POLYHEDRAL REALIZATION OF A THURSTONCOMPACTIFICATION
MATTHIEU GENDULPHE AND YOHEI KOMORI

Abstract. Let Σ−
3 be the connected sum of three real projective planes. We
realize the Thurston compactification of the Teichmüller space Teich(Σ−3 ) as a4
simplex in P(R4 ). First, we define a map L2 from Teich(Σ−
)
into
P(R
)
in
terms
3
of some geodesic-length functions. We then introduce the similar triangle flowon Teich(Σ−3 ) to control the ratios between these lengths, and show that L2 isan embedding. Finally, we study the natural extension of L2 to the Thurston
boundary using a triangulation of the projective space of measured foliations.

1. Introduction
In order to classify diffeomorphisms on a given compact surface Σ, W. P. Thurston
built a compactification of the Teichmüller space Teich(Σ) consisting in a closed balllying in an infinite dimensional projective space. This Thurston compactification is
simply defined as the closure of the image of the geodesic-length functions embedding
Teich(Σ) −→ P(RS )X 7−→ (ℓs (X))s∈S ,where S denotes the set of isotopy classes of nontrivial simple closed curves on Σ.
Although the Thurston compactification is not a polytope, its boundary carries apiecewise integral projective structure, which interest stands in the identification of
the Thurston boundary with the projective space of measured foliations. One would
simplify the complicated combinatorial structure of the Thuston boundary and, inthe same time, preserve its piecewise integral projective structure. In this direction,
we address the problem of realizing the Thurston compactification as a finite convex
integral polytope of a projective space of dimension dimTeich(Σ).
The first idea is to look at projections πF : P(RS ) → P(RF ) where F is a subset
of S of cardinal dimTeich(Σ) + 1. It is in general very hard to decide whether sucha projection defines an embedding of the Thurston compactification. Nevertheless,we know that for some F the projection πF gives an embedding of the interior of
the Thurston compactification (P. Schmutz [Sch93, Sch92]), and for some other F
an embedding of the Thurston boundary (U. Hamenstädt [Ham03]).
In this note we study the particular case of Σ−
3 , the connected sum of three realprojective planes. Its Teichmüller space Teich(Σ−3 ) is of dimension three and quite
easy to handle. We present an explicit embedding of its Thurston compactification
into P(R4 ), and describe its image. More precisely, if α, β, γ, α′ , β ′ , γ ′ in S satisfy
a precise topological configuration which will be defined in §3, then
The first author was fully supported by the Fonds National Suisse de la Recherche Scientifique(SNF projects no. 200020-121506/1 and no. 200021-131967/1). The second author was partially
supported by Grant-in-Aid for Scientific Research (19540194), Ministry of Education, Science andCulture of Japan.
1

POLYHEDRAL THURSTON COMPACTIFICATION

2Theorem. The restriction of the following map induces an embedding of the Thurston
4compactification of Teich(Σ−3 ) into P(R ) which image is a projective simplex,
L2 : P(RS ) −→ P(R4 )(xs )s∈S 7−→ (xα : xβ : xγ : xα′ + xβ ′ + xγ ′ ).
As we obtain the simplest polytope, this result is in some sense optimal. To provethat L2 is projectively injective on the Teichmüller space, we introduce the similar
triangle flow on Teich(Σ−3 ). A trajectory of this flow preserve ratios between thelength functions ℓα , ℓβ and ℓγ . Accurate estimates on the derivative of ℓα′ , ℓβ ′ andℓγ ′ along these trajectories enable us to conclude.
We also consider a projection L1 : (xs )s∈S 7→ (xα : xβ : xγ : xσ ) which restriction
does not define an embedding of the Thurston compactification of Teich(Σ−3 ). Weshow that in fact L1 gives an embedding of another compactification, which is aninteresting mixture of the Thurston compactification with the Teichmüller space ofhyperbolic structures on Σ−
3 pinched at σ (§6).
The text is organized as follows: in §2 we recall some basic facts about Teichmüller
spaces and their Thurston compactification, in §3 we describe the topology andgeometry of Σ−
3 , in §4 we introduce the similar triangle flow and study the behaviourof some length functions along its trajectories, and finaly in §6 we interpret the
projective polyhedra obtained from L1 and L2 .
Acknowledgement. Both authors are grateful to Prof. Ruth Kellerhals who makes
possible this collaboration with the support of SNF projects no. 200020-121506/1
and no. 200021-131967/1. The second author wishes to thank her for her hospitality
during his stay at the university of Fribourg.
2. Preliminaries
The reader can look at the classical references [Thu88] or to [FLP91] for moredetails on Teichmüller spaces and their Thuston compactification.
Let Σ be a compact surface of negative Euler-Poincaré characteristic χ(Σ) < 0.
A simple closed curve on Σ is indivisible if it does not bound a disk or a Mœbius
strip. We will denote S the set of isotopy classes of indivisible simple closed curveson Σ, which do not retract into a boundary component.
2.1. Thurston compactification. The Teichmüller space Teich(Σ) is the space of
isotopy classes of hyperbolic metrics on Σ. When Σ has nonempty boundary, we
assume that the lengths of the boundary components are fixed. Endowed with its
natural smooth structure, the Teichmüller space embeds into the projective space
P(RS ) via length functions of geodesics. Its image is an open ball of dimension
dimTeich(Σ) = −3χ(Σ) − n, where n denotes the number of boundary components.
The Thurston boundary of the Teichmüller space is the boundary of its image inP(RS ). It is a topological sphere of dimension dimTeich(Σ) − 1 denoted ∂Teich(Σ).
Intersection functions indentify the Thurston boundary with the projectivised spaceof measured foliations PMF(Σ), and also with the projectivised space of measured
geodesic laminations PML(Σ). The Thurston compactification of the Teichmüller
space is the closure of its image in P(RS ), which turns out to be a closed ball.

POLYHEDRAL THURSTON COMPACTIFICATION

3

2.2. Integral points in the Thurston boundary. A curve system of Σ is an
isotopy class of finite unions of disjoint two-sided simple closed curves on Σ, noneof which retracts into a point or a boundary component. We denote CS(Σ) the setof curve systems. The Thurston boundary possesses a piecewise integral projective
structure. The set of integral points correspond exactly to CS(Σ) and is dense in∂Teich(Σ).
Although ∂Teich(Σ) has no canonical simplicial decomposition, the curve complex
C(Σ) is a simplicial complex which embedds canonically into ∂Teich(Σ). The set ofvertices of C(Σ) is S , and a collection {c1 , . . . , cn } ⊂ S defines a (n − 1)-simplex if
these curves have zero intersection numbers.
2.3. Notations and conventions. We denote Σg,n (resp. Σ−g,n ) the orientable(resp. the non-orientable) compact surface of genus g with n boundary components.
A hyperbolic metric is a complete metric of constant sectional curvature −1, withtotally geodesic boundary. Unless otherwise is stated, geodesic means an indivisible
simple closed geodesic which is not a boundary component. With this convention,
the set of geodesics is in bijection with S .
Instead of two-sided (resp. one-sided), we prefer to say that a simple closed curveis orientable (resp. non-orientable) if it is transversely orientable (resp. if it is nottransversely orientable). Two isotopic non-orientable curves always intersect, but
their intersection number as measured foliations is zero, so i(c, c) = 0 for any elementc of S . This makes the intersection number multiplicative.
If we need to explicit the lengths b1 , . . . , bn of the boundary components, then we
will use the following notation: Teichb1 ,...,bn (Σ±g,n ).
3. Geometry and topology of the non-orientable surface of genus 3
In this part X is a hyperbolic connected sum of three projective planes. In thefirst paragraphs, we classify geodesics of X and look at a classical configuration
which gives a nice affine embedding of Teich(Σ−3 ). The last paragraphs are devoted
to curves systems and projective measured foliations of Σ−
3.
3.1. Simple closed geodesics. Results here are extracted from the article [Gen]of the first author. We have recently discovered that the proposition below was firstobtained by M. Scharlemann in [Sch82].
Proposition 3.1. There is a unique simple closed geodesic σ in X which produces
a one holed torus TX after cutting.
∗This induces a canonical bijection between Teich(Σ−3 ) and ∪b∈R+ Teichb (Σ1,1 ).

Proposition 3.2. Let γ be a simple closed geodesic of X distinct from σ.
(1) If γ is orientable, then γ is disjoint from σ.
(2) If γ is non-orientable, then γ intersects σ in exactly one point.
(3) There exists a unique simple closed geodesic γ ′ 6= σ disjoint from γ which
has opposite orientability. We say that γ and γ ′ are duals.
The duality defines an involution of S which has σ as a unique fixed point. The
lengths of an orientable simple closed geodesic γ and its dual γ ′ are related by
cosh

ℓ(γ)
2

= sinhℓ(γ ′ )
ℓ(σ)
sinh
.
2
2

(1)

POLYHEDRAL THURSTON COMPACTIFICATION

4

The corresponding identity on the intersection numbers isi(γ, ·) = i(γ ′ , ·) + i(σ, ·) on S \ {γ ′ , σ}.

(2)These identities appear clearly on the figure 1.
γ
γ
γ

γ′

γ′
TX
σ

σ

Figure 1. A geodesic and its dual
3.2. Triangle embedding.
Definition 3.1. A triangle is a triple (α, β, γ) of orientable simple closed geodesics
with all intersection numbers equal to one.
Remark 3.1. A triple (α, β, γ) is a triangle if and only if its dual triple (α′ , β ′ , γ ′ )consists of three disjoint simple closed geodesics. The complement in Σ−
3 of thesedual curves is a pair of pants.
Any triangle satisfies the geometric inequality (obvious after cutting the geodesics)ℓ(α) + ℓ(β) + ℓ(γ) > 2 ℓ(σ),(3)

and also the following identityℓ(α) + ℓ(β)ℓ(γ)ℓ(α) − ℓ(β)ℓ(γ)
2 ℓ(σ)= cosh− cosh− cosh
cosh
(4)
cosh
2
2
2
2
2
which comes from hyperbolic trigonometry in a right-angled hexagon.
Proposition 3.3. Let (α, β, γ) be a triangle, then the following map is an embedding
3
L : Teich(Σ−3 ) −→ R
X 7−→ (ℓα (X), ℓβ (X), ℓγ (X))

and its image is the following unbounded domain∆ = (a, b, c) ∈ R3 | b + c > a, c + a > b, a + b > c and




c
a−b
c
a+b
− coshcosh − cosh
>1 .
cosh
2
2
2
2
It is well known that L is smooth and injective (see [Sch92] for instance), one caneasily show that L is in fact a differential embedding with image ∆.
Remarks 3.1.
• Triangle inequalities imply that a, b and c are positive.
• The last inequality can be replaced by any symmetric one in a, b, c.
• ∆ is invariant under multiplication by a scalar t ≥ 1.

POLYHEDRAL THURSTON COMPACTIFICATION

5

3.3. Curve systems. Let λ be a curve system. For every c ∈ S we will denoteλc the number of components of λ isotopic to c if c is orientable, or to c2 if c is
non-orientable.
Proposition 3.4. Let (α′ , β ′ , γ ′ ) be a dual triple of a triangle. Then the following
map is a bijection
3CS(Σ−3 ) −→ Z
λ 7−→ (i(α′ , λ) − λα′ , i(β ′ , λ) − λβ ′ , i(γ ′ , λ) − λγ ′ ).

Proof. Let us consider a triple (zα , zβ , zγ ) ∈ Z3 . We assume that each coordinate isnonnegative, but our argument also works for other cases. The triple (2zα , 2zβ , 2zγ )is realized as Dehn-Thurston coordinates of some arc system λ̄ of Σ0,3 (see [LS04]
for instance). As the intersection number of λ̄ with each boundary is even, this arc
system of Σ0,3 induces a curve system λ of Σ−
3 after antipodal identification of theboundaries. Using classical surface topology, we see that the intersection numbersof λ with the curves coming from the boundaries are zα , zβ and zγ . This proves thesujectivity.
Let λ, κ be two curve systems of Σ−
3 with same image through the above map.
To simplify we assume that they have no component isotopic to α′ , β ′ or γ ′ . We
look at λ and κ as integer measured geodesic laminations of X. Let λ̄ and κ̄ be theirtraces on the pair of pants PX obtained after cutting α′ , β ′ and γ ′ . Clearly λ̄ andκ̄ determine the same arc system on PX (see [LS04]). So, for each boundary of PX ,there is a canonical bijection between the points of λ and κ on this boundary. Thisbijection can be easily realized by an isotopy of the boundary which commutes withthe antipodal map. These isotopies of the boundaries induce isotopies on PX and
X between λ and κ through geodesic arcs, proving the injectivity.
−2
Remark 3.2. There are analogous bijections CS(Σ−2,1 ) → Z and CS(Σ1,2 ) → Z.

3.4. Scharlemann’s description of PMF(Σ−3 ). M. Scharlemann gave in [Sch82] a−nice description of the embedding C(Σ3 ) ⊂ PMF(Σ−3 ), based on the classical relation
between C(Σ1,1 ) and the Farey tesselation. Let us recall that, after a choice of a
symplectic basis of the homology, simple closed geodesics in the one-holed torus
correspond bijectively to pair (p, q) of relatively prime integers, and so to rational
numbers p/q in R ∪ {∞}. The intersection number between two pairs (p, q) and(r, s) is equal to the absolute value |ps − rq| of the intersection form. Thus three
geodesics correspond to the vertices of a triangle of the Farey tesselation if and only
if they form a triangle in the one-holed torus.
The boundary ∂Teich(Σ1,1 ) embeds piecewise linearly and canonically into a cir2
cle in ∂Teich(Σ−3 ) ≃ S , and so divides it into two hemispheres. One of thesehemispheres contains only one vertex of C(Σ−3 ) in its interior, it corresponds to σ
and is related to all the vertices in ∂Teich(Σ1,1 ). The other contains all the vertices
corresponding to dual curves of orientable curves in his interior, the configurationof the curve complex in this hemisphere is a slightly modified version of the Farey
tesselation: let us look at the hemisphere as the unit disk, if z(p, q) represents theorientable geodesic with coordiantes (p, q), then the point z ′ (p, q) = z(p, q)/(1 + 1/q)
represents its dual geodesic. The segment [z(p, q), z ′ (p, q)] is an edge of the curve
complex, the other edges are between points z ′ (p, q) and z ′ (r, s) with |ps − rq| = 1.
See figure 2 for a picture where (α, β, γ) is a triangle.

POLYHEDRAL THURSTON COMPACTIFICATION

6

In fact, for any compact non-orientable surface Σ, points corresponding to nonorientable geodesics are isolated in PMF(Σ), but their closure contains all geodesics.
This was proved by M. Scharlemann in [Sch82].
γ′

β′

F
α′
E

γ′
γ

β′
α′
α

β
D
C

γ
A

σ
Figure 2. Scharlemann’s pictureβBαFigure 3

3.5. A simplicial description of PMF(Σ−3 ). Up to isotopy and multiplication bya positive constant, there are only two measured foliations on the Mœbius band:
foliations with closed leaves, and foliations with leaves transverse to the boundary.
Let α′ , β ′ and γ ′ be three disjoint indivisible simple closed curves which bound−a pair of pants in Σ−
3 . Any measured foliation on Σ3 is isotopic to a measured
foliation such that each of these curves is a leave, a cycle of leaves, or is transverseto the leaves. If one of these curves is a leave, then it admits a maximal Mœbius
neighborhood foliated by closed leaves.
The complement of these curves and their maximal Mœbius neighborhoods is apair of pants P , or eventually a graph if the support of the measure is contained in the
maximal Mœbius neighborhoods. In the first case, the measured foliation inducedon P is good, i.e. none of the boundaries is a leave. Then, using the classification ofgood foliations on P given in [FLP91] exposé 6 § II, we classify measured foliationson Σ−
3 (figure 4). The measures of the maximal Mœbius neighborhoods can change
the dimension of simplices, for instance compare simplices (4), (5) and (6) in figure 4of [FLP91] exposé 6 with their corresponding simplices in figure 4. In the case where
P is graph, the measured foliation is isotopic to the foliation F of the figure 4, it
gives a new 2-simplex. We have described a simplicial decomposition of PMF(Σ−3 ),we draw a global picture of it in figure 3.
4. Monotonicity of length functions under the similar triangle flowIn §5 we will use the similar triangle flow to show the injectivity of some map−
4from Teich(Σ−3 ) into P(R ). Two points of Teich(Σ3 ) with same image with respect
to these maps belong to the same trajectory of the similar triangle flow. So, we haveto study the behaviour of certain length functions along these trajectories to provethat distinct points have distinct images.

POLYHEDRAL THURSTON COMPACTIFICATION

7

γ′

σ

α′

α

β′

A

B

C

D

E

F

Figure 4

POLYHEDRAL THURSTON COMPACTIFICATION

8

4.1. Similar triangle flow. Let us consider the 1-parameter subgroup {x 7→ es x}s∈Rcoming from the radial linear vector field associated to the identity map of R3 . Its
composition with the logarithm defines a continuous multiplicative homomorphism
φ R∗+ −→ R −→ Diff(R3 ).
t 7−→ ln(t) 7−→ φt : x 7→ tx
Let x = (a, b, c) be a point in the positive cone. It follows from remark 3.1 that thetrace of the trajectory φt (x) on ∆ is the half-line {tx ; t > tx } where tx is the uniquepositive solution of



a+b
c
c
a−b
cosh
t − cosh tcosh t − cosh
t
= 1.
2
2
2
2
Let us fix a triangle (α, β, γ).
Definition 4.1. We call similar triangle flow with respect to (α, β, γ) the pullbackof φ by L. We denote it △(X, t), or simply △(t) when X is fixed.
The frontier of ∆ can be canonically identified with the Teichmüller space of
cusped tori, or equivalently with the Teichmüller space of hyperbolic structures onΣ−
3 noded at σ. The triangle flow can be extended to this space.
We now fix a point X in Teich(Σ−3 ). The trajectory △(t) is defined on an open
interval ]tX , +∞[⊂ R∗+ containing 1. The geometrical meaning of the similar triangle
flow can be expressed as follows: for any Y ∈ Teich(Σ−3 ) and any t ∈]tX , +∞[ ℓα (Y ) = tℓα (X)ℓβ (Y ) = tℓβ (X) .
△(t) = Y ⇐⇒
ℓγ (Y ) = tℓγ (X)In the sequel we will denote η(t) the length of any geodesic η at the point △(t).
4.2. Monotonicity of σ(t).
Proposition 4.1. For any t > tX we haveα(1) + β(1) + γ(1)
dσ
(t) >
.
dt
2
Proof. For simplicity we pose a = α(1), b = β(1) and c = γ(1). In view of (4)p
σ(t) = 2 cosh−1 f (t),
with



c
c
a−b
a+b
t − cosh tcosh t − cosht .
f (t) =
cosh
2
2
2
2
By the following simple estimationpdf ′ (t)f ′ (t)p
2 cosh−1 f (t) = p
,
>
dt
f (t)
f (t) f (t) − 1

it suffices to show thata+b+cα(1) + β(1) + γ(1)f ′ (t)
>
=
.
f (t)
2
2
In practicef ′ (t)
f (t)

=

d
d
a+b
c
c
a−b
dt (cosh 2 t − cosh 2 t)
dt (cosh 2 t − cosh 2 t)+.
ccosh a+b
cosh 2c t − cosh a−b
2 t − cosh 2 t
2 t

(5)

POLYHEDRAL THURSTON COMPACTIFICATION

9

But for any q > p > 0 and any t > 0d(cosh qt − cosh pt) = q sinh qt − p sinh pt
dt> q(sinh qt − sinh pt) = q

Z qt> q(cosh qt − cosh pt) = q

cosh(s)dspt
Z qt

sinh(s)ds.

pt

Thus, from (5) we conclude thatf ′ (t)
f (t)

>

a+b c
a+b+c
+ =
.
2
2
2


4.3. Monotonicity of α′ (t), β ′ (t) and γ ′ (t).
Proposition 4.2. For any t > tX we have
dβ ′
dγ ′
dα′
(t),
(t),
(t) < 0.
dt
dt
dt
Proof. Let us consider the case of α′ . From (1) we haveα′ (t)
σ(t)
α(t)
tα(1)
) sinh() = cosh() = cosh() (t > tX ),22
22
taking the derivative with respect to t we obtainsinh(

α′ (t)σ(t)dσ
a′ (t)
σ(t)
tα(1)
dα′
(t) cosh() sinh()+(t) sinh() cosh() = α(1) sinh(
).
dt
2
2
dt
2
2
2
Because of proposition 4.1 and triangle inequalitydσα(1) + β(1) + γ(1)(t) >> α(1).
dt
2
Hence
da′
α′ (t)
σ(t)
tα(1)
dσ
α′ (t)
σ(t)(t) cosh() sinh() = α(1) sinh()−(t) sinh() cosh(
)
dt
2
2
2
dt
2
2
α′ (t)σ(t)tα(1)) − α(1) sinh() cosh()< α(1) sinh(222α′ (t)σ(t)tα(1)) − α(1) sinh() sinh()< α(1) sinh(
2
2
2
ta
tα(1)
) − α(1) cosh( )< α(1) sinh(
2
2
< 0
′

′
′
which implies dαdt (t) < 0. Identical proofs work for β and γ .

45. Two polyhedral realizations of Teich(Σ−3 ) in P(R )4
In this part we give two embeddings of Teich(Σ−3 ) into P(R ). The injectivityof these embeddings comes from the monotonicity of ℓσ , ℓα′ , ℓβ ′ and ℓγ ′ along thesimilar triangle flow. We fix a triangle (α, β, γ) for all this part.

POLYHEDRAL THURSTON COMPACTIFICATION

10

5.1. Embeddings defined via length functions.
4
Theorem 5.1. The following maps are embeddings of Teich(Σ−3 ) into P(R ):4
L1 : Teich(Σ−3 ) −→ P(R )X 7−→ (ℓα (X) : ℓβ (X) : ℓγ (X) : ℓσ (X))
4
L2 : Teich(Σ−3 ) −→ P(R )X 7−→ (ℓα (X) : ℓβ (X) : ℓγ (X) : ℓα′ +β ′ +γ ′ (X))

Proof. Let us prove the injectivity of L1 . The system of length functions (ℓα , ℓβ , ℓγ )3defines an injective map from Teich(Σ−3 ) into R by proposition 3.3. So, it suffices−
to show that two points X, Y ∈ Teich(Σ3 ) with same image L1 (X) = L1 (Y ) havesame lengths with respect to α, β and γ.
These two points belong to the same trajectory under the similar triangle flow.
Without loss of generality, we assume Y = △(t) for some t ≥ 1. On one hand, by
definition of the similar triangle flow, we haveℓα (Y ) = α(t) = tα(1) = tℓα (X),
which implies by proportionality of lengths
σ(t) = ℓσ (Y ) = tℓσ (X) = tσ(1).
On another hand, from proposition 4.1 and (3) it comes that
Z t
dσ
(s)ds
σ(t) = σ(1) +1 dsα(1) + β(1) + γ(1)≥ σ(1) + (t − 1)2
≥ σ(1) + (t − 1)σ(1),with equality if and only if t = 1. We conclude that t = 1, thus X and Y have samelengths with respect to α, β, γ and σ.
The same proof also works for L2 . We just have to make few modifications: weconsider (α′ + β ′ + γ ′ )(t) instead of σ(t), and we use proposition 4.2 instead ofproposition 4.1.
It remains to show that the Li are submersions, or equivalently that the maps(ℓα , ℓβ , ℓγ , ℓσ ) and (ℓα , ℓβ , ℓγ , ℓα′ +β ′ +γ ′ ) are transverse to lines of R4 passing throughthe origin. Let us consider a germ of curve c = (cα , cβ , cγ , cσ ) in the image of(ℓα , ℓβ , ℓγ , ℓσ ), which is tangent to a line of R4 passing through the origin. Then3(cα , cβ , cγ ) is a germ of curve in L(Teich(Σ−3 )) tangent to a line of R passing throughthe origin, and so tangent to the image L(△(t)) of a trajectory of the similar triangle
flow. As L is an embedding (proposition 3.3) it comes that c is tangent to the imageof △(t) by (ℓα , ℓβ , ℓγ , ℓσ ). This is not possible according to proposition 4.1 andinequality (3), so L1 is a submersion. The same argument works also for L2 .

5.2. Images as convex projective polyhedra.
Corollary 5.2. The images of L1 and L2 are convex polyhedra in P(R4 ):POLYHEDRAL THURSTON COMPACTIFICATION

11• the image L1 (Teich(Σ−3 )) is the convex projective polyhedron∆1 := (a : b : c : d) ∈ P(R4 ) | a + b > c, b + c > a, c + a > b and d > 0and a + b + c > 2d .
• the image L2 (Teich(Σ−3 )) is the simplex
4∆2 := (a : b : c : d) ∈ P(R ) | b + c > a, c + a > b, a + b > c and d > 0 .
Remark 5.1. Triangle inequalities imply that a, b and c are nonnegative.
−
Proof. We clearly have L2 (Teich(Σ−3 )) ⊂ ∆2 , and also L1 (Teich(Σ3 )) ⊂ ∆1 by means−of (3). So it remains to show that ∆i ⊂ Li (Teich(Σ3 )) for i = 1, 2.
Let us consider (a, b, c, d) ∈ R4 satisfying the conditions in the definition of ∆i .
Up to multiplication by a positive scalar, we can also assume that (a, b, c) = Li (X)for some X in Teich(Σ−3 ). We claim that there exists t ∈]tX , +∞[ such that

(α : β : γ : σ)(t) = (a : b : c : d).
This will prove that (a : b : c : d) ∈ Li (Teich(Σ−3 )) and conclude the proof.
Case of L1 . By definition of tX we have σ(t) → 0 when t → tX , so
dt > σ(t)for t sufficiently close to tX . The proposition 4.1 and a + b + c > 2d leads to2σ(t) ≥ 2σ(1) + (a + b + c)(t − 1)
≥ 2σ(1) − 2d + (a + b + c − 2d)(t − 1) + 2dt≥ 2dt
for t sufficiently large. As a consequence, there is a t0 > tx such that σ(t0 ) = dt0 ,and (α, β, γ, σ)(t0 ) = t0 (a, b, c, d) as we claimed.
Case of L2 . By the collar lemma
σ(t) −−−→ 0 implies
t→tX(α′ + β ′ + γ ′ )(t) −−−→ +∞.
t→tX

So (α′ + β ′ + γ ′ )(t) > dt for t sufficiently close to tX . But (α′ + β ′ + γ ′ )(t) is adecreasing function (proposition 4.2), thus (α′ + β ′ + γ ′ )(t) < dt for t sufficiently
large. As a byproduct there exists t0 > tX such that(α, β, γ, α′ + β ′ + γ ′ )(t0 ) = t0 (a, b, c, d),as we claimed.



6. Interpretation of the boundaries of ∆1 and ∆2
Let us fix a triangle (α, β, γ). There exists a unique curve η 6= α such that (η, β, γ)
is a triangle, we denote α⋆ the dual curve of η.

POLYHEDRAL THURSTON COMPACTIFICATION

point
A
B
C
E
S

12

coordinates point coordinates point coordinates(0 : 1 : 1 : 2)A′
(0 : 1 : 1 : 0) A⋆(2 : 1 : 1 : 1)′⋆
(1 : 0 : 1 : 2)
B
(1 : 0 : 1 : 0) B
(1 : 2 : 1 : 1)
(1 : 1 : 0 : 2)C′
(1 : 1 : 0 : 0) C ⋆ (1 : 1 : 2 : 1)
(0 : 1 : 1 : 1)F
(1 : 0 : 1 : 1)G
(1 : 1 : 0 : 1)
(0 : 0 : 0 : 1)

Table 1. Some points in P(R4 ) and their coordinates

6.1. Description of the boundaries. The closure of ∆2 is the simplex spanned by
A′ , B ′ , C ′ and S. Whereas the closure of ∆1 is the truncated simplex with vertices
A′ , B ′ , C ′ , and E, F , G. All these points are defined in the table 6.1 below.
If these configurations are not obvious, one can use the projective transformation(a : b : c : d) 7→ ((b + c) − a : (a + c) − b : (a + b) − c : a + b + c + d)which
 sends respectively ∆2 and ∆1 on {(x : y : z : t) | x, y, z > 0 and t > x + y + z}
and (x : y : z : t) | x, y, z > 0 and t > x + y + z > 23 t . Then, everything becomes
clear in the affine chart {t = 1}.
6.2. Interpretation. Each embedding Li factors through L, and also through theScanonical embedding of Teich(Σ−3 ) into P(R ). The situation is represented in thecommutative diagram below, where the maps φi and ψi are quite obvious.
Teich0 (Σ1,1 )

R3
φi

L

Teich(Σ−3)

Li

P(R4 )ψi

P(RS )

PMF(Σ−
3)
6.2.1. Case of ∆1 . The map φ1 extends differentiably to the closure of ∆, andinduces an embedding of the Teichmüller space of cusped tori into P(R4 ), which issimply (ℓα : ℓβ : ℓγ : 0). The image of this embedding is exactly the interior of theface hA′ , B ′ , C ′ i of ∆1 .
The map ψ1 extends coutinuously to the set of points in the Thurston boundary
satisfying the condition (iα , iβ , iγ , iσ ) 6= 0. This set turns out to be ∂Teich(Σ−3 ) \ {σ}.
−The vertices of the triangulation of PMF(Σ−)≃∂Teich(Σ)
studiedin§3.5 are3
3
sent on vertices, and barycenters of faces, of ∆1 (figure 5). Thus, the extension

POLYHEDRAL THURSTON COMPACTIFICATION13

of ψ1 induces a piecewise integral projective isomorphism from the complement in
PMF(Σ−
3 ) of the interior of the simplex hα, β, γi, to the complement in ∂∆1 of theinterior of the simplex hA′ , B ′ , C ′ i. The extension of ψ1 mashes hα, β, γi \ {σ} onto
the boundary of hA′ , B ′ , C ′ i. More precisely, a point X in hα, β, γi \ {σ} has sameimage as the point Y on the boundary of the simplex such that X belongs to [σ, Y ].
Interpretation. The polyhedon ∆1 is a compactification of the Teichmüller spaces∪b>0 Teichb (Σ1,1 ). Its boundary decomposes into two pieces. One piece corresponds
exactly to the set of projective measured foliations of the one-holed torus, whereleaves transverse to the boundary are allowed. The other piece corresponds exactly
to the Teichmüller space of cusped tori. The frontier between these two pieces is theThurston boundary of Teichb (Σ1,1 ), which does not depend on the fixed length b > 0.
We have to take care of the way we define measured foliations. If we work with
Teichb (Σ1,1 ) (b ≥ 0 fixed) then we consider measured foliations as in [FLP91] exposé 11 §1. But if we work on ∪b>0 Teichb (Σ1,1 ), then we allowed measured foliations
to have leaves transverse to the boundary. Nevertheless, we still assume that no
closed leave is isotopic to the boundary, that’s why the simplex hα, β, γi collapses.
σ

∆2
∆1

γ

α
β
γ⋆

γ

α
β

α⋆
γ⋆
γ′

α′

γ′
α′

β′

β′
Figure 5. Polyhedral representations

α
β
γ
σ
α′
β′
γ′
α⋆
β⋆
γ⋆

iα
0
1
1
0
0
1
1
2
1
1

iβ
1
0
1
0
1
0
1
1
2
1

iγ
1
1
0
0
1
1
0
1
1
2

iσ
0
0
0
0
1
1
1
1
1
1

α⋆

iα′ +β ′ +γ ′
2
2
2
3
0
0
0
1
1
1

in ∆1 in ∆2
A′
A
B′
B
C′
C
S
E
A′
F
B′
G
C′
⋆
A
A⋆
⋆
B
B⋆
⋆
C
C⋆

Table 2. Intersection numbers

POLYHEDRAL THURSTON COMPACTIFICATION

14

6.2.2. Case of ∆2 . The map φ2 does not extend to any point of the boundary of∆, because of the collar lemma. On the contrary, ψ2 extends continuously to thewhole Thurston boundary of Teich(Σ−3 ), for no point in the Thurston boundary
satisfies (iα , iβ , iγ , iα′ +β ′ +γ ′ ) = 0. The images of the vertices of the triangulationof PMF(Σ−3 ) are sent to points in ∆2 as shown in figure 5. So the extension of ψ2
induces a piecewise integral projective isomorphism between the Thurston boundaryof Teich(Σ−3 ) and the boundary of ∆2 .
Interpretation. The polyhedron ∆2 is a convex polyhedral realization of the Thurston
compactification of Teich(Σ−3 ). The map (ℓα , ℓβ , ℓγ , ℓα′ +β ′ +γ ′ ) is an embedding of
Teich(Σ−)
ontotheinteriorof ∆2 . It has a continuous extension which induces a3piecewise integral projective isomorphism given by (iα , iβ , iγ , iα′ +β ′ +γ ′ ) between theThurston boundary of Teich(Σ−3 ) and the boundary of ∆2 .
6.3. Few words about the action of the modular group. Let Σ be a compact
surface of negative Euler-Poincaré characteristic. The mapping class group MCG(Σ)
is the group of isotopy classes of diffeomorphisms of Σ which stabilize boundary
components. We call modular group, and denote Mod(Σ), the quotient of the mapping class group by its subgroup acting trivially on the Teichmüller space Teich(Σ).
It is well known that
MCG(Σ1,1 ) ≃ GL(2, Z)

andMod(Σ1,1 ) ≃ PGL(2, Z),

and that (direct consequence of proposition 3.1)
MCG(Σ−3 ) ≃ GL(2, Z)

andMod(Σ−3 ) ≃ PGL(2, Z).

The modular group Mod(Σ−3 ) decomposes into an amalgamated product D4 ∗Z/2Z D6 ,where each factor corresponds to some stabilisator for its action on Teich(Σ−3 ).
The modular group acts also on the boundary of the Teichmüller space, and soMod(Σ−
3 ) acts on the boundary of the projective simplex ∆2 . Clearly, every element fixes the vertex associated to σ and stabilizes the subcomplex identified with
∂Teich(Σ1,1 ). The elements of Mod(Σ−3 ) acting projectively stabilize each set of′′′
curves {α, β, γ} and {α , β , γ }. But, the images of α, β and γ determine completely an element of Mod(Σ−3 ). In conclusion, the subgroup of elemets acting projectively identifies with the subroup of projective transformations of P(R4 ) which
fix the vertex associated to σ and permute the vertices associated to α′ , β ′ and γ ′ ,it corresponds to the factor D6 .
References[FLP91] Travaux de Thurston sur les surfaces, volume 66–67 of Astérisque. Société Mathématiquede France, 1991.
[Gen]
M. Gendulphe. Paysage systolique des surfaces hyperboliques compactes de caractéristique -1. available at http://matthieu.gendulphe.com.
[Ham03] U. Hamenstädt. Parametrizations of Teichmüller space and its Thurston boundary. In
Geometric analysis and nonlinear partial differential equations, pages 81–88. Springer,
2003.
[LS04] F. Luo and R. Stong. Dehn-Thurston coordinates for curves on surfaces. Comm. Anal.
Geom., 12(1-2):1–41, 2004.
[Sch82] M. Scharlemann. The complex of curves on nonorientable surfaces. J. London Math. Soc.
(2), 25(1):171–184, 1982.
[Sch92] P. Schmutz. Une paramétrisation de l’espace de Teichmüller de genre g donnée par 6g − 5géodésiques explicites. In Séminaire de Théorie Spectrale et Géométrie, No. 10, Année
1991–1992, volume 10, pages 59–64. Univ. Grenoble I, 1992.

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15[Sch93]

P. Schmutz. Die Parametrisierung des Teichmüllerraumes durch geodätischeLängenfunktionen. Comment. Math. Helv., 68(2):278–288, 1993.
[Thu88] W. P. Thurston. On the geometry and dynamics of diffeomorphisms of surfaces. Bull.
Amer. Math. Soc., 19(2):417–431, 1988.
Matthieu Gendulphe, Départment de Mathématiques, université de Fribourg, Chemin
du Musée 23, 1700 Fribourg, Suisse
E-mail address: Matthieu.Gendulphe@unifr.ch
Yohei Komori, Department of Mathematics, Osaka City University, 558-8585, Osaka,Japan
E-mail address: komori@sci.osaka-cu.ac.jp