Paths ϑ between adjacent pants

In this section we describe the holonomy of the path ϑ = ϑ(P, P0) between two

pairs of pants, sayP andP0, with initial point b=b(P)⊂∆0(P) and ending point b0 =b(P0) _⊂∆0(P0). This path, together with the pathsγ and υ, defines a set of

generators for the fundamental groupoidπ1,2k(Σ, B).

If ∂P is glued to ∂0P0, then there is an obvious path ϑ=ϑ(P, P0;, 0;σ) on Σ

from bto b0 crossing the pants curve σ=σi. As discussed in Section 2.2.2, we will

identify Σ with the quotient S_µ/∼ defined in Section 2.2.1. In particular, P ∪P0

is identified withStS0/∼, where S, S0 ⊂P are the truncated surfaces defined in Section 2.2.1 and where ∼ is the equivalence relation given by the attaching maps

along the annuli_A⊂S and A0⊂S0.

We can describeϑby defining, first, a path in_H and a path in_H0, which project

to a path in S_tS0/_∼, see Section 2.2.1 and 2.2.2.

Recall that

H_∞=H_∞(µ, ν) =_{z_∈H|=µ−ν

2 <=z < =

µ+ν

2 } ⊂H,

whereµ=µi ∈His the gluing parameter associated to the pants curve σ=σi and

1 1 0 F(b0) B0 b0 µ B0=F(B0) µ 1 2=F 21 F ₁1

Figure 2.8: The pathδ1 betweenb0 andB0, the pathδ2 =F(δ20)−1 betweenB0 and µ−B0 and the path F(δ₁−1) betweenµ−B0 and F(b0), whereF =J−1T_µ−1, in the

caseH =H0 =H_∞.

defined in Section 2.2.1. Recall also thatS⊂Pis the surface_Pwith the projection of the horocyclic neighbourhood Ω−1

{z_∈H|=z_≥ =µ₂+ν_}of deleted.

Consider in H the straight lines δ1 = δ1,∞ between b0 = 1+i

√

3

2 and B0 =

1+i(=µ−2ν)

2 and δ2 = δ2,∞ between B0 and µ−B0 = µ−

1+i(=µ−2ν) 2

; see Fig- ure 2.8. If we concatenate these two paths, we have a pathδ_∞=δ1·δ2 inHwhich

is contained in the complement of the open horoball {z _∈ H_|=z > =µ₂+ν_} around

∞. Similarly, let alsoδ = Ω−1(δ∞) andδi, = Ω−1(δi) with= 0,1,∞andi= 1,2.

Define the pathδ00 and δ_i,0 0 inH0 in a similar way, where0 = 0,1,∞ andi= 1,2.

Let ζ: H_−→ H/Γ and let ζ0: H_−→ H/Γ. Then the path δ descends to a path

δ,S =ζ(δ)⊂S ⊂P and the path δ0 descends to a path δ00_,S0 =ζ0(δ00) ⊂S0 ⊂P0.

Let alsoδi,,S =ζ(δi,) ⊂S ⊂P and δ0_i,0_,S0 =ζ0(δ0_i,0) ⊂S0 ⊂P0 with , 0 = 0,1,∞

andi= 1,2.

Note that in the stripHtH0/∼, whereH=H and H0 =H0 we have that the

projection of the pathδ2and of the path (δ02)−1(that is,δ20 with reversed orientation)

coincide, see Section 2.2.2 for understanding the identification_∼ in detail. In fact, we have that

as you can see from the fact that Ω−1

(B0) = Ω−01(µ−B₀) and Ω−1(µ−B₀) = Ω−01(B₀) in HtH0/∼.

This tells us thatδ,S ∪(δ00_,S0)−1 defines a path inStS0/∼.

With these definitions and using the notation of Section 2.2, we can see that

ϑ(P, P0;, 0;σ) is defined as the following union

ϑ(P, P0;, 0;σ) =δ,S∪(δ00_,S0)−1⊂StS0/∼.

Unless needed for clarity, we refer to all these paths asϑ(P, P0) or ϑ(P, P0;, 0).

For finding the holonomy and the developing image of ϑ(P, P0_{), we need to glue}

δ withF ◦(δ00)−1, where F = Ω−1J−1T_µ−1Ω0 is the transition function defined in

Section 2.2.2. See Sections 2.1.3.a and 2.1.3.b for the definition of the developing map and of the groupoid holonomy map. Figure 2.8 shows the developing im- age Devµ(ϑ(P, P0;∞,∞;σi)) of ϑ(P, P0;, 0;σi). Now referring to the gluing equa-

tion (2.3) and to the description of the groupoid holonomy map of Section 2.1.3.b, we see that the holonomy ofϑ(P, P0) is given by the following formula:

ρµ ϑ(P, P0;, 0;σ)

= Θ−→0 = Ω−1J−1T_µ−1Ω0, (2.4)

since the path ϑ(P, P0;, 0;σ) can be covered with only two charts and the only

transition map is the map Ω−1

J−1Tµ−1Ω0, whereµ=µ_i.

Remark 2.3.5. Note that if you look at Figure 2.8 as a picture of two superimposed

copies ofH, sayHandH0_{, as we did in Figure 2.4, then the point B}

0 ∈H, which is

the starting point ofδ2, and the point µ−B0 =F(B0)∈H0, which is the starting

point ofF◦δ₂0 (whereF = Ω−1J−1T_µ−1Ω0), project to the same point inStS0/∼

under the two different chartsψ1 _{and ψ}2 _{defined in Section 2.2.2 exactly when the}

surfacesS andS0are glued by a Fenchel–Nielsen twistT wσi,<µi−1. This agrees with the description of the marking we made in Section 2.2.3. In fact, in that section we first defined the marking for the surface Σ(µ0_{), whereµ}0 _{= (µ}0

1, . . . , µ0ξ) is defined by

<µ0_i = 1, for alli= 1, . . . , ξ, where the seams ofSand the seams ofS0match. Then,

for defining the marking on the general surface Σ(µ), we use a Fenchel–Nielsen twist T wσi,<µi−1 on the annulus aroundσi ∈ PC, for all i= 1, . . . , ξ.

As already noted in Lemma 2.2.3, the gluing parametersµare independent of the

direction of travel (fromP toP0 or vice versa). From (2.4) we have

ρµ ϑ(P0, P;0, )

= Ω−1

so that

ρµ(ϑ(P0, P;0, )−1) = Ω−1TµJΩ0.

Using the identitiesJ−1_=−J,T−1

µ =T−µ and TµJ =JT−µ this gives

ρµ ϑ(P0, P;0, )−1

=−ρµ λ(P0, P;0, )

−1

, (2.5)

as one would expect. That fact will be particularly important for our proof in Appendix B.