Bubbling complex projective structures with quasi-Fuchsian holonomy

Abstract

For a given quasi-Fuchsian representation $\rho:\pi_1(S)\to$ PSL$_2\mathbb{C}$ of the fundamental group of a surface $S$ of genus $g\geq 2$, we prove that a generic branched complex projective structure on $S$ with holonomy $\rho$ and two branch points is obtained by bubbling some unbranched structure on $S$ with the same holonomy.

Full text

arXiv:1701.03524v1 [math.GT] 12 Jan 2017
BUBBLING COMPLEX PROJECTIVE STRUCTURES
WITH QUASI-FUCHSIAN HOLONOMY
LORENZO RUFFONI
Abstract. For a given quasi-Fuchsian representation ρ:π1(S)→PSL2Cof
the fundamental group of a surface Sof genus g≥2, we prove that a generic
branched complex projective structure on Swith holonomy ρand two branch
points is obtained by bubbling some unbranched structure on Swith the same
holonomy.
Contents
1. Introduction 1
2. Branched complex projective structures 3
2.1. Grafting 5
2.2. Bubbling 6
2.3. Movements of branch points 7
2.4. Injectively developed isotopies 9
3. Geometric decomposition in quasi-Fuchsian holonomy 12
3.1. Locating branch points 15
3.2. Classification of components for BPSs with k= 2 branch points 18
4. BM-configurations 22
4.1. Standard BM-configurations 23
4.2. Taming developed images and avatars 25
4.3. Visible BM-configurations 28
5. Bubbles everywhere 31
5.1. Walking around the moduli space with bubblings 38
References 39
1. Introduction
A complex projective structure on a surface Sis a geometric structure locally
modelled on the geometry of the Riemann sphere CP1with its group of holo-
morphic automorphisms PSL2C. Since H2,E2and S2admit models in CP1, these
structures generalise the classical setting of constant curvature geometries; in par-
ticular, structures with (quasi-)Fuchsian holonomy play a central role in the theory
of (simultaneous) uniformization of Riemann surfaces of genus g≥2 (see [10],[1]).
If ρ:π1(S)→PSL2Cis a quasi-Fuchsian representation, the quotient of the
domain of discontinuity of ρby the image of ρis endowed with a natural complex
projective structure σρwith holonomy ρ, namely a hyperbolic structure. A natural
2010 Mathematics Subject Classification. 57M50, 20H10, 14H15.
1

2 LORENZO RUFFONI
problem is to try to obtain every other projective structure with holonomy ρfrom
this hyperbolic structure via some elementary geometric surgeries on it. The main
result in this direction is due to Goldman, who proved in [6] that any complex
projective structure with quasi-Fuchsian holonomy is obtained from the hyperbolic
structure σρvia grafting, i.e. by replacing some disjoint simple closed geodesic with
Hofp annuli.
We are interested in the analogous problem for branched complex projective
structures: these are a generalisation (introduced in [7]) of complex pro jective
structures in which we allow cone points of angle 2πk for k∈N. An easy way
to introduce branch points is to perform a bubbling, i.e. to replace a simple arc
with a full copy of the model space CP1(see 2.2 below for the precise definition
of this surgery). The following question was posed by Gallo-Kapovich-Marden as
Problem 12.1.2 in [5]: given two branched complex projective structures with the
same holonomy, is it possible to pass from one to the other using the operations of
grafting, degrafting, bubbling and debubbling?
Calsamiglia-Deroin-Francaviglia provided in [2] a positive answer in the case
of quasi-Fuchsian holonomy, if an additional surgery is allowed, which is known
as movement of branch points, and is a form of Schiffer variation around branch
points. The main result of this paper is the following (see 5.9 below for the precise
statement):
Theorem 1.1. Let ρ:π1(S)→PSL2Cbe quasi-Fuchsian. Then the space of
branched complex projective structures obtained by bubbling unbranched structures
with holonomy ρis connected, open and dense in the moduli space of structures with
two branch points and the same holonomy.
Combined with [6, Theorem C] by Goldman this result implies a positive answer
to the above question for a generic couple of branched complex projective structures
with the same quasi-Fuchsian holonomy and at most two branch points. Taking
into account also [2, Theorem 5.1] and [3, Theorem 1.1] by Calsamiglia-Deroin-
Francaviglia, we can see that indeed the only surgeries which are generically needed
to move around this moduli space are bubbling and debubbling. In the forthcoming
paper [11] we consider the problem of bounding the number of operations needed
in a sequence of surgeries from one structure to another.
The structure of the paper is the following: Section 2contains the first definitions
and basic lemmas, together with an example of a couple of non isomorphic struc-
tures obtained by bubbling the same unbranched structure along isotopic arcs (see
2.23); this phenomenon shows how sensitive these structures are to deformations.
In Section 3we review the geometric properties of structures with quasi-Fuchsian
holonomy, in the spirit of [6] and [2], and develop a combinatorial analysis of a
natural decomposition of such structures into hyperbolic pieces, providing an expli-
cit classification of pieces occurring for structures with at most two branch points;
this already allows to prove that many structures are obtained via bubbling, and
Section 4is concerned with the problem of deforming these structures without
breaking their bubbles. Finally Section 5contains the proof of the main theorem;
the strategy consists in two steps: first we use the analysis in 3to define a decom-
position of the moduli space into pieces and to sort out those in which it is easy to
find a bubbling, then we apply the results obtained in 4to move bubblings from
these pieces to the other ones.

BUBBLING CP1-STRUCTURES WITH QUASI-FUCHSIAN HOLONOMY 3
Acknowledgements I would like to thank Stefano Francaviglia for drawing my
attention to the study of pro jective structures and for his constant and valuable
support throughout this work. I am also very grateful to Bertrand Deroin for his
interest in this pro ject and for many useful conversations about it.
2. Branched complex projective structures
Let Sbe a closed, connected and oriented surface of genus g≥2. We will denote
by CP1=C∪{∞} the Riemann sphere and by PSL2Cthe group of its holomorphic
automorphisms acting by M¨obius transformations
PSL2C×CP1→CP1,a b
c d , z 7→ az +b
cz +d
We are interested in geometric structures locally modelled on this geometry, up to
finite branched covers. The following definition is adapted from [7].
Definition 2.1. A branched complex projective chart on Sis a couple (U, ϕ) where
U⊂Sis an open subset and ϕ:U→ϕ(U)⊆CP1is a finite degree orientation
preserving branched covering map. Two charts (U, ϕ) and (V, ψ) are compatible
if ∃g∈PSL2Csuch that ψ=gϕ on U∩V. A branched complex projective
structure σon S(BPS in the following) is the datum of a maximal atlas of
branched complex projective charts.
We will say that a structure is unbranched if all its charts are local diffeomorph-
isms. On the other hand p∈Swill be called a branch point of order ord(p) = m∈N
if a local chart at pis a branched cover of degree m+ 1, i.e. if it looks like z7→ zm+1.
Notice that a local chart (U, ϕ) can always be shrunk to ensure that it contains at
most one branch point and both Uand ϕ(U) are homeomorphic to disks. In par-
ticular branch points are isolated, hence in finite number since Sis compact.
Definition 2.2. The branching divisor of a BPS σis defined to be div(σ) =
Pp∈Sord(p)pand the branching order ord(σ) of σis defined to be the degree of its
branching divisor. We can also specify precise patterns of branching by extending
this notation: for a partition λ= (λ1,...,λn)∈Nnwe say that σhas order
ord(σ) = λif div(σ) = Pn
i=1 λipi.
Remark 2.3.A BPS on Scan be considered as a generalised (PSL2C,CP1)-structure
in the sense of [4]), for which the developing map may have critical points, corres-
ponding to branch points. A developing map for such a structure is an orientation
preserving smooth map dev :e
S→CP1with isolated critical points and equivariant
with respect to a holonomy representation ρ:π1(S)→PSL2C. As usual, for any
g∈PSL2Cthe pairs (dev, ρ) and (g dev, gρg−1) define the same BPS. Notice that
in our setting e
Sis a disk, hence dev can not be a global diffeomorphism, so that
these structures are never complete. Even worse, these structures are in general not
even uniformizable, in the sense that in general dev fails to be a diffeomorphism
onto a domain Ω ⊂CP1and is actually wildly non injective. This is of course clear
for branched structures, but it is actually already true in absence of branch points.
Let us give a few motivating examples for the study of BPSs.
Example 2.4. Every Riemann surface Xadmits a non-constant meromorphic
function f, which realizes it as a finite branched cover of CP1. This endows Xwith
a BPS with trivial holonomy and developing map given by fitself.

4 LORENZO RUFFONI
Example 2.5. Every surface Sof genus g≥2 admits a complete Riemannian
metric gof constant curvature −1, which realises it as a quotient of H2by a group
of isometries acting freely and properly discontinuously. Embedding H2as the
upper-half plane H+={I m(z)>0} ⊂ C⊂CP1shows that 2-dimensional hy-
perbolic geometry (PSL2R,H2) is a subgeometry of 1-dimensional complex pro-
jective geometry (PSL2C,CP1). Therefore these hyperbolic structures provide ex-
amples of (unbranched) complex projective structures. More generally it follows
from the work of Troyanov in [13] that, given p1,...,pn∈Sand k1,...,kn∈N, if
χ(S) + Pn
i=1 ki<0 (resp. = 0, or = 1) then there exists a hyperbolic (resp. Euc-
lidean, or spherical) metric on S\ {p1,...,pn}with a conical singularity of angle
2π(ki+ 1) at pi. These conical hyperbolic (resp. Euclidean, or spherical) structures
are examples of genuinely branched complex projective structures.
In order to define the deformation space of BPSs let us introduce a natural notion
of isomorphism for these structures.
Definition 2.6. Let σand τbe a couple of BPSs. A map f:σ→τis pro jective if
in local pro jective charts it is given by the restriction of a global holomorphic map
F:CP1→CP1. We say it is a projective isomorphism if it is also bijective.
Recalling that any global holomorphic function F:CP1→CP1is a rational
function and that the invertible elements in C(z) are exactly the fractional linear
transformations az+b
cz+dgiven by the action of PSL2C, on obtains that a projective
isomorphism is a diffeomorphism locally given by the restriction of some g∈PSL2C.
Definition 2.7. A marked branched complex projective structure on Sis a couple
(σ, f) where σis a surface endowed with a BPS and f:S→σis an orientation
preserving diffeomorphism. Two marked BPSs (σ, f ) and (τ, g) are declared to be
equivalent if gf−1:σ→τis isotopic to a projective isomorphism h:σ→τ. We
denote by BP(S) the set of marked branched complex projective structures on S
up to this equivalence relation.
Thinking of BPSs in terms of equivalence classes of development-holonomy pairs
[(dev, rho)] as in 2.3 allows us to put a natural topology (namely the compact-open
topology) on this set, and to define a natural projection to the character variety
χ(S) = Hom(π1(S),PSL2C)//PSL2Cby sending a BPS to its holonomy
hol :BP(S)→χ(S),[σ] = [(dev, ρ)] 7→ [ρ]
We are interested in the study of structures with a fixed holonomy, therefore we
introduce the following subspaces of the fibres of the holonomy map.
Definition 2.8. Let ρ∈χ(S), k∈Nand let λbe a partition of k. We define
Mk,ρ ={σ∈ BP (S)|ord(σ) = k, hol(σ) = ρ}
Mλ,ρ ={σ∈ BP(S)|ord(σ) = λ, hol(σ) = ρ}
where the order of a structure is the one defined in 2.2. We call the principal
stratum of Mk,ρ the subspace given by the partition λ= (1,...,1), i.e. the one
in which all branch points are simple.
In the Appendix of [2] Calsamiglia-Deroin-Francaviglia proved that if ρis non
elementary then the space Mk,ρ carries a natural structure of (possibly disconnec-
ted) smooth complex manifold of dimension kand that the subspace determined
by a partition λof length nis a complex submanifold of dimension n. In particular

BUBBLING CP1-STRUCTURES WITH QUASI-FUCHSIAN HOLONOMY 5
the principal stratum is an open dense complex submanifold of Mk,ρ. These com-
plex structures are locally modelled on products of Hurwitz spaces, i.e. spaces of
deformations of finite branched cover of disks, and local coordinates admit a nice
geometric description (see 2.18 below for more details).
Remark 2.9.In the following, when working with a BPS σ, we will find it conveni-
ent to fix a representative representation ρof the holonomy hol(σ), i.e. to choose
a representation in its conjugacy class. As soon as the holonomy is non element-
ary, there will be a unique developing map equivariant with respect to the chosen
representation. Indeed if dev1and dev2are developing maps for σequivariant with
respect to ρ:π1(S)→PSL2C, then ∃g∈PSL2Csuch that dev2=gdev1and for
any γ∈π1(S) we have
ρ(γ)gdev1=ρ(γ)dev2=dev2γ=gdev1γ=gρ(γ)dev1
so that (ρ(γ)g)−1gρ(γ) is an element of PSL2Cfixing every point of dev1(e
S).
Since a developing map has isolated critical points, there is some point of e
Sat
which it is a local diffeomorphism, hence its image has non empty interior. But a
M¨obius transformation fixing more than three points is the identity of CP1, hence
(ρ(γ)g)−1gρ(γ) = id. This means that gis in the centralizer of the image of ρ,
which is trivial since the holonomy is assumed to be non elementary; as a result
g=id and the two developing maps coincide.
We conclude this preliminary section by introducing three elementary geometric
surgeries which one can perform on a given BPS to obtain a new BPS with the
same holonomy.
2.1. Grafting. The first surgery consists in replacing a simple closed curve with
an annulus endowed with a projective structure determined by the structure we
begin with. It was first introduced by Maskit in [8] to produce examples of pro-
jective structures with surjective developing map; here we review it mainly to fix
terminology and notation. Let us pick σ∈ BP(S) and let (dev, ρ) be a development-
holonomy pair defining it.
Definition 2.10. Let γ⊂Sbe a simple closed curve on S. We say that γis
graftable with respect to σif ρ(γ) is loxodromic (i.e. not elliptic nor parabolic)
and γis injectively developed, i.e. the restriction of dev to any of its lifts eγ⊂e
Sis
injective.
Since dev is ρ-equivariant, if γis graftable then a developed image of it is an
embedded arc in CP1joining the two fixed points of ρ(γ). Moreover ρ(γ) acts
freely and properly discontinuously on CP1\dev(eγ) and the quotient is an annulus
endowed with a complete unbranched complex projective structure.
Definition 2.11. Let γ⊂Sbe a graftable curve with respect to σ. For any lift
eγof γwe cut e
Salong it and a copy of CP1along dev(eγ), and glue them together
equivariantly via the developing map. This gives us a simply connected surface e
S′
to which the action π1(S)ye
Sand the map dev :e
S→CP1naturally extend, so
that the quotient gives rise to a new structure σ′∈ BP(S). We call this structure
the grafting of σalong γand denote it by Gr(σ, γ). The surface σ\γpro jectively
embeds in Gr(σ, γ) and the complement is the annulus Aγ= (CP1\dev(eγ))/ρ(γ),
which we call the grafting annulus associated to γ. The inverse operation will be
called a degrafting.

6 LORENZO RUFFONI
σ
γ
Gr(σ, γ)
Figure 1. Grafting a surface
The easiest example of this construction consists in grafting a simple geodesic on
a hyperbolic surface; for such a structure every simple essential curve γis graftable,
since the holonomy is purely hyperbolic and the developing map is globally injective.
The grafting surgery preserves the holonomy and does not involve any modification
of the branching divisor, so that if σ∈ Mλ,ρ then Gr(σ, γ)∈ Mλ,ρ too. Notice
that for any structure σand any graftable curve γon it the structure Gr(σ, γ) has
surjective but non injective developing map.
2.2. Bubbling. The second surgery consists in replacing a simple arc with a disk
endowed with a projective structure determined by the structure we begin with,
hence it can be thought as a “finite version” of grafting. It was first considered
by Gallo-Kapovich-Marden in [5] as a tool to introduce new branch points on a
projective structures. As before, let us choose σ∈ BP(S) and let (dev, ρ) be a
development-holonomy pair defining it.
Definition 2.12. Let β⊂Sbe a simple arc on S. We say that βis bubbleable
with respect to σif it is injectively developed, i.e. the restriction of dev to any of
its lifts e
β⊂e
Sis injective.
The surgery is then defined as follows.
Definition 2.13. Let β⊂Sbe a bubbleable arc with respect to σ. For any lift
e
βof βwe cut e
Salong it and a copy of CP1along dev(e
β), and glue them together
equivariantly via the developing map. Once again, this gives us a simply connected
surface e
S′to which the action π1(S)ye
Sand the map dev :e
S→CP1naturally
extend, so that the quotient gives rise to a new structure σ′∈ BP(S). We call this
structure the bubbling of σalong βand denote it by Bub(σ, β). The surface σ\β
projectively embeds in Bub(σ, β) and the complement is the disk B=CP1\dev(e
β),
which we call the bubble associated to β.
σ
β+
CP1
dev(e
β)
Bub(σ, β)
∗ ∗
B
Figure 2. Bubbling a surface
The easiest example is obtained by bubbling a hyperbolic surface along an em-
bedded geodesic arc. The bubbling surgery preserves the holonomy and introduces

BUBBLING CP1-STRUCTURES WITH QUASI-FUCHSIAN HOLONOMY 7
a couple of simple branch points corresponding to the endpoints of the bubbling
arc. Therefore if σ∈ Mλ,ρ then Bub(σ, β)∈ Mλ+(1,1),ρ, where if λis a partition
of k,λ+ (1,1) is the partition of k+ 2 obtained appending (1,1) to it.
Once a bubbling is performed, we see a subsurface of Shomeomorphic to a disk
and isomorphic to CP1cut along a simple arc, the isomorphism being given by
any determination of the developing map itself. It is useful to be able to recognise
this kind of subsurface, since there is an obvious way to remove it and lower the
branching order by 2; such operation is called debubbling and is the inverse of
bubbling.
Definition 2.14. Abubble on σ∈ BP (S) is an embedded closed disk B⊂S
whose boundary decomposes as ∂B =β′∪{x, y }∪β′′ where {x, y}are simple branch
points of σand β′, β′′ are embedded injectively developed arcs which overlap once
developed; more precisely there exist a determination of the developing map on B
which injectively maps β′, β′′ to the same simple arc b
β⊂CP1and restricts to a
diffeomorphism dev :int(B)→CP1\b
β.
x∗y
∗
B
β′
β′′
Figure 3. A bubble
Notice that a BPS obtained by bubblings some unbranched structures has by
definition an even number of branch points and surjective non injective developing
map. As a consequence branched hyperbolic structures with an even number of
branch points do not arise as bubblings, as their developing maps take value only
in the upper-half plane. By the work of [13] these structures exist on every surface
of genus g≥3; this example was already mentioned in [2].
2.3. Movements of branch points. The last surgery we will use takes place
locally around a branch point and consists in a deformation of the local branched
projective chart, which can be thought as an analogue in our setting of the Schiffer
variations in the theory of Riemann surfaces (see [9]). They were introduced by
Tan in [12] for simple branch points (and then generalised in [2] for branch points
of higher order) as a tool to perform local deformations of a BPS inside the moduli
space Mk,ρ. Since we will need this surgery only for simple branch points, we
restrict here to that case and avoid the technicalities required by a more general
treatment.
Definition 2.15. Let σ∈ Mk,ρ and let p∈σbe a simple branch point. An
embedded twin pair at pis a couple of embedded arcs µ={µ1, µ2}which meet
exactly at p, are injectively developed and overlap once developed; more precisely

8 LORENZO RUFFONI
there exist a determination of the developing map around µ1∪µ2which injectively
maps µ1, µ2to the same simple arc bµ⊂CP1.
Given such a couple of arcs we can perform the following cut-and-paste surgery.
µ1µ2
∗
p◦
q2
◦
q1
∗
p1
∗
p2
◦
q2
◦
q1
µ′
1
µ′
2
∗
p1
∗
p2
◦q
Figure 4. A movement of branch point
Definition 2.16. Let σ∈ Mk,ρ , let p∈σbe a simple branch point and µ=
{µ1, µ2}an embedded twin pair at p. The BPS σ′obtained by cutting Salong
µ1∪µ2and regluing the resulting boundary with the obvious identification (as
shown in Picture 4) is said to be obtained by a movement of branch point at p
along µand is denoted by M ove(σ, µ).
This surgery preserves the holonomy and does not change the structure of the
branching divisor. Therefore if σ∈ Mλ,ρ then M ove(σ, µ)∈ Mλ,ρ. Notice that
the image of the developing map is not changed by this operation; moreover once
a movement is performed, we have an induced embedded twin pair on the new
structure, and moving points along it of course brings us back to σ.
Remark 2.17.The movement of branch points along an embedded twin pair µis
a deformation which comes in a 1-parameter family. In the above notations, if
bµ: [0,1] →CP1is a parametrization of the developed image of µ, then for t∈[0,1]
we can consider the structure σt=M ove(σ, µt), where µtis the embedded twin pair
contained in µand developing to the subarc bµ([0, t]). Following this deformation
as tvaries, we see the developed image of the branch point sliding along the arc bµ,
which motivates the name of this surgery.
Remark 2.18.As anticipated above, as soon as the holonomy ρis non elementary,
the moduli space Mλ,ρ carries a natural structure of complex manifold of dimension
equal to the length of the partition λ. It is proved in [2,§12.5] that the local neigh-
bourhoods of a BPS σ∈ Mk,ρ for this topology are obtained by local deformations
at the branch points. For simple branch points (i.e. structures in the principal
stratum) these are just the movements of branch points described above; for higher
order branch points one needs to introduce a slight generalisation of them, but we
will not need this.

BUBBLING CP1-STRUCTURES WITH QUASI-FUCHSIAN HOLONOMY 9
2.4. Injectively developed isotopies. We have so far introduced some surgeries
which can be performed on a BPS, which depend on the choice of a simple arc
which is injectively mapped to CP1by the developing map. It is natural to ask
how much this choice is relevant as far as the isomorphism class of the resulting
structure is concerned; an answer to this will be needed in the forthcoming sections.
The following turns out to be the a useful notion to consider.
Definition 2.19. Let σbe a BPS on Sand η: [0,1] →San embedded arc with
embedded developed image. An isotopy H: [0,1] ×[0,1] →Sof ηis said to be
injectively developed if ηs=H(s, .) is an embedded arc with embedded developed
image for all s∈[0,1].
Let us begin with the following lemma, which says that being injectively de-
veloped is a stable condition.
Lemma 2.20. Let σbe a BPS on S. Let γ: [0,1] →Sbe an embedded arc having
embedded developed image and not going through branch points (except possibly at
its endpoints). Then there exists an injectively developed subset U⊆Ssuch that
γ⊂Uand γ(]0,1[) ⊂int(U).
Proof. Let eγbe a lift of the arc to the universal cover. Assume first that γdoes
not go through any branch point at all. Then we can prove that it actually has
an injectively developed neighbourhood: if this were not the case, there would be
a sequence of nested open neighbourhoods Un+1 (Unof γsuch that ∀n∈Nwe
could find a couple of distinct point xn, yn∈Unwith the same developed image.
By compactness of S, these sequences subconverge to a couple of points x, y ∈γ
with the same developed image. Since the path is injectively developed, we get
x=y. But since the path does not go through branch points, the developing map
is locally injective at any of its points, so that the existence of the points xn, yn
arbitrarily close to x=y∈γis absurd.
If one endpoint, say γ(0), of γis a branch point of order k, then clearly every set
containing it in its interior is not injectively developed. Nevertheless a sufficiently
small neighbourhood Ω of γ(0) decomposes as a disjoint union of injectively de-
veloped sectors A1,...,Ak+1; an initial segment of γbelongs to one of them, say
A1; so we can simply pick a sequence of nested sets Vn+1 (Vnsuch that for every
n∈Nwe have that γ(0) ∈Vn,Vn∩Ω(A1and Vncontains γ(]0,1[) in its interior,
and apply the previous argument to obtain a couple of sequences xn6=yn∈Vn
converging to x=y∈γ. The non trivial case to discuss is the case in which the
limit is a branch point, i.e. x=y=γ(0); by construction of Vn, for nlarge enough
the points xn, ynmust lie inside A1, which is injectively developed, hence we reach
a contradiction exactly as before. 
In particular this implies that it is always possible to perform small deformations
of an injectively developed arc through an injectively developed isotopy relative to
endpoints. This applies both to bubbleable arcs and to arcs appearing in an em-
bedded twin pair. Injectively developed isotopies of bubbleable arcs and embedded
twin pairs are relevant in our discussion since they do not change the isomorph-
ism class of the structure obtained by performing a bubbling or a movement of
branch points, as established by the following statements. The next one is simply
a reformulation of [2, Lemma 2.8].

10 LORENZO RUFFONI
Lemma 2.21. Let σbe a BPS and let β, β′⊂σbe bubbleable arcs with the same
endpoints. If there exists an injectively developed isotopy relative to endpoints from
βto β′, then Bub(σ, β) = Bub(σ, β′).
The following is the statement, analogous to 2.21, for a movement of branch
points along different embedded twin pairs.
Lemma 2.22. Let σbe a BPS and let pbe a simple branch point. Let µ={µ1, µ2}
and ν={ν1, ν2}be embedded twin pairs based at pwith the same endpoints, and
let qibe the common endpoint of µiand νifor i= 1,2. Suppose that there exists an
injectively developed isotopy H: [0,1]×[−1,1] →Sfrom µto νrelative to {q1, p, q2}
and such that αs={αs
1=H(s, [−1,0]),αs
2=H(s, [0,1])}is an embedded twin pair
for all s∈[0,1]. Then M ove(σ, µ) = Move(σ, ν).
Proof. First of all notice that each path αs
iappearing in an embedded twin pair αs
is in particular an embedded arc which is injectively developed and goes through
exactly one branch point, which is p. Therefore we can pick an injectively developed
set Us
icontaining αs
i\ {p}in its interior as in 2.20. We can choose this set in such
a way that Us=Us
1∪Us
2is an open neighbourhood of αs: for instance we can
take Us
1such that its developed image is an open neighbourhood of the developed
image of αs, then pull it back via the developing map, so that Usis the domain of a
local projective chart which simply branches at pand contains the whole embedded
twin pair αs. The sets Usprovide an open cover of Im(H); by compactness we
extract a finite subcover indexed by some s0= 0, s1,...,sN= 1. Up to taking
an intermediate finite subcover between {Us0,...,UsN}and {Us|s∈[0,1]}we
can assume that the local chart Usicontains not only αsibut also αsi±1. Then
we conclude by observing that αs0=µand αsN=νand that the results in the
Appendix of [2] imply that M ove(σ, αsi) = Move(σ, αsi+1 ), because αsiand αsi+1
are contained in the domain of a single local chart. 
We conclude this preliminary section by remarking that an ordinary isotopy is in
general not enough to obtain this kind of results. In the next example we provide
an explicit construction of two bubbleable arcs which are isotopic but not isotopic
through an injectively developed isotopy, for which the resulting structures are not
isomorphic. Most of the technical parts in Section 4below are needed to avoid this
kind of phenomenon, which was already observed in [3, Remark 3.4] for the case of
graftings.
Example 2.23. Let Sbe a genus 2 surface with a hyperbolic structure σρ, with
holonomy a Fuchsian representation ρ:π1(S)→PSL2R, and let γbe a separating
oriented closed geodesic. Let ηbe an oriented embedded geodesic arc on Swith
one endpoint xon γand orthogonally intersecting γonly in x; let ybe the other
endpoint, which we assume to be on the right of γ(see Picture 5). We want
to perform a grafting of σρalong γand then show how to perform two different
bubbling on Gr(σρ, γ) along two different extensions of η. On Gr(σρ, γ ) we have
two distinguished curves γ±coming from γand bounding the grafting annulus Aγ.
We also have two marked points x±∈γ±coming from the point x, and an arc
coming from η, which we still denote by the same name, which starts at x+∈γ+
orthogonally and moves away from the annulus.
There is a natural way to extend ηby analytic continuation to an embedded arc
reaching the other point x−∈γ−: namely consider the extension of the developed

BUBBLING CP1-STRUCTURES WITH QUASI-FUCHSIAN HOLONOMY 11
image of η(which is a small geodesic arc in the upper half-plane) to a great circle
bηon CP1. This gives an embedded arc on Gr(σρ, γ ) which is not injectively de-
veloped, hence not bubbleable. To obtain bubbleable arcs we slightly perturb this
γγ−γ+
xy
ηx−x+y
η
Figure 5. Analytic extension of ηin Gr(σρ, γ)
construction; in CP1consider an embedded arc which starts at the developed image
bxof xand ends at the developed image byof y, but leaves bxwith a small angle θ
with respect to bη, stays close to it, and reaches bywith angle θon the other side,
crossing bηjust once at some point in the lower-half plane (see left side of Picture
6). This arc can be chosen to sit inside a fundamental domain for ρ(γ), so that it
gives an embedded arc on Gr(σρ, γ ) starting at x−, reaching γ+at a point z+close
to x+and ending at y. Changing the value of θin some small interval ] −ε, ε[ we
obtain a family of embedded arcs αθin Gr(σρ, γ) which are isotopic relative to the
endpoints x−, y and are all injectively developed, except α0=η.
bγ
bη
cαθ
bxby
bz
γ−γ+
x−
x+
z+y
η
αθ
Figure 6. The bubbleable arc αθin CP1and Gr(σρ, γ )
Fix now some small θand consider the BPS σ±=Bub(Gr(σρ, γ ), α±θ) obtained
by bubbling along α±θ. We now proceed to show that these two BPSs are not
isomorphic: they can be distinguished by looking at the configuration of certain
curves, which we now define. The first curve we need is the analytic continuation of
γ+: we extend it inside the bubble by following its developed image. The result is a
curve which reaches x−, and we still denote it by γ+. To define the other curve, let
us recall from [2,§3] that a BPS with Fuchsian holonomy canonically decomposes
into subsurfaces endowed with (possibly branched) complete hyperbolic metrics (see
also 3.5 below for more details). Then the curve we need is the unique geodesic δ
between x−and ywith respect to this metric, which develops isometrically onto

12 LORENZO RUFFONI
the developed image of the original geodesic segment ηof σρ. Notice that the whole
construction can be made in such a way that this is indeed the shortest geodesic
between its endpoints, just by taking the segment ηon σρto be suitably shorter
than the systole of σρ. Now we look at the tangent space at x−. The tangent vector
to γ+at x−sits on the right or on the left of the tangent vector to δ(with respect
to the underlying orientation of S) depending on the fact that we look at σ+or
at σ−. But any projective isomorphism between the two structures should be in
particular orientation preserving at x−.
3. Geometric decomposition in quasi-Fuchsian holonomy
We now restrict our attention to structures whose holonomy preserves a decom-
position of the model space CP1into two disks separated by a Jordan curve. As
observed in [6] and [2], the key feature of structures with such a representation
is the presence of a canonical decomposition of the surface into subsurfaces which
carry complete (possibly branched) hyperbolic structures with ideal boundary. The
purpose of this section is to give a description of the components that can appear
in such a decomposition, in the spirit of Goldman’s work in [6]. Let us begin by
recalling some definitions and known constructions.
Definition 3.1. AFuchsian (respectively quasi-Fuchsian)group is a subgroup
of PSL2Cwhose limit set in CP1is RP1(respectively a Jordan curve).
In particular a finitely generated quasi-Fuchsian group Γ preserves a decomposi-
tion CP1= Ω+
Γ∪ΛΓ∪Ω−
Γof the Riemann sphere into a pair of disks Ω±
Γand a Jordan
curve ΛΓ, i.e. the two components of the domain of discontinuity and the limit set
of Γ. When Γ is Fuchsian this is just the decomposition CP1=H+∪RP1∪ H+,
where H±denote the upper and lower-half plane in C.
Definition 3.2. A faithful representation ρ:π1(S)֒→PSL2Cis a Fuchsian (re-
spectively quasi-Fuchsian)representation if its image is a Fuchsian (respectively
quasi-Fuchsian) subgroup and there exists an orientation preserving ρ-equivariant
diffeomorphism f:e
S→Ω+
ρ(π1(S)). A structure σ∈ BP(S) is said to be Fuchsian
or quasi-Fuchsian when its holonomy is.
We will adopt the notation Ω±
ρ= Ω±
ρ(π1(S)) and Λρ= Λρ(π1(S)). Notice that the
action on Ω±
ρadmits an invariant complete hyperbolic metric d±
ρ, since the action
is conjugated to the action of a Fuchsian group on H2. We can therefore obtain
an extended metric dρon CP1by considering the path metric associated to d±
ρ: a
point in one disk has infinite distance from any point of the other disk.
Given a quasi-Fuchsian representation ρ, by definition we have an orientation
preserving ρ-equivariant diffeomorphism f:e
S→Ω+
ρ. This descends to an ori-
entation preserving diffeomorphism F:S→Ω+
ρ/Im(ρ), giving us a (marked)
unbranched complete hyperbolic structure on Swith holonomy ρ; we can use it as
a base point in the moduli space Mρ, so we give it a special name.
Definition 3.3. If ρ:π1(S)→PSL2Cis a quasi-Fuchsian representation, then
σρ= Ω+
ρ/Im(ρ) is called the uniformizing structure for ρ.
More generally, if dev :e
S→CP1is a developing map for a BPS on Swith quasi-
Fuchsian holonomy ρ, then the decomposition of the Riemann sphere induced by

BUBBLING CP1-STRUCTURES WITH QUASI-FUCHSIAN HOLONOMY 13
ρcan be pulled back via the dev to obtain a decomposition of e
S. Since the devel-
oping map is (π1(S), ρ)-equivariant, this decomposition is π1(S)-invariant and thus
descends to a decomposition of the surface into possibly disconnected subsurfaces
σ+and σ−and a possibly disconnected curve σRdefined as the subset of points
developing to Ω+
ρ,Ω−
ρand Λρrespectively.
Definition 3.4. We will call S=σ+∪σR∪σ−the geometric decomposition
of Swith respect to the BPS defined by the pair (dev, ρ); we will call σ±the
positive/negative part of Sand σRthe real curve of S.
We already observe at this point that, despite their apparent symmetry, the
positive and negative part play a very different role in the geometry of σ, because of
the special role played by Ω+
ρin the definition 3.2 of quasi-Fuchsian representation.
This phenomenon was already exploited by Goldman in the unbranched case (see
[6]), and we will explore the branched case below.
Notice that a priori the decomposition of the surface depends not only on the
representation, but also on the choice of a developing map. However this ambiguity
can be fixed by choosing a representation ρin its conjugacy class, as explained in
2.9 above, since quasi-Fuchsian representations are in particular non elementary
representations. As a result, the decomposition of Sdepends only on the structure
σ={(dev, ρ)}and not on the choice of particular representatives. In particular
many combinatorial properties of the geometric decomposition (such as the number
and type of components, the adjacency pattern, the location of branch points,. . . )
are well defined. The following was observed in [6,§2] for the unbranched case
and in [2,§3] for the branched case, and is the main feature of structures with
quasi-Fuchsian holonomy.
Lemma 3.5. If Sis endowed with a quasi-Fuchsian BPS σ, then σ±is a finite
union of subsurfaces carrying complete hyperbolic metrics with cone points of angle
2π(k+1) corresponding to branch points of order kof the BPS, and σRis a finite 1-
dimensional CW-complex on S; moreover if branch points are not on the real curve,
then σRis a finite union of simple closed curves with a (PSL2R,RP1)-structure.
Moreover this motivates the following terminology.
Definition 3.6. If Sis endowed with a quasi-Fuchsian BPS σ, a connected com-
ponent Cof σ\σRwill be called a geometric component of the decomposition;
a connected component Cof σ±will be called a positive/negative component. A
connected component of σRwill be called a real component.
Notice that the components of the real curve can be canonically oriented by
declaring that they have positive regions on the left and negative regions on the
right. Some examples are in order.
Example 3.7. A hyperbolic structure on Sis an example of an unbranched pro-
jective structure with Fuchsian holonomy. Any developing map is a diffeomorphism
with the upper-half plane H+. The induced decomposition is σ+=S, σ−=∅=
σR. Hence there is only one geometric component, which is the whole surface.
Example 3.8. If we graft a hyperbolic surface along a simple closed geodesic we
obtain an example of an unbranched projective structure with Fuchsian holonomy
with surjective and non injective developing map to CP1. There are a negative
geometric annulus bounded by two essential simple closed real curves and two or

14 LORENZO RUFFONI
one positive geometric components, depending on the fact that the geodesic we use
is separating or not.
The main result in [6] claims that every unbranched structure with quasi-Fuchsian
holonomy arises via a multigrafting of the uniformizing hyperbolic structure; one
of the key observations is the fact that geometric components of an unbranched
structure can not be simply connected, i.e. they can not be disks. This completely
fails for branched structures as the following easy example shows.
Example 3.9. If we bubble a hyperbolic surface along a simple arc we obtain an
example of a branched projective structure with Fuchsian holonomy and with a
negative geometric disk bounded by a contractible simple closed real curve and one
positive geometric component containing the two branch points.
+−
∗
∗
Figure 7. Geometric decomposition of a bubbling on σρ.
Our purpose here is to show that also for branched structures it is possible
to obtain a control of the behaviour of the negative components. The location of
branch points with respect to the geometric decomposition is of course something we
want to care about in the following, therefore we introduce the following definitions.
Definition 3.10. Let σbe a quasi-Fuchsian BPS. A branch point of σis said to be
geometric (respectively real) if it belongs to σ±(respectively to σR). The structure
is said to be geometrically branched if all its branch points are geometric and
it is said to be really branched if it has some real branch point. We will denote
by MR
k,ρ the subspace of really branched structures of Mk,ρ .
Notice that up to a very small movement of branch points, we can always assume
that the branch points do not belong to the real curve σR; more precisely, MR
k,ρ has
real codimension 1 inside the k-dimensional complex manifold Mk,ρ. From now on
we focus on geometrically branched structures; for these ones some index formulae
are available, which link the geometry and the topology of the components of the
geometric decomposition. We recall here the needed terminology (see [2,§3-4] for
more details.)
Definition 3.11. Let σbe a geometrically branched BPS and lbe a real component
on it. Let p∈Λρbe a fix point of ρ(l) and e
lis any lift of l. The index of the
induced real projective structure on lis the integer I(l) = # {dev−1
|e
l(p)}/ < l >.
The index of a real component can be thought as a degree of the restriction of the
developing map to it, as a map with values in the limit set of ρ, and it can a priori
assume any value. However if ρ(l) is trivial then the index must be strictly positive:
this follows by the classification of RP1-structures on S1given in [2, Proposition
3.2], which we recall for future reference.

BUBBLING CP1-STRUCTURES WITH QUASI-FUCHSIAN HOLONOMY 15
Lemma 3.12. Two unbranched RP1-structures on an oriented circle with non
elliptic holonomy are isomorphic if and only if they have the same index and
holonomy. The only case which does not occur is the case of index 0and trivial
holonomy.
Definition 3.13. For a quasi-Fuchsian representation ρlet Eρbe the induced flat
RP1-bundle on S. For any subsurface i:C ֒→Swe denote by ρCthe restriction
of ρto i∗π1(C). For any component l⊂∂C we define a section sρ:l→Eρ|lby
choosing the flat section passing through a fixed point of ρ(l). Then the Euler
class eu of ρCis defined to be the Euler class of the bundle EρC=Eρ|Cwith
respect to this choice of boundary sections.
Finally we say that a subsurface C⊂Sis incompressible if the inclusion is
injective on fundamental groups or, equivalently, if all the boundary curves are
essential (i.e. not nullhomotopic) in S. The following index formulae hold.
Theorem 3.14. ([2, Theorem 4.1-5]) Let σ∈ Mρbe geometrically branched. Let
C⊂σ±be a geometric component containing kCbranch points (counted with mul-
tiplicity) and with ∂C ={l1,...,ln} ⊂ σR. Then
±eu(ρC) = χ(C) + kC−
n
X
i=1
I(li)
Moreover if Cis incompressible (e.g. C=S) then eu(ρC) = χ(C).
Under the same hypothesis of 3.14 the following can be deduced
Corollary 3.15. If k±denotes the number of positive/negative branch points of σ,
then 2χ(σ−) = k+−k−.
In particular in quasi-Fuchsian holonomy there is always an even number of
branch points, so that M2k+1,ρ are all empty.
Example 3.16. If σis unbranched, every geometric component carries an un-
branched complete hyperbolic metric by 3.5; as a consequence all real curves have
index 0, and in particular they are essential by 3.12. On the other hand in the
branched case real curves can have positive index and be non essential; as an ex-
ample consider the simple bubbling of 3.9, where there is exactly one contractible
real component with index 1.
3.1. Locating branch points. We have observed in 3.16 that genuinely branched
structures can have real curves of positive index. Roughly speaking, if this occurs
then branch points must live in the geometric components adjacent to the real
curves of positive index. This section aims at making this statement more precise.
We begin by noticing that even if the structure has branch points, nevertheless
unbranched components are quite well behaved.
Lemma 3.17. Let σ∈ Mk,ρ be geometrically branched. If C⊂σ±is an un-
branched component then either it is a disk or it is incompressible. Moreover if it
is negative and incompressible, then it is an incompressible annulus.
Proof. We already know that unbranched disks can occur. If Cis not a disk and is
not branched, then it carries a complete hyperbolic structure such that the index
of each boundary component is zero; by 3.12 we know that it can not have trivial
holonomy. But quasi-Fuchsian representations are in particular injective, hence this

16 LORENZO RUFFONI
implies that each boundary component must be essential in the surface S, hence C
is incompressible. So we can apply the index formula and obtain χ(C) = eu(ρC) =
±χ(C) + kC−Pl⊂∂C I(l)=±χ(C), where the sign depends on the sign of C.
In the case Cis negative this implies χ(C) = 0. 
This is the first manifestation of the asymmetry between positive and negative
regions hinted at before, and which is a consequence of the special role played by
Ω+
ρin the definition 3.2 of quasi-Fuchsian representation. Notice that we get a
useless identity in the case of a positive component. The following easy observation
provides a first step to locate branch points with respect to σR.
Lemma 3.18. Let σ∈ Mk,ρ be geometrically branched. Let lbe a real component
and C, C′be the components of σ±which are adjacent along l. If I(l)≥1then
(1) at most one of C, C′is a disk;
(2) any non disk component is branched;
(3) at least one of C, C′is branched.
Proof. To prove (1) observe that C, C′can not both be disks, otherwise we would
get an embedded sphere in S, which is always assumed to have genus g≥2. To get
(2) observe that an unbranched component with ideal boundary with positive index
is necessarily a disk isometric to H2: indeed such a component carries a complete
hyperbolic structure, hence is a quotient of H2by some group Γ and as soon as
Γ6=id we see that the index of the real boundaries is 0. Finally (3) follows from
(1) and (2). 
Corollary 3.19. Let σ∈ Mk,ρ be geometrically branched. The (unique) component
adjacent to a disk of the geometric decomposition of σis branched.
Proof. The boundary of a disk has always strictly positive index. Since Sis not a
sphere, the adjacent component can not be a disk, therefore it is branched. 
Notice however that it may happen that none of the two components adjacent
to a real component with positive index is a disk.
Example 3.20. Let γbe a simple closed geodesic on the uniformizing structure
σρ, and let σ′=Gr(σρ, γ). Then pick a bubbleable arc β⊂σ′which intersects
exactly once the real curve of σ′and let σ′′ =Bub(σ′, β). Then σ′′ has an essential
real component of index 1 such that both adjacent components are non disks (both
have non positive Euler characteristic) and are branched. See Picture 8, left side.
On the other hand there are structures with negative components with essential
boundary with index 0 which are nevertheless branched.
Example 3.21. Let γbe a non separating simple closed geodesic on the uniform-
izing structure σρ, and let σ′=Gr(σρ, γ). Then let β⊂σ′be a bubbleable arc
with endpoints inside the negative annulus but which is not itself contained inside
the negative annulus and let σ′′ =Bub(σ′, β ). Then σ′′ has one real component
of index 0, a positive unbranched incompressible component and a negative incom-
pressible component of Euler characteristic -1 containing both branch points. See
Picture 8, right side.
From the above results we obtain in particular a bound on the number of branch
points contained inside a disk of the geometric decomposition of a quasi-Fuchsian
structure σ∈ Mk,ρ.

BUBBLING CP1-STRUCTURES WITH QUASI-FUCHSIAN HOLONOMY 17
Figure 8. Pictures for Examples 3.20 and 3.21.
Proposition 3.22. Let σ∈ Mk,ρ be geometrically branched and k≥2. If a
geometric component D⊂σ±is a disk of branching order kD, then kD≤k−2.
Proof. By 3.19 we already know that a disk can not contain all the branching. So
we assume by contradiction that it has branching order kD=k−1. Since Dis
contractible, by 3.14 its boundary is a real component lof index I(l) = kD+ 1 = k.
Let Cbe the component adjacent to D; then we know it is branched by 3.19, so
kC≥1. Indeed kD=k−1 implies that kC= 1. The boundary of Ca priori
could contain also mmore non essential boundary components and nessential
ones. Notice that all components of σ±different from C, D are unbranched, simply
because C∪Dcontains all the branching.
Therefore if l′6=lis a non essential component of ∂C, then ρ(l′) = id hence
I(l′)≥1 by 3.12, and then by 3.18 the geometric component after it must be an
unbranched disk D′and l′must have index 1. Let l, l′
1,...,l′
mbe the non essential
components of ∂C, and D, D ′
1,...,D′
mthe corresponding disks; then I(l) = kbut
I(l′
i) = 1, kD′
i= 0 for i= 1,...,m. On the other hand, if l′′ is an essential bound-
ary component, then the geometric component after it is a non simply connected
complete hyperbolic surface, hence I(l′′ ) = 0.
Now observe that the subsurface E=C∪D∪D′
1∪ · · · ∪ D′
mhas essential
boundary by construction, hence it is incompressible. Therefore the index formula
3.14 gives us that
eu ρ|π1(E)=χ(E) = χ(C) + χ(D) +
m
X
i=1
χ(D′
i) = χ(C) + 1 + m
On the other hand, we obtain, again by 3.14 and the fact that disks have trivial
Euler class, that
eu ρ|π1(E)=eu ρ|π1(C)+eu ρ|π1(D)+
m
X
i=1
eu ρ|π1(D′
i)=eu ρ|π1(C)=
=±
χ(C) + kC−I(l)−
m
X
i=1
I(l′
i)−
m
X
j=1
I(l′′
j)
=±(χ(C) + 1 −k−m)
where the sign depends on the sign of C(hence of that of D). We are now going
to compare the two expressions for the Euler class of E. If C⊂σ+then we
get 2m+k= 0 which is absurd since m≥0, k ≥2. If C⊂σ−then we get
2χ(C) = k−2≥0. But Ccan not be a disk, hence χ(C) = 0, i.e. Cis an
annulus. Its boundary consists of land another curve l′homotopic to it; so l′is
non essential too, hence of positive index. The component adjacent to l′can not
be a disk, otherwise Swould have genus g= 0, hence it must be branched; but by
construction all branch points live in C∪D, so we have a contradiction. 

18 LORENZO RUFFONI
Notice that so far Ccould be either positive or negative. Indeed, by performing
suitable bubbling, we can find structures with either positive or negative disks,
either branched or not. We recall the following useful lemma, which was proved
in [2, Lemma 10.3] for the positive part; here we just show that the same proof
provides an interesting equality for the negative part too.
Lemma 3.23. Let σ∈ Mk,ρ be geometrically branched. If all branch points live in
σ+and C⊂σ+is a branched component with nadjacent disks, then kC= 2n. If all
branch points live in σ−and C⊂σ−is a branched component then kC=−2χ(C).
Proof. Suppose all branch points live in the positive part or in the negative part,
and let Cbe a branched component. The hypothesis implies that all components
adjacent to Care unbranched, therefore by 3.17 we have the following dichotomy
for a real curve in the boundary of C: either it has index 0 and is essential, or it has
index 1 and bounds a disk. Let l1,...,lnbe the non essential boundary components
of Cand let D1, . . . , Dnbe the adjacent disks. The subsurface E=C∪D1∪ · · ·∪Dn
is clearly incompressible. By 3.14 and the fact that disks have trivial Euler class
we obtain
χ(C) + n=χ(E) = eu(E) = eu(C) = eu(C) +
n
X
i=1
eu(Di) =
± χ(C) + kC−
n
X
i=1
I(li)!=±(χ(C) + kC−n)
from which the statement follows. 
3.2. Classification of components for BPSs with k= 2 branch points.
When we have only two branch points, we can obtain a strong control on the
behaviour of real curves of positive index. This can be used to obtain a classification
of the components that can appear in the geometric decomposition of a structure.
As before we assume branch points are not on the real curve, so that the index
formulae 3.14 can be used.
In 3.17 we observed that in general an unbranched negative component which
is not a disk is automatically an incompressible annulus. For structures with two
branch points we can obtain a precise statement also about branched negative
incompressible components.
Lemma 3.24. Let σ∈ M2,ρ be geometrically branched. Let C⊂σ−be a branched
negative incompressible component containing kCbranch points. Then
(1) either kC= 1,Cis an annulus with ∂C =l∪l′such that I(l) = 0, I (l) = 1
(2) or kC= 2,Cis a pant or a once-holed torus and ∀l⊂∂C we have I(l) = 0
Proof. Since Cis incompressible we can applying the index formula and we get
−χ(C) = −eu ρ|π1(C)=χ(C) + kC−X
l⊂∂C
I(l)⇒2χ(C) + kC=XI(γ)≥0
and here we look for integer solutions with the constraints that χ(C)≤0 (being
incompressible, Cis not a disk) and kC≤2. We see that the only possibilities are
the following
(1) kC= 0, χ(C) = 0, so that Cis an unbranched annulus (which we discard,
since Cis assumed to be branched)

BUBBLING CP1-STRUCTURES WITH QUASI-FUCHSIAN HOLONOMY 19
(2) kC= 1, χ(C) = 0, so that Cis an annulus; we get PI(l) = 1, which means
that one boundary component has index 0 and the other has index 1
(3) kC= 2, χ(C) = 0, so that Cis again an annulus and PI(l) = 2; in partic-
ular there is a boundary with positive index and the adjacent component
should be branched, but Calready contains all the branching (so we do not
have this possibility)
(4) kC= 2, χ(C) = −1, and we have PI(l) = 0, which implies that all bound-
aries have zero index.

To do a similar study for positive branched components we need some preliminary
results. A straightforward consequence of 3.22 is that disks are always unbranched
when we have only two branch points; in particular a real component bounding a
geometric disk has index 1. We want to prove an analogous statement for essential
real components.
Lemma 3.25. Let σ∈ M2,ρ be geometrically branched. If a component C⊂σ±is
not a disk and contains a single simple branch point, then the inclusion i:C ֒→S
can not be nullhomotopic (i.e. i∗(π1(C)) ⊂π1(S)can not be the trivial subgroup).
Proof. By contradiction assume i∗(π1(C)) ⊂π1(S) is trivial. In particular Cmust
have genus 0 and its boundary must consist of m≥2 (it is not a disk) non essential
boundary components l1,...,lmwith index I(li)≥1. Since i∗(π1(C)) is trivial in
π1(S), the flat bundle associated to ρis trivial on C, hence the Euler class vanishes.
Applying the index formula we obtain
0 = ±eu ρ|π1(C)=χ(C) + kC−
m
X
i=1
I(li)≤2−m+kC−m≤kC−2
which contrasts with the fact that kC= 1. 
Proposition 3.26. Let σ∈ M2,ρ be geometrically branched. If l⊂σRis any real
component, then I(l)≤1.
Proof. Suppose by contradiction we have a real curve l0⊂SRof index I(l0)≥2.
We distinguish two cases.
In the case l0is homotopically trivial, it bounds exactly one subsurface Dhomeo-
morphic to a disk one one side and another subsurface S′which is not a disk on the
other side. This subsurface Dcan either be a geometric disk, or it can consist of
more than just one single geometric component. In the first case it is unbranched
by 3.22 hence l0should have index 1; in the second case the geometric compon-
ent Cof Dwhich has l0in its boundary is a non disk component, hence it must
be branched; since S′must be branched as well by 3.18,Ccontains exactly one
branch point, but then 3.25 applies and we get a contradiction with the fact that C
is contained in a disk (i.e. with the fact that its inclusion is homotopically trivial).
For the second case, suppose lis essential. Let us call C±the adjacent geo-
metric components. Then C±are branched by 3.18; more precisely kC±= 1, they
are not disks since l0is essential and all other components are unbranched, since
C+∪C−contains all the branching. The two components C±may have m≥0
more boundaries in common, let us call them l1,...,lm. Moreover each of them
can have more boundary components, either essential or not. Let us focus on C+;

20 LORENZO RUFFONI
its boundary consists of l0, l1,...,lmand possibly of some other non essential com-
ponents l′
1,...,l′
nand some essential ones l′′
1,...,l′′
p, for some n, p ≥0. Once again,
the non essential components l′
1,...,l′
nmust bound unbranched disks D′
1,...,D′
n
(hence they have index 1), and the essential components l′′
1,...,l′′
pmust bound
unbranched components which are not disks (hence they have index 0 and are es-
sential).
We consider the subsurface E=C+∪D′
1...D′
nand we see that it is incompressible:
l′′
1,...,l′′
pare essential by definition, l1,...,lmare non separating curves in S(C+
and C−are adjacent along l0in any case), hence they are essential as well, as soon
as m≥1. The only case we need to check is when m= 0, but we are currently
discussing the case in which l0is essential.
Then we apply the index formula and get
eu ρ|π1(E)=χ(E) = χ(C+) +
n
X
i=1
χ(D′
i) = χ(C) + n
On the other hand, as in the previous proofs, we obtain
eu ρ|π1(E)=eu ρ|π1(C)=χ(C) + kC−I(l0)−
m
X
i=1
I(li)−
n
X
j=1
I(l′
j)−
p
X
h=1
I(l′′
h)
=χ(C) + 1 −I(l0)−
m
X
i=1
I(li)−n
By comparing the two expressions we obtain that
2n+I(l0) +
m
X
i=1
I(li) = 1
Now we have that the left hand side is a sum of non negative integers and that
I(l0)≥2 by hypothesis, therefore in any case we reach an absurd. 
Now we can prove the following result about positive branched components,
which is analogous to 3.24 for the branched negative incompressible ones; a descrip-
tion of branched negative compressible components will follow from 3.28 below.
Lemma 3.27. Let σ∈ M2,ρ be geometrically branched. Let C⊂σ+be a branched
positive component. Then
(1) if Cis incompressible then kC= 1 and there is a unique boundary curve of
index 1, loxodromic holonomy and the component beyond it is branched;
(2) if Cis compressible then kC= 2 and there is a unique boundary curve of
index 1, trivial holonomy and the component beyond it is an unbranched
disk.
Proof. If Cis incompressible then we apply the index formula and get kC=
Pl⊂∂C I(l). Moreover every boundary component is essential, and by 3.26 its index
is at most 1. Therefore we have exactly kCcomponents of index 1 (and possibly
some components of index 0). Being essential, they do not bound disks, hence the
adjacent components are branched. In particular if kC= 2 then there are two
boundaries with index 1 and thus some branched component is adjacent to C; but
Calready contains all the branching, hence kC= 1 and there is a unique real com-
ponent of index 1. Since it is essential and we are in quasi-Fuchsian holonomy, the
holonomy around the curve will be loxodromic. Of course the component beyond

BUBBLING CP1-STRUCTURES WITH QUASI-FUCHSIAN HOLONOMY 21
it is branched by 3.18.
If Cis compressible, then let us say there are m≥1 non essential boundaries
l1,...,lm(which have index 1 by 3.26, since non essential curves have strictly pos-
itive index by 3.12) and n≥0 essential boundaries l′
1,...,l′
n,n0of which have
index 1 (and the others have index 0 by 3.26). Then we can cap Cwith these ad-
jacent negative disks and apply the index formula to the resulting incompressible
subsurface E
χ(C) + m=χ(E) = eu ρ|π1(E)=eu ρ|π1(C)=χ(C) + kC−m−n0
2m+n0=kC
Since m≥1 but kC≤2, this implies that indeed kC= 2, m = 1, n0= 0. 
The above study was focused on a single branched component, but now we go
global with the help of 3.23.
Theorem 3.28. Let Sbe a closed, connected and oriented surface of genus g≥2,
ρ:π1(S)→PSL2Cbe a quasi-Fuchsian representation and σ∈ M2,ρ be geomet-
rically branched. Let k±denote the number of branch points in σ±
(1) If k+= 2 then both branch points live in the same positive component;
more precisely there exists a unique negative unbranched disk and the branch
points live in the positive component which is adjacent to it.
(2) If k−= 2 then both branch points live in the same negative component;
more precisely there exists a negative component of Euler characteristic −1
containing both branch points. Moreover it has at most one non essential
boundary component (with trivial holonomy and index 1), while all essential
boundaries have loxodromic holonomy and index 0.
(3) If k+=k−= 1 then the two branched components are adjacent along
an essential real component with index 1and loxodromic holonomy; the
negative branched component is an incompressible annulus.
Moreover in each case all the other positive components are unbranched incompress-
ible and all the other negative components are unbranched incompressible annuli and
all the other real curves have index 0.
Proof. We consider the three cases.
(1) We have 2χ(σ−) = k+−k−= 2, so χ(σ−) = 1, thus there must be a
negative disk D. Let Cbe the positive component adjacent to D. By 3.23
Ccontains 2 (i.e. all) branch points and indeed there are no other negative
disks.
(2) We have 2χ(σ−) = k+−k−=−2, so χ(σ−) = −1. By 3.23 negative com-
ponents are either unbranched incompressible annuli or components with
Euler characteristic −1 and 2 branch points; hence there is exactly one of
the latter kind. If it is incompressible, then it has the required boundary
behaviour by 3.24. If it is a pair of pants and it has one non essential
boundary component, then the adjacent component is a disk (because it is
unbranched), hence the index is 1. If it had two non essential boundaries,
then also the third boundary would be non essential, but then all the com-
ponents adjacent to the three boundaries must be disk and Swould be a
sphere, so this case is absurd.

22 LORENZO RUFFONI
(3) Let Cbe the positive branched component. Since it has only one branch
point, by 3.27 it is incompressible and has a unique boundary component
of index 1 and hyperbolic holonomy. The negative component adjacent
along it can not be a disk, hence it is branched, with one branch point. By
3.24 it is an incompressible annulus and the other boundary component has
index 0. Moreover notice that the only negative disks could appear at the
boundary of C, but this is forbidden since it is incompressible.
The rest of the statement follows from the initial discussion: the non branched
components can not be disks, hence they are incompressible and with zero index
boundary by 3.17. The negative ones are annuli again by 3.17. As a consequence
all real curves have index 0, except the non essential ones in the case k±= 2 and
the curve separating the branch points in the case k+= 1, which have index 1. 
This gives a description of negative branched components also in the compressible
case, which was still missing so far.
Remark 3.29.A direct consequence of this classification is that in the case k+=
1 = k−we can always satisfy the hypothesis of [2, Theorem 7.1], hence we can
move branch points without crossing the real curves to obtain a structure which is
a bubbling over some unbranched structure. This is a key fact in the proof of the
main theorem below (see 5.9).
We conclude with the following minor but curious application of 3.28.
Corollary 3.30. Let σ∈ M2,ρ be geometrically branched. Then the number of
branch points contained in σ+and the total number of real components always sum
to an odd number.
Proof. If k+= 0,2 then there is a negative component of Euler characteristic ±1. In
both cases it has an odd number of boundary components. All the other negative
components are incompressible annuli. The total number of real components is
therefore odd. if k+= 1 then the positive branched component is incompressible
and there is exactly one index 1 real boundary, beyond which the negative branched
component sits. And it is an annulus. All other negative components are annuli
too, hence we have an even number of real components. 
4. BM-configurations
As observed above in 3.29, when a structure with two branch points and quasi-
Fuchsian holonomy has a positive branch point and a negative one, then it can be
slightly deformed inside M2,ρ without changing the induced geometric decomposi-
tion so that a bubble appears. However it is not clear a priori whether this bubble
is preserved when we keep deforming the structure to reach other regions of M2,ρ.
In this section we study what happens when we try to move branch points along
an embedded twin pair based at one of the vertices of a bubble (recall from 2.18
than moving branch points provides local coordinates on M2,ρ ). We find it useful
to introduce the following notation: if X ⊂ M2,ρ then we denote by BX the sub-
space of Xmade of BPSs which are obtained via a bubbling over some unbranched
structure from M0,ρ. At first we just pick a non elementary representation ρ; we
will specify when we will need to restrict to the quasi-Fuchsian case.

BUBBLING CP1-STRUCTURES WITH QUASI-FUCHSIAN HOLONOMY 23
4.1. Standard BM-configurations. We begin naively with the easy situation in
which points can be moved without affecting the bubble.
Definition 4.1. Let σ∈ BM(1,1),ρ. A BM-configuration (Bubbling-Movement
configuration) on σis the datum of a bubble Btogether with an embedded twin
pair µbased at a vertex pof B. We denote the configuration by (B, µ, p).
We introduce now the nicest type of BM-configuration, which will allow us to
perform local deformations of the structure preserving the bubble.
Definition 4.2. A BM-configuration (B, µ, p) on σ∈ BM(1,1),ρ is said to be a
standard BM-configuration if either all the arcs are disjoint and disjointly de-
veloped outside the obvious intersections (i.e. ∂B ∩µ={p}and dev(∂B)∩dev(µ) =
{dev(p)}) or the embedded twin pair is entirely contained in the boundary of the
bubble (i.e. µ1, µ2⊂∂B).
p•q
•
B
◦ ◦
y2y1
µ1µ2
Figure 9. A standard BM-configuration
Notice that, given a BM-configuration (B, µ, p) which is standard in the second
sense, a very tiny isotopy of the bubble (which is allowed by 2.20) reduces (B, µ, p)
to a BM-configuration which is standard in the first sense. Namely in any pro jective
coordinate we can push the developed image of the arc of bubbling slightly to the
left or right of itself; when referring to a standard BM-configuration we will really
always think of the first sense. We have the following characterisation.
Lemma 4.3. Let σ∈ BM(1,1),ρ and let (B , µ, p)be a BM-configuration on it, such
that σ=Bub(σ0, β )for some bubbleable arc β⊂σ0∈ M0,ρ. Then (B, µ, p)is a
standard BM-configuration if and only if µinduces an arc µ′on σ0such that the
concatenation of βand µ′is a bubbleable arc on σ0.
Proof. When we debubble σwith respect to Bwe naturally end up with the un-
branched structure σ0endowed with a bubbleable arc βsuch that Bub(σ0, β) = σ.
One of the two arcs contained in the embedded twin pair, let us say µ2, starts
outside the bubble, hence its germ survives in σ0, and we can try to analytically
continue it to a path µ0which has the same developed image of µ. If the BM-
configuration is standard then µ2never meets the bubble, thus µ0is a simple arc
on σ0, which does not meet βaway from p; in other words the concatenation of β
and µ0is a simple arc on σ0. Moreover the developed image of this arc is given by
the concatenation of the developed image of ∂B and µ, which are disjoint. Thus
this arc is bubbleable on σ0. Conversely, if this arc is bubbleable, then when we
perform the bubbling we can reconstruct the embedded twin pair µby looking for

24 LORENZO RUFFONI
the twin of µ0inside the bubble. Since the whole βµ0is bubbleable and we are
bubbling only along the subarc β, we see that the developed image of the remaining
part does not cross that of β. This means exactly that the twin starting inside the
bubble will not leave it. Therefore the induced BM-configuration is standard. 
The interest in standard BM-configurations is motivated by the following lemma.
Lemma 4.4. Let σ0∈ BM(1,1),ρ and (B, µ, p)be a standard BM-configuration on
it; let σtbe the BPS obtained by moving branch points on σ0along µup to time t,
where t∈[0,1] is a parameter along the developed image of µ. Then σt∈ BM(1,1),ρ
for all t∈[0,1].
Proof. This directly follows from the characterisation in 4.3 together with [2, Lemma
2.9]. In the above notations we have that σt=Move(σ0, µt) = Bub(σ0, βµ′t), where
µtand µ′tare the subarcs of µand µ′respectively from time 0 to time t.
We are now ready to prove the following result.
Theorem 4.5. Let ρ:π1(S)→PSL2Cbe a non elementary representation. Then
BM(1,1),ρ is open in M(1,1),ρ (hence in M2,ρ).
Proof. By 4.4 it is enough to show that given σ0∈ BM(1,1),ρ there is a small
neighbourhood Uof it such that any structure in Uis obtained by moving branch
points along a standard BM-configuration on σ0. This easily follows from the fact
the moving branch points gives a full neighbourhood of σ0in the moduli space,
because local movement of branch points can always be performed along embedded
twin pairs which are in standard BM-configuration with a given bubble on σ0.
Notice however that a priori more complicated BM-configurations might arise,
which can not be used to move branch points preserving the bubble; namely if the
embedded twin pair intersects the boundary of the bubble (or if this holds for their
developed images), then moving branch points results in the break of the bubble:
the aspiring bubbleable arc is either not embedded or not injectively developed. In
this case it is not clear if it is possible to find another bubble on the spot.
• •
◦◦
Figure 10. A non standard BM-configuration
This heuristic argument can be made more precise by the following observation:
moving branch points on a standard BM-configuration preserves the isotopy class
(relative to endpoints) of the bubble; in particular it does not change the underlying
unbranched structure. On the other hand it is not difficult to produce examples
of movements of branch points which do not preserve the underlying unbranched
structure.

BUBBLING CP1-STRUCTURES WITH QUASI-FUCHSIAN HOLONOMY 25
Example 4.6. Let ρ:π1(S)→PSL2Cbe a Fuchsian representation, βbe a
bubbleable arc on the hyperbolic surface σρ=H2/ρ and σ=Bub(σρ, β ). Notice
that if dev is a developing map for σand x, y are its branch points, then it is not
possible to find a couple of developed images bxand byof them such that bx=by; as
we will say below, such a structure is said to be simply developed: the developed
images of the branch point have disjoint ρ-orbits. On the other hand it is possible
to move branch points on σalong suitable embedded twin pairs µand νwith both
endpoints inside the bubble in such a way that the resulting structure does not have
this property (see Picture 11). This of course prevents the structure Move(σ, µ)
from being a bubbling over σρ. Of course the BM-configuration on σis not standard.
•
x•y
∗
∗
∗
∗•
•
•
∗
∗
∗
•
•
•
bxby
CP1
Figure 11. Picture for Example 4.6.
4.2. Taming developed images and avatars. One of the main technical is-
sues about CP1-structures is that the developing map is dramatically non injective
(already in the case of unbranched structures), hence it is quite difficult to control
the relative behaviour of the developed images of some configuration of objects on
the surface, even when the configuration is well behaved on the surface, as seen in
Example 4.6.
Definition 4.7. Let H≤π1(S) be a subgroup and U⊂Sbe any subset. Let σ
be a BPS on S. We say that Uis H-tame (with respect to σ) if for some lift e
U
of Uwe have that a developing map for σis injective when restricted to ∪h∈Hh. e
U.
We will just say Uis tame if it is π1(S)-tame.
Notice that a tame simple arc is in particular bubbleable, and that a tame simple
closed curve is in particular graftable as soon as the holonomy is loxodromic.
Example 4.8. Any subset of a hyperbolic surface is tame, simply because the
developing map is globally injective. More generally, if σis a quasi-Fuchsian BPS
and C⊂σ±is an unbranched geometric component, then any subset of Cis
π1(C)-tame, and any subset of the convex core of Cis tame.
Being able to control the collection of developed images of a given object on the
surface (e.g. a curve) will not be enough in the following. For example, even if we
start with a very well behaved structure σ0(e.g. a hyperbolic surface), when we
perform a bubbling or a grafting we introduce in our structure σ0a region Rwhose

26 LORENZO RUFFONI
full developed image is the whole model space CP1; as a result, inside Rwe “see” a
lot of developed images of any given subset U⊂σ0. The following definition aims
at making this more precise.
Definition 4.9. Let ρ:π1(S)→PSL2Cbe a representation, σ∈ Mk,ρ and U⊂σ
be any subset. An avatar of Uis any subset V⊂σsuch that there exist a lift e
U
of Uand a lift e
Vof Vsuch that dev(e
U) = dev(e
V). A structure σ∈ M(1,1),ρ is said
to be simply developed if the two branch points are not avatars of each other.
Example 4.10. If a structure has an injective developing map, then having the
same developed image means being the same set, so that there are no non-trivial
avatars. This happens for a hyperbolic surface, and more generally for the uniform-
izing structure σρof a quasi-Fuchsian representation ρ:π1(S)→PSL2C.
In quasi-Fuchsian holonomy we have a well-defined notion of size for subsets
avoiding the real curve, which allows us to control the collection of avatars of a
small set, as the following result shows. Let us denote by sys(ρ) the systole of the
uniformizing structure σρor, equivalently, the minimum of the translation lengths
of the elements in ρ(π1(S)).
Lemma 4.11. Let ρ:π1(S)→PSL2Cbe quasi-Fuchsian and σ∈ M0,ρ. Let U⊂σ
be a connected set with diam(U)< sys(ρ)and which is π1-trivial (i.e. i∗(π1(U)) ⊂
π1(S)is the trivial subgroup). Then Usits inside a geometric component, it is tame
and its avatars are disjoint.
Proof. Recall that when the holonomy is quasi-Fuchsian there is a well defined
hyperbolic metric on the complement of the real curve, which blows up in a neigh-
bourhood of it; hence we can define a generalised path metric on the whole surface.
Any connected subset of σwhich intersects the real curve must have infinite dia-
meter with respect to this metric, because any path intersecting the real curve has
infinite length. Therefore Ucan not intersect the real curve, hence it is contained
in some geometric component.
Since Uis π1-trivial, it lifts homeomorphically to the universal cover. To prove
tameness, assume that there are two lifts e
U1and e
U2which overlap once developed,
i.e. ∃xi∈e
Uisuch that dev(x1) = dev(x2). Let γ∈π1(S) be the unique deck
transformation such that γe
U1=e
U2. Then we have the following absurd chain of
inequalities
sys(ρ)≤d(ρ(γ)dev(x1), dev (x1)) = d(ρ(γ)dev(x1), dev(x2)) =
=d(dev(γx1), dev(x2)) ≤diam(dev(e
U2)) = diam(U)< sys(ρ)
where ddenotes the hyperbolic distance on CP1\Λρand the last equality fol-
lows from the fact that the restriction of the developing map to each geometric
component is an isometry.
Finally let us prove that the avatars in each geometric component are disjoint.
Let Cbe a geometric component, and choose a lift e
Cof it and a lift e
Uof U. The
collection of avatars of Uin Cis given by
π|Cdev−1
|Cdev π1(S)e
U
So we want to prove that this is a disjoint collection. By tameness we know that the
collection dev π1(S)e
Uis disjoint, and the same is true for dev−1
|Cdev π1(S)e
U,

BUBBLING CP1-STRUCTURES WITH QUASI-FUCHSIAN HOLONOMY 27
since the restriction of the developing map to each geometric component is a dif-
feomorphism and dev π1(S)e
Usits in the upper-half plane because Uis entirely
contained in a positive geometric component. So we only need to prove that the
projection πdoes not overlap things too much. Let us introduce the following
notation: if γ∈π1(S) then
γ∗e
U:= dev−1
|Cdev γe
U
With this notation what we want to prove now is that if there exist γ1, γ2∈π1(S)
such that πγ1∗e
U∩πγ2∗e
U6=∅then actually πγ1∗e
U=πγ2∗e
U. So
let xi∈γi∗e
Usuch that π(x1) = π(x2). Then ∃γ∈π1(C) such that γx1=x2. If
we develop these points we see that
dev(x2) = dev(γx1) = ρ(γ)dev(x1)
and that dev(x2)∈dev(γ2∗e
U) = ρ(γ2)dev(e
U) and ρ(γ)dev(x1)∈ρ(γ)dev(γ1∗
e
U) = ρ(γγ1)dev(e
U). Since we already know that Uis tame, we can conclude that
ρ(γγ1) = ρ(γ2), hence that γγ1=γ2, because quasi-Fuchsian representations are
faithful. But then we have that
γ2∗e
U= (γγ1)∗e
U=dev−1
|Cdev γγ1e
U=dev−1
|Cρ(γ)dev γ1e
U
The last term is indeed equal to γdev−1
|Cdev γ1e
U, because γ∈π1(C). So we
have proved that γ2∗e
U=γγ1∗e
Ufor γ∈π1(C), which of course implies that
πγ1∗e
U=πγ2∗e
Uas desired. 
Notice that the proof above shows that in the collection dev−1
|C(dev(π1(S)e
U))
either two elements differ by an automorphism of the universal cover π:e
C→C
and project to the same set on C, or they pro ject to disjoint sets on C. In other
words the avatars of Uin Ccan be labelled by the cosets of π1(C) in π1(S); the
index of π1(C) in π1(S) is 1 in the case of the uniformizing structure (where there
are no non-trivial avatars, as already observed), and infinite otherwise, because in
all the other cases any geometric component is a non compact (incompressible)
subsurface and free subgroups of surface groups have infinite index. We conclude
with the following technical lemma which says that it is always possible to nicely
isotope a bubbleable arc in order to minimise its intersections with a sufficiently
small neighbourhood of its endpoints.
Lemma 4.12. Let ρ:π1(S)→PSL2Cbe quasi-Fuchsian and σ∈ M0,ρ. Let
β⊂σbe a bubbleable arc with endpoints x, y such that x6∈ σR. Let U⊂σbe a
connected π1-trivial neighbourhood of xwith diam(U)< sys(ρ)and not containing
any avatar of y. Then there is an injectively developed isotopy (relative to xand y)
from βto another bubbleable arc β′, such that β′does not intersect any non-trivial
avatar of Uand β′∩Uis connected (i.e. β′does not come back to Uafter the first
time it leaves it).
Proof. First of all notice that if Udoes not contain avatars of y, then in particular
yis not an avatar of x. Moreover no avatar of Ucontains avatars of y; in particular
no avatar of Ucontains y. We also know by 4.11 that Uis geometric (i.e it avoids

28 LORENZO RUFFONI
the real curve), tame and its avatars are disjoint. Since Uis geometric, for ε > 0
small enough the ε-neighbourhood Nε(U) of Uenjoys the same properties.
Let {Ui}i∈Ibe the collection of avatars of Ucrossed by β. Going along βfrom
xto ywe see that, apart from the initial segment starting at xinside U, every time
βenters one of the Ui’s it crosses it and leaves it (this is exactly because no avatar
contains the second endpoint y). Therefore we can isotope all the arcs given by
β∩Uito arcs living in Nε(Ui)\Ui, for each i∈I, without touching the first segment
starting at x; since the chosen neighbourhood is tame this can be done in such a
way that the isotopy is injectively developed. Since all the Nε(Ui) are disjoint, this
gives an isotopy on σfrom βto an arc β′which intersects the whole collection of
avatars only in the initial segment starting at xin U. It is still a bubbleable arc
because it coincides with β(which is bubbleable) outside the Nε(Ui)’s, and the
deformations inside these sets do not produce any new intersection because Nε(U)
is tame. 
•
•
β
•
•
β′
Figure 12. Pushing an arc outside the avatars of a neighbourhood
of one endpoint.
To get an intuition of what can go wrong, consider for instance the following
picture; the second endpoint ybelongs to one of the avatars, hence there is no
guarantee that the deformation that we want to perform is an injectively developed
isotopy.
•
•
β
•
•
β′
Figure 13. Avoiding avatars may result in self-intersections.
4.3. Visible BM-configurations. This section is about a class of BM-configura-
tions with the property that, roughly speaking, the embedded twin pair survives
after debubbling the structure, as in the proof of 4.3; these should be thought as
a strict generalisation of standard BM-configurations, which can still be dealt with

BUBBLING CP1-STRUCTURES WITH QUASI-FUCHSIAN HOLONOMY 29
by exploiting the underlying unbranched structure, where deformations are more
easily defined and controlled.
Definition 4.13. Let σ0∈ M0,ρ ,β⊂σ0a bubbleable arc and σ=Bub(σ0, β)∈
M2,ρ with distinguished bubble Bcoming from β. Let pbe a branch point of σ
and µan embedded twin pair based at pwith developed image bµ. Notice that the
germ of µis well-defined on σ0. We say that the BM-configuration (B, µ, p) is a
visible BM-configuration if we can take the analytic continuation of this germ
on σ0to obtain a properly embedded path µ0on σ0which still develops to bµ.
•
p
µ
•
B
◦
◦
• •
p◦
µ0
β0
Figure 14. A visible BM-configuration.
Example 4.14. A standard BM-configuration is visible: as already observed in
4.3, the boundary of the bubble and the embedded twin pair of a standard BM-
configuration induce a pair of adjacent embedded arcs on the debubbled structure,
whose concatenation is actually a bubbleable arc itself.
Example 4.15. If σis a standard bubbling over a hyperbolic surface and we
pick an embedded twin pair µwhich intersects the real curve, then the resulting
configuration is not visible: the debubbled structure is the uniformizing hyperbolic
structure, which has no real curve, so there can be no path on it developing as
needed; the analytic continuation of the germ of µis an arc which wraps around
the surface without converging to a compact embedded arc.
The next result shows that in quasi-Fuchsian holonomy visible BM-configurations
can be deformed to standard BM-configurations in a controlled way.
Proposition 4.16. Let ρ:π1(S)→PSL2Cbe quasi-Fuchsian, σ0∈ M0,ρ,β⊂
σ0a bubbleable arc. Let x, y be the branch points of σ=Bub(σ0, β )and Bthe
bubble coming from β. Assume σis simply developed and x6∈ σR. Let K=
infγ∈π1(S)d(dev(x), ρ(γ)dev(y)) ∈]0,+∞]and let µbe an embedded twin pair based
at xsuch that (B, µ, x)is a visible BM-configuration and the length of µis less than
min{sys(ρ), K }. Then there is another bubble B′⊂σsuch that Debub(σ, B′) = σ0
and (B′, µ, x)is a standard BM-configuration.
Proof. Since the BM-configuration is visible, after debubbling σwe can define an
arc µ0on σ0starting at xand developing as µ. By hypothesis this arc is shorter
than sys(ρ) and K; in particular it can be put inside a connected contractible

30 LORENZO RUFFONI
neighbourhood Uof xwith diam(U)< sys(ρ) and which does not contain any
avatar of y. By 4.12 there is an injectively developed isotopy from βto a bubbleable
arc β′which avoids all non trivial avatars of Uand intersects Ujust once at the
starting segment at x. Since this isotopy is injectively developed, bubbling σ0along
β′gives a structure isomorphic to σby 2.21. Moreover the fact that µ⊂Uand
that β′avoids all non trivial avatars of Uand does not come back to it after the
first time it leaves it implies that the concatenation of µand β′is a bubbleable
arc; this is equivalent to saying that the resulting BM-configuration (B′, µ, x) is
standard by the characterisation in 4.3.
•
x
µ
•
◦
◦
• •
x◦
µ0
Figure 15. Deforming a visible BM-configuration into a standard one.
compare to 4.6
Remark 4.17.The above result means that moving branch points on a given bub-
bling by a very small displacement (with respect to the representation) does pre-
serve the [isotopy class of the] given bubble. In particular the underlying un-
branched structure can be left unchanged throughout the movement. Notice that
the hypothesis on the length is indeed necessary, as shown in Example 4.6: the
BM-configuration therein is of course a visible one, but the embedded twin pairs
are too long with respect both to the representation (the constant sys(ρ) in 4.16)
and to the relative distance between the developed image of the branch points (the
constant Kin 4.16).
The condition of being visible is a bit obscure, if compared to that of being
standard, in the sense that we have to debubble the structure to check visibility,
and we do not have a simple characterisation as the one in 4.3 for standard BM-
configurations; but visibility is always at least locally available at geometric branch
points, as shown by the following result.
Lemma 4.18. Let ρ:π1(S)→PSL2Cbe quasi-Fuchsian, σ0∈ M0,ρ,β⊂σ0a
bubbleable arc such that σ=Bub(σ0, β )∈ M2,ρ has a branch point xnot on the
real curve. Let µbe an embedded twin pair based at xof length smaller than sys(ρ).
Then the resulting BM-configuration is visible.
Proof. Let us fix a developed image bxof xand bµfor µ. Since l(bµ)< sys(ρ), it is
contained in a fundamental domain for ρ, and a fortiori in a fundamental domain for
ρ|H, for any subgroup H≤π1(S). Therefore it pro jects injectively to every quotient

BUBBLING CP1-STRUCTURES WITH QUASI-FUCHSIAN HOLONOMY 31
H2/ρ(H). Now consider the debubbled structure σ0, and let Cbe the geometric
component of σ0containing x; since σ0is unbranched, Cis incompressible and
carries a complete unbranched hyperbolic structure, so that the developing map
induces an isometry DC:C→H2/ρ(π1(C)), where bµprojects injectively to an
embedded arc. Pulling that arc back by DCgives the desired arc on σ0, which
proves the visibility of the BM-configuration. 
Remark 4.19.We want to remark that it is not possible to apply these ideas to a
movement of a branch point which sits on the real curve. Indeed, here geometri-
city is used to produce neighbourhoods of the relevant objects which have disjoint
avatars. On the other hand if a point belongs to the real curve, then any of its
neighbourhoods will contain infinitely many avatars of both branch points, and ac-
tually of whatever object we want to consider. This follows from the fact that if Γ
is a Fuchsian group, then the collection of fixed points of its hyperbolic elements is
dense in the limit set ΛΓ=RP1.
5. Bubbles everywhere
In 4.5 we have proved that if ρ:π1(S)→PSL2Cis a non elementary repres-
entation, then the space of bubblings BM2,ρ ⊂ M2,ρ is an open subspace of the
moduli space M2,ρ . In this section we prove the main result of this paper, i.e. that
in quasi-Fuchsian holonomy it is also dense. The strategy will be to consider a
decomposition of the moduli space M2,ρ obtained by looking at the position of the
branch points with respect to the real curves: first of all we will show that if a piece
of this decomposition contains a bubbling, then the bubblings fill a dense subspace
in it, and then we will see that actually every piece of the decomposition contains a
bubbling. The key step is the combination of Theorem 3.28 above with [2, Theorem
7.1]. Let us begin by defining this decomposition: recall from 3.10 that if σis a BPS
with quasi-Fuchsian holonomy, then a branch point is geometric (respectively real)
if it is outside (respectively on) the real curve; σitself is said to be really branched
if it has some real branch points, and geometrically branched otherwise.
Lemma 5.1. The space MR
2,ρ of really branched structures has real codimension 1
in M2,ρ.
Proof. This is a direct consequence of the fact that moving branch points on a
structure provides local neighbourhoods for the manifold topology of M2,ρ (see
2.18), and the fact that the real curve σRis an analytic curve on the surface. 
As a result the moduli space M2,ρ decomposes as the union of the real hyper-
surfaces of MR
2,ρ and the remaining open pieces ∪Xi=M2,ρ \ MR
2,ρ consisting of
geometrically branched structures.
Definition 5.2. We will refer to the decomposition M2,ρ =MR
2,ρ ∪Si∈IXias the
real decomposition of M2,ρ; any connected component Xiof M2,ρ \ MR
2,ρ will
be called a geometric piece of the real decomposition of M2,ρ.
Moving a geometrically branched structure σfrom a geometric piece Xito an
adjacent one Xjis a quite dramatic deformation, since it involves crossing MR
2,ρ, i.e.
moving branch points beyond the real curve σRof σ. This forces the combinatorial
properties of the geometric decomposition to change abruptly; on the other hand,
moving branch points on σinside their own geometric components keeps σinside

32 LORENZO RUFFONI
the piece Xiit belongs to. This can actually be done in a quantitatively controlled
way in order to preserve existing bubbles, as the following result shows.
Theorem 5.3. Let ρ:π1(S)→PSL2Cbe quasi-Fuchsian. Let X ⊂ M2,ρ \ MR
2,ρ
be a geometric piece of the real decomposition and let Y ⊂ X be the subspace of
simply developed structures. If Xcontains a bubbling, then every structure in Yis
a bubbling; in particular Y=BY is open and dense in X.
Proof. First of all notice that Yis an open dense connected submanifold of X, since
its complement is a complex analytic subspace of complex (co)dimension 1. As a
consequence of 4.5 BX is open; since Yis dense, it contains a bubbling too, i.e. BY
is an open non empty subset of Y. We will prove that BY is also closed in Yand
conclude by connectedness of Y.
Let σ∞∈ Y ∩ BY. By hypothesis the branch points x∞and y∞of σ∞are outside
the real curve of σ∞and not avatar of each other. Fix any developed image bx∞of
x∞and by∞of y∞; then define K∞= inf γ∈π1(S)d(bx∞, ρ(γ)by∞). The distance here
is the one induced by the hyperbolic metrics on the domain of discontinuity of ρ;
K∞is strictly positive since the branch points of σ∞are not avatars, but can be
+∞in the case they have opposite sign. Then let A= min{sys(ρ),1
3K∞}.
Choose L < A and consider the neighbourhood NL(σ∞) of σ∞in Yobtained
by moving branch points by a distance L < A (this is well defined since σ∞is
geometrically branched). Since σ∞is in the closure of BY,NL(σ∞) will contain
a bubbling σ∈ BY . Let ζbe an embedded twin pair based at x∞and ξbe an
embedded twin pair based at y∞such that M ove(σ∞, ζ, ξ ) = σ. By construction
they can be chosen to have length smaller than L. Let b
ζand b
ξbe the developed
images based at bx∞,by∞, and let σ0∈ M0,ρ and β⊂σ0be such that σ=Bub(σ0, β).
Also let x, y be the branch points of σcorresponding to x∞, y∞respectively.
σ∞σ
σ′
NL(σ∞)⊂ Y ⊂ M2,ρ
•
bx∞
•
bx
b
ζ•
by
•
by∞
b
ξ
CP1
Figure 16. The neighbourhood NL(σ∞) and the movement of
points in CP1.
We are now going to show that σ∞is actually a bubbling over the same σ0. First
of all notice that by 4.18 both BM-configurations are visible, since both embedded
twin pairs are shorter than the systole of the representation. Moreover, by definition
of A, the two movements are independent from each other, i.e. commute; more
precisely they do not interfere with each other in the sense that each twin pair

BUBBLING CP1-STRUCTURES WITH QUASI-FUCHSIAN HOLONOMY 33
avoids all avatars of the other twin pair. We begin by focusing at x; let us denote
by bx, bythe developed images of x, y which are seen at the endpoints of b
ζand b
ξ. We
have that for any γ∈π1(S)
K∞≤d(bx∞, ρ(γ)by∞)≤d(bx∞,bx)+d(bx, ρ(γ)by)+d(ρ(γ)by, ρ(γ)by∞) = 2L+d(bx, ρ(γ)by)
so that
d(bx, ρ(γ)by)≥K∞−2L
and we get
inf
γ∈π1(S)(d(bx, ρ(γ)by)) ≥K∞−2L > 3L−2L=L=l(b
ζ)
by definition of A. Therefore we can apply 4.16 and replace βby a new bubbleable
arc which is in standard BM-configuration on σwith respect to ζ. We now let
σ′=Move(σ, ζ ), which is still a bubbling over σ0by 4.4. We now want to use
the same strategy again at yto get back to σ∞; to do so, we just have to check
that the movement is small enough with respect to the distance between the two
branch points of σ′, which now develop to bx∞and by. But this is easily checked: if
γ∈π1(S) then
K∞≤d(bx∞, ρ(γ)by∞)≤d(bx∞, ρ(γ)by) + d(ρ(γ)by, ρ(γ)by∞) = L+d(bx∞, ρ(γ)by)
so that
d(bx∞, ρ(γ)by)≥K∞−L
and we get
inf
γ∈π1(S)(d(bx∞, ρ(γ)by)) ≥K∞−L > 3L−L > L =l(b
ξ)
So we can apply 4.16 again and replace the bubbleable arc with one which is in
standard BM-configuration and safely move branch points along ξ. This movement
results in our structure σ∞and does not break the bubble by 4.4. In other words
this proves that σ∞∈ BY (and indeed the underlying unbranched structure is the
same as that of σand σ′), so that BY is closed. 
Let us denote by k±the number of positive and negative branch points of a
structure as before. Notice that the value of k±is constant on every geometric
piece of the real decomposition, so that it makes sense to say that a piece Xhas
a given value of k+. Combining all the results obtained so far, we can prove the
following.
Corollary 5.4. Let ρ:π1(S)→PSL2Cbe quasi-Fuchsian. Let Xbe a geometric
piece of the real decomposition of M2,ρ with k+= 1. Then X=BX i.e. is entirely
made of bubblings.
Proof. First observe that every structure in Xis simply developed, since branch
points have different sign. Let σ∈ X . By 3.28 it satisfies the hypothesis of [2, The-
orem 7.1]; therefore it is possible to move branch points inside their own geometric
components so that a bubble appears, which proves that Xcontains a bubbling.
The statement then follows from 5.3.
We now have to care about geometric pieces of the real decomposition of M2,ρ
with k+= 0,2. We will prove that any such piece actually contains a bubbling. It
should be said that the results in [2, Proposition 8.1, Lemma 10.5-6] imply that in
some cases branch points can be moved inside their own geometric components so

34 LORENZO RUFFONI
that a bubble appears, but it is not clear how to verify a priori when this occurs.
Our strategy here will be to look for bubblings in the geometric pieces adjacent to
Xand drag them from there back into X. In trying to do so, two problems occur:
on one side if we naively take a bubbling in some piece adjacent to Xand move
branch points on it beyond the real curve, then it is quite difficult to control that
we are actually moving to the chosen piece X; on the other hand if we start with
a structure σ∈ X and move branch points on it across the real curve to get to a
bubbling, then it is quite difficult to check that when we move branch points to get
back to σwe do not break the bubble. Some lemmas are in order to guarantee that
we can handle these issues.
Lemma 5.5. Let Xbe a geometric piece of the real decomposition of M2,ρ with
k+= 0 or 2. Then there exists a geometric piece of the real decomposition Y
adjacent to Xand such that k+= 1.
Proof. This is just a reformulation of the results in [2,§9], which say that it is always
possible to move a branch point along a geodesic embedded twin pair crossing the
real curve. 
We remark that in the process of moving a branch point towards the real curve
with the techniques of [2,§9] a bubble might appear before actually crossing the
real curve; this would be fine for us, since our ultimate goal now is to prove that
Xcontains a bubbling; therefore we will forget about this detail in the following.
The following lemma is needed to guarantee that it is always possible to go from
one piece to an adjacent one by moving along a geodesic embedded twin pair.
Lemma 5.6. Let σ∈ M2,ρ be a geometrically branched BPS and µ={µ1, µ2}
an embedded twin pair on σ. Suppose that µicrosses σRat only one point ri.
Then there exists a geodesic embedded twin pair νon σsuch that Move(σ, µ) =
Move(σ, ν ).
Proof. Let pbe the base point of the embedded twin pair µand yibe the endpoint
of µi. By hypothesis the subarcs µ1
i⊂µifrom pto riare entirely contained in a
geometric component C. We let ν1
ibe the unique geodesic in Cfrom pto riwhich
is isotopic to µ1
irelatively to {p, ri}. Then we can do the same in the adjacent
components to obtain geodesic arcs ν2
iisotopic to the subarcs µ2
i⊂µifrom rito
yi. The concatenation of these paths gives rise to a couple of geodesic paths νifrom
••• • •
p
r1r2
y1
y2
Figure 17. Straightening the embedded twin pair.
pto yiwhich are isotopic to µirelatively to {p, ri, yi}. Each geometric subarc µj
i

BUBBLING CP1-STRUCTURES WITH QUASI-FUCHSIAN HOLONOMY 35
is the geodesic representative of an embedded and injectively developed arc, hence
it is embedded and injectively developed; moreover the two geometric subarcs of
νilive in two adjacent component, hence their developed images are disjoint and
νis thus actually a geodesic embedded twin pair. The isotopy from µto νcan be
chosen to be an isotopy of embedded twin pairs, so that 2.22 applies and gives us
that Move(σ, µ) = M ove(σ, ν). 
Notice that this result does not hold for paths which cross more components,
because subarcs of νcontained in geometric components of the same sign can over-
lap once developed, even if µis an embedded twin pair. We are now ready to prove
that all the pieces of the real decomposition of M2,ρ contain a bubbling. We will
need the following terminology.
Definition 5.7. Let σ∈ Mk,ρ be geometrically branched. Let C⊂σ±be a
geometric component and l⊂∂C a real component in its boundary. We call the
peripheral geodesic of lin Cthe unique geodesic representative γin the free
homotopy class of l. The end of lin Cis the connected component Elof C\γ
which has lin its boundary.
It is shown in [2,§3.3] that ends are embedded open annuli, that ends associated
to different real components are disjoint, and that if a geodesic enters an end, then
it can not leave it and must necessarily reach the associated real curve.
Theorem 5.8. Let ρ:π1(S)→PSL2Cbe a quasi-Fuchsian representation. Let X
be a geometric piece of the real decomposition of M2,ρ. Then Xcontains at least
one structure which is a bubbling over an unbranched structure in M0,ρ .
Proof. If Xhas k+= 1 then this follows directly from 5.4. So let us assume that
k+= 2, the case k+= 0 being the same up to switching the signs of the branch
points. We choose some σ1∈ X and move branch points along an embedded twin
pair µto get to a structure σ2=Move(σ1, µ) in some adjacent piece Ywith k+= 1,
which can be done by 5.5. By 3.28 we know the combinatorial properties of the
geometric decomposition of σ2: all real curves are essential, one has index 1 and
the others have index 0. Let us call lthe unique real curve of index 1; the branch
points p±live in the two geometric components C±adjacent to l. By construction
we have an induced embedded twin pair νat p−on σ2such that M ove(σ2, ν) = σ1.
Here we know by [2, Theorem 7.1] that we can move both branch points inside their
own components to get to a structure σ3∈ Y such that the peripheral geodesics of l
go through the branch points q±of σ3with angles {π, 3π}and also such that it has
a geodesic bubble B(such a bubble can indeed be chosen in many ways, which will
be exploited below). Of course we have an induced couple of embedded twin pairs
ζ±⊂σ3based at q±such that Move(σ3, ζ +, ζ−) = σ1, and we would like to operate
this movement of branch points on σ3without breaking the bubble B; unfortunately
there is no reason why (B, ζ ±, q±) should be a standard BM-configuration.
However for our purposes we do not actually need to move branch points to
go back to σ1: it is enough to move to a structure in the same piece Xwithout
breaking the bubble B. Therefore we can forget about the embedded twin pair
ζ+, since we only need to move q−to go back to that piece. Since ζ−crosses the
real curve just once, by 5.6 we can replace it with a geodesic embedded twin pair
ξ−which is such that σ4=Move(σ3, ξ−) = Move(σ3, ζ −)∈ X . As mentioned
above, the bubble Bon σ3can be chosen in a quite free way, and our aim now is to

36 LORENZO RUFFONI
σ1
σ4σ5
σ2
σ3
XY
M2,ρ
µ
ζ±
ζ−, ξ−
Figure 18. The structures σ1, σ2, σ3, σ4and σ5involved in the proof.
prove that it is always possible to choose the bubble so that the BM-configuration
(B, ξ−
cut, q−) is standard, for some suitable truncation ξ−
cut of the embedded twin
pair ξ−; of course we still have that σ5=M ove(σ3, ξ−
cut)∈ X .
First of all we recall from [2,§7] that the real curve lcarries a natural action of the
infinite cyclic group generated by ρ(l) and a natural ρ(l)-invariant decomposition
l={0} ∪ l+∪ {∞} ∪ l−, corresponding to the decomposition of the limit set of ρ
given by the fixed points of ρ(l); according to [2, Proposition 7.8] for any u∈l+we
can find a geodesic bubble Buintersecting lexactly at uand ρ(l)−1u. Suppose we
pick one of these geodesic bubbles Buand look at the situation on C−, neglecting
for a moment what happens beyond the real curve l. Since the embedded twin pair
ξ−and the bubble Buare both geodesic, when one of the paths of ξ−enters the
bubble it can never leave it, and must reach the real curve l. One of them, let us
say ξ−
1starts inside Bu(up to an arbitrarily small displacement of u), hence hits
lat some point v1. If the BM-configuration (Bu, ξ−, q−) is not already standard,
it means that the twin ξ−
2starting outside Bugoes somewhere around the surface
and then comes back to intersect Buat some point x, and finally hits the real curve
lat some point v2, distinct from v1, because ξ−is an embedded twin pair. Now,
let us show that v2must live in l+. To do this, we choose uso that the bubble Bu
is orthogonal at q−to the peripheral geodesic of l. Since ξ−
2is a geodesic from q−
to l, once it enters the end relative to lit constantly increases its distance from the
peripheral geodesic; in particular, when it intersects the bubble at xit forms an
angle smaller than π
2with the boundary of Bu. Since uis in l+, this forces v2∈l+
as well. But then it is now possible to choose a different u′in such a way that the
arc α⊂lfrom u′to ρ(l)−1u′containing 0 and ∞(i.e. the part of lcontained in
Bu′) does not contain v2. This choice guarantees that v2is outside the bubble Bu′,
hence that ξ−
2does not intersect Bu′before crossing the real curve l. We have no
tools to control what happens beyond l, but we can truncate ξ−to a sub-embedded
twin pair ξ−
cut which ends beyond land which is in standard BM-configuration with
respect to the bubble Bu′. By 4.4 σ5=M ove(σ3, ξ−
cut) is still a bubbling. But we
can clearly keep moving branch points on σ5along what is left of ξ−to reach the
structure σ4=Move(σ3, ξ −), which, as we already know, lives in the same piece
Xcontaining σ1. Since this movement does not cross the real curve, the structure
σ5lives in Xtoo, which proves that Xcontains a bubbling. 

BUBBLING CP1-STRUCTURES WITH QUASI-FUCHSIAN HOLONOMY 37
l
γ
ξ−
1ξ−
2
•q−
•
v1
•
u/λ
•
u
•
v2
•
∞
•
0
•
x
0bγ
•
bq−
•
bu
•
bx
•
bv
Figure 19. The configuration in CP1and C−⊂σ3when Buis
the bubble orthogonal to the peripheral geodesic.
•
u′/λ
•
u′
l
γ
ξ−
1
ξ−
2
•q−
•
v1
•
v2
•
∞
•
0
Figure 20. A bubble in standard BM-configuration.
We can finally prove the main result.
Theorem 5.9. Let ρ:π1(S)→PSL2Cbe quasi-Fuchsian. Then any simply
developed structure with at most one real branch point is a bubbling. In particular
the space of bubblings is connected, open and dense in M2,ρ.
Proof. At first let σbe a geometrically branched and simply developed structure.
Since its branch points are outside the real curve it belongs to some geometric piece
Xof the real decomposition. By 5.8 we know that Xcontains a bubbling. Moreover
σavoids the subspace of Xmade of non simply developed structures. Then by 5.3 σ
is a bubbling. In the case σhas one real branch point, we can perform a movement
of that branch point to go from σto some structure σ′in some geometric piece of
the real decomposition with k+= 1. Then the previous arguments apply verbatim,

38 LORENZO RUFFONI
because the isotopy in 5.6 fixes the points of intersection between the embedded
twin pair and the real curve, so that we are able to pick a bubble on σ′and move
back to σas in 5.8. The subspace of structures left outside by this approach is the
union of the subspaces of non simply developed structures and the one of structures
with both branch points on the real curve; each of them has real codimension 2 in
M2,ρ, which is a connected manifold of real dimension 4 by [2], so that the last
statement follows. 
5.1. Walking around the moduli space with bubblings. As a consequence
of the results obtained in this paper we get a generically positive answer in our
setting to the question asked by Gallo-Kapovich-Marden as Problem 12.1.2 in [5],
i.e. if any two BPS with the same holonomy are related by a sequence of grafting,
degrafting, bubbling and debubbling. More precisely Theorem 5.9 shows that, if
σand τare a generic couple of BPS with at most two branch points and a fixed
quasi-Fuchsian holonomy, then we can apply one debubbling to each of them (if
needed), to reduce to a couple of unbranched structures σ0and τ0with the same
holonomy. By Goldman’s theorem in [6] we can then apply mdegraftings on σ0to
obtain the uniformizing structure σρand then ngraftings on σρto obtain τ0, for
suitable m, n ∈N.
σ
1 debub
σ0
mdegraft
σρ
ngraft
τ0
1 bub
τ
Actually it is possible to do even better, since we can remove the need for de-
graftings; by the proof of [3, Theorem 11], there exists a simple closed geodesic γ
on σρsuch that σγ=Gr(σρ, γ) can be obtained by m′graftings on σ0and τ0can
be obtained by n′graftings on σγ, for suitable m′, n′∈N.
Finally, according to [2, Theorem 5.1] every simple grafting can be realised by
a sequence of one bubbling and one debubbling. This implies the following, which
shows that it is generically possible to move around the moduli space only via
bubblings and debubblings.
Corollary 5.10. Let ρ:π1(S)→PSL2Cbe quasi-Fuchsian. There is a connected,
open and dense subspace B ⊂ M2,ρ such that if σ, τ ∈ M0,ρ ∪ B then τis obtained
from σby a finite sequence of bubblings and debubblings.
σ
1 debub
σ0
m′bub
m′debub
σγ
n′bub
n′debub
τ0
1 bub
τ
Notice that the length of this sequence depends on the choice of the unbranched
structures σ0and τ0(i.e. the choice of the bubbles on σand τ), which are not
uniquely determined: a BPS with two branch points can in general be realised
as a bubbling over different unbranched structures along different arcs. This phe-
nomenon is outside the point of view of this paper, which was concerned with the
preservation of the underlying unbranched structure during all the deformations,
and is dealt with in a forthcoming paper of the author (see [11]).

REFERENCES 39
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