Top terms of polynomial traces in Kra's plumbing construction

0.81 MB·English

by  Sara Maloni

TOP TERMS OF POLYNOMIAL TRACES IN KRA’S
PLUMBING CONSTRUCTION.
SARA MALONI AND CAROLINE SERIES
0
1
0 Abstract. Let Σ be a surface of negative Euler characteristic
2 together with a pants decomposition . Kra’s plumbing construc-
n tionendowsΣwithaprojectivestrucPtureasfollows. Replaceeach
a pair of pants by a triply punctured sphere and glue, or ‘plumb’,
J adjacent pants by gluing punctured disk neighbourhoods of the

4 punctures. The gluing across the ith pants curve is defined by a
1 complex parameter τ C. The associated holonomy representa-
i
tion ρ : π (Σ) PS∈L(2,C) gives a projective structure on Σ
] 1 −→
N whichdependsholomorphicallyontheτ . Inparticular, thetraces
i
G of all elements ρ(γ),γ π1(Σ), are polynomials in the τi.
∈
Generalising results proved in [5, 16] for the once and twice
.
h punctured torus respectively, we prove a formula giving a simple
t
a linearrelationshipbetweenthecoefficientsofthetoptermsofρ(γ),
m as polynomials in the τ , and the Dehn-Thurston coordinates of γ
i
[ relative to .
P
This will be applied elsewhere to give a formula for the asymp-
1
totic directions of pleating rays in the Maskit embedding of Σ as
v
5 the bending measure tends to zero, see [9].
1
MSC classification: 30F40, 57M50
5
2
.
1
0
1. Introduction
0
1
: Let Σ be a surface of negative Euler characteristic together with a
v
pants decomposition . Kra’s plumbing construction endows Σ with a
i
X P
projective structure as follows. Replace each pair of pants by a triply
r
a punctured sphere and glue, or ‘plumb’, adjacent pants by gluing punc-
tured disk neighbourhoods of the punctures. The gluing across the ith
pants curve is defined by a complex parameter τ C. More precisely,
i
∈
zw = τ where z,w are standard holomophic coordinates on punctured
i
disk neighbourhoods of the two punctures. The associated holonomy
representation ρ : π (Σ) PSL(2,C) gives a projective structure
1
−→
on Σ which depends holomorphically on the τ , and in which the pants
i
curvesthemselvesareautomaticallyparabolic. Inparticular, thetraces
of all elements ρ(γ),γ π (Σ), are polynomials in the τ .
1 i
∈
Date: January 15, 2010.
1
2 SARA MALONI AND CAROLINE SERIES
The main result of this paper is a very simple relationship between
the coefficients of the top terms of ρ(γ), as polynomials in the τ , and
i
the Dehn-Thurston coordinates of γ relative to . This generalises
P
resultsof[5,16]fortheonceandthetwicepuncturedtorusrespectively.
Our formula is as follows. Let denote the set of homotopy classes
S
of multiple loops on Σ, and let the pants curves defining be σ ,i =
i
P
1,...,ξ. (For brevity we usually refer to elements of as curves,
S
see Section 2.) The Dehn-Thurston coordinates of γ are i(γ) =
(q ,p ),i = 1,...,ξ, where q = i(γ,σ ) N 0 is∈tShe geometric
i i i i
intersection number between γ and σ an∈d p ∪ {Z}is the twist of γ
i i
∈
about σ . We prove:
i
Theorem A. Let γ be a connected simple closed curve on Σ, not par-
allel to any of the pants curves σ . Then Trρ(γ) is a polynomial in
i
τ , ,τ whose top terms are given by:
1 ξ
···
Trρ(γ) = iq2h(cid:16)τ + (p1 −q1)(cid:17)q1 (cid:16)τ + (pξ −qξ)(cid:17)qξ +R,
1 ξ
± q ··· q
1 ξ
(cid:32) (cid:33)
ξ
(cid:88)
= iq2h τq1 τqξ + (p q )τq1 τqi−1 τqξ +R
± 1 ··· ξ i − i 1 ··· i ··· ξ
i=1
where
q = (cid:80)ξ q > 0;
• i=1 i
R represents terms with total degree in τ τ at most q 2
1 ξ
• ··· −
and of degree at most q in the variable τ ;
i i
h = h(γ) is the total number of scc-arcs in the standard repre-
•
sentation of γ relative to , see below.
P
If q = 0, then γ = σ for some i, ρ(γ) is parabolic, and Trρ(γ) = 2.
i
±
The non-negative integer h = h(γ) is defined as follows. The curve
γ is first arranged to intersect each pants curve minimally. In this
position, it intersects a pair of pants P in a number of arcs joining
boundary loops of P. We call one of these an scc-arc (short for same-
(boundary)-component-connector, called an archetype in Penner [14])
if it joins one boundary component to itself, and denote by h the total
number of scc-arcs, taken over all pants in .
P
The precise definition of the twist coordinates p in Theorem A re-
i
quires some care; we use essentially the standard definition implied
in [3] and explained in detail in [17] (called here the DT-twist, see
Section 3.1), although for the proof we find useful the form given by
Penner [14] (called here the P-twist and denoted pˆ, see Section 3.2).
i
TOP TERMS OF POLYNOMIAL TRACES 3
We remark that the formula in Theorem A could of course be made
neater by replacing the parameter τ by τ 1; we use τ to be in accor-
−
dance with the conventions of [5, 16], see also Section 4.2.
We believe the formula in Theorem A noteworthy in its own right.
However the main motivation for this work was the following. If the
representation ρ constructed in the above manner is free and discrete,
then the resulting hyperbolic 3-manifold M = H3/ρ(π (Σ)) lies on the
1
boundary of quasifuchsian space (Σ). One end of M consists of a
QF
union of triply punctured spheres obtained by pinching in Σ the curves
σ defining . Suppose that, in addition, ρ(π (Σ)) is geometrically
i 1
P
finite and that the other end Ω+/ρ(π (Σ)) of M is a Riemann sur-
1
face homeomorphic to Σ. Since the triply punctured spheres are rigid,
it follows from Ahlfors-Bers’ measurable Riemann mapping theorem
that the Riemann surface structure of Ω+/ρ(π (Σ)) runs over the Te-
1
ichmu¨ller space (Σ) of Σ. The image of the space of all such groups
T
in the character variety of Σ is called the Maskit embedding of (Σ).
R T
In [5, 16], special cases of the trace formula were important in con-
structing a computational method of locating the image of (Σ) in
M T
. In those papers we defined a pleating ray to be a line in along
R R
which the projective class of the bending measure was kept constant.
The trace formulae enabled us to find the asymptotic directions of
pleating rays in as the bending measure tends to zero. Theorem A
M
allows the extension of these results to the general case, see [9].
The plan of this paper is as follows. After establishing preliminaries
in Section 2, in Section 3 we review the Dehn-Thurston coordinates
and, in particular, the definition of twists. In Section 4 we discuss the
gluing construction which leads to the family of projective structures
andtheirholonomyrepresentation. InSection5weexplainindetailthe
holonomy representation in various special cases, starting with arcs in
asinglepairofpantsandgoingontotheoneholedtorusandfourholed
sphere. Finally,inSection6,wemakeexplicitthegeneralcombinatorial
pattern of matrix products obtained in the holonomy, and use this to
give an inductive proof of Theorem A.
2. Background and Notation
Suppose given a surface Σ = Σb of finite type and negative Euler
g
characteristic, and choose a maximal set = σ ,...,σ of homo-
1 ξ
PC { }
topically distinct and non-boundary parallel loops in Σ called pants
curves, where ξ = ξ(Σ) = 3g 3 + b is the complexity of the sur-
−
face. These connected curves split the surface into k = 2g 2 + b
−
three-holed spheres P ,...,P , called pairs of pants. (Note that the
1 k
4 SARA MALONI AND CAROLINE SERIES
boundary of P may include punctures of Σ.) We refer to both the set
i
= P ,...,P , and the set , as a pants decomposition of Σ.
1 k
P { } PC
We take P to be a closed three-holed sphere whose interior Int(P ) is
i i
embedded in Σ; the closure of Int(P ) fails to be embedded precisely in
i
thecaseinwhichtwoofitsboundarycurvesareidentifiedinΣ, forming
an embedded one-holed torus Σ . Thus each pants curve σ = σ is
1,1 i
the common boundary of one or two pants whose union we refer to
as the modular surface associated to σ, denoted M(σ). If the closure
of Int(P ) fails to be embedded then M(σ) is a one-holed torus Σ ,
i 1,1
otherwise it is a four-holed sphere Σ .
0,4
Any hyperbolic pair of pants P is made by gluing two right angled
hexagons along three alternate edges which we call its seams. In much
of what follows, it will be convenient to designate one of these hexagons
as ‘white’ and one as ‘black’. A properly embedded arc in P, that
is, an arc with its endpoints on ∂P, is called scc (same component
connector) if it has both its endpoints on the same component of ∂P
and dcc (different component connector) otherwise.
Let = (Σ) denote the set of free homotopy classes of connected
0 0
S S
closed simple non-boundary parallel loops on Σ, and let = (Σ)
S S
be the set of curves on Σ, that is, the set of finite unions of non-
homotopic curves in . For simplicity we usually refer to elements of
0
S
as ‘curves’ rather than ‘multi-curves’, in other words, a curve is not
S
required to be connected. The geometric intersection number i(α,β)
between α,β is the least number of intersections between curves
∈ S
representing the two homotopy classes, that is
i(α,β) = min a b .
a∈α,b∈β| ∩ |
2.0.1. Conventionondualcurves. Weshallneedtoconsiderdualcurves
to σ , that is, curves which intersect σ minimally and which are
i i
∈ PC
completely contained in M(σ ), the union of the pants P,P(cid:48) adjacent
i
to σ . The intersection number of such a connected curve with σ is
i i
1 if M(σ ) a one-holed torus and 2 otherwise. In the first case, the
i
curve is made by identifying the endpoints of a single dcc-arc in the
pair of pants adjacent to σ and, in the second, it is the union of two
i
scc-arcs, one in each of the two pants whose union is M(σ ). We adopt
i
a useful convention introduced in [17] which simplifies the formulae in
such a way as to avoid the need to distinguish between these two cases.
Namely, for those σ for which M(σ ) is Σ , we define the dual curve
i i 1,1
D to be two parallel copies of the connected curve intersecting
i
∈ S
σ once, while if M(σ ) is Σ we take a single copy. In this way we
i i 0,4
TOP TERMS OF POLYNOMIAL TRACES 5
always have, by definition, i(σ ,D ) = 2, where i(α,β) is the geometric
i i
intersection number as above.
A marking on Σ is the specification of a fixed base surface Σ , to-
0
gether with a homeomorphism Ψ : Σ Σ. Markings can be defined
0
−→
in various equivalent ways, for example by specifying the choice of dual
curves, see Section 3.1 below.
2.0.2. Convention on twists. Our convention will always be to measure
twists to the right as positive. We denote by Tw (γ) the right Dehn
σ
twist of the curve γ about the curve σ.
3. Dehn-Thurston coordinates
Suppose we are given a surface Σ together with a pants decomposi-
tion as above. Let γ and for i = 1,...,ξ, let q = i(γ,σ ) Z .
i i (cid:62)0
P ∈ S ∈
Notice that if σ ,σ ,σ are pants curves which together bound a pair
i1 i2 i3
of pants whose interior is embedded in Σ, then the sum q +q +q
i1 i2 i3
of the corresponding intersection numbers is even. The q = q (γ) are
i i
sometimes called the length parameters of γ.
To define the twist parameter tw = tw (γ) Z of γ about σ , we
i i i
∈
first have to fix a marking on Σ, for example by fixing a specific choice
of dual curve D to each pants curve σ , see Section 3.1 below. Then,
i i
after isotoping γ into a well-defined standard position relative to
P
and to the marking, the twist tw is the signed number of times that γ
i
intersects a short arc transverse to σ . We make the convention that if
i
i(γ,σ ) = 0, then tw (γ) (cid:62) 0 is the number of components in γ freely
i i
homotopic to σ .
i
There are various ways of defining the standard position of γ, leading
to differing definitions of the twist. The parameter tw (γ) = p (γ)
i i
which occurs in the statement of Theorem A is the one defined by
Dylan Thurston [17], however in the proof of the formula we will find
it convenient to use a slightly different definition tw (γ) = pˆ(γ) given
i i
by Penner [14]. Both of these definitions are explained in detail below,
as is the precise relationship between them. With either definition,
a classical theorem of Dehn [2], see also [14] (p.12), asserts that the
length and twist parameters uniquely determine γ:
Theorem 3.1. (Dehn’s theorem)
The map Ψ : (Σ) Zξ Zξ which sends γ (Σ) to
(cid:62)0
S −→ × ∈ S
(q (γ),...,q (γ);tw (γ),...,tw (γ)) is an injection. The point
1 ξ 1 ξ
(q ,...,q ,tw ,...,tw ) is in the image of Ψ (and hence corresponds
1 ξ 1 ξ
to a curve) if and only if:
(i) if q = 0, then tw (cid:62) 0, for each i = 1,...,ξ.
i i
6 SARA MALONI AND CAROLINE SERIES
(ii) if σ ,σ ,σ are pants curves which together bound a pair of
i1 i2 i3
pants whose interior is embedded in Σ, then the sum q +q +q
i1 i2 i3
of the corresponding intersection numbers is even.
We remark that as a special case of (ii), the intersection number
with a pants curve which bounds an embedded once-punctured torus
or twice-punctured disk in Σ is even.
One can think of this theorem in the following way. Suppose given
a curve γ , whose length parameters q (γ) necessarily satisfy the
i
∈ S
parity condition (ii). Then the q (γ) uniquely determine γ P for
i j
∩
each pair of pants P , j = 1,...,k, in accordance with the possible
j
arrangements of arcs in a pair of pants, see for example [14]. Now
given two pants adjacent along the curve σ , we have q (γ) points of
i i
intersection coming from each side and we have only to decide how to
match them together to recover γ. The matching takes place in the
cyclic cover of an annular neighbourhood of σ . The twist parameter
i
tw (γ)specifieswhichoftheZpossiblechoicesisusedforthematching.
i
3.1. The DT-twist. In [17], Dylan Thurston gives a careful definition
of the twist tw (γ) = p (γ) of γ which is essentially the ‘folk’
i i
∈ S
definition and the same as that implied in [3]. He observes that this
definition has a nice intrinsic characterisation, see Section 3.1.4 below.
Furthermore, it turns out to be the correct definition for our formula
in Theorem A.
3.1.1. The marking. Given the pants decomposition of Σ, we note,
P
following [17], that we can fix a marking on Σ in three equivalent ways.
These are:
(a) a reversing map: an orientation-reversing map R : Σ Σ so
−→
that for each i = 1,...,ξ we have R(σ ) = σ ;
i i
(b) ahexagonal decomposition: thiscanbedefinedbyacurvewhich
meets each pants curve twice, decomposing each pair of pants
into two hexagons;
(c) dual curves: for each i, a curve D so that i(D ,σ ) = 2δ .
i i j ij
The characterisations (a) and (b) are most easily understood in con-
nection with a particular choice of hyperbolic metric on Σ. Recall that
a pair of pants P is the union of two right angle hexagons glued along
its seams. There is an orientation reversing symmetry of P which fixes
the seams. The endpoints of exactly two seams meet each component
of the boundary ∂P. Now let Σ be a hyperbolic surface formed by glu-
0
ing pants P ,...,P in such a way that the seams are exactly matched
1 k
TOP TERMS OF POLYNOMIAL TRACES 7
on either side of each common boundary curve σ . In this case the ex-
i
istence of the orientation reversing map R as in (a) and the hexagonal
decomposition as in (b) are clear and are clearly equivalent.
If the modular surface associated to σ is made up of two distinct
i
pants P,P(cid:48), then, as explained above, the dual curve D to σ is ob-
i i
tained by gluing the two scc-arcs in P and in P(cid:48) which run from σ
i
to itself. Each arc meets σ orthogonally so that in the metric Σ the
i 0
two endpoints on each side of σ are exactly matched by the gluing.
i
If the modular surface is a single pair of pants P, then the dual curve
is obtained by gluing the single dcc-arc in P which runs from σ to
i
itself. Once again both ends of this arc meet σ orthogonally and in
i
the metric Σ are exactly matched by the gluing. In this case, follow-
0
ing the convention explained in Section 2, we take the dual curve D
i
to be two parallel copies of the loop just described. Thus in all cases
i(D ,σ ) = 2δ and furthermore the curves D are fixed by R.
i j ij i
AgeneralsurfaceΣcanbeobtainedfromΣ byperformingaFenchel-
0
Nielsen twist about each σ . Namely, if A = σ [0,1] is an annulus
i i i
×
around σ and if we parameterise σ as s σ (s) Σ for s [0,1),
i i i
(cid:55)→ ∈ ∈
then the distance t twist, denoted FN : Σ Σ, maps A to itself
t 0 i
−→
by (σ (s),θ) (σ (s + θt),θ) and is the identity elsewhere. Clearly
i i
(cid:55)→
FN induces a reversing map, a hexagonal decomposition, and dual
t
curves on the surface FN (Σ), showing that each of (a), (b) and (c)
t
equivalently define a marking on an arbitrary surface Σ.
3.1.2. The twist. Having defined the marking, we can now define the
twist p (γ) for any γ . Arrange, as above, the dual curves D to
i i
∈ S
be fixed by R, so that, in particular, if σ is the boundary of a single
i
pair of pants P, then the two parallel components of the curve D are
i
contained one in each of the two hexagons making up P. For each
i = 1,...,ξ, choose a small annular neighbourhood A of σ , in such
i i
a way that the complement Σ ξ Int(A ) of the interiors of these
− ∪i=1 i
ˆ ˆ
annuli in Σ are pants P ,...,P . Arrange γ so that its intersection
1 k
ˆ
with each P is fixed by R and so that it is transverse to D . Also push
i i
any component of γ parallel to any σ into A .
i i
If q = i(γ,σ ) = 0, define p 0 to be the number of components of
i i i
≥
γ parallel to σ . Otherwise, q = i(γ,σ ) > 0. In this case, orient both
i i i
γ A and D A to run consistently from one boundary component
i i i
∩ ∩
of A to the other. (If M(σ ) is Σ , then the two arcs of D A will
i i 0,4 i i
∩
be oriented in opposite directions relative to the connected curve D .)
i
Then define
p =ˆi(γ A ,D A ),
i i i i
∩ ∩
8 SARA MALONI AND CAROLINE SERIES
where ˆi(α,β) is the algebraic intersection number between the curves
α and β, namely the sum of the indices of the intersection points of α
and β, where an intersection point is of index +1 when the orientation
of the intersection agrees with the orientation of Σ and 1 otherwise.
−
Note that this definition is independent of both the choice of the
orientationsofγ A andD A ,andofthechoiceofthearrangementof
i i i
∩ ∩
γ in the pants adjacent to σ . Also note that, following the convention
i
about dual curves in Section 2.0.1, p is always even. Two simple
i
examples are illustrated in Figures 1 and 2.
3.1.3. An alternative definition. The twist p can also be described in
i
a slightly different way as follows. Lift A to its Z-cover which is an
i
infinite strip H. As shown in Figures 1 and 2, the lifts of D A are
i i
∩
arcs joining the two boundaries ∂ H and ∂ H of H. They are equally
0 1
spaced like rungs of a ladder in such a way that there are exactly two
lifts in any period of the translation corresponding to σ . Any arc of
i
γ enters H on one side and leaves on the other. Fix such a rung D
∗
say and number the strands of γ meeting ∂ H in order as X ,n Z,
0 n
∈
where X is the first arc to the right of D and n increases moving to
0 ∗
the right along ∂ H, relative to the orientation of the incoming strand
0
of γ. Label the endpoints of γ on ∂ H by X(cid:48),n Z correspondingly,
1 n ∈
as shown in Figure 1. Since γ is simple, if X is matched to X(cid:48), then
0 r
X is matched to X(cid:48) for all n Z. Then it is not hard to see that
n n+r ∈
r = p /2.
i
X X X X X
2 1 0 1 2
− −
∂ H
0
γ D γ D γ D γ D γ
! !
∂ H
1
X X X X X
2! 1! 0! !1 !2
− −
Figure 1. A curve γ with p (γ) = 0. The arcs D,D(cid:48)
i
together project to the dual curve D .
i
3.1.4. Intrinsic characterisation. The intrinsic characterisation of the
twist in [17] uses the Luo product α β of curves α,β on an oriented
· ∈ S
surface Σ. This is defined as follows [8, 17]:
If a b = , then α β = α β .
• ∩ ∅ · ∪ ∈ S
Otherwise, arrange α and β in minimal position, that is, such
•
that i(α β) = α β . In a neighbourhood of each intersection
∩ | ∩ |
TOP TERMS OF POLYNOMIAL TRACES 9
X X X X X
2 1 0 1 2
− −
∂ H
0
γ D γ D γ D γ D
! !
∂ H
1
X X X X X
2! 1! 0! !1 !2
− −
Figure 2. A curve γ with p (γ) = 2.
i
−
point x α β, replace α β by the union of the two arcs
j
∈ ∩ ∪
which turn left from α to β relative to the orientation of Σ, see
Figure 3. (In [8] this is called the resolution of α β from α
∪
to β at x .) Then α β is the curve made up from α β away
j
· ∪
from the points x , and the replacement arcs near each x .
j j
β β
xj
α α
Figure 3. The Luo product: the resolution of α β at x .
j
∪
Proposition 3.2 ([17] Definition 15). The function p : (Σ) Z is
i
S −→
the unique function such that for all γ :
∈ S
(i) p (σ γ) = p (γ)+2δ ;
i j i ij
·
(ii) p depends only on the restriction of γ to the pants adjacent to
i
σ ;
i
(iii) p (R(γ)) = p (γ), where R is the orientation reversing invo-
i i
−
lution of Σ defined above.
We call p (γ) the DT-twist parameter of γ about σ . Property (i)
i i
fixes our convention noted above that the right twist is taken positive.
Notice that p (D ) = 0. We also observe:
i i
Proposition 3.3. Let γ . Then
∈ S
p (Tw (γ)) = p (γ)+2q .
i σi i i
10 SARA MALONI AND CAROLINE SERIES
3.1.5. Relation to [3]. In [3], a curve γ is parameterized by three
∈ S
non-negative integers (m ,s ,t ). These are defined as the intersection
i i i
numbers of γ with the three curves K , K(cid:48) and K(cid:48)(cid:48), namely the pants
i i i
curve σ , its dual curve D , and Tw (D ), the right Dehn twist of D
i i σi i i
about σ , see Figure 4 on p. 62 in [3]. In particular:
i
m (γ) = i(γ,K ) = i(γ,σ ) = q (γ)
i i i i
•
p (γ)
s (γ) = i(γ,K(cid:48)) = i(γ,D ) = | i |
• i i i 2
t (γ) = i(γ,K(cid:48)(cid:48)) = i(γ,Tw (D )) = (cid:12)(cid:12)pi(γ) q (γ)(cid:12)(cid:12)
• i i σi i 2 − i
As proved in [3], the three numbers m ,s and t satisfy one of the
i i i
three relations m = s +t ; s = m +t ; t = m +s . As it is easily
i i i i i i i i i
verified by a case-by-case analysis, we have:
Theorem 3.4. Each triple (m ,s ,t ) uniquely determines and is deter-
i i i
mined by the parameters q and p . In fact, q = m and p = 2sign(p )s
i i i i i i i
where
(cid:40)
+1 if m = s +t or s = m +t ;
i i i i i i
sign(p ) =
i
1 if t = m +s .
i i i
−
Proof. If sign(p ) = 1, then p /2 q 0. So
i i i
− − ≤
p p p
i i i
t = q = ( q ) = | | +q = s +m ;
i i i i i i
|2 − | − 2 − 2
If sign(p ) = +1, then:
i
p p p
(1) if i (cid:54) q , then t = i q = q | i| = m s ;
i i i i i i
2 |2 − | − 2 −
p p p
(2) if q (cid:54) i, then t = i q = | i| q = s m ,
i i i i i i
2 |2 − | 2 − −
as we wanted to prove. (cid:3)
3.2. The P-Twist. WenowsummarisePenner’sdefinitionofthetwist
parameter following [14] Section 1.2. Instead of arranging the arcs of γ
transversetoσ symmetricallywithrespecttotheinvolutionR, wenow
i
arrange them to cross σ through a short closed arc w σ . There is
i i i
⊂
some choice to be made in how we do this, which leads to the difference
with the definition of the previous section. It is convenient to think
of w as contained in the two ‘front’ hexagons of the pants P and P(cid:48)
i
glued along σ , which we will also refer to as the ‘white’ hexagons.
i
Precisely, foreachpantscurveσ , fixashortclosedarcw σ ,
i i i
∈ PC ⊂
which we take to be symmetrically placed in the white hexagon of one
of the adjacent pants P, midway between the two seams which meet
σ ∂P. For each σ , fix an annular neighbourhood A and extend
i i i
⊂
w into a ‘rectangle’ R A with one edge on each component of ∂A
i i i i
⊂