The action of the mapping class group on the pants complex
Ute Wolf
Abstract: For any compact surface S = Σg,b of genus g with
b boundary components, there is a cell complex attached to S,
namely the pants complex P(S). The mapping class group
Mod(S) of S acts cocompactly on this complex. In this article
we calculate the orbits and stabilizers of this action.
1
Introduction
Let S = Σg,b be a compact orientable surface of genus g with b boundary components.
The mapping class group Mod(Σg,b ) is the group of all isotopy classes of orientationpreserving homeomorphisms Σg,b → Σg,b . In the literature, other versions/definitions of
the mapping class group can be found. For example, the extended mapping class group
Mod∗ (Σg,b ) is the group of isotopy classes of all (orientation-preserving and orientationreversing) homeomorphisms Σg,b → Σg,b . On the other hand, Putman [P] defines the
mapping class group as the group of all isotopy classes of orientation-preserving homeomorphisms Σg,b → Σg,b which fix the boundary ∂Σg,b up to isotopy fixing the boundary.
A pants decomposition of Σg,b is a collection of isotopy classes of 3g − 3 + b pairwise
disjoint simple closed curves. Any such collection is maximal and cuts the surface into
pairs of pants, i.e. subsurfaces homeomorphic to Σ0,3 .
We assume that S is a surface with negative Euler characteristic χ(S) = 2 − 2g − b,
i.e. S is none of the surfaces Σ0,0 , Σ0,1 , Σ0,2 , Σ1,0 . In this case pants decompositions of
S do exist, and each pants decomposition consists of 3g − 3 + b curves and divides S
into 2g − 2 + b pairs of pants.
Two pants decompositions P, Q differ by a move if there are curves α ∈ P, β ∈ Q,
α 6= β, such that P \ {α} = Q \ {β} and α, β intersect minimally. If i(α, β) = 1 then
the move P — Q is called a simple move or S-move. This is the case when both sides
of α (resp. β) belong to the same pair of pants. If the sides of α (resp. β) belong to
different pairs of pants, then i(α, β) = 2 and P — Q is called an associativity move or
A-move. Compare Figure 1.
The pants graph P 1 (S) is by definition a graph having all possible pants decompositions
as vertices and all possible moves as edges. It was first defined in 1980 by Hatcher
and Thurston [HT]. In 2000 Hatcher, Lochak and Schneps defined the pants complex
P(S) by inserting certain 2-cells into P 1 (S) and showed that the resulting 2-complex
is connected and simply connected [HLS].
There are 2-cells of type (3S), (3A), (4C), (5A), (6AS). The definition of these cells can
be found in [HLS] or in the thesis [W]. For further use, we recall the definition of a
2-cell of type (6AS). Namely, a 2-cell is inserted in any hexagon of the form
1
Figure 1: S-move and A-move
{γ1 , γ2} — {γ1 , γ3} — {γ3, γ4 } — {γ4, γ5 } — {γ4, γ6 } — {γ2, γ6 }
where all curves γ1 , . . . , γ6 lie in a common subsurface of type Σ1,2 , γ1 — γ4 and γ2 — γ4
being S-moves and all other moves being A-moves, compare Figure 2.
γ1
γ4 γ3
γ3
P2 =
P3 =
γ1
γ5
P1 =
P4 = γ4
γ2
P6 =
P5 =
γ2
γ6
γ4 γ6
Figure 2: A hexagonal 2-cell
The mapping class group Mod(S) acts on the set of all isotopy classes of simple closed
curves via [f ] · [α]≃ := [f (α)]≃ . This defines an action of Mod(S) on P(S) because any
homeomorphism preserves the intersection number of two curves. In this paper we will
investigate this action by calculating its orbits and stabilizers.
This paper is an excerpt of the autor’s thesis “Die Aktion der Abbildungsklassengruppe
auf dem Hosenkomplex” [W]. In that thesis, the stabilizers and orbits of the action are
used to give a new presentation for Mod(S). The details of this presentation will be
given in a further paper.
The author would like to thank Frank Herrlich and Gabriela Schmithüsen for their
advice for both the thesis and this paper.
2
2
Orbits
An important tool for the calculation of the orbits under the action of Mod(S) on P(S)
is the graph Γ(P) which is associated to a pants decomposition P.
Definition 2.1:
Let P be a pants decomposition of S. The vertices of the graph Γ(P) are by definition all
pairs of pants of P together with all boundary components of S. Two pairs of pants are
connected by an edge if they are bounded by the same curve. A boundary component
of S is connected by an edge to that pair of pants which it bounds. The resulting
graph Γ(P) is connected with trivalent vertices corresponding to the pairs of pants and
univalent vertices corresponding to the boundary components of S. See Figure 3 for an
example.
Figure 3: Example of P and Γ(P)
Theorem 2.2:
Two pants decompositions P, Q of S are in the same orbit under the action of the
mapping class group (i.e. there is an f ∈ Mod(S) such that f (P) = Q) if and only if
Γ(P) ∼
= Γ(Q).
Proof:
One direction is given by the definition of the graphs Γ(P), Γ(Q): Any mapping class f ∈
Mod(S) satisfying f (P) = Q induces an isomorphism ϕf : Γ(P) → Γ(Q) by mapping
a vertex of P (= pair of pants or boundary component) to its image under f . On the
other hand, let ϕ : Γ(P) → Γ(Q) be an isomorphism. ϕ induces a homeomorphism
[
[
β ∪ ∂S
F̃ :
α ∪ ∂S →
α∈P
β∈Q
in a straightforward way. We only have to be careful choosing the correct orientation of
mapping a curve onto its image. F̃ can be extended to a homeomorphism F : S → S in
the same straightforward way. The mapping class f = [F ] obviously fulfills f (P) = Q.
xxx
The orbit of an edge K = (P — Q) of the pants complex P(S) is also given by combinatorial properties of the graphs Γ(P) and Γ(Q). But here the situation is slightly
more complicated.
3
Definition 2.3:
x
a) Let K = (P — Q) be an edge in P(S). As in Definition 2.1, the system of
curves P ∩ Q defines a graph Γ(P ∩ Q) whose vertices are the components of the
complement of the curves and the components of the boundary of S.
b) Now let K = (P — Q) and K′ = (P ′ — Q′ ) be two edges in P(S) where in K
(resp. K′ ) the curve α ∈ P (resp. α′ ∈ P ′ ) is moved to the curve β ∈ Q (resp. β ′ ∈
Q′ ). If Γ(P) and Γ(P ′ ) are isomorphic then any isomorphism ϕ : Γ(P) → Γ(P ′ )
with ϕ(α) = α′ induces an isomorphism ϕ̂ : Γ(P ∩ Q) → Γ(P ′ ∩ Q′ ). The same is
valid for Q and Q′ .
Theorem 2.4:
Two oriented edges K = (P — Q), K′ = (P ′ — Q′ ) as in Definition 2.3b) are in the
same orbit under the action of the mapping class group (i.e. there is an f ∈ Mod(S) such
that f (P) = P ′ and f (Q) = Q′ ) if and only if there are isomorphisms ϕ : Γ(P) → Γ(P ′ )
and ψ : Γ(Q) → Γ(Q′ ) satisfying ϕ(α) = α′ , ψ(β) = β ′ and ϕ̂ = ψ̂.
Proof:
If f ∈ Mod(S) maps P and Q to P ′ and Q′ respectively then we have f (α) = α′ ,
f (β) = β ′ and f (P \ {α}) = P ′ \ {α′ }, so f induces an isomorphism ϕf : Γ(P) → Γ(P ′ )
such that ϕf (α) = α′ and an isomorphism ψf : Γ(Q) → Γ(Q′ ) such that ψf (β) = β ′ .
The induced isomorphisms ϕ̂f , ψ̂f fulfill ϕ̂f = ψ̂f .
For the other direction, let ϕ : Γ(P) → Γ(P ′ ) and ψ : Γ(Q) → Γ(Q′ ) be as required.
Of course K and K′ are moves of the same type because otherwise the graphs Γ(P ∩ Q)
and Γ(P ′ ∩ Q′ ) would not be isomorphic.
Case 1: K and K′ are S-moves.
There is a realization g ∈ Mod(S) of ϕ such that g(P) = P ′ and g(α) = α′ . The curve
g(β) is supported in the same subsurface Σ′1,1 in which α′ is supported. Furthermore
α′ — g(β) is also an S-move. In the surface Σ′1,1 all edges lie in the same orbit. This is
because Mod(Σ′1,1 ) ∼
= SL2 (Z) and edges are of the form [p, q] — [r, s] with ps − qr = ±1
and the action is given by matrix multiplication, see [W, Chapter 2] for details. We find
a mapping class h ∈ Mod(S) such that h|S\Σ′1,1 = id and h(α′ ) = α′ and h(g(β)) = β ′ .
Now we set f := h ◦ g and derive f (α) = α′ , f (β) = β ′ , f (P \ {α}) = P ′ \ {α′ },
i.e. f (P) = P ′ and f (Q) = Q′ .
Case 2: K and K′ are A-moves.
This case is similar to the previous one; we have to use the properties of the mapping
class group and the pants complex of a 4-punctured sphere. The details of this case can
be found in [W, proof of 4.9].
Corollary 2.5:
The quotient of the pants graph P 1 (S) modulo Mod(S) is finite.
4
Proof:
The orbits of all pants decompositions of S under the action of Mod(S) correspond
bijectively to the isomorphism classes of all finite graphs having 2g − 2 + b trivalent
vertices, b univalent vertices and 3g − 3 + 2b edges, and this set is finite. (Unfortunately
the number of elements of this set as a function of g and b is not known.)
Now we suppose that there are infinitely many edges in P 1 (S)/ Mod(S). Because there
are only finitely many vertices, there is a vertex with infinitely many incident edges. So
in P 1 (S) there are infinitely many edges Ki = (P — Qi ) (i ∈ N) which are pairwise
not in the same orbit and which are all incident to some pants decomposition P. Any
edge is given by a move along one of the curves of P. Since P has only finitely many
curves, infinitely many of these edges are given by a move along the same curve α ∈ P.
But the moves along a given curve lie in at most two orbits which gives a contradiction.
xxx
It is also possible to calculate the orbits of the 2-cells of P(S) and to show that there
are only finitely many of these. This has been done in detail in [W] and will be largely
omitted here. To illustrate the methods we will calculate the orbit of a 2-cell of type
(6AS). For such a 2-cell F = (P1 — . . . — P6 ), the graph Γ(F ) is defined similarly to
Definitions 2.1 and 2.3a), with P in 2.1 replaced by the set P1 ∩ . . . ∩ P6 .
Lemma 2.6:
Two unoriented 2-cells F , F ′ of type (6AS) are in the same orbit under the action
of the mapping class group (meaning here the existence of an f ∈ Mod(S) with
f ({P1 , . . . , P6 }) = {P1′ , . . . , P6′ }) if and only if Γ(F ) ∼
= Γ(F ′ ).
Proof:
The “only if” direction is obvious since every f ∈ Mod(S) with f (F ) = F ′ induces an
isomorphism Γ(F ) → Γ(F ′ ). For the “if” direction let
F = {γ1, γ2 } — {γ1, γ3 } — {γ3, γ4 } — {γ4, γ5 } — {γ4, γ6 } — {γ2, γ6 }
= P1 — . . . — P6 ,
′
F = {γ1′ , γ2′ } — {γ1′ , γ3′ } — {γ3′ , γ4′ } — {γ4′ , γ5′ } — {γ4′ , γ6′ } — {γ2′ , γ6′ }
= P1′ — . . . — P6′
and ϕ : Γ(F ) → Γ(F ′ ) be an isomorphism. There is a realization g ∈ Mod(S) of ϕ with
g(P1 ) = P1′ , g(γ1) = γ1′ and g(γ2) = γ2′ . Let Σ′1,2 be the subsurface of S in which γ1′ and
γ2′ are supported and Σ′0,4 ⊆ Σ′1,2 be the subsurface of S in which γ2′ is supported. The
curves γ3′ and g(γ3) both lie in Σ′0,4 and are separating. Analyzing the surface Σ′0,4 we
find a mapping class h2 ∈ Mod(S) which is a power of a Dehn twist about γ2′ mapping
g(γ3) to γ3′ . (For the definition of a Dehn twist, compare Section 3.) As such a power
of a Dehn twist, h2 fulfills h2 (γ1′ ) = γ1′ , h2 (γ2′ ) = γ2′ , h2 (γ6′ ) = γ6′ , h2 (g(γ6)) = g(γ6) and
h2 |S\Σ′1,2 = id. With the same arguments we get a mapping class h1 ∈ Mod(S) with
h1 (g(γ6)) = γ6′ and h1 (γ1′ ) = γ1′ , h1 (γ2′ ) = γ2′ , h1 (γ3′ ) = γ3′ , h1 |S\Σ′1,2 = id. Therefore the
composition h1 ◦ h2 ◦ g fulfills h1 ◦ h2 ◦ g(γi′ ) = γi′ for i = 1, 2, 3, 6.
5
There is exactly one curve in Σ′1,2 which is nontrivial and not isotopic to a boundary
curve and which is disjoint from γ3′ and γ6′ . Since γ4′ and h1 ◦ h2 ◦ g(γ4 ) both have these
properties, we get h1 ◦ h2 ◦ g(γ4) = γ4′ .
There are exactly two curves in Σ′1,2 which are nontrivial and not isotopic to a boundary
curve and which are disjoint from γ4′ and have minimal intersection number with both
γ3′ and γ6′ . One of these curves is γ5′ , call the other one γ5′′ . Since h1 ◦ h2 ◦ g(γ5) has the
mentioned properties it coincides with γ5′ or γ5′′ .
Case 1: h1 ◦ h2 ◦ g(γ5 ) = γ5′
Then h1 ◦ h2 ◦ g(γi) = γi′ for all i = 1, . . . , 6 and therefore h1 ◦ h2 ◦ g(F ) = F ′ .
Case 2: h1 ◦ h2 ◦ g(γ5 ) = γ5′′
There is a mapping class  ∈ Mod(S) satisfying |S\Σ′1,2 = id, interchanging γ1′ and
γ2′ , γ3′ and γ6′ , γ5′ and γ5′′ , and with (γ4′ ) = γ4′ , compare Figure 4. Then the mapping
class f :=  ◦ h1 ◦ h2 ◦ g fulfills f (γ1 ) = γ2′ , f (γ2) = γ1′ , f (γ3 ) = γ6′ , f (γ4 ) = γ4′ ,
f (γ5 ) = (γ5′′ ) = γ5′ , f (γ6 ) = γ3′ and hence f (F ) = F ′ .
180◦
Figure 4: The mapping class 
Example 2.7:
We give a detailed description of the quotient P(S)/ Mod(S) for the surface S = Σ1,3 .
There are three orbits of pants decompositions corresponding to the three graphs depicted in Figure 5.
Γ1 =
Γ2 =
Γ3 =
Figure 5: Graphs of signature (1,3)
An easy calculation shows that two oriented edges K1 = (P1 — Q1 ), K2 = (P2 — Q2 )
lie in the same orbit if and only if Γ(P1 ) ∼
= Γ(P2 ) and Γ(Q1 ) ∼
= Γ(Q2 ). In particular, any
edge K = (P — Q) corresponding to an S-move is inverted because Γ(P) ∼
= Γ1 ,
= Γ(Q) ∼
∼
and any edge K = (P — Q) corresponding to an A-move with Γ(P) = Γ(Q) is inverted.
These edges are therefore barycentrically subdivided so that the quotient can be defined.
All other edges connect pants decompositions of different types and are therefore not
inverted by the action. Altogether the quotient graph P 1 (Σ1,3 )/ Mod(Σ1,3 ) has the form
shown in Figure 6.
6
BS
BA
B2
B3
Figure 6: The quotient graph P 1 (Σ1,3 )/ Mod(Σ1,3 )
The calculation of the orbits of all 2-cells is done in detail in the thesis [W]. We will
skip the development here and restrict ourselves to the enumeration of these cells. Most
of the 2-cells have to be barycentrically subdivided leading to 16 orbits of 2-cells. It is
virtually impossible to put all of them into a single picture, therefore we split them in
two complexes. The result can be found in Figure 7. The new vertices labelled 3S, 4C,
3A, 5A1 and 5A2 arise from the barycentric subdivision of some of the 2-cells.
4C
BS
BA
3S
B2
B3
3A
BS
BA
B2
5A1
B3
5A2
Figure 7: The quotient complex P(Σ1,3 )/ Mod(Σ1,3 )
7
3
Stabilizers
In this section we discuss the stabilizers of the action of Mod(S) on P(S). That means,
we calculate the stabilizers of all vertices, edges and 2-cells of the pants complex. For
this purpose let S be a surface which (has negative Euler characteristic and) is none of
the surfaces Σ0,3 , Σ0,4 , Σ1,2 , Σ2,0 .
Let P = {α1 , . . . , αn } be a pants decomposition of S. By choosing a suitable numeration
we can assume that α1 , . . . , αr are all genus-1-separating curves and αr+1 , . . . , αs are all
2-separating curves. Here genus-k-separating resp. k-separating means that the curve is
separating and that one component of its complement is a Σk,1 resp. a Σ0,k+1 . For any
genus-1-separating curve and any 2-separating curve α we have an associated half twist
σα , that is a mapping class twisting the Σ1,1 resp. Σ0,3 by 180 degrees and fixing the
rest of the surface. The square of any such half twist is a Dehn twist. A Dehn twist tα
about a simple closed curve α is defined by cutting S open along α, twisting one side
of an appropriate neighbourhood of α by 360 degrees, and reglueing the surface.
Definition 3.1:
Stab(P) := {f ∈ Mod(S) : f (P) = P}
Stabpw (P) := {f ∈ Mod(S) : f (αi) = αi for all αi ∈ P}
Stab∗pw (P) := {f ∈ Mod(S) : f (αi) = αi for all αi ∈ P and f (∂) = ∂ for all components
∂ of ∂S}
Lemma 3.2:
Stabpw (P) = hσα1 , . . . , σαr , σαr+1 , . . . , σαs , tαs+1 , . . . , tαn i,
Stab∗pw (P) = hσα1 , . . . , σαr , tαr+1 , . . . , tαs , tαs+1 , . . . , tαn i.
Proof:
In both cases the inclusion “⊇” is obvious. Let f ∈ Stabpw (P). Suppose first that
there is a curve αi ∈ P which is mapped onto itself by f , but with reversed orientation.
Consider the following cases:
Case 1: Both sides of αi belong to the same pair of pants H. Then we find a curve
β ∈ P such that ∂H = αi ∪ β. The mapping class f ◦ σβ preserves the orientation of αi ,
and all other curves are not changed by σβ .
Case 2: αi bounds two different pairs of pants H1 and H2 . H1 and H2 are interchanged
by f because f reverses the orientation of αi . f is an element of Stabpw (P), i.e. all
curves of P are mapped to itself. These two facts lead to the conclusion that S is one
of the surfaces Σ0,4 , Σ1,2 , Σ2,0 which are all excluded.
So we may assume that all curves of P are mapped orientation-preservingly to themselves
by f .
If f ∈ Stab∗pw (P) then every boundary component is mapped to itself by f . If f ∈
Stabpw (P), suppose that ∂1 , ∂2 ⊆ ∂S are different boundary components of S such that
f (∂1 ) = ∂2 . Then the pair of pants H1 next to ∂1 is mapped onto the pair of pants
8
H2 next to ∂2 . Due to the assumptions on S there are curves αi , αj ∈ P such that αi
bounds H1 and αj bounds H2 and f (αi ) = αj , hence αj = f (αi ) = αi and H1 = H2 . In
this case the mapping class f ◦ σαi maps ∂1 to ∂1 and ∂2 to ∂2 . Therefore we can assume
that f maps all boundary components of S to themselves.
Altogether f is a mapping class which maps every curve of P and ∂S orientationpreservingly to itself. That means that f is a composition of Dehn twists about the
curves of P and ∂S. Because Dehn twists about the curves of ∂S are trivial, f is an
element of htα1 , . . . , tαn i as required.
Every element f of the stabilizer of a pants decomposition P induces an automorphism
f˜ of the associated graph Γ(P). (For the definition of Γ(P), compare definition 2.1.)
Furthermore any automorphism γ of Γ(P) induces a permutation of the edges of Γ(P),
hence an element of the symmetric group S3g−3+2b . This gives rise to a group homomorphism
γP : Stab(P) → Aut(Γ(P)) → S3g−3+2b .
Let TP be the image of γP . Then we have
Proposition 3.3:
The following sequence is exact:
γP
1 −→ Stab∗pw (P) ֒→ Stab(P) −→ TP −→ 1
Proof:
f ∈ ker(γP ) ⇔ f induces the trivial permutation on the edges of Γ(P)
⇔ f (αi ) = αi for any αi ∈ P and f (∂) = ∂ for any boundary component ∂
⇔ f ∈ Stab∗pw (P)
By Lemma 3.2 and Proposition 3.3 the stabilizer Stab(P) is a group extension of a finite
group by a free abelian group. Be careful that the exact sequence in 3.3 in general does
not split (it is easy to find counterexamples).
At the beginning of the section we excluded some special surfaces. In those cases
there are more elements in the pointwise stabilizers Stabpw (P) and Stab∗pw (P). For
example, there are nontrivial mapping classes which map every curve on S to itself,
called hyperelliptic involutions. It is an easy exercise to calculate the stabilizers and find
the appropriate short exact sequences in these cases (see [W]).
Now we turn to the stabilizer of an edge of the pants complex. So let K = (P — Q) =
(α — β) be such an edge with P = {α, α2 , . . . , αn } and Q = {β, α2, . . . , αn }. We define
the following special stabilizer groups.
9
Definition 3.4:
x
a) Stab+
or (K) := {f ∈ Mod(S) : f (P) = P and f (Q) = Q}
−
Stabor (K) := {f ∈ Mod(S) : f (P) = Q and f (Q) = P}
−
Stabnor (K) := {f ∈ Mod(S) : f ({P, Q}) = {P, Q}} = Stab+
or (K) ∪ Stabor (K)
b) Stab({α2 , . . . , αn }) := {f ∈ Mod(S) : f ({α2 , . . . , αn }) = {α2 , . . . , αn }}
Stabpw ({α2 , . . . , αn }) := {f ∈ Mod(S) : f (αi) = αi for all i = 2, . . . , n}
Stab∗pw ({α2 , . . . , αn }) := {f ∈ Mod(S) : f (αi) = αi for all i = 2, . . . , n and
f (∂) = ∂ for all components ∂ of ∂S}
+
c) Fα,β
:= {f ∈ Mod(S) : f (α) = α and f (β) = β}
−
Fα,β := {f ∈ Mod(S) : f (α) = β and f (β) = α}
+
−
Stab({α, β}) := {f ∈ Mod(S) : f ({α, β}) = {α, β}} = Fα,β
∪ Fα,β
The following lemma is very easy to prove, so we omit the proof.
Lemma 3.5:
+
Stab+
or (K) = Fα,β ∩ Stab({α2 , . . . , αn })
−
Stab−
or (K) = Fα,β ∩ Stab({α2 , . . . , αn })
Stabnor (K) = Stab({α, β}) ∩ Stab({α2 , . . . , αn })
Proposition 3.6:
As above, let α2 , . . . , αr be all genus-1-separating curves and αr+1 , . . . , αs be all 2separating curves. Then
+
Fα,β
∩ Stabpw ({α2 , . . . , αn }) = hσα2 , . . . , σαr , σαr+1 , . . . , σαs , tαs+1 , . . . , tαn i
+
Fα,β ∩ Stab∗pw ({α2 , . . . , αn }) = hσα2 , . . . , σαr , tαr+1 , . . . , tαs , tαs+1 , . . . , tαn i
Proof:
+
Proposition 3.6 is proved similarly to Lemma 3.2. For a given mapping class f ∈ Fα,β
∩
Stabpw ({α2 , . . . , αn }) we first show that without loss of generality we may assume that
every curve αi (i = 2, . . . , n) and every boundary curve ∂ ⊆ ∂S is mapped orientationpreservingly to itself. (To achieve this assumption, we possibly have to modify f by
half twists about some appropriate curves.) As in 3.2, f is a composition of Dehn
twists about the curves α2 , . . . , αn and of a special mapping class supported in a regular
neighbourhood Σ of α ∪ β. Σ is one of the surfaces Σ1,1 or Σ0,4 . In both cases, the
identity is the only mapping class preserving the boundary of Σ and the curves α, β
(compare [W, section 2]). This means that f is a composition of Dehn twists about the
curves α2 , . . . , αn .
Proposition 3.7:
Again, let α2 , . . . , αr be all genus-1-separating curves and αr+1 , . . . , αs be all 2-separating
curves.
10
−
a) Let K be an S-move. Then Fα,β
∩ Stab∗pw ({α2 , . . . , αn }) 6= ∅ and for any fα,β ∈
−
Fα,β
∩ Stab∗pw ({α2 , . . . , αn }) we have
−
Fα,β
∩ Stabpw ({α2 , . . . , αn }) = fα,β ◦ hσα2 , . . . , σαr , σαr+1 , . . . , σαs , tαs+1 , . . . , tαn i,
−
Fα,β ∩ Stab∗pw ({α2 , . . . , αn }) = fα,β ◦ hσα2 , . . . , σαr , tαr+1 , . . . , tαs , tαs+1 , . . . , tαn i.
b) Let K be an A-move and let α, β be nonseparating. If there is a subsurface
−
Σ1,2 ⊆ S and a curve αi ∈ {α2 , . . . , αn } such that αi ∪ α ∪ β ⊆ Σ1,2 then Fα,β
∩
∗
∗
−
Stabpw ({α2 , . . . , αn }) 6= ∅ and for any fα,β ∈ Fα,β ∩ Stabpw ({α2 , . . . , αn }) we have
−
Fα,β
∩ Stabpw ({α2 , . . . , αn }) = fα,β ◦ hσα2 , . . . , σαr , σαr+1 , . . . , σαs , tαs+1 , . . . , tαn i,
−
Fα,β ∩ Stab∗pw ({α2 , . . . , αn }) = fα,β ◦ hσα2 , . . . , σαr , tαr+1 , . . . , tαs , tαs+1 , . . . , tαn i.
αi
α
β
c) Let K be an A-move and let Σ0,4 ⊆ S be the component of S \ (α2 ∪ . . . ∪ αn )
containing α and β. Furthermore, assume that one of the following properties is
fulfilled:
• Exactly two of the boundary components of Σ0,4 are also boundary components of S, and these two components neither lie in the same component of
Σ0,4 \ α nor in the same component of Σ0,4 \ β.
∂j
α
β
∂i
• Exactly three of the boundary components of Σ0,4 are also boundary components of S.
∂i
α
∂k
β
∂j
−
−
Then Fα,β
∩Stabpw ({α2 , . . . , αn }) 6= ∅ and for any fα,β ∈ Fα,β
∩Stabpw ({α2 , . . . , αn })
we have
−
Fα,β
∩ Stabpw ({α2 , . . . , αn }) = fα,β ◦ hσα2 , . . . , σαr , σαr+1 , . . . , σαs , tαs+1 , . . . , tαn i,
−
Fα,β ∩ Stab∗pw ({α2 , . . . , αn }) = ∅.
11
d) In all other cases
−
Fα,β
∩ Stabpw ({α2 , . . . , αn }) = ∅,
−
Fα,β ∩ Stab∗pw ({α2 , . . . , αn }) = ∅.
The proof is a little tedious and will be skipped here. It can be found in [W, Proposition
3.11].
As for vertices, it is possible to arrange the stabilizer groups of an edge K of P(S) into
suitable short exact sequences. Namely any element of Stabnor (K) induces an automorphism of Γ(P ∩ Q), and any automorphism of the latter graph induces a permutation of
the edges, hence an element of the symmetric group S3g−4+2b . This gives rise to a group
homomorphism
γP,Q : Stabnor (K) → Aut(Γ(P ∩ Q)) → S3g−4+2b .
nor
or
Let TP,Q
and TP,Q
be the images of Stabnor (K) and Stab+
or (K) respectively under γP,Q .
Like in Proposition 3.3, we get the following short exact sequences:
nor
1 −→ Stab({α, β}) ∩ Stab∗pw ({α2 , . . . , αn }) ֒→ Stabnor (K) −→ TP,Q
−→ 1
or
1 −→ hσα1 , . . . , σαr , tαr+1 , . . . , tαs , tαs+1 , . . . , tαn i ֒→ Stab+
or (K) −→ TP,Q −→ 1
The details can be found in the thesis [W].
In the last part of this section we treat the stabilizers of 2-cells.
Definition 3.8:
For a 2-cell F of P(S) we define
Stabpw (F ) := {f ∈ Mod(S) : f (P) = P for all P ∈ F },
Stabor (F ) := {f ∈ Mod(S) : f (F ) = F and the orientation of F is preserved},
Stabnor (F ) := {f ∈ Mod(S) : f (F ) = F }.
Clearly Stabpw (F ) ⊆ Stabor (F ) ⊆ Stabnor (F ).
We follow the policy of Section 2 and skip a detailed analysis of these stabilizer groups.
Like in Section 2, we illustrate the methods by proving the following lemma.
Lemma 3.9:
Let F be a 2-cell of P(S) of type (6AS). Then
Stabpw (F ) = Stabor (F ) = Stabnor (F ).
12
Proof:
Let us write
F = {γ1 , γ2 } — {γ1 , γ3 } — {γ3 , γ4 } — {γ4 , γ5 } — {γ4 , γ6 } — {γ2 , γ6 }
= P1 — P2 — P3 — P4 — P5 — P6
as in Figure 2. Let f ∈ Stabnor (F ). Since γ5 is the only curve which occurs in precisely
one of the pants decompositions of F , we conclude f (γ5) = γ5 and hence f (γ4 ) = γ4 .
Therefore we also have f ({γ1 , γ2}) = {γ1 , γ2 } and f ({γ3, γ6 }) = {γ3 , γ6 }. If f (γ1) = γ1
then f (γi ) = γi for all i = 1, . . . , 6 and f ∈ Stabpw (F ).
Now assume that f (γ1 ) = γ2 and hence f (γ2 ) = γ1 , f (γ3) = γ6 , f (γ6 ) = γ3 . As in
case 2 of the proof of Lemma 2.6 there is a mapping class  ∈ Mod(S) with the following
properties:
• |S\Σ1,2 = id where Σ1,2 is the subsurface of S in which F is supported,
• (γ1 ) = γ2 , (γ2 ) = γ1 , (γ3 ) = γ6 , (γ4 ) = γ4 , (γ6 ) = γ3 .
• 2 = t∂1 ◦ t−1
∂2 where ∂1 , ∂2 are the boundary curves of the subsurface Σ1,2 .
For an illustration of , compare again Figure 4.  is defined in such a way that ◦f (γ1) =
γ1 ,  ◦ f (γ2 ) = γ2 ,  ◦ f (γ3 ) = γ3 ,  ◦ f (γ4 ) = γ4 ,  ◦ f (γ5 ) = (γ5 ) 6= γ5 ,  ◦ f (γ6 ) = γ6 .
Case 1: f (∂1 ) = ∂1 and f (∂2 ) = ∂2
In this case also  ◦ f (∂1 ) = ∂1 and  ◦ f (∂2 ) = ∂2 . This means that the restriction
 ◦ f |Σ1,2 of  ◦ f to the subsurface Σ1,2 is a mapping class of Σ1,2 which preserves the
curves γ1 , γ2 and the boundary curves ∂1 , ∂2 of Σ1,2 . As in the proof of Lemma 3.2
this means that  ◦ f |Σ1,2 is a composition of Dehn twists about the curves γ1 , γ2, ∂1 , ∂2 .
Because of  ◦ f (γ3) = γ3 and  ◦ f (γ6 ) = γ6 we conclude that  ◦ f |Σ1,2 is a composition
of Dehn twists about the curves ∂1 , ∂2 , whence  ◦ f (γ5 ) = γ5 , a contradiction.
Case 2: f (∂1 ) = ∂2 and f (∂2 ) = ∂1
In this second case there is a mapping class ı ∈ Mod(S) satisfying the following, compare
Figure 8:
• ı|S\Σ1,2 = f |S\Σ1,2 ,
• ı(α) = α for any curve α in Σ1,2 ,
• ı2 |Σ1,2 = id.
For this mapping class we have f ◦ ı(∂1 ) = ∂1 and f ◦ ı(∂2 ) = ∂2 and f ◦ ı(α) = f (α) for
any curve α in Σ1,2 . Replacing f by f ◦ ı and applying case 1 gives the contradiction. 13
180◦
Figure 8: The mapping class ı|Σ1,2
β1
α1
P1 =
e12
γ1
P2 =
β2
γ1
α1
e23
P3 =
β2
α1
γ3
k1
β1
α1′
k2
k3
β1
α1
α1
γ1
k4
β2′
γ1
β2
α1
γ3′
γ1′
Figure 9: A subtree of P(S)
Example 3.10:
We resume the example of 2.7 and calculate the stabilizers of representatives of all orbits.
To begin with, we consider the subtree of P(S) depicted in Figure 9.
In the parametrization of Figure 9, we also consider two special mapping classes of finite
order, compare Figure 10.
180◦
120◦
=
δ=
Figure 10: The mapping classes  and δ
Then we have:
Stab(P1 ) = htα1 , σβ1 , σγ1 i
Stab(P2 ) = htα1 , tβ2 , σγ1 , i
Stab(P3 ) = htα1 , tβ2 , tγ3 , , δi
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Furthermore:
Stab+
or (e12 ) = Stabnor (e12 ) = htα1 , σγ1 i
Stab+
or (e23 ) = Stabnor (e23 ) = htα1 , tβ2 , i
There is a mapping class a ∈ Mod(Σ1,3 ) which is a root of σβ1 , interchanging the curves
α1 and α1′ , namely a := tα1 ◦ tα′1 ◦ tα1 . Also there is a mapping class b ∈ Mod(Σ1,3 )
with b2 = id, interchanging β2 and β2′ , namely b := σγ−1
◦ t−1
α1 ◦ σβ1 ◦ tβ2 . Furthermore,
1
′
there is a half twist c1 about β1 interchanging γ1 and γ1 with c21 = tβ1 . Finally c3 :=
′
tα1 ◦ tβ2 ◦ σγ−1
◦ t−1
γ3 interchanges γ3 and γ3 and is a square root of the mapping class
1
tα3 ◦ tβ2 . With these four mapping classes we get for the stabilizers of the inverted edges
k1 , . . . , k4 :
Stabnor (k1 ) = ha, σγ1 i
Stab+
or (k1 ) = hσβ1 , σγ1 i
Stabnor (k2 ) = htα1 , σβ1 , c1 i
Stab+
or (k2 ) = htα1 , σβ1 i
Stabnor (k3 ) = htα1 , σγ1 , bi
Stab+
or (k3 ) = htα1 , σγ1 i
Stabnor (k4 ) = htα1 , tβ2 , , c3 i
Stab+
or (k4 ) = htα1 , tβ2 , i
At the end of our example consider Figure 11 where a 2-cell F of type (6AS) is sketched.
γ1
γ3
γ4 γ3
β
β
γ1
γ2
γ5
γ4
β
γ2
β
β
γ6
γ4 γ6
β
Figure 11: A 2-cell of type (6AS) in P(Σ1,3 )
An easy calculation shows
Stabpw (F ) = Stabor (F ) = Stabnor (F ) = Stab({γ1 , γ3 , β}) ∩ Stab({γ2, γ6 , β}) = hσβ i.
15
References
[HLS] Allen Hatcher, Pierre Lochak, Leila Schneps: On the Teichmüller tower of mapping class groups. Journal für die reine und angewandte Mathematik 521 (2000),
1-24.
[HT]
Allen Hatcher, William Thurston: A presentation for the mapping class group of
a closed orientable surface. Topology 19 (1980), 221-237.
[P]
Andrew Putman: Cutting and Pasting in the Torelli Group.
http://www-math.mit.edu/~andyp/papers/CutPasteTorelli.pdf
[W]
Ute Wolf: Die Aktion der Abbildungsklassengruppe auf dem Hosenkomplex. Dissertation, Karlsruhe 2009,
http://digbib.ubka.uni-karlsruhe.de/volltexte/1000012215
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