# Bounded geodesic image theorem via bicorn curves

Xifeng Jin  Xifeng Jin
Department of Mathematics
University at Buffalo–SUNY
Buffalo, NY 14260-2900, USA
<EMAIL_ADDRESS>
###### Abstract.

We give a uniform bound of the bounded geodesic image theorem for the closed
oriented surfaces. The proof utilizes the bicorn curves introduced by
Przytycki and Sisto [13]. With the uniformly bounded Hausdorff distance of the
bicorn paths and 1-slimness of the bicorn curve triangles, we are able to show
the bound is 44 for both non-annular and annular subsurfaces. In a particular
case when the distance between a geodesic and an essential boundary component
of subsurface (or core if it is annular) is $\geq$ 18, then the bound can be
as small as 3, which is comparable to the bound 4 in the motivating examples
by Masur and Minsky [11], and is same as the bound given by Webb [18] for non-
annular subsurfaces.

###### Key words and phrases:

curve graphs, bounded geodesic image theorem, bicorn curves.

###### 2010 Mathematics Subject Classification:

Primary: 57M60. Secondary: 57M20.

##  1\. Introduction

Let $S=S_{g,b}$ be a compact oriented surface with genus $g$ and $b$ boundary
components, the _complexity_ of the surface is $\xi(S)=3g+b-3$. A simple
closed curve on the surface $S$ is _essential_ if it does not bound a disk or
an annulus. A simple properly embedded arc on the surface $S$ is _essential_
if it does not cut off a disk.

###  1.1. Curve complex

Suppose that $\xi(S)\geq 2$, the _arc and curve graph_ $\mathcal{AC}(S)$ is
the graph whose vertices are the isotopy classes of essential properly
embedded arcs and essential curves on the surface $S$, where the isotopy
allows the endpoints of the arcs move around in the corresponding boundary
components. Two vertices can be connected by an edge in $\mathcal{AC}(S)$ if
they can be realized disjointly. The _curve graph_ $\mathcal{C}(S)$ is the
full subgraph of $\mathcal{AC}(S)$ whose vertices are the isotopy classes of
essential curves. The _curve graph_ is the 1-skeleton of the curve complex
introduced by Harvey [7] with the aim to study the mapping class group.
$\mathcal{C}_{0}(S)$ and $\mathcal{AC}_{0}(S)$ are used to denote the vertices
of $\mathcal{C}(S)$ and $\mathcal{AC}(S)$, respectively.

For the surfaces with $\xi(S)\leq 1$, we need to modify the definition
slightly. If $\xi(S)=1$, the surface $S$ is either a one-holed torus $S_{1,1}$
or a four-holed sphere $S_{0,4}$. In both cases, two vertices are connected by
an edge if some of their curve representatives intersect exactly once for
$S_{1,1}$ or twice for $S_{0,4}$. The resulting curve graphs are isomorphic to
the Farey graph. If $\xi(S)=0$, then $S=S_{1}$ is a torus or $S=S_{0,3}$ is a
pair of pants. Similar to the once-punctured torus, the curve graph of torus
is isomorphic to the Farey graph if two curve representatives intersect
exactly once are joined by an edge. The curve graph of a pair of pants is
empty. If $S=S_{0,2}$ is an annulus, we define the vertices of
$\mathcal{C}(S)$ to be the essential arcs up to isotopy fixing the boundary
component pointwise, and two vertices can be joined by an edge if they can be
realized with disjoint interiors. In this case, we let
$\mathcal{C}(S)=\mathcal{AC}(S)$.

For any two vertices $x$ and $y$ in the $\mathcal{C}_{0}(S)$, the _distance_
$d_{\mathcal{C}(S)}(x,y)$ between $x$ and $y$ is the minimal number of edges
in $\mathcal{C}(S)$ joining $x$ and $y$. A _geodesic_ in the curve graph
$\mathcal{C}(S)$ is a sequence of vertices $\Gamma=(\gamma_{i})_{i\in I}$ such
that $d_{\mathcal{C}(S)}(\gamma_{i},\gamma_{j})=|i-j|$ for all $i,j$ in $I$.
These notions can be defined on the arc and curve graph $\mathcal{AC}(S)$ in
the same manner. A metric space is called _$\delta$ -hyperbolic_ if for every
geodesic triangle, each side lies in a $\delta$-neighborhood of the other two
sides. A seminal result of the curve graphs is its hyperbilicity proved by
Masur and Minsky.

######  Theorem 1.1.

_(Masur-Minsky[10] Theorem 1.1)_ Let $S$ be a compact oriented surface with
$\xi(S)\geq 1$, the curve graph $\mathcal{C}(S)$ is a $\delta$-hyperbolic
metric space with infinite diameter for some $\delta$, where $\delta$ depends
on the surface.

Alternative proofs were given by Bowditch [2] and Hamenstädt [6]. Moreover,
the hyperbilicity constant $\delta$ can be chosen to be independent of the
surface. The existence of such uniform constant has been proved independently
by Aougab [1], Bowditch [3], Clay-Rafi-Schleimer [5], Hensel-Przytycki-Webb
[8]. Przytycki and Sisto [13] also proved it for the closed oriented surfaces.

###  1.2. Subsurface projection

A subsurface $Y$ is an _essential subsurface_ of $X$, if $Y$ is a compact,
connected, oriented, proper subsurface such that each component of $\partial
Y-\partial X$ is essential in $X$. Suppose $\xi(Y)\geq 1$, we define a map
$\pi_{A}:\mathcal{C}_{0}(X)\rightarrow\mathcal{P}(\mathcal{AC}_{0}(Y))$, where
$\mathcal{P}(\mathcal{AC}_{0}(Y))$ is the power set of $\mathcal{AC}_{0}(Y)$.
Take any $\alpha$ in $\mathcal{C}_{0}(X)$, then consider the representative of
$\alpha$ such that it intersects $Y$ minimally, $\pi_{A}(\alpha)$ is the set
of all isotopy classes of $\alpha\cap Y$ relative to the boundary of $Y$.
$\pi_{A}(\alpha)$ is empty if $\alpha$ can be realized disjointly from $Y$. We
say $\alpha$ _cuts_ $Y$ if $\alpha\cap Y\neq\varnothing$, and $\alpha$
_misses_ $Y$ if $\alpha\cap Y=\varnothing$.

There is a natural way to send the arcs back to the curves in
$\mathcal{C}_{0}(Y)$. We can define
$\pi_{0}:\mathcal{AC}_{0}(Y)\rightarrow\mathcal{C}_{0}(Y)$ as follows. If
$\alpha$ is in the $Y$, then $\pi_{A}(\alpha)=\alpha$ in $\mathcal{AC}_{0}(Y)$
and $\pi_{0}(\pi_{A}(\alpha))=\alpha$ in $\mathcal{C}_{0}(Y)$. Otherwise,
$\pi_{A}(\alpha)=\\{\alpha_{1},\alpha_{2},\cdots,\alpha_{n}\\}$ is a
collection of isotopy classes of essential properly embedded arcs in $Y$. The
set $\pi_{0}(\pi_{A}(\alpha))$ is the isotopy classes of the essential
components of $\partial N(\alpha_{i}\cup\partial Y)$ in $Y$, where
$N(\alpha_{i}\cup Y)$ is a regular neighborhood of $\alpha_{i}\cup\partial Y$
in $Y$. The composition
$\pi_{0}\circ\pi_{A}=\pi_{Y}:\mathcal{C}_{0}(X)\rightarrow\mathcal{C}_{0}(Y)$
is called the _subsurface projection_.

If $Y$ is an annulus, an alternative definition is needed. Let $X$ be endowed
with a complete hyperbolic metric with finite area and $Y\subset X$ is an
essential annulus. Let $p_{Y}:\widetilde{X}_{Y}\rightarrow X$ be the annular
covering map such that $Y$ lifts homeomorphically to
$Y^{\prime}\subset\widetilde{X}_{Y}$ and
$\widetilde{X}_{Y}\cong\text{interior}(Y)$. The compactification of
$\widetilde{X}_{Y}$ with the hyperbolic metric induced from $X$ is a closed
annulus and it is denoted by $\overline{X}_{Y}$. If a curve $\alpha$ cuts $Y$,
the subsurface projection $\pi_{Y}(\alpha)$ is the set of arcs of the full
preimage $\widetilde{\alpha}=p_{Y}^{-1}(\alpha)$ that connect the two boundary
components of $\overline{X}_{Y}$. If a curve $\alpha$ misses $Y$, then
$\pi_{Y}(\alpha)=\varnothing$. Suppose that $v$ is the core of $Y$, then we
also use $\pi_{v}$ to denote the projection and $d_{v}$ to denote the
distance. For more details about the subsurface projection and bounded
geodesic image theorem, see the paper of Masur and Minsky [11].

######  Theorem 1.2.

_(Bounded Geodesic Image Theorem, Masur-Minsky[11] Theorem 3.1)_ Let $Y$ be a
subsurface of $X$ with $\xi(Y)\neq 0$ and let $\Gamma=(\gamma_{i})_{i\in I}$
be a geodesic in $\mathcal{C}(X)$. If each $\gamma_{i}$ cuts $Y$, then there
is a constant $M$ depending only on the surface so that $d_{Y}(\Gamma)\leq M$.

The notation
$d_{Y}(\Gamma):=\text{diam}_{\mathcal{C}(Y)}(\underset{\gamma_{i}\in\Gamma}{\cup}\pi_{Y}(\gamma_{i}))$
is used above.

######  Lemma 1.3.

_(Masur-Minsky[11] Lemma 2.2)_ Let $\xi(Y)\geq 1$, and
$d_{\mathcal{AC}(Y)}(\alpha,\beta)\leq 1$ for
$\alpha,\beta\in\mathcal{AC}(Y)$, then
$d_{\mathcal{C}(Y)}(\pi_{0}(\alpha),\pi_{0}(\beta))\leq 2$.

By the Lemma 1.3, one only needs to consider the diameter of the projection
$\pi_{A}:\mathcal{C}_{0}(X)\rightarrow\mathcal{P}(\mathcal{AC}_{0}(Y))$,
because $\pi_{0}:\mathcal{AC}_{0}(Y)\rightarrow\mathcal{C}_{0}(Y)$ is
2-Lipschitz. Moreover, the projection $\pi_{A}$ satisfies the 1-Lipschitz
property for the surfaces with $\xi(X)\geq 2$.

######  Lemma 1.4.

_(Webb[17] Lemma 1.2)_ Let $Y$ be an essential subsurface of $X$ and let
$\gamma_{1},\gamma_{2}$ be curves on $X$. Suppose that $\gamma_{1}$ cuts $Y$,
$\gamma_{2}$ cuts $Y$ and $\gamma_{1}$ misses $\gamma_{2}$. Then
$d_{\mathcal{AC}(Y)}(\pi_{A}(\gamma_{1}),\pi_{A}(\gamma_{2}))\leq 1$.

Using the notation
$d_{\mathcal{AC}(Y)}(\Gamma):=\text{diam}_{\mathcal{AC}(Y)}(\underset{\gamma_{i}\in\Gamma}{\cup}\pi_{A}(\gamma_{i}))$,
we restate the theorem for closed oriented surfaces.

######  Theorem 1.5.

Suppose that $X$ is a closed oriented surface with genus $\geq 2$. Let $Y$ be
an essential subsurface of $X$ with $\xi(Y)\neq 0$ and let
$\Gamma=(\gamma_{i})_{i\in I}$ be a geodesic in $\mathcal{C}(X)$ such that
each $\gamma_{i}$ cuts $Y$, then $d_{\mathcal{AC}(Y)}(\Gamma)\leq 44$ for both
non-annular and annular subsurfaces $Y$.

The bounded geodesic image theorem was originally proved by Masur and Minsky
[11]. With the uniform hyperbilicity of the curve graphs, Webb [17] proved
that the constant $M$ is independent of the surface. Later on, he gave
explicit bounds in his dissertation [18] using the unicorn arcs, where the
bounds is 62 for annular subsurfaces, and 50 for non-annular subsurfaces. In
[17, 18], Webb remarked that an unpublished proof of Chris Leininger combined
with the work of Bowditch [3] also provided a uniform bound. Our proof relies
on a uniformly bounded Hausdorff distance of the bicorn paths and 1-slimness
of the bicorn curve triangles. The outline of the proof is similar to Webb’s,
while the proof is simple and the bound is smaller for the closed oriented
surfaces. The approach can be generalized to the surfaces with boundary,
possibly with larger bounds.

The paper is organized as follows. In the Section 2, we briefly describle the
motivating example by Masur and Minsky. We recall the definition of bicorn
curves and bicorn paths between two curves in the Section 3. In the Section 4,
we obtain a uniform bound for the bounded geodesic image theorem for closed
oriented surfaces.

## Acknowledgements

The author would like to thank his advisor, Professor William W. Menasco, for
alerting the author to the small bound in the motivating examples by Masur and
Minsky [11], and the encouragement of obtain small bounds for the bounded
geodesic image theorem. The author would also like to thank Professor Johanna
Mangahas for reading a draft of this paper and offering helpful comments.
Finally, the author is grateful to the support of the Dissertation Fellowship
for Fall 2020 from the Department of Mathematics at the State University of
New York at Buffalo.

##  2\. A motivating example

Figure 1. A typical Farey graph. Each vertex is labeled with a number in
$\hat{\mathbb{Q}}$ that is a slope of the curve representative under a fixed
homology basis. The figure was generated by Wolfram Mathematica, and the code
snippet is availabe on the $\href
https://mathematica.stackexchange.com/questions/194838/how-can-i-plot-a-farey-
diagram$.

In this section, we will briefly describe a motivating example for the bounded
geodesic image theorem from the Section 1.5 in [11]. The curve graph
$\mathcal{C}(S_{1})$, $\mathcal{C}(S_{1,1})$ and $\mathcal{C}(S_{0,4})$ are
isomorphic to the _Farey graph_ , as roughly illustrated in the Figure 1. It
can be embedded in a unit disk, in which the rational numbers and the infinity
$\hat{\mathbb{Q}}=\mathbb{Q}\cup\\{\infty\\}$ are located on the boundary of
the unit disk as the vertices of the graph. Any two vertices
$\frac{p_{1}}{q_{1}}$ and $\frac{p_{2}}{q_{2}}$ in lowest terms are connected
by an edge if and only if the determinant $p_{1}q_{2}-p_{2}q_{1}=\pm 1$. Let
$v$ be a vertex, then the _link_ of $v$, $\text{Link}(v)$, is a set of
vertices that share an edge with $v$, which can be identified with the
integers $\mathbb{Z}$. For example, if $v=\frac{0}{1}$, then the
$\text{Link}(v)=\\{\cdots,-\frac{1}{2},-\frac{1}{1},\frac{1}{0},\frac{1}{1},\frac{1}{2},\cdots\\}$.

Let $u$ and $w$ be two vertices of a geodesic $h$ in the Farey graph, and a
vertex $v$ of $h$ follows $u$ and followed by $w$, the subsurface projection
$d_{v}(u,w)$ defines a distance in the $\text{Link}(v)$, where $v$ is the core
of an annular subsurface. Let $h^{\prime}$ be another geodesic with the same
endpoints as $h$. If $d_{v}(u,w)\geq 5$, then $h^{\prime}$ must contain the
vertex $v$. In other words, if a geodesic does not contain $v$ (each vertex of
the geodesic cuts $v$), then $d_{v}(u,w)\leq 4$ for any $u,w$ in the geodesic.

##  3\. Bicorn curves

Przytycki and Sisto [13] introduced the bicorn curves to give a simple proof
of the hyperbilicity of curve graphs for the closed oriented surfaces
$S_{g\geq 2}$. For more applications of the bicorn curves, see [4, 9, 12, 14,
15, 16].

######  Definition 3.1 (Bicorn curves).

Let $\alpha,\beta\subset S_{g\geq 2}$ be two essential simple closed curves
that intersect minimally. An essential simple closed curve $\gamma$ is a
_bicorn curve_ between $\alpha$ and $\beta$ if either $\gamma=\alpha$,
$\gamma=\beta$, or $\gamma$ is represented by the union of an arc
$a\subset\alpha$ and an arc $b\subset\beta$, which we call the $\alpha$-arc
and the $\beta$-arc of $\gamma$, and $a$ only intersects $b$ at the endpoints.
If $\gamma=\alpha$, then its $\alpha$-arc is $\alpha$ and its $\beta$-arc is
empty, similarly if $\gamma=\beta$, then its $\beta$-arc is $\beta$ and its
$\alpha$-arc is empty.

Figure 2. Two configurations of a bicorn curve $\gamma$ between $\alpha$ and
$\beta$; The two intersection points have opposite orientations on the left
and same orientation on the right.

Based on the orientations of two intersection points, there are two
configurations of the bicorn curves illustrated in the Figure 2. If the
surface is closed, any bicorn curves defined above are essential, as $\alpha$
and $\beta$ intersect minimally. The intersection number of $\alpha$ and
$\beta$ is finite, so the number of bicorn curves is finite.

The collection of bicorn curves can be partially ordered. Two bicorn curves
$\gamma<\gamma^{\prime}$ if the $\beta$-arc of $\gamma^{\prime}$ strictly
contains the $\beta$-arc of $\gamma$. Take a minimal subarc
$b^{\prime}\subset\beta$ that does not intersect $\alpha$ except for the
endpoints. Denote the bicorn curve $\alpha_{1}=a^{\prime}\cap b^{\prime}$ as
the union of the minimal subarc $b^{\prime}\subset\beta$ and the subarc
$a^{\prime}$ of $\alpha$ determined by the endpoints. Then, $\alpha$
intersects $\alpha_{1}$ at most once. One can extend the minimal subarc
$b^{\prime}$ to the next intersection point with $a^{\prime}$. The extended
subarc $b^{\prime\prime}$ of $\beta$ intersects $a^{\prime}$ on the endpoint,
the bicorn curve is denoted as $\alpha_{2}=a^{\prime\prime}\cup
b^{\prime\prime}$. See the Figure 3. As we can see, $\alpha_{1}$ intersects
$\alpha_{2}$ at most once.

Next, we extend the subarc $b^{\prime\prime}$ to the minimal subarc
$b^{\prime\prime\prime}$ such that $b^{\prime\prime\prime}$ intersects
$a^{\prime\prime}$ right on the endpoint, the subarc of $a^{\prime\prime}$
with bounded by the new intersection point is denoted by
$a^{\prime\prime\prime}$. The bicorn curve
$\alpha_{3}=a^{\prime\prime\prime}\cup b^{\prime\prime\prime}$ intersects
$\alpha_{2}$ at most once.

Figure 3. Extend the bicorn curve $\alpha_{1}$ along $\beta$ to obtain bicorn
curves $\alpha_{2}$ and $\alpha_{3}$.

Continue in this way, one will be able to construct a sequence of bicorn
curves $\alpha=\alpha_{0},\alpha_{1},\cdots,\alpha_{n}$, where the adjacent
curves $\alpha_{i}$, $\alpha_{i+1}$ intersect at most once. Since the
intersection number of $\alpha$ and $\beta$ is finite, the sequence must
terminate at $\beta$, that is, $\alpha_{n}=\beta$. The sequence of bicorn
curves constructed above is called a _bicorn path_. We will use
$B(\alpha,\beta)$ to denote one bicorn path between $\alpha$ and $\beta$, and
all the bicorn paths are denoted by $\mathcal{B}(\alpha,\beta)$. Note that a
bicorn path is not a real path, as the adjacent curves are not necessarily
disjoint.

The bicorn paths $\mathcal{B}(\alpha,\beta)$ have uniformly bounded Hausdorff
distance to the geodesics between $\alpha$ and $\beta$.

######  Proposition 3.2.

Let $\Gamma$ be a geodesic connecting two curves $\alpha$ and $\beta$, then a
bicorn path $B(\alpha,\beta)$ between $\alpha$ and $\beta$ stay in the
14-neighborhood of $\Gamma$ in the curve graph.

In [4], a proof was given by Chang, Menasco and the author for the curves
$\alpha$ and $\beta$ with coherent intersection. The proof proceeds without
any change for the general cases.

##  4\. The proof

In this section, we will utilize the bicorn curves to prove the Theorem 1.5
for the closed oriented surfaces $X$ with genus $\geq 2$. With the uniformly
bounded Hausdorff distance of the bicorn paths and 1-slimness of the bicorn
curve triangles, the proof follows immediately from Webb’s strategy to the
proof of the Theorem 3.2 in [17]. To start off, let us deal with a particular
case when the geodesic is away from the curves in the subsurface.

######  Lemma 4.1.

Suppose that $X$ is a closed oriented surface with genus $\geq 2$ and $Y$ is
an essential subsurface of $X$ with $\xi(Y)\neq 0$. Let $\Gamma$ be a geodesic
in $\mathcal{C}(X)$ such that each $x\in\Gamma$ cuts $Y$, and $\alpha$ be an
essential boundary component of the non-annular subsurface $Y$ or the core of
$Y$ if it is annular. If $d_{\mathcal{C}(X)}(x,\alpha)\geq 18$, for any vertex
$x\in\Gamma$, then $d_{\mathcal{AC}(Y)}(\Gamma)\leq 3$.

###### Proof.

Figure 4. The bicorn paths between $\gamma_{i}$, $\gamma_{j}$ and $\alpha$
form a bicorn curve triangle. $b_{k}$ and $b_{k+1}$ are two adjacent bicorn
curves in $B(\gamma_{i},\gamma_{j})$, there exists $b^{\prime}_{k}\in
B(\gamma_{i},\alpha)$ and $b^{\prime}_{k+1}\in B(\gamma_{j},\alpha)$ due to
the 1-slimness of the bicorn curve triangle.

Let $\gamma_{i}$ and $\gamma_{j}$ be any two distinct vertices in the geodesic
$\Gamma$, we need to show that $d_{\mathcal{AC}(Y)}(\gamma_{i},\gamma_{j})\leq
3$. By the Proposition 3.2, each bicorn curve of $B(\gamma_{i},\gamma_{j})$
stays in the 14-neighborhood of the geodesic segment
$[\gamma_{i},\gamma_{j}]\subset\Gamma$. Since
$d_{\mathcal{C}(X)}(x,\alpha)\geq 18$ for any $x\in\Gamma$, then
$d_{\mathcal{C}(X)}(\gamma,\alpha)\geq 4$ for any bicorn curve $\gamma\in
B(\gamma_{i},\gamma_{j})$. The minimal intersection number of a filling pair
on a closed oriented surface $X$ with genus $\geq 2$ is 4, so
$|\gamma\cap\alpha|\geq 4$.

Suppose that $\gamma_{i}$, $\gamma_{j}$ and $\alpha$ are in pairwise minimal
intersection without any triple points. For any bicorn curve $\gamma=a\cup b$,
where $a$ is $\gamma_{i}$-arc and $b$ is $\gamma_{j}$-arc, we consider the
intersection points of $\alpha$ with the $a$ and $b$. One can take a minimal
arc $c$ of $\alpha$ intersecting $a$ only at the two endpoints and $c$
intersects $b$ at most once (or intersecting $b$ only at the two endpoints and
intersects $a$ at most once). This is the key observation to prove the Lemma
2.6 [13] that the bicorn curve triangles are 1-slim, see the Figure 5 for an
illustration. Recall that $|\gamma\cap\alpha|=|(a\cup b)\cap\alpha|\geq 4$,
then either $|a\cap\alpha|\geq 2$ or $|b\cap\alpha|\geq 2$, so the curve
surgery is allowed.

Figure 5.  Either the $\gamma_{i}$-arc $a$ intersects $\alpha$ at least twice
or the $\gamma_{j}$-arc $b$ intersects $\alpha$ at least twice. The minimal
arc $c\subset\alpha$ intersects $a$ (or $b$) at the two endpoints
(intersection number $\leq$ 1) while intersecting the $b$ (or $a$) at most
once.

It follows that either one can construct a bicorn curve in
$B(\gamma_{i},\alpha)$ from $a$ and $c$ or a bicorn curve in
$B(\gamma_{j},\alpha)$ from $b$ and $c$. Since $\gamma_{i}\in
B(\gamma_{i},\gamma_{j})$ and $\alpha$ can only produce bicorn curves in
$B(\gamma_{i},\alpha)$, and $\gamma_{j}\in B(\gamma_{i},\gamma_{j})$ and
$\alpha$ can only produce bicorn curves in $B(\gamma_{j},\alpha)$, one can
conclude that there must be two adjacent bicorn curves $b_{k},b_{k+1}$
(possibly same) in $B(\gamma_{i},\gamma_{j})$ such that $b_{k}$ will produce a
bicorn curve $b^{\prime}_{k}\in B(\gamma_{i},\alpha)$ and $b_{k+1}$ will
produce a bicorn curve $b^{\prime}_{k+1}\in B(\gamma_{j},\alpha)$, as
illustrated in the Figure 4.

By the construction of the curves
$b_{k},b^{\prime}_{k},b_{k+1},b^{\prime}_{k+1}$, one can find one subarc from
$\pi_{A}(\gamma_{i})$ and another subarc from $\pi_{A}(\gamma_{j})$ that are
disjoint from each other. More precisely, such subarcs can be chosen from
$\pi_{A}(\delta_{i})$ and $\pi_{A}(\delta_{j})$, where $\delta_{i}$ belongs to
the $\gamma_{i}$-arc of $b_{k}$ that is bounded by the endpoints of one arc
$c\subset\alpha$, and $\delta_{j}$ belongs to the $\gamma_{j}$-arc of
$b_{k+1}$ that is bounded by the endpoints of another arc $c\subset\alpha$.
One issue is that the $\pi_{A}(b^{\prime}_{k})$ and
$\pi_{A}(b^{\prime}_{k+1})$ might be empty. It only occurs when the arcs
bounded by the endpoints of arc $c\subset\alpha$ does not have any
intersection with $\alpha$ in the interior of the arcs and the two
intersection points have opposite orientations. It implies that the
$b^{\prime}_{k}$ (or $b^{\prime}_{k+1}$) is the bicorn curve is disjoint from
$\alpha$. Then,

$d_{\mathcal{C}(X)}(b_{k},\alpha)\leq
d_{\mathcal{C}(X)}(b_{k},b^{\prime}_{k})+d_{\mathcal{C}(X)}(b^{\prime}_{k},\alpha)\leq
2+1=3,$

which contradicts to $d_{\mathcal{C}(X)}(b_{k},\alpha)\geq 4$. With the
1-Lipschitz property, one will obtain

$d_{\mathcal{AC}(Y)}(\gamma_{i},\gamma_{j})\leq 1+1+1=3.$

∎

######  Remark 4.2.

The bound in the Lemma 4.1 is comparable to the bound 4 in the motivating
examples by Masur and Minsky [10]. For the non-annular subsurfaces, the bound
is same as the bound given by Webb using the unicorn arcs in the Theorem 4.1.7
[18], while it is slightly better for annular subsurfaces.

###### Proof of the Theorem 1.5.

Let $\alpha$ be an essential boundary component of the non-annular subsurface
$Y$ and the core of $Y$ if it is annular. Let $I=N_{18}(\alpha)\cap\Gamma$ be
the intersection (possibly empty) of the 18-neighborhood of the $\alpha$ and
the geodesic $\Gamma$, then there exists a geodesic segment $g$ of length at
most 36 with $I\subset g\subset\Gamma$. Denote $\gamma^{\prime}_{i}$ and
$\gamma^{\prime}_{j}$ as the two ends of the geodesic segment $g$. Suppose at
least one of $\gamma_{i}$ and $\gamma_{j}$ is not in $g$. Otherwise, since
each vertex of $g$ cuts $Y$, then
$d_{\mathcal{AC}(Y)}(\gamma_{i},\gamma_{j})\leq 36$ follows from the Lemma
1.4.

Figure 6. The geodesic $\Gamma$ is divided into three pieces. The middle
geodesic segment $g=[\gamma^{\prime}_{i},\gamma^{\prime}_{j}]$ (possibly
empty) contains the vertices of $\Gamma$ within the 18-neighborhood of
$\alpha$ in the $\mathcal{C}(X)$.

Assume $\gamma_{i}$, $\gamma_{j}$ or both are not in the geodesic segment
$g\subset\Gamma$, see the Figure 6. Using the Lemma 1.4 and the Lemma 4.1, one
will have

$d_{\mathcal{AC}(Y)}(\gamma_{i},\gamma_{j})\leq
d_{\mathcal{AC}(Y)}(\gamma_{i},\gamma^{\prime}_{i})+d_{\mathcal{AC}(Y)}(\gamma^{\prime}_{i},\gamma^{\prime}_{j})+d_{\mathcal{AC}(Y)}(\gamma^{\prime}_{j},\gamma_{j})\leq
3+1+36+1+3=44.$

∎

######  Remark 4.3.

The strategy can be applied to the surfaces with boundary, because the bicorn
curves can be defined in the same manner as long as the bicorn curves are
essential. Restricting to the nonseparating curves, the bicorn curves
traingles have been used to prove the uniform hyperbilicity of the
nonseparating curve graphs by Rasmussen [15]. Combined with the uniformly
bounded Hausdorff distance of the bicorn paths on the surfaces with boundary
(Corollary 2.17, Rasmussen [14]), one will be able to prove a similar result,
possibly with larger bounds.

## References

  * [1] Tarik Aougab, _Uniform hyperbolicity of the graphs of curves_ , Geom. Topol. 17 (2013), no. 5, 2855 - 2875.
  * [2] Brian H. Bowditch. _Intersection numbers and the hyperbolicity of the curve complex._ J. Reine Angew. Math., 598: 105 - 129, 2006.
  * [3] Brian H. Bowditch. _Uniform hyperbolicity of the curve graphs._ Pacific J. Math., 269(2): 269 - 280, 2014.
  * [4] Hong Chang, Xifeng Jin, William W. Menasco. _Origami edge-paths in the curve graph._ Available at https://arxiv.org/abs/2008.09179
  * [5] Matt Clay, Kasra Rafi, and Saul Schleimer. _Uniform hyperbolicity of the curve graph via surgery sequences_. Algebr. Geom. Topol., 14(6): 3325 - 3344, 2014.
  * [6] Ursula Hamenstädt. _Geometry of the complex of curves and of Teichmüller space_. In Handbook of Teichmüller Theory, Vol. I, IRMA Lect. Math. Theor. Phys. 11, Eur. Math. Soc., Zürich, 447 - 467 (2007).
  * [7] William J. Harvey. _Boundary structure of the modular group._ in: Riemann surfaces and related topics. Proceedings of the 1978 Stony Brook Conference (Stony Brook 1978), Ann. of Math. Stud. 97, Princeton University Press, Princeton (1981), 245 - 251.
  * [8] Sebastian Hensel, Piotr Przytycki, and Richard C. H. Webb. _1-slim triangles and uniform hyperbolicity for arc graphs and curve graphs._ J. Eur. Math. Soc. (JEMS), 17(4): 755 - 762, 2015.
  * [9] Joseph Maher, Saul Schleimer. _The compression body graph has infinite diameter._ Available at https://arxiv.org/abs/1803.06065
  * [10] Howard A. Masur, Yair N. Minsky. _Geometry of the complex of curves I: Hyperbilicity._ Invent. Math. 138 (1999), no. 1, 103 - 149.
  * [11] Howard A. Masur, Yair N. Minsky. _Geometry of the complex of curves II: Hierarchical Structure._ Geom. Funct. Anal. 10 (2000), no. 4, 902 - 974.
  * [12] Witsarut Pho-On. _Gromov boundaries of complexes associated to surfaces._ Ph.D. thesis, University of Illinois at Urbana-Champaign, 2017.
  * [13] Piotr Przytycki, Alessandro Sisto. _A note on acylindrical hyperbolicity of mapping class groups_. In Hyperbolic geometry and geometric group theory, volume 73 of Adv. Stud. Pure Math., pages 255 - 264. Math. Soc. Japan, Tokyo, 2017.
  * [14] Alexander J. Rasmussen. _Geometry of the graphs of nonseparating curves: covers and boundaries._ Geom. Dedicata (2020).
  * [15] Alexander J. Rasmussen. _Uniform hyperbolicity of the graphs of nonseparating curves via bicorn curves._ Proc. Amer. Math. Soc. 148 (2020), 2345 - 2357.
  * [16] Kate M. Vokes. _Uniform quasiconvexity of the disc graphs in the curve graphs_ : in "Beyond Hyperbolicity", ed. M. Hagen, R. Webb, H. Wilton, London Math. Soc. Lecture Note Ser. 454, Cambridge Univ. Press (2019) 223 - 231.
  * [17] Richard C. H. Webb. _Uniform bounds for bounded geodesic image theorems._ J. Reine Angew. Math., 709: 219 - 228, 2015.
  * [18] Richard C. H. Webb. _Effective geometry of curve graphs_. Thesis (Ph.D.), University of Warwick, (2014).