A dual braid monoid for the free group

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by  David Bessis

A DUAL BRAID MONOID FOR THE FREE GROUP
DAVID BESSIS
Abstract. We construct a quasi-Garside monoid structure for the free group. This
4
monoid should be thought of as a dual braid monoid for the free group, generalisingthe
0
constructions by Birman-Ko-Lee and by the author of new Garside monoids for Artin
0
2 groups of spherical type. Conjecturally, an analog construction should be available for
arbitraryArtingroupsandforbraidgroupsofwell-generatedcomplex reflectiongroups.
n
a
J

3 This article continues the exploration of the theory of Artin groups and generalised
2
braid groups from the new point of view introduced by Birman-Ko-Lee in [BKL] for the

] classical braid group on n strings. In [B1], we generalised their construction to Artin
R
groups of spherical type. In the current article, we study the case of the free group, which
G
is the Artin group associated with the universal Coxeter group. The formal analogs of
.
h the main statements in [B1] turn out to be elementary consequences of classical material
t
a (some of which was known to Hurwitz and Artin). In an attempt to interpolate some
m
recent generalisations of the dual monoid construction (by Digne for the Artin group of
[
type A , [D]; by Corran and the author for the braid group of the complex reflection
n

1 group G(e,e,n), [BC]), we propose two conjectures describing properties of a generalised
v f
dual braid monoid, in the contexts of
4
2 (a) arbitrary Artin groups and
3
(b) braid groups of well-generated finite complex reflection groups.
1
0
This would provide the first uniform combinatorial approach to these objects. The initial
4
0 motivation for the current work was to understand the situation (b) from a natural geo-
/ metric viewpoint; the conjectures about complex reflection groups will be studied in the
h
t sequel [B2], answering some questions raised in [BMR].
a
m
: 1. Hurwitz action
v
i
X For any positive integer n, the “usual” braid group is the abstractly presented group
r
a Bn := hσ1,...,σn−1|σiσi+1σi = σi+1σiσi+1, σiσj = σjσi if |i−j| > 1i.
In the problems we are interested in, two “braid groups” simultaneously come into play:
this “usual” braid group, and the Artin group associated with a Coxeter system (or the
generalised braid group associated with a complex reflection group). Except in the final
conjectures, this Artin group will be the free group.
Let G be a group. For any sequence (g ,...,g ) ∈ Gn, set
1 n
σi ·(g1,...,gn) := (g1,...,gi−1,gigi+1gi−1,gi,gi+2,...,gn).
This article is the fruit of an inspiring visit to KIAS (Seoul) in June 2003. I thank Sang Jin Lee for
his hospitality, for stimulating discussions and for important suggestions.
1
2 DAVIDBESSIS
It is straightforward (and well-known) that this assignment extends to a left-action of B
n
on Gn.
Definition 1.1. This action is called Hurwitz action of B on Gn.
n
This actioncan beviewed asaparticular example of amoregeneral construction, where
the important property of Gn is that it is an automorphic set (in the sense of [Br]) or
equivalently a rack (in the sense, for example, of [DDRW]).
In [Br], Brieskorn considers several problems about braid group actions on automorphic
sets. One of these problems is to characterise orbits. A very naive invariant of Hurwitz
action is the product
π : (g ,...,g ) 7→ g ...g .
1 n 1 n
We will be interested in situations where π−1(g) is a single Hurwitz orbit, for a specific
g ∈ G.
2. Non-crossing loops
In all this section, we fix n + 1 distinct points x ,...,x in C. The complex line is
0 n
endowed with an orientation called “positive” or “direct”.
We set
F := π (C−{x ,...,x },x ).
n 1 1 n 0
This group is isomorphic to a (“the”) free group on n generators, but its geometric
definition gives additional structure, which is what matters here. For example, we may
consider the following natural elements in F :
n
Definition 2.1. A non-crossing loop is a continuous embedding λ : S1 ֒→ C−{x ,...,x }
1 n
whose image contains x .
0
To any non-crossing loop λ, we associate the element f ∈ F obtained by following
λ n
λ with the positive orientation (coming from the orientation of C). Elements f ∈ F
λ n
which may be obtained this are said to be non-crossing. We denote by NC the set of
non-crossing elements in F
n
We consider the length function
l : F −→ Z
n
n
1 dz
f 7−→
2iπ Z z −x
X f j
j=1
For any non-crossing loop λ, we may consider the set Int(λ) of points of C which are
“inside” λ (in the weak sense: we consider the support of λ to be “inside”). Clearly,
the index of f around x is 1 if x ∈ Int(λ), 0 otherwise. Setting ht(λ) := |Int(λ) ∩
λ i i
{x ,...,x }|, we have the relation
1 n
l(f ) = ht(λ).
λ
Definition 2.2. We define a relation ⊆ in NC by
def
∀f,g ∈ NC,f ⊆ g ⇐⇒ ∃ non-crossing loops λ,µ,f = f ,g = f ,Int(λ) ⊆ Int(µ).
λ µ
We leave to the reader the following easy topological lemma:
A DUAL BRAID MONOID FOR THE FREE GROUP 3
Lemma 2.3. For all f,g ∈ NC, the following assertions are equivalent:
(i) f ⊆ g;
(ii) for any non-crossing loop λ such that f = f , there exists a non-crossing loop µ
λ
such that g = f and Int(λ) ⊆ Int(µ);
µ
(iii) for any non-crossing loop µ such that g = f , there exists a non-crossing loop λ
µ
such that f = f and Int(λ) ⊆ Int(µ).
λ
Lemma 2.4. (i) For all f,g ∈ NC, f ⊆ g implies l(f) ≤ l(g). If f ⊆ g and
l(f) = l(g), then f = g.
(ii) The relation ⊆ is an order relation.
Proof. (i): The first statement is trivial. For the second statement, choose λ and µ such
that λ,µ,f = f ,g = f . Since ht(λ) = ht(µ), the annulus “between” λ and µ contains
λ µ
no point in {x ,...,x }, thus λ and µ are isotopic.
1 n
(ii): The relation is clearly reflexive. Antisymmetry follows from (i). Transitivity
follows from Lemma 2.3. (cid:3)
The main result of this section says that certain subposets of NC are lattices. Before
stating it, let us observe that NC as a whole is not a lattice. A first obstruction is that
one may find non-isotopic height n non-crossing loops. Clearly, they do not even have
a common upper bound (let alone a least common upper bound). For n = 2, two such
loops are illustrated below (one with a full line, the other one with a dotted line):
.........................
. .
. .
.... • ..•.... • ....
.. x . . x ..
..........1.......x0.........2........
We may also observe that the corresponding elements in NC do not have a largest
common lower bound: the two height 1 non-crossing loops represented below are distinct
maximal common lower bounds to the above height 2 non-crossing loops:
x•1 x•..0..............x..•..2............
Definition 2.5. For any g ∈ NC, we set NC := {f ∈ NC|f ⊆ g}.
g
Theorem 2.6. For any g ∈ NC, the poset (NC ,⊆) is a lattice.
g
The author thanks Sang Jin Lee, for suggesting to use hyperbolic geometry in the
following proof.
Proof. First, it is easy to reduce the question to the case when l(g) = n.
Up to isotopy, we may assume that x = −1, and that g is represented by the unit
0
circle. We set
D := {z ∈ C | |z| ≤ 1}.
4 DAVIDBESSIS
Using Lemma 2.3 (ii) and (iii), we observe that we may forget the outside of D: any
element of f ∈ NC is represented by non-crossing loops λ with Int(λ) ⊆ D, and in NC
g g
the relation ⊆ could be equivalently redefined using only such loops.
If n = 1, the result is straightforward.
Assume now that n > 1. We may endow D := D − {x ,...,x } with a complete
n 1 n
hyperbolic metric (see, for example, [DDRW], Chapter 7) Let D be the universal cover
n
of D may be viewed as a subset of the hyperbolic plane (see the nice picture on page
n f
114, loc. cit.).
Any element f ∈ F may be represented by a (possibly self-intersecting) loop in the
n
pointed space (D ,x ), thus be a path in D ; among such paths, there is a unique geo-
n 0 n
desic. The corresponding loop in (D ,x ) is called the geodesic loop of f. Geodesic loops
n 0 f
minimise self-intersections and mutual intersections; in particular:
• For all f ∈ F , then f ∈ NC if and only if its geodesic loop is non-crossing.
n g
′ ′ ′ ′
• For all f,f ∈ NC with geodesic loops λ,λ, then f ⊆ f ⇔ Int(λ) ⊆ Int(λ).
g
′
The theorem is a trivial consequence of the last statement: Let f,f ∈ NC with geodesic
g
′
loops λ,λ.
•
•
x0•.............•...................•......... •
•
′
Any h ∈ NC such that f ⊆ h and f ⊆ h may be represented by a non-crossing loop
g
′ ′
ν such that Int(λ) ⊆ Int(ν) and Int(λ) ⊆ Int(ν). Consider the loop λ ∨ λ obtained
by glueing the successive “outermost” portions of the two loops (in the above example,
this element is made with three successive portions of loops). Clearly, any non-crossing
′ ′
loop containing Int(λ)∪Int(λ) in its interior must also contain λ∨λ in its interior: the
′ ′
element represented by λ∨λ is the minimal least upper bound of f and f .
′
Similarly, considering the connected component of Int(λ) ∩ Int(λ) containing x , we
0
′
obtain a maximum lower bound. An illustration with the above f,f is given below (the
original loops are the dotted curves, the inf and the sup are the full curves).
•
x0•..........................................•.........................•...................•..................•................... •
(cid:3)
Remark. In the last proof, instead of using hyperbolic geometry, one could use a more
computationalviewpoint, whichmayalsobeusedtoimplementtheinf andsupoperations.
Say that two non-crossing loops are tight if their number of intersections is minimal
A DUAL BRAID MONOID FOR THE FREE GROUP 5
(within their homotopy classes). A first observation is that tight representatives for a
pair of elements of NC may be obtained by successive “bigon eliminations”: a bigon is
g
portion of the picture looking like
.................
withnomarkedpointx intheinsideportion;eliminatingsuchabigonconsistsofreplacing
i
this portion of the picture by something like
..................
Tightness may be detected by the absence of bigons. One may actually prove (by bigon
elimination) the stronger result: for any triple of non-crossing loops, one may find homo-
topicloopswhich arepairwise tight. Theonlyproperty ofhyperbolic geodesics used above
is that they are pairwise tight, thus that they solve the latter problem. However, for prac-
tical use, it is very efficient to perform bigon elimination without relying on hyperbolic
geometry.
3. Braid reflections and coordinate systems
Since F is the fundamental group of the complement in C of a complex algebraic
n
hypersurface (a finite set), we may consider special elements usually called generators-of-
the-monodromy or meridiens (we prefer here to call them braid reflections).
These elements may be described as follows. A connecting path is a continuous map
γ : [0,1] → C such that γ(0) = x , γ(1) ∈ {x ,...,x } and t 6= 1 ⇒ γ(t) ∈/ {x ,...,x }.
0 1 n 1 n
Onemayassociatetosuch aγ anelement r asfollows: startingfromx , followγ; arriving
γ 0
close to γ(1), make a positive turn around a small circle centered on γ(1); return to x
0
following γ backwards.
Definition 3.1. An element r ∈ F is a braid reflection if there exists a connecting path
n
γ such that r = r . The set of reflections in F is denoted by R.
γ n
Lemma 3.2. The set R coincides with the set of non-crossing elements of height 1.
Proof. If r is non-crossing of height 1, then choose a non-crossing loop λ representing r.
We have Int(λ) ∩ {x ,...,x } = {x }. Since Int(λ) is path connected, we may draw
1 n i0
inside λ a path γ connecting x and x . It is clear that r = r .
0 i0 γ
To prove the converse statement, one may check that for any path γ connecting x
0
and some x , there exists γ˜ without self-intersections such that r = r (it is clear by
i γ γ˜
construction that r is non-crossing of height 1). To find such a γ˜, one may remove
γ˜
self-intersections by “sliding” them past x . [Alternatively, one could observe that the
0
conjugacy classes in R are indexed by the irreducible components of the hypersurface;
that each conjugacy class contains a non-crossing element; and finally that NC is stable
under conjugacy.] (cid:3)
The standard way to see F as an abstractly presented group (with n generators and
n
no relation) is by means of a coordinate system:
Definition 3.3. Consider a planar graph Γ, whose vertices are x ,...,x , and with n
0 n
edges γ ,...,γ , each γ being a connecting path form x to x . We assume that the γ ’s
1 n i 0 i i
have no self-intersections and no mutual intersection (except at x ).
0
6 DAVIDBESSIS
To each γ , we associate f := r .
i i γi
A coordinate system is the (unordered) n-tuple of reflections {f ,...,f } obtained this
1 n
way.
We say that a coordinate system is compatible with an element g ∈ NC if there exists
a non-crossing loop γ representing g, such that Γ is drawn inside Int(γ).
Coordinate systems are in bijection with isotopy classes of planar graphs Γ as above
(isotopy with fixed vertices).
Saying that g ∈ NC is compatible with {f ,...,f } is equivalent to the existence
1 n
n
of a permutation σ such that g = f . The planar structure around x endows
i=1 σ(i) 0
{f ,...,f } with a natural cyclic orQdering. Once {f ,...,f } is fixed, choosing a com-
1 n 1 n
patible g is equivalent to the choice of a total ordering refining the cyclic ordering (there
are n such choices).
Up to isotopy and relabelling of the marked points, we may assume that the situation
looks like:
x0•LYreLrYeLrYeLrYeLrYeLrYeLrYeLrYeLrYeLrYeLrYeLr•••xxx234
•
x
1
More explicitly, our assumption is that x = −1, that the x are purely imaginary with
0 j
−1 < ℑ(x ) < ℑ(x ) < ··· < ℑ(x ) < 1,
1 2 n
and, for each j, we consider the affine connecting path [x ,x ] and the associated braid re-
0 j
flection f . The coordinate system is then compatible with the element of NC represented
j
by the unit circle.
We have F = hf ,...,f i. For any f ∈ F , an expression f = m fεi, with ε = ±1,
n 1 n n i=1 ji i
may be obtained as follows. First, find a (possibly self-intersectingQ) loop γ representing f
and drawn inside D. Then, following γ, write f each time it crosses some [x ,1] moving
j j
upwards, andf−1 each time it crosses some [x ,1]moving downwards (up to perturbation,
j j
we may assume that γ is transversal to these segments).
x0• oo 44 ••••..............................................................................
((
In the above example, the word is f f f−1f−1.
1 2 3 2
A word in the {f ,f−1,...,f ,f−1} is reduced if the patterns f f−1 and f−1f never oc-
1 1 n n j j j j
cur. Anyf ∈ F admitsauniqueexpressionasareducedwordinthe{f ,f−1,...,f ,f−1}.
n 1 1 n n
A DUAL BRAID MONOID FOR THE FREE GROUP 7
A loop is reduced if the associated word is reduced. Clearly, any loop in D admits, in
its homotopy class, a reduced loop. More precisely, this reduced loop may be obtained
by a certain “bigon elimination” procedure, during which one may avoid introducing self-
intersections. In particular, any non-crossing loop is homotopic to a non-crossing reduced
loop.
In the next two results, we denote by g the (maximal) element of NC represented by
D.
Lemma 3.4. Let f ∈ NC . The reduced word associated with f is “quadratfrei”: it does
g
not contain the patterns f f and f−1f−1.
j j j j
Proof. A picture is worth a thousand words:
x0• ••••....22--..........................................................................
**
(cid:3)
4. Simple transitivity of Hurwitz actions
The material in this section is certainly classical, except the interpretation in terms of
Coxeter elements in the universal Coxeter group.
Choose g a maximal non-crossing element of F . As we have noted earlier, it is possible
n
to find a coordinate system f ,...,f such that F = hf ,...,f i and g = f ...f . To
1 n n 1 n 1 n
fix the notations, we make the standard choice for g and f ,...,f , already used in the
1 n
previous section:
g
x0•LrYeLrYeLrYeLrYeLrYeLrYeLrYeLrYeLrffYeLrf23YeLrYe4Lr•••xxx234
f •
1 x
1
Clearly, any expression of g as a product of elements of R must be of length n (consider
the largest abelian quotient of F ).
n
Thus
(f ,...,f ) ∈ Red (g).
1 n R
Lemma 4.1. Let β ∈ B , let (r ,...,r ) := β · (f ,...,f ). Consider a sequence of
n 1 n 1 n
integers j ,...,j such that 1 ≤ j < j < ··· < j ≤ n. Then r ...r ∈ NC .
1 k 1 2 k j1 jk g
Proof. The elements r ,...,r form a coordinate system (the B -action sends coordinate
1 n n
systems to coordinate systems). Up to isotopy, all coordinate systems look the same.
This reduces the problem to the case when β = 1, for which the lemma is obvious. (cid:3)
8 DAVIDBESSIS
Definition 4.2. The universal Coxeter group W is defined by the presentation:
n
W := s ,...,s |s2 = 1 .
n 1 n i
(cid:10) (cid:11)
We consider the epimorphism π : F ։ W ,f 7→ s . We set T := π(R). Elements of
n n j j
T are called reflections.
We set c := π(g). It is again easy to see that
(s ,...,s ) ∈ Red (c).
1 n T
The map πn : Red (g) → Red (c) isa morphism of B -sets (where bothsets are equipped
R T n
with Hurwitz action).
Theorem 4.3. (1) The Hurwitz action is simply transitive on Red (g).
R
(2) The Hurwitz action is simply transitive on Red (c).
T
(3) The map πn : Red (g) → Red (c) is an isomorphism of B -sets.
R T n
The author is grateful to Sang Jin Lee for pointing out that (1) was already contained
in Artin’s 1947 article [A].
Proof. The transitivity statement in (1) is Theorem 16 in [A] (although it appears in a
formulation closer to ours at the top of p. 114 of loc. cit.).
Let us prove the transitivity statement in (2) – our argument is so similar to Artin’s
that we could have omitted the proof, but we include it for the convenience of the reader,
who will easily reconstruct the proof of Artin’s Theorem 16. We start with a remark
about normal forms in W . This group is a free product of n cyclic groups of order 2.
n
Consider a finite sequence w := (a ,...,a ), where each a is taken in {s ,...,s }. We
1 m i 1 n
say that w represents the element a ...a ∈ W . We say that w is the normal form of
1 m n
a ...a if it does not contain a pattern s s of consecutive equal terms. When w is a
1 m j j
normal form, we say that m is the length of a ...a . Clearly, the normal form always
1 m
exists and is unique. It may actually be computed with the following non-deterministic
procedure. Start from an arbitrary w.
(I) If w is a normal form, return w.
(II) Otherwise, a least a pattern s s appears. Choose an occurence and remove the
j j
involved terms. Start again with the new (shorter) sequence.
A sequence of successive choices in (II) is called an execution of the procedure. Though
there are usually several executions, the end result is always the (unique) normal form.
The surviving terms in the output come from terms in the input. If we choose a particular
execution, we say that a given term of w is untouched by the execution if it survives it.
Any t ∈ T, being a reflection, may be written
(∗) t = u u ...u s u ...u u
1 2 k t k 2 1
where the u ’s and s are in {s ,...,s }. We may clearly assume that (∗) is a normal
i t 1 n
form. We say that s is the content term of t.
t
Let (t ,...,t ) ∈ Red (c). Considering the largest abelian quotient of W , one may
1 n T n
observe that, the content terms s satisfy {s ,...,s } = {s ,...,s }.
tj t1 tn 1 n
The normalformof (t ,...,t ) is (s ,...,s ). Let w be theconcatenation of thenormal
1 n 1 n
forms of t ,...,t . Choose an execution of the normal form procedure, applied to w. The
1 n
output is (s ,...,s ). We distinguish two cases:
1 n
A DUAL BRAID MONOID FOR THE FREE GROUP 9
Case 1. The content terms of the normal forms of the t ’s are untouched by the
j
execution. Write
w = (u ,...,u ,s ,u ,...,u ,v ,...,v ,s ,v ,...,v ,...,...)
1 k 1 k 1 1 l 2 l 1
(since they are untouched, the content terms must already be in the order s ,...,s in
1 n
w). The execution rewrites w to (s ,...,s ) while leaving s untouched. Thus it rewrites
1 n 1
(u ,...,u ) to (). Since (u ,...,u ) is normal, this implies that k = 0. Considering the
1 k 1 k
fragment v ,...,v between the unaffected terms s and s , we conclude that l = 0, and
1 l 1 2
so on... Thus (t ,...,t ) = (s ,...,s ).
1 n 1 n
Case 2. At least one content term of one the t is destructed. Consider the first
j
iteration of the execution where this happens: a certain pattern appears, involving (the
descendant of) a content term of at least one of the t ’s: denoting by (a ,...,a ) the
j 1 m
word just before this particular iteration, we have a = a for some i, with a or a
i i+1 i i+1
being the (until then untouched) content term of one of t ’s. Note that a and a may
j i i+1
not both be content terms, because distinct t ’s have distinct contents. Let us assume
j
that a is the content term of some t . (The case when a is the content term may be
i j i+1
dealt with symmetrically). Inside w, we are interested in the portion involving t and
j
t :
j+1
′
w = (...,u ,...,u ,s,u ,...,u ,v ,...,v ,s,v ,...,v ,...,),
1 k k 1 1 l l 1
′
where s is the content of t and s the content of t .
j j+1
Lemma 4.4. The length of su−1...u−1v ...v is < l−k (in particular, k < l).
k 1 1 l
Proof of the lemma. From the assumptions, it is easy to see that the first term s is mod-
ified in any execution with input (s,u ,...,u ,v ,...,v ); in particular, this sequence is
k 1 1 l
not a normal from. Consider an execution with this input.
Ifk = 0, we observe that(v ,...,v )isanormalform. Since (s,v ,...,v )isnotnormal,
1 l 1 l
we must have s = v . The claim holds.
1
If k > 0, we observe that both (s,u ,...,u ) and (v ,...,v ) are normal forms. We
k 1 1 l
must have u and v , and the first step of the execution leads to (s,u ,...,u ,v ,...,v).
1 1 k 2 2 l
We conclude by an easy induction. (cid:3)
Consider the pair (t t t−1,t ). The first reflection is represented by
j j+1 j j
′
(u ,...,u ,s,u ,...,u ,v ,...,v ,s,v ,...,v ,u ,...,u ,s,u ,...,u ).
1 k k 1 1 l l 1 1 k k 1
By the lemma, the length of su ...u v ...v is < l−k. The same property holds for its
k 1 1 l
inverse v ...v u ...u s. Thus the length L of t t t−1 satisfies L < k+(l−k)+1+(l−
l 1 1 k j j+1 j
k)+k = 2l+1. The totallength of (t1,...,tj−1,tjtj+1t−j 1,tj,tj+2,...,tn) is strictly smaller
than the total length of (t ,...,t ). These two decompositions lie in the same Hurwitz
1 n
orbit. One may prove the transitivity part of (2) by induction on the total length.
The simplicity statement in (1) says that
∀β ∈ B ,∀(t ,...,t ) ∈ Red (g),β ·(t ,...,t ) = (t ,...,t ) ⇒ β = 1.
n 1 n R 1 n 1 n
Using the transitivity, this statement is equivalent to
∀β ∈ B ,β ·(f ,...,f ) = (f ,...,f ) ⇒ β = 1,
n 1 n 1 n
10 DAVIDBESSIS
which is nothing but the faithfullness of the standard representation of B in Aut(F ),
n n
already known to Hurwitz.
Let us now prove the simplicity statement in (2). Using transitivity, it is enough to
prove that
∀β ∈ B ,β ·(s ,...,s ) = (s ,...,s ) ⇒ β = 1.
n 1 n 1 n
Let β ∈ B such that β · (s ,...,s ) = (s ,...,s ). Let (r ,...,r ) := β · (f ,...,f ).
n 1 n 1 n 1 n 1 n
Since πn commutes with Hurwitz action, we have πn((r ,...,r )) = (s ,...,s ), thus
1 n 1 n
s = π(r ) for all j. Fix j ∈ {1,...,n}. By Lemma 4.1, we know that r ∈ NC .
j j j g
Consider the normal form fε1...fεm of r in F . By Lemma 3.4, this normal form is
j1 jm j n
“quadratfrei”. Thus s ...s is the normal form of s in W . Thus m = 1 and r = fε.
j1 jm j n j j
Since r ∈ R, we have ε = 1. This holds for any j, thus (r ,...,r ) = (f ,...,f ). By
j 1 n 1 n
(1), we must have β = 1.
(3) follows trivially. (cid:3)
Corollary 4.5. There are natural bijections between:
(i) Maximal strict chains of NC .
g
(ii) Elements of Red (g).
R
(iii) Coordinate systems compatible with g.
More precisely, the map from (i) to (ii) sends a maximal chain 1 = a < a < ··· < a = g
0 1 n
to (a−1a ,...,a−1 a ), and the map from (ii) to (iii) send (t ,...,t ) to {t ,...,t }.
0 1 n−1 n 1 n 1 n
Proof. Consider the classical interpretation of B as the mapping class group of the n-
n
punctured disk, fixing the outer circle.
By Lemma 2.3, maximal strict chains of NC are represented by chains of concentric
g
non-crossing loops in D, of stricly increasing height. Isotopy classes of such data clearly
form a single B -orbit.
n
Similarly, coordinates systems drawn inside D form a single B -orbit.
n
The corollary then follows from the fact that Red (g) is a single Hurwitz orbit, and
R
that the natural maps with the above objects are B -equivariant. (cid:3)
n
Corollary 4.6. Denote by R the subset of R consisting of elements which may appear
g
in some sequence in Red (g). Denote by T the subset of T consisting of elements which
R c
may appear in some sequence in Red (c). Then R = R∩NC . Moreover, π induces a
T g g
bijection R ≃ T .
g c
Note that π does not induce a bijection from R to T. Also, the injectivity of R ≃ T
g c
is a priori stronger than the injectivity of πn : Red (g) → Red (c) from the theorem.
R T
Proof. The statement R = R∩NC isalready in Lemma 4.1. Using the theorem, we note
g g
that T = π(R ). We are left with having to prove the injectivity. First, we observe that
c g
the fiber of R → T over s is a singleton (it follows from Lemma 3.4). By transitivity,
g c 1
all fibers have the same cardinal. (cid:3)
5. Quasi-Garside structure
Definition 5.1. We denote by F+ the submonoid of F generated by R. We endow F+
n n n
def
with the divisibility partial ordering: for all f,g ∈ F+, f 4 g ⇐⇒ ∃h ∈ F+,fh = g.
n n