The maximum injectivity radius of hyperbolic orbifolds

Abstract

For two-dimensional orientable hyperbolic orbifolds, we show that the radius of a maximal embedded disk is greater or equal to an explicit constant $\rho _T$, with equality if and only if the orbifold is a sphere with three cone points of order 2, 3 and 7.

Appendix

In this Appendix we use the same techniques of the rest of the article to give a new proof of the following theorem by Yamada:

Theorem 5.3

(Yamada [12]) For every hyperbolic surface $S$

$$\begin{aligned} r(S)\ge \rho _S={{\mathrm{arcsinh}}}(2/\sqrt{3}) \end{aligned}$$

with equality if and only if $S$ is the thrice-punctured sphere.

Note that another proof of this theorem has been given by Gendulphe in [8].

To simplify the notation, given a simple closed geodesic $\alpha$, we will denote its length simply by $\alpha$ (instead of $l(\alpha )$).

Remark 5.4

We will use the collar lemma (cf. [3]) and in particular the fact that if two simple closed geodesics $\alpha$ and $\beta$ intersect $n$ times, then

$$\begin{aligned} \alpha \ge 2 n w(\beta ) \end{aligned}$$

where $w(\beta )={{\mathrm{arcsinh}}}\left( \frac{1}{\sinh (\beta /2)}\right)$ is half the width of the collar of $\beta$.

Proof

Fix $p\in S$ with $r_p=r(S)$; the disk $B_{r_p}(p)$ determines at least two loops based at $p$ of length $2r_p$. If there are exactly two loops of length $2r_p$ such that either at least one is homotopic to a cusp, or their geodesic representatives do not intersect, we are in the situation of Fig. 18, where $\alpha$, $\beta$ or $\gamma$ can be cusps:

Fig. 18

The point $p$ with the two loops

By choosing a point $q$ in a small enough neighborhood of $p$ on the orthogonal to the third curve, if $\gamma$ is not a cusps, or on the geodesic from $p$ to $\gamma$, if $\gamma$ is a cusp, we increase the lengths of the two loops around $\alpha$ and $\beta$ and by continuity all other loops are still longer. So $r_q>r_p$, a contradiction. $\square$

We have then two possibilities:

• (a) there are two loops of length $2r_p$ whose geodesic representatives intersect each other once, or

• (b) there are at least three loops such that, if two are not homotopic to cusps, their geodesic representatives do not intersect.

Case (a) consider three loops of length $2r_p$; they determine a 3- or a 4-holed sphere.

If there exists three loops determining a 3-holed sphere, we can write equations for $p$. Denote by $\alpha$, $\beta$ and $\gamma$ the three boundary curves or cusps, by $\tilde{\alpha }$, $\tilde{\beta }$ and $\tilde{\gamma }$ the angles of the three loops at $p$ and by $\theta$ the angle of the (equilateral) triangle whose sides are the three loops (see Fig. 19).

Fig. 19

Three loops determining a Y-piece

We have:

$$\begin{aligned} \left\{ \begin{array}{l} \cosh \left( \frac{\alpha }{2}\right) =\cosh (r_p)\sin \left( \frac{\tilde{\alpha }}{2}\right) \\ \cosh \left( \frac{\beta }{2}\right) =\cosh (r_p)\sin \left( \frac{\tilde{\beta }}{2}\right) \\ \cosh \left( \frac{\gamma }{2}\right) =\cosh (r_p)\sin \left( \frac{\tilde{\gamma }}{2}\right) \\ \cos (\theta )=1-\frac{1}{2\cosh ^2(r_p)}\\ \tilde{\alpha }+\tilde{\beta }+\tilde{\gamma }+3\theta =2\pi \end{array}\right. \end{aligned}$$

(4)

Using $r_p<\rho _S$, we get

$$\begin{aligned} \tilde{\alpha },\tilde{b},\tilde{\gamma }>2\arcsin \left( \sqrt{\frac{3}{7}}\right) \end{aligned}$$

and

$$\begin{aligned} \theta >\arccos \left( \frac{11}{14}\right) , \end{aligned}$$

so

$$\begin{aligned} \tilde{\alpha }+\tilde{\beta }+\tilde{\gamma }+3\theta >2\pi \end{aligned}$$

which is a contradiction. Moreover, $r_p=\rho _S$ is a solution if and only if $\alpha =\beta =\gamma =0$, i.e. if we are on a thrice-punctured sphere. So in this case, $r(S)\ge \rho _S$, with equality if and only if $S$ is a thrice-punctured sphere.

If no three loops determine a 3-holed sphere, fix three loops (with corresponding geodesic representatives or cusps $\alpha$, $\beta$ and $\gamma$) and the associated four-holed sphere. Denote by $\delta$ the fourth boundary curve or cusp (Fig. 20). The loop based at $p$ and homotopic to $\delta$ has length at least $2r_p$. We can again write down equations satisfied by the pieces we obtain by cutting the four-holed sphere along the loops.

Fig. 20

Four loops in a four-holed sphere

If we assume that $r_p\le \rho _S$, we get (similarly to before)

$$\begin{aligned} \tilde{\alpha },\tilde{\beta },\tilde{\gamma }\ge 2\arcsin \left( \sqrt{\frac{3}{7}}\right) . \end{aligned}$$

Consider the quadrilateral with the four loops as sides; three sides have the same length $2r_p$ and the fourth has length at least $2r_p$. The two diagonals of the quadrilateral are longer than $2r_p$, otherwise we have three loops of length $2r_p$ determining a 3-holed sphere. Let’s denote the angles as in Fig. 21.

Fig. 21

The quadrilateral with the four loops as sides

By hyperbolic trigonometry we get

$$\begin{aligned} \varphi _1,\varphi _2 >2\arcsin \left( \frac{\sqrt{3}}{2\sqrt{7}}\right) \end{aligned}$$

and

$$\begin{aligned} \varphi _3,\varphi _4 >\arcsin \left( \frac{\sqrt{3}}{2\sqrt{7}}\right) . \end{aligned}$$

So

$$\begin{aligned} \tilde{\alpha }+\tilde{b}+\tilde{\gamma }+\tilde{\delta }+2\theta +2\varphi >2\pi , \end{aligned}$$

a contradiction. Thus $r(S)>\rho _S$.

Case (b) assume $r_p\le \rho _S$. Consider the one-holed torus determined by the geodesic representatives $\alpha$ and $\beta$ of the two loops of length $2r_p$. Denote its boundary curve or cusp by $\gamma$. Since $\alpha ,\beta \le 2\rho _S$, by the collar lemma we have $\alpha ,\beta \ge 2{{\mathrm{arcsinh}}}\left( \frac{\sqrt{3}}{2}\right)$.

Cut along $\alpha$ and denote by $d$ the shortest path between the two copies of $\alpha$. We have (again, by the collar lemma) $2{{\mathrm{arcsinh}}}\left( \frac{\sqrt{3}}{2}\right) \le d\le \beta \le 2\rho _S$. If $\gamma$ is not a cusp, then

$$\begin{aligned} \cosh (\gamma /2)&= \sinh ^2(\alpha /2)\cosh (d)-\cosh ^2(\alpha /2)\\&= \sinh ^2(\alpha /2)(\cosh (d)-1)+1. \end{aligned}$$

So $\frac{17}{8}\le \cosh (\gamma )\le \frac{41}{9}$ and the width $w(\gamma )$ of the collar around $\gamma$ satisfies

$$\begin{aligned} w(\gamma )>\log (5/4)>\log (2\sqrt{3}). \end{aligned}$$

Remark 5.5

By hyperbolic trigonometry, if a point $p$ has distance at least $\log (2/\sqrt{3})$ from the collar around a curve $\alpha <2\rho _S$, then the loop based at $p$ and homotopic to $\alpha$ is at least $2\rho _S$.

If $\gamma >2\rho _S$, fix $q\in \gamma$. Consider a loop based at $q$ of length $2r_q$ and its geodesic representative $\delta$. There are three possibilities:

• if $\delta \cap \gamma =\emptyset$, by Remark 5.5 we get $2r_q=\delta >2\rho _S$;

• if $\delta \cap \gamma \ne \emptyset$ and $\delta \ne \gamma$, then $\delta$ crosses the one-holed torus, so it crosses $\alpha$ or $\beta$ at least once and $\gamma$ at least twice. Thus

  $$\begin{aligned} 2r_q=\delta \ge 2{{\mathrm{arcsinh}}}(\sqrt{3}/2)+4\log (5/4)>2\rho _S; \end{aligned}$$

• if $\delta =\gamma$, then $r_q>\rho _S$.

In all cases, $r_q>\rho _S>r_q$, a contradiction.

Suppose then that $\gamma$ is a curve with $\gamma \le 2\rho _S$. One can show that in these conditions there exists a solution to the system (4), determining a point $q$ with loops of length $\ell >2{{\mathrm{arccosh}}}(\frac{\sqrt{37}}{3})>2\rho _S$. Moreover $\cosh (d(q,\alpha ))=\frac{\sinh (\ell /2)}{\sinh (\alpha /2)}>\rho _S$. So there exists $r>\rho _S$ such that all loops based at $q$ have length at least $2r$, thus again $r(S)>r_p$, a contradiction.

If $\gamma$ is a cusp, cut along $\alpha$ and consider a point $q$ which is equidistant from the two copies of $\alpha$ and at distance $\log (2/\sqrt{3})+\varepsilon$ (for $\varepsilon >0$ small) from the horoball of area 2. By explicit computations, $r_q>\rho _S>r_p$, contradiction again.

So in case (b), $r(S)>\rho _S$.$\square$