Cook-hats and Crowns
Yohei Komori
This paper is dedicated to Professor Cliﬀord Earle.

Abstract. We will study hyperbolic structures on a torus with a hole (named
as a “cook-hat”), and on a thrice-punctured sphere with a hole (named as a“crown”). See Figure 1 below. Both of them have three simple closed geodesics
called canonical triples, whose hyperbolic lengths and the hyperbolic lengthof the boundary geodesic deﬁne homogeneous coordinates of the Teichmüller
space for each cases. We will show that their Teichmüller spaces are realized
as convex polyhedra in the three-dimensional real projective space P (R4 ), bymeans of the canonical isomorphism between them.

Figure 1. a cook-hat and a crown
1. Introduction
Let X be an orientable surface of genus g with n punctures whose Euler number
is negative, χ(X) := 2 − 2g − n < 0. Then the Teichmüller space T (X) is the spaceof isotopy classes of hyperbolic metrics on X which has a metric space structure
homeomorphic to the real aﬃne space R6g−6+2n .
By using hyperbolic lengths of simple closed geodesics we can embed T (X)into the inﬁnite-dimensional real aﬃne space as follows: Let S be the non-trivial
and non-peripheral free homotopy classes of simple closed curves on X. For any
hyperbolic structure m ∈ T (X) and any free homotopy class α ∈ S, we denotethe hyperbolic length of a unique simple closed geodesic belonging to α by l(m, α).
Then the mapping l∗ : Tg,n → RS+ deﬁned by l∗ (m) = (l(m, α))α∈S is injective.
1991 Mathematics Subject Classiﬁcation. Primary 51M10, 32G15; Secondary 14H15, 30F60.
Key words and phrases. Teichmüller spaces, hyperbolic geometry.
The author was partially supported by Grant-in-Aid for Scientiﬁc Research (19540194), Ministry of Education, Science and Culture of Japan.
1

2

YOHEI KOMORIIn practice we can embed T (X) into R9g−9+3n : Fix a pants decomposition P on
X, i.e. a multicurve such that X \ P is homeomorphic to the disjoint union ofthrice punctured spheres. P consists of 3g − 3 + n numbers of disjoint simple close
curves. The Fenchel-Nielsen coordinates associate to each m ∈ T (X) the lengthand the twist of each components of P, which is a diﬀeomorphism from T (X) onto
× R3g−3+n (see [IT]). On the other hand the twist of each components
R3g−3+n+of P can be determined by the lengths of two more curves for each components
so that T (X) can be embedded into R9g−9+3n by length functions of 9g − 9 + 3n
number of simple closed geodesics. It should be remarked that the minimal numberof simple closed geodesics whose hyperbolic lengths globally parametrize T (X) isequal to dimR T (X) + 1 = 6g − 5 + 2n (see [S1]).
Let π be the projection from RS \ {0} to the inﬁnite-dimensional real projective
space P (RS ). In Proposition 6 of Exposé 7 [FLP] Kerckhoﬀ showed that thecomposition map π ◦ l∗ : T (X) → P (RS ) is also injective: In his argument, it
is essential that the surface X has at least one handle, because he used the factthat for the case g ≥ 1 we can ﬁnd two simple closed curves γ1 and γ2 whose
intersection number is equal to one. Then simple closed curves γ3 and γ4 which arefreely homotopic to γ1 · γ2 and γ1−1 · γ2 respectively satisfy the key identity for his
proof:
l1 + l2
l 1 − l2
l3
l4
cosh(
) + cosh() = cosh( ) + cosh( ).
2222where li := l(m, [γi ]) for m ∈ T (X) and i = 1, 2, 3, 4. Hence for the case g = 0, weshould look for other ideas to claim that the composition map π ◦l∗ : Tg,n → P (RS )is also injective (see Corollary 3.7 in Section 3).
The composition map π ◦ l∗ : Tg,n → P (RS ) is the basic ingredient for theThurston compactiﬁcation of T (X): The image π ◦ l∗ (T (X)) is relatively compact
in P (RS ) and its compactiﬁcation π ◦ l∗ (T (X)) in P (RS ) is homeomorphic to theclosed ball of dimension 6g − 6 + 2n. The relative boundary of π ◦ l∗ (T (X)) coincides with PMF(X) the projective image of the space of measured foliations
on X under the intersection number functions, which has a PL-manifold structure
homeomorphic to the sphere of dimension 6g − 7 + 2n (see Exposé 8 [FLP]).
Now we have the following natural question:
Can we ﬁnd dimR T (X) + 1-number of simple closed geodesics
whose hyperbolic lengths embed T (X) into the ﬁnite dimensional
real projective space P (RdimR T (X)+1 )?
Because of the PL-Structure of the Thurston boundary, we might expect that theimage should be the interior of some convex polyhedron in P (RdimR T (X)+1 ).
For this question, Schmutz proved aﬃrmatively for the case (g, n) = (2, 0) (see[S2]). Hamenstädt also consider the similar question by using non-simple geodesics
for the case n ≥ 1 (see [H]). Gendulphe and the author solved this questionaﬃrmatively for non-orientable genus 3 surfaces (see [GK]). They also showedthat the image of the Teichmüller space in P (R4 ) becomes a convex polyhedron.
To attack this question in general, in this paper we will consider the case of
a torus with a hole in section 2, and the case of a thrice punctured sphere with ahole in section 3, since any surface X contains one of these surface as an essential
subsurface. In practice we will answer this question for surfaces with at least one
hole with few exceptional cases (see Corollaries 2.7 and 3.8). In section 2 we willshow that the Teichmüller space of a torus with a hole can be realized as a convex

COOK-HATS AND CROWNS

3

polyhedron in P (R4 ) via hyperbolic length functions, which is a key idea for themain results of [GK]. Then in section 3 we will show the geometric bijection
between the Teichmüller space of a torus with a hole and the Teichmüller space ofa thrice punctured sphere with a hole, which itself seems interesting. By means of
this bijection we can also realize the Teichmüller space of a thrice punctured spherewith a hole as a convex polyhedron in P (R4 ). And as an application of this result,
we will prove that the composition map π ◦ l∗ : Tg,n → P (RS ) is injective also forthe case g = 0.
Acknowledgements. The author is grateful to Professor Ruth Kellerhals forher hospitality during his stay at the university of Fribourg, who suggested him
such charming names, Cook-hats and Crowns. He also thanks Doctor Matthieu
Gendulphe for his critical comments on a draft version of this paper.

2. Cook-hats
In this section we will consider complete hyperbolic structures on a torus witha hole. We call a hyperbolic torus with a hole a cook-hat.
Definition 2.1. Three simple closed geodesics (α, β, γ) on a cook-hat is calleda canonical triple if each pair of them has the intersection number equal to one.
We remark that the hyperbolic lengths of a canonical triple (α, β, γ) satisfy
triangle inequalities.
For the hyperbolic lengths of a canonical triple (α, β, γ) and the boundary
geodesic δ on a cook-hat, we have the following equality and inequality.
Proposition 2.2. For any cook-hat with the boundary geodesic δ and a canonical triple (α, β, γ), their hyperbolic lengths l(α), l(β), l(γ) and l(δ) satisfy the following equality and inequality:
(2.1)

cosh2

l(δ)
l(β) + l(γ)
l(α)
l(α)
l(β) − l(γ)= (cosh− cosh)(cosh− cosh
).
4
2
2
2
2

(2.2)

l(α) + l(β) + l(γ) > l(δ).

Proof. We uniformize a cook-hat by a Fuchsian group Γ ⊂ SL(2, R), anddenote the traces of elements representing α, β, γ and δ by t(α), t(β), t(γ) and t(δ).
Then they satisfy(2.3)

t(δ) − 2 = t(α)t(β)t(γ) − (t(α)2 + t(β)2 + t(γ)2 ).

By means of the relation between trace functions and length functions(2.4)|t(α)| = 2 coshl(α)
2and the equality
2 cosh x cosh y = cosh(x + y) + cosh(x − y),4

YOHEI KOMORI

we can rewrite (2.3) in terms of length functionsl(δ)− 2 = t(δ) − 22= t(α)t(β)t(γ) − (t(α)2 + t(β)2 + t(γ)2 )
l(α)
l(β)
l(γ)
l(α)
l(β)
l(γ)
= 4(2 cosh
cosh
cosh
− cosh2− cosh2− cosh2
)
2
2
2
2
2
2
l(α)
l(α)
l(β) − l(γ)l(β) + l(γ)− cosh)(cosh− cosh) − 4.
= 4(cosh2222Therefore
l(δ)
1
l(δ)
cosh2
=
(cosh
+ 1)
4
2
2
l(β) + l(γ)
l(α)
l(α)
l(β) − l(γ)= (cosh− cosh)(cosh− cosh)2222which is the equality (2.1).
Since cosh x, hence cosh2 x is monotonely increasing function of x, the equality(2.1) implies that it is enough to show that
2 coshl(β) + l(γ)
l(α)
l(α)
l(β) − l(γ)l(α) + l(β) + l(γ)−cosh)(cosh−cosh) < cosh222224
for the proof of the inequality (2.2). In practice(coshl(α) + l(β) + l(γ)4l(β) + l(γ)
l(α)
l(α)
l(β) − l(γ)−(cosh− cosh)(cosh− cosh)2222l(α) + l(β) + l(γ)l(α)l(β) + l(γ)l(β) − l(γ)= cosh2+ cosh2+ cosh
cosh
4
2
2
2
l(α)
l(β) + l(γ)l(α)l(β) − l(γ)− coshcosh− coshcosh
2222l(β)+l(γ)−l(α)l(γ)+l(α)−l(β)l(α)+l(β)−l(γ)
1 l(α)l(β)222) + (e−e) + (el(γ) − e
)
=
{(e
−e
4
cosh2

+(1 − e−l(α)+el(α)−l(β)−l(γ)
2

−l(β)

+e

) + (1 − e
−l(γ)

+e

l(β)−l(γ)−l(α)2) + (1 − el(γ)−l(α)−l(β)2)+ 1} > 0.
¤

Remark 2.3.
(1) The equality (2.1) also follows from the plane hyperbolic geometry of the right angled hexagon which is the symmetric half ofthe pair of pants T \ α.
(2) The inequality (2.2) also comes from the fact that the curve α ∪ β ∪ γ isfreely homotopic to the geodesic δ.
By means of the equality (2.1) in Proposition 2.2, we can embed the Teichmüller
space T (T ) of a torus with a hole into the 3-dimensional real projective space P (R4 ).
Theorem 2.4. For a cook hat with a canonical triple (α, β, γ) and the boundary
geodesic δ, their hyperbolic lengths l(α), l(β), l(γ) and l(δ) satisfy
cosh2

sl(β) + sl(γ)
sl(α)
sl(α)
sl(β) − sl(γ)sl(δ)< (cosh− cosh)(cosh− cosh)42222COOK-HATS AND CROWNS

5

for any s > 1. In particular the system of length functions L := (l(α), l(β), l(γ), l(δ))
gives a homogeneous coordinate of the Teichmüller space T (T ) of a torus with ahole into P (R4 ).
Proof. For simplicity we will writea = l(α), b = l(β), c = l(γ), d = l(δ).
Then our claim is rewritten as√ds < cosh−1 f (s), ∀s > 1
4
where
b+c
a
a
b−c
f (s) := (coshs − cosh s)(cosh s − cosh
s),
2
2
2
2
for which it is enough to show that√ddcosh−1 f (s) > , ∀s > 1.
ds
4
By the inequality (2.2), it is enough to show that
√
d
a+b+c
cosh−1 f (s) >, ∀s > 1.
ds
4
By the following simple estimation√f ′ (s)f ′ (s)d√cosh−1 f (s) = √>ds
2f (s)
2 f (s) f (s) − 1
we will show thatf ′ (s)
a+b+c>, ∀s > 1.
f (s)2

In practicef ′ (s)
f (s)

=

b+c
a
d
a
b−c
d
ds (cosh 2 s − cosh 2 s)
ds (cosh 2 s − cosh 2 s)+acosh b+c
cosh a2 s − cosh b−c
2 s − cosh 2 s
2 s

b+c a
a+b+c
+ =
.
2
2
2
Here we use the following lemma:>

Lemma 2.5. For 0 < p < q,d(cosh qs − cosh ps)
q sinh qs − p sinh ps=> q, ∀s > 1.
g(s) := ds
cosh qs − cosh ps
cosh qs − cosh ps

Proof. It is enough to show that the derivative of g(s) is negative for ∀s > 1,since
q sinh qs − p sinh ps
lim g(s) = lim= q.
s→∞s→∞ cosh qs − cosh psHence we will show the negativity of the numerator of g ′ (s):g ′ (s)=(q 2 cosh qs − p2 cosh ps)(cosh qs − cosh ps) − (q sinh qs − p sinh ps)2.
(cosh qs − cosh ps)2

6

YOHEI KOMORIIn practice(q 2 cosh qs − p2 cosh ps)(cosh qs − cosh ps) − (q sinh qs − p sinh ps)2
= q 2 cosh2 qs + p2 cosh2 ps − (q 2 + p2 ) cosh qs cosh ps−q 2 sinh2 qs − p2 sinh2 ps + 2pq sinh qs sinh ps
1
1= q 2 + p2 − (q + p)2 cosh(q − p)s − (q − p)2 cosh(q + p)s
2
2
1
1
2
2
2
2
< q + p − (q + p) − (q − p) = 0.
2
2¤¤
By means of the triangle inequalities of l(α), l(β), l(γ) and the inequality (2.2)
in Proposition 2.2, we can determine the image of T (T ) in P(R4 ) as follows.
Theorem 2.6. The image of T (T ) the Teichmüller space of a cook-hat underthe map L := (l(α) : l(β) : l(γ) : l(δ)) is the convex polyhedron ∆ in P(R4 ) deﬁnedby∆ :={(a : b : c : d) ∈ P(R4 ) | a > 0, b > 0, c > 0, d > 0,a < b + c, b < c + a, c < a + b, d < a + b + c}.

Proof. By means of the inequality (2.2) in Proposition 2.2, we have L(T ) ⊂ ∆.
Hence we will prove that ∆ ⊂ L(T ). Take any point p ∈ ∆ and four positive real
numbers (a, b, c, d) ∈ R4+ satisfying p = (a : b : c : d). Then there exist s > 0 and a
hyperbolic structure m ∈ T (T ) such that(l(α), l(β), l(γ), l(δ)) = (as, bs, cs, ds )where l(α) = l(m, α) and ds > 0 is deﬁned by√sb + scsa
sa
sb − sc−1
ds := 4 cosh(cosh
− cosh )(cosh− cosh).
222
2
To conclude that L(m) = p, It is enough to show that there is s > 0 such thatds = sd. We will show that ds /s takes any value between 0 and a + b + c when svaries. In practice ds /s is a continuous function on s and(cosh

sb + scsa
sa
sb − sc− cosh )(cosh− cosh)→12222when s decreases, hence ds /s → 0. On the other hand,
(cosh
= e

sa
sa
sb − scsb + sc− cosh )(cosh− cosh)2222(a+b+c)s2O(1), s → ∞

and
ds
ds
= e 4 O(1), s → ∞4
imply that lims→∞ ds /s = a + b + c. Hence ds /s takes any value between 0 anda + b + c.
¤

cosh

COOK-HATS AND CROWNS

7

As an application, let us consider the Teichmüller space T (Y ) of a orientablesurface Y of genus g with n punctures and r holes. Assuming that r ≥ 1, thenit is known that the minimal number of simple closed geodesics whose hyperbolic
lengths globally parametrize T (Y ) is dimR T (X) = 6g − 6 + 2n + 3r (see [S1]).
Moreover suppose that Y has at least one handle i.e. g ≥ 1, there is a subsurface
homeomorphic to a torus with a hole. Therefore Theorem 2.4 implies the aﬃrmative
answer to our question in section 1 for the case that r ≥ 1 and g ≥ 1:
Corollary 2.7. Assume that r ≥ 1 and g ≥ 1. Then via length functions ofsimple closed geodesics, the Teichmüller space T (Y ) of a orientable surface Y ofgenus g with n punctures and r holes can be embedded into P (RdimR T (X)+1 ).
3. Crowns
In this section we will consider complete hyperbolic structures on a thricepunctured sphere with a hole. We call a hyperbolic thrice-punctured sphere with ahole a crown.
Definition 3.1. Three simple closed geodesics (α, β, γ) on a crown is called acanonical triple if each pair of them has the intersection number equal to two.
We will show that similar results in section 2 also hold for T (S) the Teichmüller
space of a thrice-punctured sphere with a hole with the help of the geometricbijection between T (T ) and T (S) explained below. For this purpose we realize
T (T ) and T (S) as hypersurfaces in R4 in terms of trace functions:
Theorem 3.2. (Theorem 2 of [L] and Proposition 3.1 of [NN])(1) We uniformize a cook-hat m ∈ T (T ) by a Fuchsian group and denotethe traces of elements representing a canonical triple α, β, γ and boundary
geodesic δ by tα (m), tβ (m), tγ (m) and tδ (m). Then the map ϕT : T (T ) →R4 deﬁned by ϕT (m) := (tα (m), tβ (m), tγ (m), tδ (m)) is injective and theimage ϕT (T (T )) is described as follows:{(a, b, c, d) ∈ R4

| a > 2, b > 2, c > 2, d > 2,
abc − a2 − b2 − c2 + 2 = d}.

(2) We uniformize a crown m ∈ T (S) by a Fuchsian group and denote thetraces of elements representing a canonical triple α, β, γ and boundary
geodesic δ by tα (m), tβ (m), tγ (m) and tδ (m). Then the map ϕS : T (S) →R4 deﬁned by ϕS (m) := (tα (m), tβ (m), tγ (m), tδ (m)) is injective and theimage ϕS (T (S)) is described as follows:{(p, q, r, s) ∈ R4

| p > 2, q > 2, r > 2, s > 2, s2 + 2(p + q + r + 4)s+4(p + q + r) + p2 + q 2 + r2 − pqr + 8 = 0}.

Than by means of trace functions, we have the following geometric bijection
between T (T ) and T (S):Theorem 3.3. There is a bijection from T (T ) to T (S) which sends a cookhat T with the lengths of a canonical triple and the boundary geodesic equal to(l1 , l2 , l3 , l4 ) to a crown S with the lengths of a canonical triple and the boundary
geodesic equal to (2l1 , 2l2 , 2l3 , l4 ).

8

YOHEI KOMORIProof. When we substitute (a2 −2, b2 −2, c2 −2, d) for (p, q, r, s), the equations + 2(p + q + r + 4)s + 4(p + q + r) + p2 + q 2 + r2 − pqr + 8 factorizes as2d2 + 2(p + q + r + 4)d + 4(p + q + r) + p2 + q 2 + r2 − pqr + 8=(d − (abc − a2 − b2 − c2 + 2))(d − (−abc − a2 − b2 − c2 + 2)).

Hence the map Ψ : ϕT (T (T )) → ϕS (T (S)) deﬁned by Ψ(a, b, c, d) := (a2 − 2, b2 −2, c2 − 2, d) is bijective. Also the relation between trace functions and length functionsl(α)|t(α)| = 2 cosh2
tells us the length relations between m ∈ T (T ) and ϕ−1
S ◦ Ψ ◦ ϕT (m) ∈ T (S).

¤Remark 3.4. For the limiting case l(δ) = 0, this bijection reduces to the wellknown correspondence between punctured tori and forth-punctured spheres, which
follows from the commensurability of uniformizing Fuchsian groups (see [ASWY]).
This bijection induces the next corollaries: The following inequality is the counterpart of the inequality (2.2) in Proposition 2.2 for crowns.
Corollary 3.5. For any crown with the boundary geodesic δ and a canonicaltriple (α, β, γ), their hyperbolic lengths l(α), l(β), l(γ) and l(δ) satisfy the following
inequality:l(α) + l(β) + l(γ) > 2l(δ).
Next result is the counterpart of Theorem 2.4 and 2.6 for crowns.
Corollary 3.6. For a crown with a canonical triple (α, β, γ) and the boundary
geodesic δ, the system of length functions (l(α), l(β), l(γ), l(δ)) gives a homogeneous
coordinate of the Teichmüller space T (S) into P (R4 ). The image of T (S) is theconvex polyhedron in P(R4 ) deﬁned by{(a : b : c : d) ∈ P (R4 ) | a > 0, b > 0, c > 0, d > 0,a < b + c, b < c + a, c < a + b, 2d < a + b + c}.
As an application of Corollary 3.6,
Corollary 3.7. The composition map π ◦ l∗ : T (X) → P (RS ) is also injective
for the case g = 0.
For the ﬁnal application of Corollary 3.6, let us consider the Teichmüller space
T (Y ) of a orientable surface Y of genus g with n punctures and r holes for thecase that g = 0, n ≥ 3 and r ≥ 1. Then there is a subsurface homeomorphic toa thrice-punctured sphere with a hole, hence Corollary 3.6 implies the aﬃrmative
answer to our question in section 1 for this case also:
Corollary 3.8. Assume that g = 0, n ≥ 3 and r ≥ 1. Then via length functions of simple closed geodesics, the Teichmüller space T (Y ) of a orientable surface
Y of genus g with n punctures and r holes can be embedded into P (RdimR T (X)+1 ).
For a sphere (i.e., g = 0) with holes (i.e., r ≥ 1), this question is still open forthe cases n = 0, 1, 2.

COOK-HATS AND CROWNS

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[S1] P. Schmutz,
Die Parametrisierung des Teichmüllerraumes durch geodätischeLängenfunktionen, Comment. Math. Helv. 68 (1993), no. 2, 278?288.
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Osaka City University Advanced Mathematical Institute and Department of Mathematics, Osaka City University, 558-8585, Osaka, Japan
E-mail address: komori@sci.osaka-cu.ac.jp