}^{X}(L)|.$

Theorem 8.1 follows now from Theorem 1.3. ∎

##  9\. Comments

In this section we discuss where and how much we use the assumption that
$\Sigma$ is a closed orientable surface.

### First, do the results here apply if we replace $\Sigma$ by a compact
2-dimensional orbifold $\mathcal{O}=\Gamma\backslash\mathbb{H}^{2}$?

The answer is yes for Theorem 1.3, with exactly the same proof. One should
just do everything equivariantly. For example, a realization in $\mathcal{O}$
of a graph $X$ should be a map $\tilde{\phi}:\tilde{X}\to\mathbb{H}^{2}$ from
the universal cover of $X$ to $\mathbb{H}^{2}$ which is equivariant under a
homomorphism $\tilde{\phi}_{*}:\pi_{1}(X)\to\Gamma$, and where two such pairs
$(\tilde{\phi},\tilde{\phi}_{*})$ and $(\tilde{\psi},\tilde{\psi}_{*})$ are
identified if they differ by an element of $\Gamma$. Once we rephrase the
situation in those terms, everything extends in the obvious way. For example,
the space $\mathcal{G}^{X}$ of realizations of a graph in $\mathcal{O}$ is now
an orbifold: the map
$\mathcal{G}^{X}\to\mathcal{O}^{\operatorname{\mathbf{vert}}X}$ sending each
realization to the images of the vertices is a covering in the category of
orbifolds.

On the other hand, to prove Theorem 1.1 one needs to be a little bit careful
because in Section 5 and Section 6 we used repeatedly that every sufficiently
short curve in $\Sigma$ is homotopically trivial, and this is no longer true
if we are working in $\mathcal{O}$.

### What about allowing $\Sigma$ to have cusps?

Here we again have problems with the discussion in Section 5 and Section 6,
but this time it is much worse. In some sense, the results of Section 5 and
Section 6 are just generalizations of Lemma 4.1, and this lemma fails in the
presence of cusps: indeed, suppose that $\Sigma$ is a once punctured torus and
$X$ is a graph with two vertices $x$ and $x^{\prime}$ (if you wish you can
make $X$ trivalent while essentially keeping the same reasoning as we will
present, but doing so might obscure things slightly) and with $6$ edges
$f_{1},f_{2},e_{1},e_{2},e_{3}$ and $h$ such that

  * •

the edges $f_{1},f_{2}$ are incident on both ends to $x$,

  * •

the edges $e_{1},e_{2},e_{3}$ are incidents on both ends to $x^{\prime}$, and

  * •

the edge $h$ runs from $x$ to $x^{\prime}$.

Fix now a horospherical neighborhood of the cusps in $\Sigma$, fix a point
${\bf x}_{0}$ and let ${\bf x}_{t}$ be the point at distance $t$ of ${\bf
x}_{0}$ along the ray pointing directly into the cusp. Now, if we are given a
vector $\vec{L}=(F_{1},F_{2},E_{1},E_{3},E_{3},H)$ with positive real
coefficients consider realizations $\phi:X\to\Sigma$ with $\phi(x)={\bf
x}_{0}$, with $\phi(x^{\prime})={\bf x}_{H}$, and with $\phi(h)$ equal to the
segment of length $H$ joining ${\bf x}_{0}$ and ${\bf x}_{H}$. Now, we have
$\operatorname{\mathbf{const}}e^{F_{1}+F_{2}}$ choices for the images of
$f_{1}$ and $f_{2}$ subject to the restriction that $\phi(f_{i})$ is for
$i=1,2$ a geodesic segment of length at most $F_{i}$. Note also that the
horospherical simple loop based at ${\bf x}_{H}$ has length
$\operatorname{\mathbf{const}}\cdot e^{-H}$ and that geodesic loops in the
cusp, based at ${\bf x}_{H}$ and with length $\ell$ are homotopic to
horospherical segments of length at most $\operatorname{\mathbf{const}}\cdot
e^{\frac{\ell}{2}}$. This implies that, if we want to map $e_{i}$ to a loop in
the cusp and of length at most $E_{i}$, then we have at least
$\operatorname{\mathbf{const}}\cdot e^{\frac{1}{2}E_{i}}\cdot e^{H}$ choices.
Altogether we have at least

$\operatorname{\mathbf{const}}\cdot
e^{F_{1}+F_{2}+\frac{1}{2}E_{1}+\frac{1}{2}E_{2}+\frac{1}{2}E_{3}+3H}$

choices of (homotopy classes of) realizations of $X$ into $\Sigma$ with
$\ell(\phi(f_{1}))\leqslant F_{1}$, $\ell(\phi(f_{2}))\leqslant F_{2}$,
$\ell(\phi(e_{1}))\leqslant E_{1}$, $\ell(\phi(e_{2}))\leqslant E_{2}$
$\ell(\phi(e_{3}))\leqslant E_{3}$, $\ell(\phi(H))\leqslant H$. In particular
if we set

$\vec{L}=(F_{1},F_{2},E_{1},E_{2},E_{3},H)=(n,n,\frac{1}{2}n,\frac{1}{2}n,\frac{1}{2}n,\frac{5}{2}n)$

we have at least $\operatorname{\mathbf{const}}\cdot e^{\frac{61}{4}n}$ such
realizations, and this is a much larger number than
$\operatorname{\mathbf{const}}e^{6n}=\operatorname{\mathbf{const}}e^{\|L\|}$.
This proves that the analogue of Lemma 4.1 fails if $\Sigma$ is not compact.

In summary, if $\Sigma$ has finite volume but is not compact, then we do not
even know whether Theorem 1.3 holds. Although we suspect that the answer is
yes.

### Can $\Sigma$ be non-orientable?

If $\Sigma$ is a compact hyperbolic surface which however is non-orientable
then we know that both Theorem 1.3 and Theorem 1.1 hold true: we never used
orientability during their proofs. We did however in the proof of Theorem 1.2:
unless $\Sigma$ is oriented it makes little sense to speak about the induced
fat graph structure. We do not know what happens with Theorem 1.2 when the
ambient surface is not oriented.

### And what about higher dimensions?

If we replace $\Sigma$ by a closed hyperbolic manifold of other dimension than
$2$, then Theorem 1.3 and Theorem 1.1 should still hold, and with proofs
which, if not identical, keep the same spirit. It is however less clear
whether there should be an interesting analogue of Theorem 1.2: at least if
$\dim\geqslant 5$, where every map of a surface can be deformed to an
embedding.

##  Appendix A The geometric prime number theorem

The argument we used to prove Theorem 1.3 can be used to recover Huber’s
geometric primer number theorem, and this is what we do here. Besides giving a
simple proof of this theorem, it might help the reader understand the logic of
the proof of Theorem 1.3.

######  The Geometric Prime Number Theorem (Huber).

Let $\Sigma$ be a closed, connected, orientable, hyperbolic surface and let
$\mathbf{C}(L)$ be the set of closed non-trivial oriented geodesics in
$\Sigma$ of length at most $L$. We have

$|\mathbf{C}(L)|\sim\frac{e^{L}}{L}$

as $L\to\infty$.

######  Remark.

In Section 9 we discussed some of the difficulties one would face when
extending the main results to the case that $\Sigma$ is a finite volume
surface or an orbifold. None of these problems really arise when proving the
geometric prime number theorem, and indeed the proof we present here works
with only minimal changes if $\Sigma$ is replaced by an arbitrary finite are
orbifold $\Gamma\backslash\mathbb{H}^{2}$. We decided to just deal with the
compact case to avoid hiding the structure of the argument.

The proof of Huber’s theorem would be cleaner and nicer if all closed
geodesics were primitive. Luckily, this is almost true. Indeed it follows for
example from the work of Coornaert and Knieper [6] that there is some $C>0$
with

(A.41) $\frac{1}{C}\cdot\frac{e^{L}}{L}\leqslant|\mathbf{C}(L)|\leqslant
C\cdot e^{L}$

for all $L>0$. We stress that the Coornaert-Knieper argument is pretty coarse:
what they actually prove is a statement about groups acting isometrically,
discretely and cocompactly on Gromov-hyperbolic spaces.

The point for us is that (A.41) implies that most geodesics of length at most
$L$ are primitive. Indeed, every non-primitive geodesic of length at most $L$
is a multiple of a primitive geodesic of length at most $\frac{1}{2}L$. On the
other hand, if $s_{0}$ is the systole of $\Sigma$ then at most
$\frac{L}{s_{0}}$ geodesics of length at most $L$ arise as multiples of any
given geodesic. These two observations, together with the right side of
(A.41), imply that there are at most
$\frac{L}{s_{0}}\mathbf{C}(\frac{1}{2}L)\leqslant\frac{C}{s_{0}}\cdot L\cdot
e^{L/2}$ non-primitive geodesics of length at most $L$. Taking the left side
of (A.41) into consideration we deduce that, as we had claimed above, most
geodesics are primitive.

######  Lemma A.1.

Fix $h$ and let $\mathbf{C}(L,h)$ and $\mathbf{P}(L,h)$ be, respectively, the
sets of all geodesics and of all primitive geodesics of length in $[L,L+h]$.
Then we have

$|\mathbf{C}(L,h)|\sim|\mathbf{P}(L,h)|$

as $L\to\infty$.∎

After this preparatory comment let us start the real business. Let
$\mathcal{L}$ be the space of all geodesic loops in $\Sigma$ and consider the
map $\Pi:\mathcal{L}\to\Sigma$ mapping each loop to its base point. As was the
case for more general graphs, the map $\Pi$ is a covering, meaning that when
we pull-back the hyperbolic metric using $\Pi$ we can think of $\mathcal{L}$
as being a hyperbolic surface. Note also that the set of connected components
of $\mathcal{L}$ agrees with the set of free homotopy classes of loops. It
follows that for every closed geodesic $\gamma$ we have a connected component
$\mathcal{L}^{\gamma}$.

Now, given $\varepsilon>0$ small let $\mathcal{L}_{\varepsilon}$ be the set of
all geodesic loops with angle defect at most $\varepsilon$, that is, the set
of geodesic loops whose initial and terminal velocity vectors meet with
unoriented angle in $[0,\varepsilon]$. For $L>0$ let
$\mathcal{L}_{\varepsilon}(L,L+h)$ be the elements in
$\mathcal{L}_{\varepsilon}$ with length between $L$ and $L+h$. Accordingly,
set
$\mathcal{L}^{\gamma}_{\varepsilon}(L,L+h)=\mathcal{L}^{\gamma}\cap\mathcal{L}_{\varepsilon}(L,L+h)$.

Let us establish some basic properties of $\mathcal{L}_{\varepsilon}(L,L+h)$
and $\mathcal{L}_{\varepsilon}^{\gamma}(L,L+h)$:

######  Lemma A.2.

Fix $h>0$ and $\varepsilon>0$. We have that

$\operatorname{vol}(\mathcal{L}_{\varepsilon}(L,L+h))\sim\varepsilon\cdot(e^{L+h}-e^{L})\text{
as }L\to\infty.$

Moreover, there is a function $C(\varepsilon)$ with $\lim_{\varepsilon\to
0}C(\varepsilon)=1$ such that for every sufficiently long closed geodesic
$\gamma$ we have that

  1. (1)

$\mathcal{L}_{\varepsilon}^{\gamma}(L,L+h)=\varnothing$ unless
$\ell(\gamma)\in[L-\varepsilon,L+h]$,

  2. (2)

$\operatorname{vol}(\mathcal{L}_{\varepsilon}^{\gamma}(L,L+h))\leqslant
C(\varepsilon)\cdot\varepsilon\cdot L$, and

  3. (3)

$C(\varepsilon)^{-1}\leqslant\frac{1}{\varepsilon\cdot
L}\operatorname{vol}(\mathcal{L}_{\gamma}^{\varepsilon}(L,L+h))\leqslant
C(\varepsilon)$ if $\gamma$ is primitive and
$\ell(\gamma)\in[L,L+h-\varepsilon]$.

###### Proof.

As in the proof of Proposition 4.2 we exploit the cover
$\Pi:\mathcal{L}\to\Sigma$ to compute the volume of
$\mathcal{L}_{\varepsilon}(L,L+h)$:

$\operatorname{vol}(\mathcal{L}_{\varepsilon}(L,L+h))=\int_{\Sigma}|\Pi^{-1}(x)\cap\mathcal{L}_{\varepsilon}(L,L+h)|dx$

Now, $\Pi^{-1}(x)\cap\mathcal{L}_{\varepsilon}(L,L+h)$ is nothing other than
the number of geodesic arcs going from $x$ to $x$ with length within $[L,L+h]$
and such that the initial and terminal velocities make at most angle
$\varepsilon$. Once we fix $x$, the set of admissive pairs of initial and
terminal velocities is a subset $T^{1}_{x}\Sigma\times T^{1}_{x}\Sigma$ with
volume $4\pi\varepsilon$. We thus get from Theorem 3.2 that

$|\Pi^{-1}(x)\cap\mathcal{L}_{\varepsilon}(L,L+h)|\sim\varepsilon\cdot\frac{e^{L+h}-e^{L}}{\operatorname{vol}(\Sigma)}$

Since we are assuming that $\Sigma$ is closed, this is uniform in $x$, meaning
that we get

$\operatorname{vol}(\mathcal{L}_{\varepsilon}(L,L+h))\sim\int_{\Sigma}\varepsilon\cdot\frac{e^{L+h}-e^{L}}{\operatorname{vol}(\Sigma)}=\varepsilon\cdot(e^{L+h}-e^{L})$

We have proved the first claim.

$d$ $\gamma_{x}$ $x$

Figure 6. The cylinder $\langle\gamma\rangle\backslash\mathbb{H}^{2}$. The
marked angles each measures one half of the angle defect of $\gamma_{x}$.

To prove the rest, let us give a concrete description of
$\mathcal{L}^{\gamma}$. Well, writing $\gamma=\eta^{k}$ for some $\eta$
primitive and $k\geqslant 1$ consider the cover

$\pi_{\eta}:\langle\eta\rangle\backslash\mathbb{H}^{2}\to\Sigma$

of $\Sigma$ corresponding to $\eta$. Now, for each
$x\in\langle\eta\rangle\backslash\mathbb{H}^{2}$ there is a unique hyperbolic
loop $\gamma_{x}$ based at $x$ and freely homotopic to running $k$ times over
$\eta$. The basic observation is that the map

$\langle\eta\rangle\backslash\mathbb{H}^{2}\to\mathcal{L}^{\gamma},\
x\mapsto\pi_{\eta}\circ\gamma_{x}$

is an isometry between $\langle\eta\rangle\backslash\mathbb{H}^{2}$ and the
connected component $\mathcal{L}^{\gamma}$.

Now, if one denotes by $d$ the distance in
$\langle\eta\rangle\backslash\mathbb{H}^{2}$ between $x$ and the central
geodesic (see Figure 6) one gets from formula 2.3.1(vi) in the final page in
[3] that

$\displaystyle\frac{1}{2}\cdot(\text{angle defect of }\gamma_{x})$
$\displaystyle\sim\tan\left(\frac{1}{2}\cdot(\text{angle defect of
}\gamma_{x})\right)$
$\displaystyle=\sinh(d)\cdot\tanh\left(\frac{\ell(\gamma)}{2}\right)\sim d$

where the asymptotics hold when $\ell(\gamma)$ is large and the angle defect
is small. Now, claims (1), (2) and (3) follow from elementary considerations.
∎

Armed with these two lemmas we are ready to conclude the proof of the
geometric prime number theorem. Note that it suffices to prove that for $h$
positive and fixed we have

(A.42) $|\mathbf{C}(L,h)|\sim\frac{e^{L+h}-e^{L}}{L}$

as $L\to\infty$. Here $\mathbf{C}(L,h)$ is as in Lemma A.1.

Anyways, with notation as in both lemmas we have for $\varepsilon$ positive
and small and for $L$ large that

$\displaystyle|\mathbf{P}(L,h)|$
$\displaystyle\stackrel{{\scriptstyle\text{\ref{lem
ap2}(3)}}}{{\preceq}}\frac{C(\varepsilon)}{\varepsilon\cdot
L}\operatorname{vol}\left(\cup_{\gamma\in\mathbf{P}(L,h)}\mathcal{L}_{\varepsilon}^{\gamma}(L,L+h+\varepsilon)\right)$
$\displaystyle\leqslant\frac{C(\varepsilon)}{\varepsilon\cdot
L}\operatorname{vol}(\mathcal{L}_{\varepsilon}(L,L+h+\varepsilon))$
$\displaystyle\stackrel{{\scriptstyle\text{\ref{lem
ap2}}}}{{\sim}}C(\varepsilon)\cdot\frac{e^{L+h+\varepsilon}-e^{L}}{L}$

On the other hand we also have that

$\displaystyle|\mathbf{C}(L,h)|$
$\displaystyle\stackrel{{\scriptstyle\text{\ref{lem
ap2}(2)}}}{{\geqslant}}\frac{1}{C(\varepsilon)\cdot\varepsilon\cdot
L}\operatorname{vol}\left(\cup_{\gamma\in\mathbf{C}(L,h)}\mathcal{L}_{\varepsilon}(L,h)\right)$
$\displaystyle=\frac{1}{C(\varepsilon)\cdot\varepsilon\cdot
L}\operatorname{vol}(\mathcal{L}_{\varepsilon}(L,h))$
$\displaystyle\stackrel{{\scriptstyle\text{\ref{lem
ap2}}}}{{\sim}}\frac{1}{C(\varepsilon)}\frac{e^{L+h}-e^{L}}{L}$

Since this holds for all $\varepsilon$, and since $C(\varepsilon)$ tends to
$1$ when $\varepsilon\to 0$ we have that

$|\mathbf{P}(L,h)|\preceq\frac{e^{L+h}-e^{L}}{L}\preceq|\mathbf{C}(L,h)|$

Now, Lemma A.1 implies (A.42), and Bob’s your uncle.

## References

  * [1] R. Bacher and A. Vdovina, Counting 1-vertex triangulations of oriented surfaces, Discrete Math. 246 (2002).
  * [2] B. Bekka and M. Mayer, Ergodic theory and topological dynamics of group actions on homogeneous spaces, London Mathematical Society Lecture Note Series, 269. Cambridge University Press, (2000).
  * [3] P. Buser, Geometry and spectra of compact Riemann surfaces, Progress in Mathematics, 106. Birkhäuser, (1992)
  * [4] D. Calegari, scl, MSJ Memoirs, 20. Mathematical Society of Japan, (2009).
  * [5] R. Canary, D. Epstein, and P. Green, Notes on notes of Thurston, in Analytical and geometric aspects of hyperbolic space (Coventry/Durham, 1984), London Math. Soc. Lecture Note Ser., 111, Cambridge Univ. Press, (1987).
  * [6] M. Coornaert and G. Knieper, Growth of conjugacy classes in Gromov hyperbolic groups, GAFA 12, (2002).
  * [7] J. Delsarte, Sur le gitter fuchsien, C. R. Acad. Sci. Paris 214 (1942), 147–179.
  * [8] V. Erlandsson and J. Souto, Mirzakhani’s curve counting and geodesic currents, Progress in Mathematics, 345, Birkhäuser (2002).
  * [9] D. Hejhal, The Selberg Trace Formula for $\operatorname{SL}_{2}\mathbb{R}$, Vol. II, Lecture Notes in Mathematics, Volume 1001. Springer-Verlag, Berlin-Heidelberg (1983).
  * [10] H. Huber, Zur analytischen Theorie hyperbolischen Raumformen und Bewegungsgruppen, Math. Ann. 138 (1959).
  * [11] A. Katsuda and T. Sunada, Homology and closed geodesics in a compact Riemann surface, Amer. J. Math. 110 (1988).
  * [12] G. Margulis, On some aspects of the theory of Anosov systems. With a survey by Richard Sharp: Periodic orbits of hyperbolic flows, Springer Monographs in Mathematics. Springer-Verlag, (2004).
  * [13] M. Mirzakhani, Growth of the number of simple closed geodesics on hyperbolic surfaces, Ann. of Math. (2) 168 (2008)
  * [14] M. Mirzakhani, Counting mapping class group orbits on hyperbolic surfaces, arXiv:1601.03342 (2016).
  * [15] P. Park, Conjugacy growth of commutators, Journal of Algebra 526 (2019).
  * [16] R. Phillips and P. Sarnak, Geodesics in homology classes, Duke Math. J. 55 (1987).
  * [17] T. Roblin, Ergodicité et équidistribution en courbure négative, Mém. Soc. Math. Fr. No. 95 (2003).
  * [18] W. Thurston, Geometry and topology of 3-manifolds, http://library.msri.org/books/gt3m/, Lecture notes (1980).