2.17 Interpretación de los resultados
2.17.2 Análisis del sitio
To describe the gluing in detail, first recall (see for example [37] p. 207) that any triply punctured sphere is homeomorphic to the standard triply punctured sphere P=H/Γ, where Γ =D 1 2 0 1 ! , 1 0 2 1 ! E .
Fix a standard fundamental set ∆ for Γ as shown in Figure 2.2, so that the three punctures of P are naturally labelled 0,1,∞. (A fundamental set for the group G
is a subset of Ω(G) which contains exactly one point from each equivalence class of
points of Ω(G); see p.32 of Maskit [32].) In detail let ∆ be defined by the following:
∆ ={z∈H:−1<<z≤1,|z−12| ≥ 12,|z+1
2|> 1 2}.
From now on, we will useto denote element of the cyclically ordered set{0,1,∞}.
Let ∆0 be the (closed) ideal triangle with vertices{0,1,∞}, and ∆1 be the interior
of its reflection in the imaginary axis. We sometimes refer to ∆0as the white triangle
and ∆1 as the black. The set ∆ is the union of ∆0 and ∆1.
With our usual pants decompositionP, we define bijections ˆ
1
1 0
0 1
1
Figure 2.2: The standard fundamental set for Γ. The white triangle ∆0is unshaded
and the black one ∆1 is shaded.
from each (open) pair of pantsPj to the fundamental set ∆ by
ˆ
Φj := (ζ|∆)−1◦Φj,
where
Φj: Pj −→P
is the homeomorphism which identifiesPj toP, and where
ζ: H−→P=H/Γ
is the natural quotient map (which is a bijection when restricted to ∆). Note that ˆΦj
restricts to a homeomorphism betweenPj minus the seams to ∆ minusλ1∪λ0∪λ∞,
whereλ is the geodesic joining + 1 and + 2 with in the cyclically ordered set
{0,1,∞}.
This identifications induce a labelling of the three boundary components ofPj as
0,1,∞ in some order, fixed from now on. We denote the boundary of Pj labelled
∈ {0,1,∞}by ∂Pj. The identifications also induce a colouring of the two right
angled hexagons whose union isPj, one being white and one being black. We will
call∂Pthe boundary component ofPcorresponding to∂P under the identification
Φ: P −→P.
Suppose that the pants P, P0 ∈ P are adjacent along the pants curveσ meeting
along boundaries∂P and ∂0P0. (If P =P0 then clearly 6=0.) The gluing across
σ is described by a complex parameterµwith=µ >0, called thegluing parameter,
as already defined above. We first discuss the gluing in the case =0=∞.
Tµ ˆ ˆ0 P @✏(P) P0 @✏0(P0) z H1 ⌦0 J z0 H00 ⌦1 ⌦1(z) ⌦0(z0) J ⌦0(z)
Figure 2.3: The gluing construction when = 1 and 0 = 0. Only the parts of H1
andH00 in ∆0 and ∆00 are shown.
illustration in the figure explains the more general case= 1 and 0 = 0.)
Take two copiesP,P0ofP. Each of these is identified withH/Γ as described above.
We refer to the copy ofHassociated toP0 asH0 and denote the natural parameters
in H,H0 by z, z0 respectively. Let ζ and ζ0 be the projections ζ: H −→ P and
ζ0: H0 −→P0 respectively.
Leth∞=h∞(µ) be the loop on Pwhich lifts to the horocycle
h∞,H={z∈H|=z=
=µ
on H. For a small positive numberν >0, we define H∞=H∞(µ, ν) ={z∈H|=µ−ν
2 <=z < =
µ+ν
2 } ⊂H
to be the horizontal strip which projects to the annular neighbourhoodA∞=A∞(µ)
of h∞ ⊂ P. Let S ⊂ P be the surface P with the projection of the horocyclic
neighbourhood{z∈H|=z≥ =µ2+ν}of ∞ deleted. Note that S is open. Defineh0∞, S0 and A0∞ in a similar way. We are going to glue S to S0 by matchingA∞ toA0∞
in such a way thath∞ is identified toh0∞ with orientation reversed, see Figure 2.3.
The resulting homotopy class of the looph∞on the glued up surface (the quotient of
the disjoint union of the surfacesSj by the attaching maps across theAi=A(σi)) is
in the homotopy class ofσ. To keep track of the marking on Σ, we do the gluing on
the level of theZ–covers ofS, S0 corresponding toh∞, h0∞, that is, we actually glue
the strips H∞ and H∞0 . See Section 2.2.3 for a detailed discussion of the marking.
As shown in Figure 2.3, the deleted punctured disks are on opposite sides of h∞
in S and h0∞ in S0. Thus we first need to reverse the direction in one of the two
stripsH∞ and H∞0 . Set
J = −i 0 0 i ! , Tµ= 1 µ 0 1 ! . (2.1)
We reverse the direction in H∞ by applying the map J(z) = −z to H. We then
glue H∞ to H∞0 by identifying z ∈ H∞ to z0 = TµJ(z) ∈ H∞0 . Since both J
and Tµ commute with the conjugacy classes (with respect to the action of Γ) of
the holonomies z 7→ z + 2 and z0 7→ z0 + 2 of the curves h∞, h0∞, this identi-
fication descends to a well defined identification of A∞ with A0
∞, in which the
‘outer’ boundary ζ({z ∈ H|=z = =µ2+ν}) of A∞ is identified to the ‘inner’ bound-
ary ζ0({z0 ∈ H|=z0 = =µ2−ν}) of A0∞. In particular, h∞ is glued to h0∞ reversing
orientation.
Now we treat the general case in which P and P0 meet along punctures with
arbitrary labels, 0 ∈ {0,1,∞}. As above, let ∆0 ⊂H be the ideal ‘white’ triangle
with vertices 0,1,∞. Notice that there is a unique orientation preserving symmetry
Ω of ∆0 which sends the vertex∈ {0,1,∞}to∞:
Ω0 = 1 −1 1 0 ! , Ω1 = 0 −1 1 −1 ! , Ω∞= Id = 1 0 0 1 ! . (2.2)
Leth be the loop onP which lifts to the horocycle
h,H = Ω−1({z∈H|=z= =
µ
2 })
onH, so thath is a loop round ∂(P) inP. Also letH=H(µ, ν) be the region in
Hdefined by H= Ω−1({z∈H|= µ−ν 2 <=z < = µ+ν 2 }) = Ω−1(H∞).
The stripH projects to annular neighbourhoodsA =A(µ) ofh⊂P. Define h00,
H00 andA00 in a similar way.
To do the gluing, first move and 0 to∞ using the maps Ω and Ω0 and then
proceed as before. Thus the gluing identifiesz∈H toz0∈H0 by the formula
Ω0(z0) =Tµ◦J(Ω(z)), (2.3)
see Figure 2.3.
Finally, we carry out the above construction for each pants curveσi∈ PC. To do
this, we need to ensure that the annuli corresponding to the three different punctures of a given Pj are disjoint. (Note that, for example, the condition =µi >2, for all
i = 1, . . . , ξ, ensures that the three curves h0, h1 and h∞ associated to the three
punctures ofPj are disjoint inP.) Under this condition, we can clearly chooseν >0
so that their annular neighbourhoodsA,j,+ ⊂Sj are disjoint. In what follows, we
shall usually write h and H forh and H provided the subscript is clear from the
context.
Hence, we are defining the quotient Sµ/ ∼ (homeomorphic to Σ), where Sµ = S1t. . .tSkis the disjoint union of the truncated surfacesSj ⊂Pdefined above and
the equivalence relation∼ is given by the attaching maps along the annuli A(σi).
Let Π: Sµ −→ Sµ/ ∼ be the quotient map. In Section 2.2.2 we will see that
this quotient is endowed with a complex projective structure Σ(µ) coming from our
gluing construction.
Remark 2.2.1. In the above construction, we glued a curve exiting from the white tri-
angles ∆0(P) to one entering the white triangle ∆0(P0), where we denote ∆0(Pj) =
ˆ Φ−1
j (∆0) ⊂ Pj and ∆1(Pj) = ˆΦ−j1(∆1) ⊂ Pj the white and the black hexagons
in Pj, respectively. On the other hand, suppose we wanted to glue the two black
triangles ∆1(P) and ∆1(P0). This can be achieved, when gluing∂∞(P) to ∂∞(P0),
by replacing the parameter µ with µ−2. However, following our recipe, it is not
because the black triangle is to the right of both the outgoing and incoming lines, while the white triangle is to the left.
Remark 2.2.2. In our construction we require the parameterµto be inHξ, but the
gluing construction makes sense also for gluing parametersµ∈Lξ, whereL={z∈
C|=z <0}. In fact, if in Theorem 2.2.5 we substitute the hypothesis µ∈Hξ with
µ∈Lξ, we get 3–manifolds where the pants curvesσ
1, . . . , σξ are pinched in the top
components Ω+, rather than in the bottom one Ω−.